UNIVERSITÉ DE BRETAGNE OCCIDENTALE Océanographie Physique Pierrick Penven Alain Colin de Verdière Directeur de thèse Bernard Barnier Rapporteur Geoff Brundrit Rapporteur Xavier Carton Examinateur Philippe Gros Examinateur Robert Mazé Examinateur Claude Roy Examinateur 1$^{\hbox{er}}$ décembre 2000

A numerical study of the Southern Benguela circulation with an application to fish recruitment.


Contents

Introduction

The Benguela ecosystem, along the South-west coast of Africa is, with the California Current, the Peru-Chile and the North African upwelling systems, one of the world's 4 major ecosystems driven by an upwelling along the eastern margin of the Oceans. Their combined total area accounts only for 0.1 % of the total surface of the world oceans, but they provide almost 30 % of the world's total fish catch [Durand et al., 1998]. Furthermore, their yearly fluctuations explain most of the inter-annual variability of the total marine fish catch. These fluctuations, showing years of high abundance and dramatic collapses, result from the variability of the recruitment (which is the number of young fish produced each year). The vulnerability of the fish larvae during the first weeks of their lives when their displacement capabilities are limited, leaving them at the mercy of the ocean for food accessibility or transportation, explains this large variability in recruitment. This critical period implies that the number in a year class is determined at a very early stage [Hjort, 1914,Hjort, 1926]. During this period, the environment has a major impact on the survival rate of larvae. Bakun [1993] has identified 3 classes of environmental processes that combine together to create a favorable environment for recruitment:

In upwelling areas, the existence of multi-variable and non-linear relationships between recruitment and upwelling intensity is a recurrent pattern resulting from the interaction between several environmental process [Cury and Roy, 1989,Cury et al., 1995,Durand et al., 1998]. The competition between these different processes (enrichment, mixing, dispersion...) leads to an "Optimal Environmental Window" that gives a maximum for pelagic fish recruitment success in upwelling areas for a limited averaged wind range ($\sim$ 5-7 m.s$^{-1}$) [Cury and Roy, 1989].

The Benguela upwelling system is a highly dispersive environment, where a strong equatorward wind along the coast induces an offshore displacement of the surface waters. Although important for the enrichment of the ecosystem in nutrients, this divergence can have a detrimental effect on the recruitment: eggs and larvae are then carried offshore, away from their coastal habitat. In the Southern Benguela, sardines and anchovies, the most abundant pelagic species, have adapted their reproductive strategies to the environmental constraints. They migrate to spawn on the western Agulhas Bank, upstream of the food sources. Eggs and larvae are advected by the currents towards the productive areas of the West Coast of South Africa. St Helena Bay, in the North of Cape Columbine, is recognized as the most important nursery ground of the West Coast of South Africa [Hutchings, 1992]. This area shelters the biggest fishing industry of the country. The loss of biological material during transport from the Agulhas Bank to the West Coast and the retention inside the nursery ground of St Helena Bay are supposed to be the principal factors affecting the recruitment of sardines and anchovies [Hutchings et al., 1998].

The work presented in this manuscript is part of the VIBES (Viability of exploited pelagic fish resources in the Benguela Ecosystems and Stocks in relation with the environment) project. VIBES is a pluridisciplinary research project involving IRD (Institut de Recherche pour le Développement, France), UCT (University of Cape Town, South Africa), MCM (Marine and Coastal Management, South Africa) and LPO (Laboratoire de Physique des Océans, France). One of the scientific goals of VIBES is to improve our understanding of the spatial dynamics of the pelagic marine resources, the fisheries and the environment through modeling. The present work concentrates on the modeling and the understanding of the physical oceanic processes affecting pelagic fish recruitment in the Southern Benguela upwelling system.

To carry out this study, we use numerical tools in order to simulate the complex physical patterns observed in the Southern Benguela. We follow a step by step approach. We start by setting up idealized experiments in order to provide an understanding of the peculiarities of the circulation in St Helena Bay. At a later stage, a 3-dimensional realistic model is implemented to reproduce the dynamics of the ocean around the South western corner of Africa. The key processes of the dynamics of the Southern Benguela will be identified from idealized and realistic experiments. An analysis of these processes and a quantification of their impact on the transport, retention and dispersion of the biological material are performed in order to obtain the characteristic patterns affecting recruitment. The South western corner of Africa has been much studied because of the global climate implication of the inter-ocean exchanges that occur in this area. A high resolution model of this region might also give new insights on the physical processes involved in the South Atlantic-Indian Ocean exchange of properties.

The first part of the thesis concentrates on the description of the characteristic elements of the Benguela dynamics. Numerous articles related to surveys conducted in the Benguela upwelling system have been published during the last 30 years. Several reviews [Nelson and Hutchings, 1983,Shannon, 1985,Shannon and Nelson, 1996,Shillington, 1998] provide a broad outline of the observed dynamics of the Benguela. The bibliographic study conducted in this first part of the manuscript provides a general description of the actual understanding of the system and leads to the identification of key questions relevant to the thesis.

The second chapter presents the idealized experiments conducted to analyze the peculiarities of the shelf circulation in St Helena Bay. The bay is situated North of Cape Columbine, a step like variation of 100 km in the coastline. Associated with the cape, the shelf broadens from 50 to 150 km. These topographic variations should considerably alter the shelf dynamics. Two hypothesis are used to simplify the problem. Firstly, the gentle slope of the shelf should allow the neglect of processes related to stratification in the simulation of the shelf dynamics [Clark and Brink, 1985]. Secondly, spatial and temporal wind variations are assumed to be of secondary importance in comparison to the processes related to topography. Hence, barotropic experiments are conducted, forced by a constant wind. These experiments are conducted to test if a topographically induced process can balance the dispersion caused by the wind forced coastal currents. Diagnostic tools are designed to help in the understanding of the simulated process and a sensitivity analysis will explore the shelf dynamics response to a range of wind forcing, bottom friction parameter and size of the cape. An analytical solution in the form of standing shelf waves, illuminates this important behavior of the shelf dynamics. A tracer of water age is integrated into the model to quantify retention.

For the third chapter, a realistic regional model is implemented in order to produce a high resolution portrayal of the ocean dynamics surrounding the South-western corner of Africa and to explore the physical processes involved in the different biological stages leading to recruitment, from eggs to larvae and juveniles. A meeting organized at the beginning of the project and discussions with the different partners of the project allowed the selection of model requirements:

The Benguela upwelling system is unique in a way that the African continent ends at around 34$^o$ S. This induces the highly energetic poleward termination of the western boundary current of the Indian Ocean, the Agulhas Current, to flow along the Agulhas Bank and somehow to interact with the Benguela upwelling system. It retroflects South of the Agulhas Bank to flow back into the Indian Ocean. One should note that the anticyclonic eddies shed at the retroflection area, the Agulhas rings, are the biggest coherent structures observed in the Ocean. The handling of these highly energetic structures and currents by a regional oceanic model of finite dimension is a challenge that require specific treatments. Recently, long term simulations (of more than 10 years) have been conducted using a regional oceanic model for the California Current System [Marchesiello et al., 2000]. The model employed is ROMS, the Regional Ocean Modeling System. It uses a generalized nonlinear terrain-following coordinate, high order schemes and new parameterizations that have been especially implemented to resolve with a high level of accuracy the primitive equation of momentum along the shelf and the slope on a regional scale. Though there is no equivalent of the Agulhas Current along the West Coast of the United States, we expect to obtain long-term meaningful results using the same tool for the Benguela upwelling system. The validation of the model results will be done through comparison with data. The study of the variability of the system and of typical mesoscale processes will give insights for the understanding of the Benguela dynamics. Special attention is given to the model solution on the shelves along the South and West coasts, and comparison is also made with the results of the idealized experiments. If the realistic model solution is satisfactory, it will be possible to use the model to explore transport mechanisms from the Agulhas Bank to West Coast and retention processes in the coastal domain. This is done by introducing a passive tracer that simulates eggs and larvae transport behavior.

Following this approach, we expect to provide a better understanding of the dynamics of the Southern Benguela as well as necessary tools for the ongoing study of the dynamics of the recruitment.


1 The Benguela




The fisheries of the South African West Coast being of large economical importance, an important effort has been directed by South African marine research institutes to analyze the ecosystem. Thus, numerous studies have been undertaken in the last 30 years, involving for the physical part: hydrological samplings, current meters deployments, aerial atmospheric and sea surface temperature measurements, ADCP current measurements, drifters deployments, satellite data analysis and theoretical studies. As a result, a thorough description of the system is available and the understanding of many important processes has significantly progressed. These results have been summarized in several reviews [Nelson and Hutchings, 1983,Shannon, 1985,Shannon and Nelson, 1996,Shillington, 1998]. The aim of this chapter is to produce a general description of the Benguela system and its peculiarities. A more specific goal is to identify the characteristic patterns of the Benguela dynamics and to extract the key processes that affect the recruitment of sardines and anchovies along the South African West Coast. This analysis leads to the identification of a few key questions relevant for this study.


Les pêcheries le long de la Côte Ouest de l'Afrique du Sud étant d'une importance économique majeure, un effort considérable a été réalisé par les instituts de recherches marines Sud-africains pour analyser l'écosystême du Benguela. Ainsi, de nombreuses études ont été conduites durant les 30 dernières années, comprenant pour la partie physique: des échantillonages hydrologiques, le déploiement de mouillages courantométriques, des mesures aériennes des composantes atmosphériques et de la température de surface de l'eau, des mesures courantométriques par ADCP, le larguage de flotteurs dérivants, l'analyse d'images satellitales, et des études théoriques. Il en découle une descrition détaillée du système; et des progrès significatifs ont été obtenus dans la compréhension des principaux processus. Ces résultats ont été résumés dans différentes revues d'articles [Nelson and Hutchings, 1983,Shannon, 1985,Shannon and Nelson, 1996,Shillington, 1998]. L'objectif de ce chapitre est de produire une descripton générale du système du Benguela et de ses particularités. Un but plus précis est l'identification des motifs caractéristiques de la dynamique du Benguela et d'extraire les processus clefs pouvant affecter le recrutement des sardines et des anchois le long de la Côte Ouest de l'Afrique du Sud. Cette analyse conduit à la formulation de quelques questions clefs, pertinantes pour cette étude.


1 Geographical settings

Figure 1.1: Surface currents of the South Atlantic Ocean. Abbreviations are used for the Angola-Benguela Front (ABF), Brazil Current Front (BCF), Sub-tropical Front (STF), Sub-antarctic Front (SAF), Polar Front (PF) and Continental Water Boundary / Wedell Gyre Boundary (CWB/WGB). Adapted from Tomczak and Godfrey [1994].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/satl.eps,width=14cm}} \end{figure}

The Benguela Current is the eastern boundary current of the South Atlantic sub-tropical gyre
[Peterson and Stramma, 1987] (figure 1.1). It can be described as a broad northward flow that follows the west coast of southern Africa from the southern tip of Africa (i.e. the Cape Agulhas at 35$^o$ S) to Cape Frio ($18.4^o$ S) near the border between Angola and Namibia [Garzoli and Gordon, 1996]. The similar paths of 2 drifters released near Cape Peninsula with an interval of two years exhibit the coherent equatorward surface movement of the current (figure 1.2) [Nelson and Hutchings, 1983].

Figure 1.2: Paths of two satellite-tracked drogues released west of Cape Town in March 1977 and in February 1979. Adapted from Nelson and Hutchings [1983].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/drogues2.eps,width=14cm}} \end{figure}

The Benguela system is bounded in the North at about $16^o$ S by the warm Angolan current, which flows poleward. It is bounded in the South by the warm Agulhas Current, the western boundary current of the Indian Ocean that follows the South Coast of South Africa [Shillington, 1998]. The terminology "Benguela Current" describes as well the coastal upwelling system and the large scale eastern limb of the sub-tropical gyre [Peterson and Stramma, 1987], thus no precise offshore boundary of the system is defined. For Garzoli and Gordon [1996], at $30^o$ S, the entire Benguela Current is confined between the South African West Coast and the Walvis ridge ($\sim 1200$ km from the coast). In this manuscript, we will limit our definition of the Benguela Current to the part of the current that flows over the shelf and the continental slope.

Figure 1.3: Bathymetry of the South-east Atlantic Ocean derived from the ETOPO2 dataset.
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/topo_beng.eps,width=15cm}} \end{figure}

The West Coast of Southern Africa is a narrow coastal plain which rises to the main continental escarpment situated between 50 and 200 km inland. North of $32^o$ S (Cape Columbine), the coastline is regular and runs in a north-westward direction. South of $32^o$ S the coastline is irregular, with several capes (Cape Columbine $32^o$ S, Cape Peninsula $34^o$ S, Cape Agulhas $35^o$ S) and bays (St Helena bay, Saldanha Bay, Table Bay, False Bay). One thousand meters-high mountains ranging along the Cape Peninsula can play an important role in perturbing the local wind field [Shannon, 1985]. Most of the coastal region is arid with the Namib Desert that extends between $14^o$ S and $31^o$ S. The southern region has a cooler Mediterranean type climate. The continental shelf is highly variable in width (figure 1.3). It can be narrow with minimums located South of Lüderitz (75 km) and in front of Cape Peninsula (40 km), and it can be relatively wide with maximums located off the Orange river (180 km) and on the Agulhas Bank (230 km). The shelf break is deep (200 m) and quasi-rectilinear, running North westward roughly parallel to the coast. It is cut in a north-south direction at 60 km offshore of Cape Columbine by the Cape Canyon. The Agulhas Bank is a wide and shallow feature that forms the southernmost margin of the African continent [Shannon and Nelson, 1996].


2 Large scale

The Benguela current flows northward from the Cape of Good Hope. It bends towards the northwest to separate from the coast at around $30^o$ S while widening rapidly [Peterson and Stramma, 1987]. Three currents are feeding the Benguela Current: the South Atlantic Current, which is the southern part of the sub-tropical gyre, the Agulhas current and the Antarctic Circumpolar Current. The composition of the Benguela Current water is as follows: 50 $\%$ of Atlantic water, 25 $\%$ of water from the Indian Ocean and 25 $\%$ of a blend of Agulhas and tropical Atlantic water [Garzoli and Gordon, 1996].

Figure 1.4: The principal water masses and potential temperature - salinity characteristics of the South-east Atlantic and Benguela system. Adapted from Shannon and Nelson [1996].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/diagts2.eps,width=10cm}} \end{figure}

A T-S diagram exposes the hydrological characteristics of the principal water masses in the Benguela system (figure 1.4). The surface waters are composed of tropical surface water and subtropic surface water. Three kinds of thermocline waters are present: the South Atlantic Central Water (SACW), the South Indian Central Water (SICW) and the Tropical Atlantic Central Water (TACW). Under these, the fresh Antarctic Intermediate Water (AAIW), characteristic of the South-East Atlantic Ocean by its core of minimum of salinity around 700-800 m, flows toward the equator [Shannon and Nelson, 1996]. 4-5 Sv of AAIW is carried this way northward in the Benguela. Underneath, the relatively warm and saline North Atlantic Deep Water (NADW) spreads southward from the North Atlantic between 1000 m and 3500 m. It generates a poleward current along the African continental margin. The Antarctic Bottom Water (AABW) lies below the NADW under 3800 m. Blocked in the North by the Walvis ridge (figure 1.3) , the circulation of the AABW in the Cape Basin is cyclonic. Thus the AABW produces as well a poleward current along the African continental margin with typical speeds of 5 to 10 cm.s$^{-1}$ [Nelson, 1989].

Figure 1.5: Schematic flow field of surface and thermocline waters. Current speeds refer to surface values. Transports (circled) refers to total transport above 1500 db (i.e. includes AAIW). Adapted from Shannon and Nelson [1996].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/schema_beng.eps,width=10cm}} \end{figure}

Equatorward flow occurs in the surface to depth of several hundred of meters. In the surface friction layer, the Ekman drift is typically 20 to 35 cm.s$^{-1}$ [Nelson, 1989]. Recent measurements showed that in the upper 1000 m, the Benguela current carries 13 Sv towards the equator across $30^o$ S [Garzoli and Gordon, 1996]. The upper layers averaged circulation and transport is summarized in figure 1.5. One can note in figure (1.5) the large transport (75 Sv) carried by to the Agulhas Current, the western boundary current of the Indian Ocean subtropical gyre, just South-East of the Benguela system. As it flows along the South and East coasts of South Africa, the Agulhas Current reaches an intensity unmatched by any other western boundary currents [Beal and Bryden, 1997] at up to 6 knots (300 cm.s$^{-1}$) [Boyd and Oberholster, 1994]. Its follows the shelf break along the eastern margin of the Agulhas Bank, developing meanders, reverse plumes and counter current on the shelf [Lutjeharms et al., 1989]. It generally leaves the coast just East of the tip of the Agulhas Bank before turning back on itself and flowing eastward into the South Indian Ocean. In this retroflection area, the largest eddies found in the oceans (with diameters up to 300 km), the Agulhas rings, detach from the current and transport warm and salty waters into the South Atlantic. They account for a time-averaged transfer of 10-15 Sv in the upper 1000 m and thus play an important part in the global thermohaline circulation [Peterson and Stramma, 1987]. Regularly, warm filaments detach from the southern tip of the Agulhas Bank and develop into the Benguela system [Lutjeharms and Cooper, 1996]. Along the East coast of South Africa ($32^o$ S) the presence of a persistent Agulhas undercurrent with velocities as large as 30 cm.s$^{-1}$ and a core at depth around 1200 m has been recently measured [Beal and Bryden, 1997].


3 Atmospheric forcing

1 Large scale

Figure 1.6: Mean atmospheric sea level pressure (hpa) for (a) January, and (b) July. Adapted from Peterson and Stramma [1987].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/atm2.eps,width=14cm}} \end{figure}

The wind field in the Benguela is mainly controlled by the South Atlantic high pressure system (figure 1.6). This anticyclone oscillates seasonally along a North-West (austral fall) / South-East (austral spring) axis. It generates equatorward, upwelling favorable wind stress all around the year in the Northern Benguela and mostly in summer in the Southern Benguela. The flow is steered equatorward along the coast by a thermal barrier set-up by the desert in the North and by the mountain range along the Cape Peninsula in the South. In winter, the Southern Benguela system is under the control of westerly moving depressions that travel past the southern tip of Africa, the dominant winds being more North-Westerly [Shillington, 1998]. The upwelling season in the Southern Benguela occurs between September to March. The along shore wind maximum is situated offshore, inducing a cyclonic wind stress curl along the coast [Shannon and Nelson, 1996].

2 Mesoscale modulations

Figure 1.7: Cyclic weather pattern over the Benguela system, typical of summer conditions. (a) South Atlantic high established - coastal low at Lüderitz - southerly winds at Cape Town. (b) South Atlantic High ridging - gale force winds at Cape Town - coastal low moves south. (c) South Atlantic High weakens - North West winds at Cape Town, following the passage of the coastal low. (d) South Atlantic High strengthens - southerly winds along the west coast. (e) Berg wind conditions. Adapted from Nelson and Hutchings [1983].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/atm_cycle2.eps,width=14cm}} \end{figure}

Low pressure cells propagate freely south of the African Continent over a typical period of 1 to 2 weeks. Coastal cells of low pressure develop in association with the approach of the cyclonic systems. These features, named coastal lows, form near Lüderitz and travel South around the continent as coastal trapped waves [Nelson and Hutchings, 1983]. Sometimes, a flow of dry adiabatically heated air blows of the western escarpment when high pressure cells form over the subcontinent: the so called "berg" winds [Shannon and Nelson, 1996]. The typical cyclic summer weather pattern is portrayed on figure (1.7). It has been proposed that this cycle induces a strong variability in coastal upwelling and shelf currents of a period of 3 to 6 days [Nelson and Hutchings, 1983]. Pulsing of the Benguela ecosystem has been related to the resonance between shelf waves and the passage of coastal lows [Jury et al., 1990] and an optimum resonant pulse interval of 10 days has been suggested [Jury and Brundrit, 1992]. Strong diurnal rotary winds induced by land-sea breezes occur north of Cape Columbine [Shannon and Nelson, 1996].

3 Local structures

Numerous studies have been conducted on local wind structures with the aim of relating them to mesoscale structures observed in the Benguela upwelling system [Jury, 1985a,Jury, 1985b,Jury et al., 1985,Jury, 1986,Jury, 1988,Kamstra, 1985,Taunton-Clark, 1985]. Areas of cyclonic wind stress curl have been identified. They are induced by land topography in the lee of Cape Columbine (i. e. in St. Helena Bay) and in the lee of Cape Peninsula or by locally intensified atmospheric thermal front between warm land and cool sea along the Namaqualand coastline ($30^o$ S) [Jury, 1988]. The cyclonic wind stress curl induced by the wake in the lee of Cape Columbine has been measured during typical events for the vertical atmospheric structure [Jury, 1985a]. The wake (and then the cyclonic curl) is stronger in shallow events (low inversion layer) than in deep events (high inversion layer). The presence of upwelling plumes in the lee of Cape Peninsula and Cape Columbine has been related to those topographically induced cyclonic wind stress curls [Jury, 1985a,Jury, 1985b,Jury et al., 1985,Jury, 1986,Jury and Taunton-Clark, 1986,Jury, 1988,Kamstra, 1985,Taunton-Clark, 1985]. However, a complete dynamical demonstration of the plume / wind stress curl relationship is missing. In the same way, the presence of cyclonic eddies in the vicinity of Cape Peninsula has been related to mesoscale temporal and spatial variations in surface winds [Jury et al., 1985], but the formation of these eddies does not appear to be correlated to local winds [Lutjeharms and Matthysen, 1995].


4 Along the West Coast

1 Upwelling

The wind induced coastal upwelling is characterized by a pronounced negative sea surface temperature anomaly found mainly within the 150-200 km off the West Coast of southern Africa [Shannon, 1985,Lutjeharms and Stockton, 1987]. Four major semi-permanent upwelling centers are present in the Southern Benguela: the Lüderitz cell at $27^o$ S, the Namaqualand cell at around $30^o$ S, the Cape Columbine upwelling plume at $33^o$ S and the Cape Peninsula upwelling plume at $34^o$ S. The presence of these cells has been related to local maximums in wind stress curl [Jury, 1988], change in coastline orientation [Shannon and Nelson, 1996] or narrowing of the shelf [Nelson and Hutchings, 1983]. The upwelled water originates from 200-300 m [Nelson and Hutchings, 1983]. It is separated all along the coast from the offshore warmer water by a well developed oceanic front [Brundrit, 1981]. Although highly convoluted and variable, the front coincides approximately with the shelf break. It shows large scale stationary features that have been related to the existence of propagating barotropic shelf waves, although the latter cannot explain the standing nature of the process [Shannon, 1985].


2 Circulation

Figure 1.8: Schematic flow-field of near-surface currents based on ADCP data collected between November 1989 and January 1992. Velocity ranges indicated are typical values. Adapted from Boyd and Shillington [1994].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/adcp2.eps,width=14cm}} \end{figure}

A strong equatorward surface baroclinic jet is in geostrophic balance with the upwelling front
[Shillington, 1998] and follows the 200-300 m isobath [Nelson and Hutchings, 1983]. The speed of this semi-permanent jet ranges from 40 cm.s$^{-1}$ to 80 cm.s$^{-1}$ west of Cape Town [Boyd and Nelson, 1998] and is typically in excess of 50 cm.s$^{-1}$ offshore of Cape Columbine (figure 1.8) [Nelson and Hutchings, 1983]. It can be strengthened near Cape Peninsula by the vicinity of Agulhas waters with high steric height offshore [Strub et al. 1998]. Its width has been estimated at some 20-30 km [Nelson and Hutchings, 1983]. As represented on figure (1.8), this jet separates in two branches just north of Cape Columbine, one branch bending into St. Helena Bay, the other, with a stronger intensity, flowing offshore. Altimeter data show a convoluted equatorward jet several hundred kilometers offshore North of $33^o$ S [Strub et al. 1998]. A net subsurface poleward flow, with averaged velocities of 4.5 cm.s$^{-1}$ to 6 cm.s$^{-1}$, has been observed close to the shore along the entire west coast [Boyd and Oberholster, 1994,Nelson, 1989]. It exhibits a wave-like motion with periodicities of approximately three days. The cause of this flow along the inner shelf is still unknown [Nelson, 1989]. Another characteristic feature of the Benguela system is the existence of the shelf edge poleward undercurrent [Nelson and Hutchings, 1983]. It has been directly observed in a number of cross sections and an average speed of 5-6 cm.s$^{-1}$ is given [Shannon and Nelson, 1996]. It is part of a more extensive poleward motion stretching from the coast across the bottom of the shelf to the Cape Basin [Nelson, 1989]. The observation of oxygen-deficient water coming from a source area off Angola along the shelf edge and onto the the shelf is another confirmation of the existence of the poleward undercurrent [Dingle and Nelson, 1993]. Recent current measurements off Cape Columbine have revealed a deep poleward undercurrent of 11 cm.s$^{-1}$ in autumn, 6.8 cm.s$^{-1}$ in winter, 7.4 cm.s$^{-1}$ in spring and 8.3 cm.s$^{-1}$ in summer [Nelson et al., 1999]. The averaged bottom temperature shows a cross-isobath trend on the shelf and numerous hot-spots on coastal locations. It exhibits a flooded area North of Cape Columbine, where the $8^o$C isotherm intrudes onto the shelf. On the shelf, the bottom mixed layer has seldom a thickness less than 3 $\%$ of the total depth and it can be 2 or 3 times thicker on the shelf edge [Dingle and Nelson, 1993]. Tides along the West Coast are semi-diurnal, with a maximum spring range of about 2 m. The phase arrives almost almost simultaneously everywhere along the West Coast. Tides induces small oscillations in the current in the order of 10-15 cm.s$^{-1}$ [Shillington, 1998].

3 Mesoscale features

The most impressive aspect of the Benguela system, and surely of the most importance either from both a physical or a biological point of view, is the high mesoscale activity that develops all along the coast. Four classes of characteristic mesoscale features have been extracted from satellite images of sea surface temperature [Lutjeharms and Stockton, 1987]: upwelling plumes, upwelling filaments, upwelling eddies and Agulhas current filaments. The impact of mesoscale activity on primary production in the Southern Benguela has been illustrated by satellite imagery [Shannon et al., 1985]. Mesoscale activity can also affect the transport pattern of fish larvae [Lutjeharms and Stockton, 1987]. The Agulhas rings that have been described in section 1.2 might interfere sporadically with the Benguela system.

Figure 1.9: A schematic sea surface temperature ($^o$ C) map of a typical developed Cape Peninsula upwelling tongue. Adapted from Taunton-Clark [1985].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/capepoint_plume.eps,width=7cm}} \end{figure}

Figure 1.10: Sea surface temperature ($^o$ C) distribution in St Helena Bay for 1 November 1980. An upwelling plume extends from Cape Columbine. Adapted from Jury [1985a].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/columbine_plume.eps,width=8cm}} \end{figure}

Figure 1.11: Namaqualand sea surface temperature ($^o$ C) and wind streamlines for 25 November 1980 (Max. = area of highest speed). The Namaqualand upwelling plume develops from the coast. Adapted from Jury and Taunton-Clark [1986].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/namaqua_plume.eps,width=8cm}} \end{figure}

Upwelling plumes are variable, semi-permanent tongues of cold water spreading from the major upwelling centers. Four sites of generation off upwelling plumes have been recognized in the Benguela: Cape Peninsula, Cape Columbine, Hondeklip Bay-Namaqualand and Lüderitz. The Cape Peninsula upwelling plume is present during the summer months [Taunton-Clark, 1985], whereas it can be masked during Northerly winds. It shows an elongated shape extending north-westward (figure 1.9) enclosing cooler water at the coast. The funneling of cold water through a canyon in the south-west (the Cape Point Valley) [Shannon et al., 1981] and the influence of the coastal mountains on local winds [Jury, 1988] has been advanced has explanations of the presence of this plume. A similar tongue of cold water has been observed extending from Cape Columbine (figure 1.10) [Shannon, 1985,Jury, 1985a,Jury, 1985b]. It has an inverted "S" characteristic shape, suggesting topographic control [Shannon, 1985]. Whereas upwelling in Namaqualand is confined into a coastal strip, a broad plume of cold water extends offshore near $30^o$ S (figure 1.11). The base of this plume coincides with a maximum in along shore winds and a broadening in the continental shelf [Jury and Taunton-Clark, 1986]. The Lüderitz plume doesn't grow from a fixed location on the coastline, but as it develops, it makes roughly always the same angle with the coast [Lutjeharms and Stockton, 1987]. In comparison with upwelling plumes, upwelling filaments are narrower, not standing, and short-lived features (between five days and five weeks) extending from the upwelling front
[Lutjeharms and Stockton, 1987]. An in-situ investigation off a filament have shown that it is a relatively shallow feature that is confined in the upper 50 m. Filaments have typical elongations of 200 km [Shannon and Nelson, 1996], ranging from 50 km to 600 km [Lutjeharms and Stockton, 1987]. In extreme cases, they may extend 1000 km or more offshore [Lutjeharms et al., 1991]. On average, two times more filaments develop from Lüderitz than from the Cape Peninsula [Lutjeharms and Stockton, 1987]. ADCP measurements have revealed the interaction between eddies and filaments North of the Cape Peninsula [Nelson et al., 1999].

Figure 1.12: Location of frontal eddies on the upwelling front from (a) February 1985, (b) August 1985 and (c) the whole of 1985, according to imagery from NOAA-9 satellite. Adapted from Lutjeharms and Stockton [1987].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/eddies_bengu.eps,width=14cm}} \end{figure}

Eddies are numerous in the system with a preponderance off-shore and downstream of the four major upwelling centers (figure 1.12). Their distributions show no clear seasonal patterns
[Lutjeharms and Stockton, 1987]. The Cape Peninsula has been recognized as a highly productive area of cyclonic eddies with averaged diameters of 42 km $\pm$ 16 km. No correlation exists between the formation of these eddies and the local winds [Lutjeharms and Matthysen, 1995]. Vortex dipoles composed of eddies of about 50 km diameter has been observed near Cape Columbine and Lüderitz [Stockton and Lutjeharms, 1988]. Altimeter data has revealed the generation of cyclonic eddies from the coast and their propagation offshore to the west [Strub et al. 1998]. One possible link between the Agulhas and the Benguela systems is the spreading of long streak of warm water from the Agulhas current along the western edge of the Agulhas Bank: the Agulhas filaments. They can interact with the Cape Peninsula upwelling front and catalyse eddy formation from the cape [Lutjeharms and Stockton, 1987] or increase steric height gradient at the front [Strub et al. 1998]. Intrusion of Agulhas water within 30 km of Cape Peninsula, flowing northward at around 40-60 cm.s$^{-1}$ has been regularly observed [Boyd and Nelson, 1998]. Most of the Agulhas filaments detach from the Agulhas Current just downstream of the southern tip of the Agulhas Bank. Six or seven are formed per year, each usually lasting 3-4 weeks. Their average width is around 50 $\pm$ 16 km and their average length 530 $\pm$ 166 km. They do not appear to extend deeper than 85 m, but they can be responsible of between 5 $\%$ and 15 $\%$ of the total interbasin salt flux generated by the Agulhas Current [Lutjeharms and Cooper, 1996].


4 The nursery ground of the West Coast: St Helena Bay

St Helena bay can be defined in a broad sense as the wide shelf area extending 200 km North of Cape Columbine (figure 1.3). South of Cape Columbine, the width of the shelf is narrow (50 km) and becomes broader further north (up to 150 km). The size repartition of anchovy larvae have shown that the area North of Cape Columbine is the major nursery ground of the West Coast, with large numbers of small anchovy larvae being advected into the area from the South [Boyd and Hewitson, 1983]. High concentrations of juvenile fish have been observed in the area [Hutchings, 1992].

Figure 1.13: Schematic representation of currents in the Cape Columbine-St Helena Bay area. Adapted from Shannon [1985].
\begin{figure} \centerline{\psfig{figure=Figures/Chap1/schema_helena.eps,width=10cm}} \end{figure}

The Cape Columbine upwelling plume develops during upwelling events. Kamstra [1985] and Jury [1985a] have related the generation of the plume to the cyclonic wind stress curl in the vicinity of Cape Columbine. This curl is generated by topographic effects on wind around the cape and it appears to be pronounced during "shallow southeasterly events" (marine layer thickness comparable to land elevation). Using radio tracked drifters, Holden [1985] shows that whereas the flow is predominantly northward and perturbed by small eddies, a cyclonic vortex remains in St. Helena Bay. It connects to a southward current flowing along the coast (figure 1.13). ADCP measurments show the same pattern in St Helena Bay with an inshore curvature of the surface currents, bounding a broad area of weak mean currents [Boyd and Oberholster, 1994]. Whereas stratification might be important in the bay [Bailey and Chapman, 1985], current meter moorings near Cape Columbine [Lamberth and Nelson, 1987] have demonstrated the transient barotropic nature of the flow. Offshore and associated with a subsurface front, the baroclinic jet described in section (1.4.2) follows the shelf edge with estimated surface velocities of 60 cm.s$^{-1}$ (figure 1.13) [Shannon, 1985].

5 The spawning area: the Agulhas Bank

The Agulhas Bank forms the southernmost extremity of the African continent. It is a wide triangular shelf extending up to 230 km south from the coast (figure 1.3). It has been recognized as the main spawning area for sardines and anchovies in the Southern Benguela. The transport of eggs and larvae between the western part of the Bank and the nursery grounds of the West Coast is a major factor for the success of sardines and anchovies recruitment [Fowler and Boyd, 1998]. The coastal boundary of the Agulhas Bank is forced by the wind. Easterly winds can drive episodic coastal upwelling in summer. Inshore of the 100 m isobath, the currents are weak and/or variable in speed and direction, with a net North-West flow West of Cape Agulhas. This convergent North-West current system funnels into the shelf edge jet of Cape Peninsula [Boyd and Oberholster, 1994], it is supposed to be the path followed by the larvae to reach the West Coast [Fowler and Boyd, 1998]. Within the Bank, the summer vertical structure shows a strong stratification, whereas it is well mixed in winter due to erosion by winter storms. Coastal sea level and coastal current reversals reveal the passage of eastward traveling coastal trapped waves [Boyd and Shilligton, 1994]. The eastern and offshore part of the Agulhas Bank is highly influenced by the Agulhas current. It flows along the shelf edge, developing meanders, shear edge features, borders eddies and reverse plumes on the Bank [Lutjeharms et al., 1989]. A ridge of cool water surrounded by a cyclonic circulation has been observed from the eastern to the central Bank [Boyd and Shilligton, 1994]. Vertical cross sections have shown that uplift of cold water can be associated with reverse plumes and border eddies [Lutjeharms et al., 1989]. There is not yet an explanation on how this feature is formed [Boyd and Shilligton, 1994].

6 Variability

Typical currents spectra on the Benguela shelf show significant peaks between 2.5 and 4 days [Nelson, 1989]. This have been related to modulations in atmospheric forcing [Jury, 1986] or the passing of coastal trapped waves [Shillington, 1998]. Jury and Brundrit [1992] have suggested a resonant optimum pulse between oceanic and atmospheric coastal trapped waves of 10 days. Pulsing in the upwelling cycle has been found to range between 10 days and more than 20 days [Jury et al., 1990]. A spectral analysis of tide gauge measurements in Cape Town has exhibited a wide peak at around 10-15 days [Schumann and Brink, 1990]. At the seasonal scale, upwelling is less variable in the Northern Benguela than in the South where it stops during winter. Garzoli and Gordon [1996] have found the strongest seasonal pattern in transport near the shelf edge at $30^o$ S. The system shows definite interannual variability with the occurrence of cold and warm events [Shannon and Nelson, 1996]. The possibility of high sea-level events propagating poleward from the equatorial Atlantic in the manner of the Pacific El Niño has been confirmed [Brundrit et al., 1987]. Less intense and less frequent than Pacific El Niño, the warm Benguela Niños events are characterized by the advection of tropical water southwards along the coast of Namibia [Shannon and Nelson, 1996]. Benguela Niños are not necessarily in phase with the El Niño Southern Oscillation.

7 Summary

The Benguela current differs from the other eastern boundary systems by the poleward limitation of the coastal boundary at $34^o$ S. This allows the South Indian western boundary current to approach closely and to interact with the system. The dominant equatorward wind regime induces a strong coastal upwelling separated from the open ocean by a well developed oceanic front. This front is highly convoluted and follows roughly the shelf edge. It is associated with a strong surface baroclinic jet that is present from Cape Peninsula to Cape Columbine. After dividing near Cape Columbine, the outer branch of the jet is found further offshore northward. Whereas the wind forcing is mainly equatorward, poleward motion occurs in the Benguela in the form of a poleward coastal counter current, a poleward undercurrent and a deep poleward motion at the base of the shelf edge. High mesoscale activity is a major characteristic of the system. It includes localized upwelling plumes, upwelling filaments extending sometimes far offshore from the front, upwelling eddies that can carry coastal products offshore in the ocean, Agulhas filaments that sometimes interact with the upwelling front, coastal trapped waves, and the famous Agulhas rings that are shed from the Agulhas Current. This variability is exhibited on spatial scales ranging from around ten kilometers to hundreds of kilometers and temporal scales ranging from a few days to several months. Sardines and anchovies have adapted their life strategy to the complexity and peculiarities of the system by spawning on the Agulhas Bank, upstream of the upwelling centers. St Helena bay, in the lee of Cape Columbine and hundreds of kilometers away from the spawning grounds, appears to be the most important nursery ground of the South African West Coast. The Benguela has been extensively studied, but the complexity of the system is such that numerous questions are still open. I would like to present here a few that seem to be relevant for this manuscript: There are several ways to explore these questions. During the last 30 years, a large quantity of data has been collected, providing numerous insights regarding the dynamics of the Benguela upwelling system. More data and new oceanographic cruises could be set up to answer the questions listed to the previous paragraph. However, numerical tools are now widely available and become more and more relevant to explore coastal processes. In the Benguela, modeling is in its infancy and we have taken the opportunity to explore the dynamics of the system using an approach, as well as tools, that have never been used in the region.


2 Recirculation and retention on the shelf in St. Helena Bay




In this chapter, we will concentrate on the first question listed in the summary of the first chapter: "Why is St. Helena Bay such a successful nursery ground ?". The shelf being large in St. Helena Bay, idealised barotropic numerical experiments are conducted in order to explore the interactions between an equatorward, upwelling favorable, wind forced current and the topogaphy of the Bay. Diagnostic analysis and analytical calculations bring to light the dynamics involved in the simulations. The impact of the circulation on the retention of biological material in the Bay is explored through a tracer marking the age of the water masses.


Dans ce chapitre, nous nous concentrerons sur la première question énoncée dans le résumé du chapitre pr'ecédent: "Quelle est la cause du succès de la nourricerie de la Baie de Ste Hélène ?". La Baie de Ste Hélène présentant un large plateau, des expériences idéalisées barotropes sont mises en place afin d'explorer les interactions entre un courant vers l'équateur, forcé par un vent favorable a l'upwelling, et la topographie de la baie. Des analyses diagnostiques, et des calculs analytiques éclairent la dynamique impliquée durant les simulations. L'impact de la circulation sur la rétention des composantes biologiques est quantifiée à l'aide d'un traceur représentant l'age des masses d'eau. The upwelling of the West Coast of South Africa provides the necessary enrichment for the recruitment. But the driving mechanism of coastal upwelling, the offshore Ekman transport, at the same time, advects the larvae away from the productive area. Hence, the success of recruitment requires the presence of a retention process that keeps the larvae in the favorable area [Bakun, 1998]. If enrichment by upwelling occurs all along the West Coast, the success of St. Helena Bay should be related to presence of retention in the bay. St. Helena Bay is located just North of Cape Columbine, one the two major capes of the West Coast. It can be seen as a step-like indentation of 100 km in the coastline. Associated with this topographic feature, the shelf broadens dramatically, to reach a width of 150 km (figure 1.3). This topographic configuration should alter the coastal circulation in a favorable way for the recruitment. To test this last statement, 2 hypotheses are assumed. Firstly, on a broad relatively flat shelf like St. Helena Bay, following the criterion of Clark and Brink [1985], baroclinic processes should be of less importance than barotropic dynamics. Secondly, spatial and temporal wind variations, although important, should not be necessary to produce a favorable environment during upwelling events. Following these hypotheses, a set of idealized numerical experiments are undertaken to explore the influence of a cape and a broadening shelf on the retention during the upwelling season (e. g. for a coastal circulation forced by an equatorward wind). The outline of this chapter is as follow. After a review of the interaction between coastal currents and capes, a description of the numerical model is provided. An analytical model of the barotropic processes gives characteristic values for velocities and free surface elevation. Outputs of a reference numerical experiment are analyzed and sensitivity tests are conducted, using a range of values for wind forcing, bottom friction or different size of capes. Different mechanisms, such as control by bottom friction or the generation of standing waves, are tested to explain the flow patterns observed in the experiments. Finally, a tracer showing the age of the water is integrated into the model to quantify retention.

1 Interactions between coastal currents and capes

Interactions between capes and coastal currents are complex and remain poorly understood; although they have been studied in many ways. Crepon et al. [1984] solved analytically a linear upwelling two-layer model around a rectangular promontory. Baroclinic and barotropic Kelvin waves generated at the corner of the cape propagate poleward and can lead to upwelling fluctuations independent of local winds. Further, they relate the poleward undercurrent to the difference of the phase speeds between baroclinic and barotropic waves. They found numerically the same pattern with different shapes of cape. Batteen [1997] explains the enhancement of upwelling equatorward of capes by conservation of potential vorticity in equatorward flows. Downstream and inshore of the plume of upwelled water, an "upwelling shadow" can be found such as that described by Graham and Largier [1997] for Northern Monterey Bay where warm water is trapped at the coast behind a narrow oceanic front. Several laboratory experiments involved flow past capes. Davies et al. [1990] introduced stratification in the case of a flat bottom and no rotation. Whereas stratification determines all aspects of eddy generation or eddy shedding from the capes, bottom friction seems to be crucial during the decay of the eddy [Davies et al., 1990]. By introducing a counterclockwise rotation, Boyer and Tao [1987a] showed that the response of the flow differs dramatically if the cape is on the left or on the right looking downstream in the Northern hemisphere. Their setting corresponds to respectively an equatorward and poleward flow along an oceanic Eastern boundary. The poleward current passes through three regimes, depending on the Burger number ( $S=\frac{N^2H^2}{f^2L^2}$; where $N^2$ is the Brunt-Väisälä frequency, $H$ is the water depth, $f$ is the Coriolis parameter and $L$ a characteristic length scale):
Small S: flow fully attached, no eddy generated;
Medium S: generation of an attached anticyclonic eddy;
Larger S: shedding of anticyclonic eddies.
For the equatorward current, there is no fully attached regime:
Small S: generation of an attached cyclonic eddy (quickly formed but subsequently spins down);
Larger S: shedding of cyclonic eddies.
Boyer et al. [1987b] have performed the same kind of experiments with an obstacle on the left or on right of the flow, but involving this time an homogeneous fluid. They found a complex wake motion for a certain range of Rossby and Ekman parameters, and again strong differences if the cape is on the left or on the right. In the case "cape on the left", the vortex shedding is more regular, but in both cases, eddies can merge into larger structures that can be, depending of the parameters, attached, shed or advected downstream. Klinger [1983] has tested the influence of the Rossby and Ekman numbers on the formation of anticyclones on slopes, by concentrating on a barotropic flow past a corner in a rotating tank (poleward flow along an eastern boundary). In this case, whereas the gyre size is approximately proportional to the Rossby number, it is not strongly influenced by bottom friction. Narimousa and Maxworthy [1989] have built a more realistic laboratory model to interpret satellite observations of coastal upwelling. This experiment shows the effects of ridges and capes on the generation of standing waves, meanders, filaments and eddies. The capes produce cyclones inshore and filaments offshore. These experimental results are in good agreement with satellite images of sea surface temperature off the West Coast of the North American continent. To describe the patterns measured in the lee of islands, Wolanski et al. [1984] have introduced an "Island wake parameter": P from an Ekman pumping model for the control of wake eddies. This parameter can predict if friction dominates the flow (P$<$1), if there is a stable wake (P$\sim$1) or if there is apparition of instabilities (P$>$1). This result has been found to be in good agreement with the flow patterns derived from remotely sensed imagery by Pattiaratchi et al. [1986], but has been in bad agreement when bottom topography is complex. Similar studies have been conducted using numerical models. Becker [1991] built a numerical model of a viscous flow past a cylinder in a rotating frame, when the Rossby (Ro) and the Ekman (Ek) numbers are small. She has found two key parameters for the boundary layer separation: $\lambda=\frac{Ro}{2\sqrt{Ek}}$, which is an equivalent of the "Island wake parameter" and $\delta$ the boundary layer thickness. The flow starts to detach when $\lambda>1$, and the bubble length increases linearly with $\lambda$ and with decreasing $\delta$. The generation and evolution of eddies around headlands by a tidal flow have been studied analytically and numerically by Signell and Geyer [1991]. In a boundary layer model, detachment occurs because of bottom friction as soon as an adverse pressure gradient is established. They found using a 2D numerical model that the extent of vorticity is limited by the frictional length scale: $l_f=\frac{H}{2C_D}$. In a two layer realistic numerical model of Oregon coast, Peffley and O'Brien [1975] showed that bottom topography overwhelms coastline irregularities in the generation of mesoscale upwelling features. On the contrary, using a realistic 3D numerical model of the California upwelling system, Batteen [1997] found that wind forcing and coastline irregularities are key mechanisms for the generation of meanders, eddies, jets and filaments. She showed that capes "anchor" filaments and generate cyclonic eddies. The process of generation and control of a cyclonic eddy past Point Conception (California) has been studied by Oey [1996]. In a one and a half layer, reduced-gravity model (infinite bottom layer), equatorward wind forced currents generate a cyclonic eddy past the cape by advection of vorticity at the corner. Viscosity and the Rossby number control it. This eddy is found again in a 3D realistic model of the Santa Barbara Channel. It seems to follow the same processes of formation, although bottom topography and beta effect are shown to become important for longer time period ($>30$ days). A main discrepancy between our study and Oey's [1996] work is the width of the shelf in St Helena Bay that extends from 50 km to 150 km, while the extension of the shelf in front of Point Conception is limited to 20 km. The presence of those shallow waters invalidate the use of a one layer and a half reduced gravity model, and bottom effects should be important in the generation and control of cyclonic eddies. It appears then that the presence and the shape of the bottom topography should overwhelm the effects related to stratification. To quantify this, the Brunt-Väisälä frequency $N^2$ has been calculated using recent temperature and salinity measurements in St. Helena Bay. It has a typical value of $7 \times 10^{-3}s^{-1}$. Whereas stratification is significant, the gentle bottom slope of St. Helena Bay (slope coefficient: $\alpha \sim 0.2$ %) satisfies the dynamic criterion for barotropic shelf water response [Clark and Brink, 1985]: $\frac{N^2\alpha^2}{f^2} \ll 1$, where $f$ is the Coriolis parameter. In St Helena Bay, $\frac{N^2 \alpha^2}{f^2}\sim 0.05$. This 'bottom slope' Burger number allows us to state that whereas density related processes such as upwelling and associated baroclinic coastal jets might be important, the major characteristics of the circulation can be described by barotropic dynamics. This is consistent with the barotropic nature of the flow measured by Lamberth and Nelson [1987]. In this chapter, we will only concentrate on the barotropic response of a coastal ocean to equatorward wind forcing. Baroclinic effects are expected to be secondary or localized [Graham and Largier, 1997] and will be investigated in future work. The aim of this chapter is to understand the processes controlling the pattern of flow detachment and eddy generation in the vicinity of Cape Columbine.


2 Model description

The numerical code is the barotropic part of the SCRUM 3D oceanic model from Rutgers University [Song and Haidvogel, 1994,Hedström, 1997]. The model is based on the hydrostatic and Boussinesq approximations. The barotropic component solves the vertically integrated momentum equation [Hedström, 1997] and in our case density variations are not taken into account. SCRUM conserves the first moments of u and v. This is accomplished by using the flux form of the momentum equations [Hedström, 1997]:
\begin{displaymath} \frac{\partial}{\partial t} (D\bar{u}) + \frac{\partial}{... ...t(\bar{u}\right)}{D}\right)+ \frac{\tau_x}{\rho} - r \bar{u} \end{displaymath} (1)


\begin{displaymath} \frac{\partial}{\partial t} (D\bar{v}) + \frac{\partial}{... ...t(\bar{v}\right)}{D}\right)+ \frac{\tau_y}{\rho} - r \bar{v} \end{displaymath} (2)

Where $\hat{\bigtriangleup}$ is the operator:
\begin{displaymath} \hat{\bigtriangleup}(A)=\frac{\partial}{\partial x}D\frac{... ...}+ \frac{\partial}{\partial y}D\frac{\partial A}{\partial y} \end{displaymath} (3)

The continuity equation takes the from:
\begin{displaymath} \frac{\partial \zeta}{\partial t} + \frac{\partial}{\part... ...ght) + \frac{\partial}{\partial y} \left(D\bar{v}\right) = 0 \end{displaymath} (4)

Where:
x is the along shore coordinate (positive towards the equator).
y is the cross-shore coordinate (positive towards the open ocean).
$\bar{u}$ and $\bar{v}$ are the vertically averaged flow velocity respectively in each coordinate direction.
$\zeta$ is the free surface elevation.
$D$ is the total water column depth, $D=H+\zeta$, where $H$ is the ocean depth.
$f$ is the Coriolis parameter, $f=2\Omega sin \phi$, where $\Omega$ is the Earth angular velocity and $\phi$ is the latitude. In our case, because the time scale O(10 days) and the length scales O(100 km) are small enough, we can assume a constant Coriolis parameter as explained by Kundu [1990]. $f=-7.707 \times 10^{-5}$ s$^{-1}$ at Cape Columbine.
$g$ is the Earth gravity acceleration, $g=9.81$ m.s$^{-2}$.
$\nu_4$ is the lateral biharmonic constant mixing coefficient (m$^4$.s$^{-1}$).
$r$ is the linear bottom drag coefficient (m.s$^{-1}$).
$\frac{\tau_x}{\rho}$ and $\frac{\tau_y}{\rho}$ are the kinematic surface momentum fluxes (wind stress) respectively in each coordinate direction (m$^2$.s$^{-2}$).
In order to preserve the mesoscale structures, a bi-harmonic operator parameterizes the horizontal viscosity. For the sake of simplicity and as suggested by Csanady [1982], the weak tidal currents are not resolved and the bottom stress is chosen to be proportional to the barotropic velocities. The linear bottom friction coefficient is initially fixed at a typical value found in the literature ( $r=3 \times 10^{-4}$ m.s$^{-1}$), but sensitivity tests have been conducted to explore the strong impact of this coefficient on the circulation.

Figure 2.1: The periodic analytical bathymetry implemented in the model.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/topoperiodic.eps,width=15cm}} \end{figure}

The regular grid has a 5 km resolution along shore and cross-shore. The coastline is represented by a free-slip wall in x=0 and its variations are modelized by masking the inshore grid points where the depth is less than 50 m [Hedström, 1997]. The most straightforward way to close the domain offshore and on the sides is the use of a periodical channel: a free-slip wall far beyond the shelf break and all the outflows (inflows) at the southern boundary being inflows (outflows) for the northern boundary. Those boundary conditions allow an along shore wind forced circulation and conserve mass. The presence of the shelf break should isolate the shelf circulation from the effects of the offshore wall [Csanady, 1978]. The bottom topography is represented by a set of analytical functions which retains the main topographical features, thus focusing attention on the effects of Cape Columbine and the widening shelf on the circulation. The bathymetry has been made periodic to allow the use of the periodic channel (figure 2.1). Nevertheless, this topography is still comparable for the first 300 km to the topography of St Helena Bay (see figure 1.3). The second cape on the right might perturb the solution at distances up to 300 km (the external Rossby radius) upwind of the cape, thus interfering with our area of interest. For this reason, the experiments are run with an along shore domain of 900 km. The atmospheric forcing is a constant wind stress parallel to the x-axis, accounting for the summer southeaster wind. The wind stress is uniform in space and constant in time for a given experiment. A set of runs with wind stress ranging from 0.02 N.m$^{-2}$ to 0.2 N.m$^{-2}$ are performed to investigate the effects of the intensity of the wind forcing.

3 Analytical expectations

To illustrate the basic mechanism for wind forcing in the coastal ocean, Brink [1998] solved a linearized along shore wind forced model where the spatial scales are small enough compared to the external Rossby radius of deformation to neglect divergence, and where the along shore scales are large compared to the cross-shore ones (boundary layer approximation, [Csanady, 1998]). This implies that the along shore flows are much stronger than the cross-shelf ones [Brink, 1998]. This is not true near Cape Columbine but it can be a good approximation further North. Thus, the results from this model can give us a scale for the mean along shore velocities and the sea surface slope to compare with the numerical experiment outputs. If the wind forcing is uniform, the along shore variations can be neglected and the barotropic equations of motion with linear bottom friction become [Brink, 1998,Csanady, 1998], using the same notations as in the previous paragraph:
\begin{displaymath} \frac{\partial \bar{u}}{\partial t} - f\bar{v}= \frac{\tau_x}{H \rho} -\frac{r\bar{u}}{H} \end{displaymath} (5)


\begin{displaymath} f\bar{u}= -g \frac{\partial \zeta}{\partial y} \end{displaymath} (6)


\begin{displaymath} \frac{\partial H \bar{v}}{\partial y}=0 \end{displaymath} (7)

The along shore velocities are forced by the along shore wind and the free surface remains in geostrophic equilibrium with the along shore velocities. In order to satisfy equation (2.7) and the fact that there is no cross-shore flow at the coastal boundary, the cross-shelf transport has to vanish everywhere [Brink, 1998]. Starting from rest at t=0 with a constant wind stress, we obtain from equation (2.5):
\begin{displaymath} \bar{u}=\frac{\tau_x}{\rho r}(1-e^{-\frac{rt}{H}}) \end{displaymath} (8)

If the maximum depth is 500 m, the solution is nearly stationary after 40 days with along shore velocities:
\begin{displaymath} \bar{u}=\frac{\tau_x}{\rho r} \end{displaymath} (9)

This result shows that bottom friction allows us to expect for the numerical experiments a steady solution after nearly 50 days with along shore velocities of the order of magnitude: $O(\frac{\tau_x}{\rho r})$, resulting from a balance between the wind stress and the bottom friction. For example if the wind stress value equal 0.1 N.m$^{-2}$, after 50 days the mean along shore velocities should be $O(0.33)$ m.s$^{-1}$.


4 The reference experiment

For the reference experiment, the constant wind stress value is fixed at $\tau=0.1$ N.m$^{-2}$ corresponding to the average wind stress measured in the area during upwelling seasons ( $\tau \sim 0.098 $ N.m$^{-2}$). The linear bottom friction coefficient value is r = $3 \times 10^{-4}$ m.s$^{-1}$ and the viscosity parameter is set to the lowest possible value to avoid numerical noise ( $\nu_4=1.5 \times 10^9$ m$^4$s$^{-1}$).

Figure 2.2: Barotropic velocities and free surface elevation for the reference experiment in the vicinity of Cape Columbine: a) day 10 (maximum velocity: 62 cm.s$^{-1}$, averaged velocity: 16 cm.s$^{-1}$), b) day 30 (maximum velocity: 69 cm.s$^{-1}$, averaged velocity: 25 cm.s$^{-1}$), c) day 50 (maximum velocity: 70 cm.s$^{-1}$, averaged velocity: 28 cm.s$^{-1}$ ),) d) day 50 for the same experiment with no shelf-edge (maximum velocity: 71 cm.s$^{-1}$, averaged velocity: 30 cm.s$^{-1}$). The horizontal coordinates are in kilometers and the greyscale range for the free surface elevation is in centimeters. The interval between the isolines is 5 cm.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/refexperiment.eps,width=15cm}} \end{figure}

Starting from rest, an equatorward current develops in response to the equatorward wind forcing. In balance with the along shore velocities, a cross shelf slope of the free surface is set up (figures 2.2-a, 2.2-b and 2.2-c). The solution becomes steady after 50 days (figure 2.2-c), the average sea surface slope value is between 0.17 and 0.38 centimeters per kilometer and the mean along shore velocities are approximately equal to $0.28 $ m.s$^{-1}$. Although the presence of a coastline and bottom topography variations induces a drag that reduces the along shore velocity, its value stays in the same order of magnitude as in the analytical solution. Four scaling parameters allow us to compare the results with other studies:
  1. the Rossby number,
    \begin{displaymath} Ro=\frac{\hbox{advection}}{\hbox{Coriolis}}=\frac{U}{f L}. \end{displaymath} (10)

  2. Because of the use of a biharmonic operator, the Reynolds number takes the form:
    \begin{displaymath} Re_{biharm}=\frac{\hbox{advection}}{\hbox{viscosity}}=\frac{UL^3}{\nu_4} \end{displaymath} (11)

  3. The Reynolds number associated with bottom friction,
    \begin{displaymath} Re_{friction}=\frac{\hbox{advection}}{\hbox{bottom friction}}=\frac{UH_0}{rL}, \end{displaymath} (12)

    which is equivalent to the 'island wake parameter' P introduced by Wolanski et al. [1984]. A scaling analysis shows that it is equivalent to the $\lambda$ parameter defined by Becker [1991] (The same analysis shows that the lateral boundary layer thickness defined by Becker [1991] is in our case: $\delta=\frac{H^\frac{1}{4}\nu_4^\frac{1}{4}}{r^\frac{1}{4}L}$).
  4. The Ekman number,
    \begin{displaymath} E_V=\frac{\hbox{bottom friction}}{\hbox{Coriolis}}=\frac{r}{f H_0}. \end{displaymath} (13)

Where,
U is a characteristic velocity, $O(0.28)$ m.s$^{-1}$.
f is the Coriolis parameter, $-7.707 \times 10^{-5}$ $ s^{-1}$.
L is a characteristic length scale, for example the size of the cape: $100$ km.
$\nu_4$ is the viscosity parameter, $1.5 \times 10^{9}$ $ m^{4}.s^{-1}$.
$H_0$ is a characteristic depth, $150$ m on the shelf.
$r$ is the bottom friction parameter, $3 \times 10^{-4}$ $ m.s^{-1}$.
From these values we obtain: $Ro=3.6 \times 10^{-2}$, $Re_{biharm}=1.9 \times 10^{5}$, $Re_{friction}=1.4$ and $E_V=2.6 \times 10^{-2}$. Hence, Coriolis acceleration and the resultant pressure gradient are the main driving forces and at the first order, advection and bottom friction are the two important terms controlling the flow. Further, those terms show that this experiment is in the attached-cyclonic-eddy regime [small Rossby number, small Burger number (null in our case)] when the cape is on the left (in the Northern Hemisphere) of the study of Boyer and Tao [1987a]. $Re_{friction}=O(1)$ shows that this regime corresponds to the stable wake regime of Wolanski et al. [1984a]. $\lambda=1.4$ and $\delta=5.2 \times 10^{-2}$ are at the limit between the detachment and the no-detachment regime of Becker [1991, see figure 7]. The small Rossby and Ekman numbers correspond to the stable wake regime described by Boyer at al. [1987b, see figure 3]. The reference experiment shows that the bottom topography associated with Cape Columbine generates three main features:
  1. Attached cyclonic eddy: before day 10, the flow detaches from Cape Columbine and generates an attached cyclonic eddy (figure 2.2-a, x=50 km y=100 km). The size of the eddy is approximately 60 km at day 10 and expands to a size of 110 km by day 50 (figures 2.2-b and 2.2-c). The presence of this stationary attached-cyclonic-eddy is in agreement with the in-situ measurements of Holden [1985] and the recent averaged ADCP data of Boyd and Oberlholster [1994]. It can be compared with the schematic representation of the currents in St Helena Bay (figure 1.13) made by Shannon [1985]. This is also in agreement with the experimental results of Boyer and Tao [1987a] for the cape on the left (Northern Hemisphere), the smallest Burger number and $Ro=O(0.02)$. Further, the presence of the cyclone and the strength of the velocities match the results of the 3D model of Oey [1996, see figure 12 and figure 13]. In the Santa Barbara Channel model, which includes baroclinic processes, Oey [1996] applied an equatorward wind stress similar to the forcing of the reference experiment. The bottom fiction is quadratic ( $C_d=2.5 \times 10^{-3}$) and the grid resolution is 5/3 km. The equatorward flow associated with a pronounced coastal upwelling is comparable with the barotropic velocities obtained here, and forms a stationary cyclonic eddy in the lee of Point Conception. The size of this eddy is approximately half the size of our barotropic eddy. This discrepancy might be due to the depth of the shelf (300 m versus 150 m in our experiment), the width of the shelf (50 km versus 150 km in our experiment) or the presence of the Santa Barbara islands that might block the cyclone extension.
  2. Influence of the shelf break: the steep shelf edge offshore (figure 2-b, y=150 km) apply a strong topographic constraint on the flow, prohibiting cross-topographic currents. Thus, mass conservation implies that the velocities between Cape Columbine and the shelf break (figures 3-a, 3-b and 3-c, for x=0 km and y=0 to 150 km) are stronger than in the other parts of the shelf. This can affect the detachment process. An experiment with no shelf break (figure 2.2-d) shows that at day 50, the size of the cyclonic eddy is approximately 60 $\%$ the size of the eddy in the reference experiment. The value of the along shore velocities near the tip of the cape is 80 $\%$ the value of the velocities in the reference experiment. Further, vortex squashing produces an anticyclonic bend on the shelf edge (figure 2.2-c, x=180 km, y=150 km) and a divide in the currents (figure 2.2-c, x=110 km, y=150 km). Its location corresponds approximately to the location of the Columbine divide described by Shannon [1985a], and it can be an explanation of this phenomenon.
  3. Upstream blocking: the artificial cape on the right seems to have no influence on the detachment processes, but it produces weak near shore velocities on the right of the shelf (figure not shown). This effect can be felt up to 300 km [O(external Rossby radius of deformation)] upstream of the cape. Because we use a large domain (900 km along shore), this phenomenon does not affect our area of interest. This has been tested using a smaller domain (600 km along shore) and the similarities between the solutions validate the use of the periodic channel.

5 Diagnostic analysis

1 Dynamical balance

To understand the processes involved in the cyclone generation, the dynamical terms have been computed from the model outputs using equations (2.1) and (2.2). They have been rewritten in the form of a sum of acceleration vectors as follow:
\begin{displaymath} \underbrace{\frac{1}{D}\frac{\partial D\vec{\bar{u}}}{\par... ...brace{+ \frac{\vec{\tau}}{\rho D}}_{\overrightarrow{Forcing}} \end{displaymath} (14)

Where $\vec{\bar{u}}$ is the vertically averaged flow velocity, $\vec{\tau}$ is the wind stress and $\vec{k}$ is a vertical unit vector. Because $\frac{\zeta}{H}\sim O(10^{-3})\ll 1$ and $\frac{\frac{\partial D}{\partial x}}{D}\sim O(10^{-5})\ll 1$, the terms of equation (2.14) are respectively equivalent to the terms of the vertically averaged momentum equation:
\begin{displaymath} \underbrace{\frac{\partial \vec{\bar{u}}}{\partial t}}_{\... ...brace{+ \frac{\vec{\tau}}{\rho D}}_{\overrightarrow{Forcing}} \end{displaymath} (15)

Figure 2.3: Accelerations (m.s$^{-2}$) applied to the water particles for the reference experiment at day 50 (the contours represent the free surface elevation). a) At zero order, the balance between pressure gradient and Coriolis acceleration dominates the dynamics. b) First order tendency terms: by summing the pressure gradient and the Coriolis acceleration vectors, the zero order balance can be filtered out and the first order terms can be studied. Note that this terms are 20 times smaller than the zero order ones.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/balance.eps,width=15cm}} \end{figure}

It appears (figure 2.3-a) that, at zero order, the balance between pressure gradient and Coriolis acceleration overwhelms the other accelerations. This is in agreement with the small Rossby and Ekman numbers found previously. The Coriolis acceleration pushes the flow offshore, creating an Ekman transport away from the coast, in agreement with the upwelling models. Because of mass conservation, this transport produces a down slope toward the coast in the free surface, generating a pressure gradient in balance with the Coriolis acceleration [Csanady, 1982]. By summing the pressure gradient and the Coriolis acceleration vectors, we can filter out the zero order balance and get the equilibrium at the first order [figure 2.3-b, note that these accelerations are 20 times smaller than in figure 2.3-a, in agreement with Ro and Ek $\sim O(10^{-2})$]. At this order, most of the terms start to be important. In the following, we call the sum of the pressure gradient and the Coriolis acceleration: the ageostrophic pressure gradient.
  1. Viscous terms are of an order of magnitude lower than the other ones. They have some relative importance near the tip of the cape (figure 2.3-b, x=40 km, y=100 km). As explained by Becker [1991], viscosity has to be small for detachment to occur.
  2. Around the external part of the eddy (figure 2.3-b, x=50 to 130 km, y=70 to 130 km), there is a competition between advection and ageostrophic pressure gradient. It appears that around the eddy (figure 2.3-b, x=140 km, y=100 km), the radius of curvature of the flow is approximately $R=85$ km and the tangential velocities are about $V=0.5$ m.s$^{-1}$. Then the normal acceleration is $\gamma=\frac{V^2}{R}\simeq 2.9 \times 10^{-6}$ m.s$^{-2}$, which is close to the value of the advective acceleration ( $\simeq 3.4 \times 10^{-6}$ m.s$^{-2}$). This shows a cyclo-geostrophic equilibrium around the eddy. This advective acceleration, forcing the water particles away from the cape, seems to be responsible for the detachment.
  3. Away from perturbations (figure 2.3-b, for example y$>$220 km), the wind forcing and bottom friction relative equilibrium controls the along shore velocities as in the analytical solution.
  4. Another equilibrium occurs where the velocities are weak: inside the eddy (figure 2.3-b, x=75 km, y=70 km) and in the upstream blocking area (x=650 to 850 km, y=0 to 60 km, figure not shown). In those places, the Coriolis acceleration is weak and a static wind stress-pressure gradient balance prevails.


2 Vorticity balance

To refine the study, curls of the acceleration vector fields extracted from equation (2.14) are computed to obtain the terms of the equation for the vertically averaged vertical component of vorticity, $\bar{\xi}$. Their physical meaning can be illustrated by deriving the curl of equation (2.15) and introducing the continuity equation (2.4):
$\displaystyle \underbrace{\frac{\partial \bar{\xi}}{\partial t}}_{ \nabla \wedge \overrightarrow{Tendency}}$ $\textstyle =$ $\displaystyle \underbrace{-\bar{u}\frac{\partial \bar{\xi}} {\partial x} -\bar{... ...+ \frac{\zeta}{\partial t} \right) }_{\nabla \wedge \overrightarrow{Advection}}$  
    $\displaystyle \underbrace{+\frac{f}{D} \left( \bar{u}\frac{\partial D}{\partial... ...}+ \frac{\zeta}{\partial t} \right) }_{\nabla \wedge \overrightarrow{Coriolis}}$  
    $\displaystyle \underbrace{+ \nu_4 \left(\frac{\partial^4 \bar{\xi}}{\partial x^... ...4 \bar{\xi}}{\partial y^4} \right) }_{\nabla \wedge \overrightarrow{Viscosity}}$  
    $\displaystyle \underbrace{-\frac{r \bar{\xi}}{D}+\frac{r}{D^2} \left(\bar{v}\fr... ...frac{\partial D}{\partial y}\right) }_{\nabla \wedge \overrightarrow{Friction}}$  
    $\displaystyle \underbrace{+ \frac{1}{\rho D} \left(\frac{\partial \tau_y}{\part... ...frac{\partial D}{\partial y} \right) }_{\nabla \wedge \overrightarrow{Forcing}}$ (16)

Figure 2.4: Tendency terms of the vorticity equation ($s^{-2}$), computed from the dynamical terms at day 50 for the reference experiment: a) advection of vorticity, b) curl of bottom friction, c) vortex stretching, d) lateral viscous dissipation, e) wind stress slope torque, f) curl of the pressure gradient. The horizontal coordinates are in kilometers and the greyscale range represents the values in $s^{-2}$, multiplied by $10^{-10}$, except for the values of curl of the pressure gradient which are multiplied by $10^{-24}$ to show the validity of the computation.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/vorticity.eps,width=15cm}} \end{figure}

Figure (2.4) exhibits the relative importance of the vorticity equation terms in the vicinity of Cape Columbine. The advection of vorticity (figure 2.4-a) contains as well the vortex stretching associated to relative vorticity, but this term is one order of magnitude smaller than the other terms. The curl of bottom friction (figure 2.4-b) contains both the bottom dissipation and the 'friction slope torque' and is thus not proportional to relative vorticity in shallow waters. The vortex stretching (figure 2.4-c) is the vortex stretching associated to planetary vorticity. The unit of these terms is $s^{-2}$ and for clarity the values have been multiplied by $10^{10}$ (figures 2.4-a, 2.4-b, 2.4-c, 2.4-d and 2.4-e). To validate the approximations made in deriving the tendency terms, the curl of the pressure gradient has been also portrayed, multiplied by $10^{24}$ (figure 2.4-f). We can then extract a number of equilibriums:
  1. The balance between advection of vorticity and vortex stretching occurs almost everywhere where the slopes are strong: on the shelf break (figures 2.4-a and 2.4-c, y=150 to 200 km) and in the bay (figures 2.4-a and 2.4-c, y=0 to 60 km)
  2. Around the external part of the eddy, where the shelf is relatively flat and the dynamics are cyclo-geostrophic (see previous section), the curl of bottom friction balances the advection of vorticity (figures 2.4-a and 2.4-b, x=50 to 130 km, y=70 to 130 km). This balance seems to follow the contour of the eddy, and might control its extension.
  3. As pointed out in the previous section, the lateral viscosity has some importance near the tip of Cape Columbine. In figure 2.4-d, the viscous boundary layer is clearly seen for x=0 to 50 km and y=100 to 110 km. Past the cape, there is a detachment of this boundary layer (figure 2.4-d, x=50 to 80 km, y=115 km).
  4. In shallow waters, the friction and wind stress slope torques start to have a strong influence and seem to be in balance for x$>$80 km and y=0 to 10 km (figures 2.4-b and 2.4-e).
We can expect from the second point that the size of the eddy is controlled by the balance between advection of vorticity and curl of bottom friction. From this, we can extract a characteristic length scale l:
\begin{displaymath} \frac{H_0U^2}{l^2}\hbox{ (advection of vorticity)} \sim \frac{r U}{l}\hbox{ (curl of bottom friction)} \end{displaymath} (19)


\begin{displaymath} \Rightarrow l\sim \frac{H_0 U}{r} \end{displaymath} (20)

This length scale can be seen as a frictional e-folding distance and is equivalent to the eddy length scale $R_l$ described by Pattiaratchi et al. [1986] and Wolanski et al. [1984]. It is also equivalent to the frictional length scale $l_f$ tested by Signell and Geyer [1991]. Taking the analytical result (equation 2.9) for the characteristic velocities, the equation (2.20) becomes:
\begin{displaymath} l\sim \frac{H_0 \tau}{\rho r^2} \end{displaymath} (21)

That gives us a characteristic eddy length scale which is a function of the wind stress and of the linear bottom friction parameter. This length scale can be compared to the size of the eddy. In this example, $H_0$ = 150 m, $\frac{\tau}{\rho}=10^{-4} $ m$^2$.s$^{-2}$, and $r=3 \times 10^{-4}$ m.s$^{-1}$ so that $l=167$ km (the difference with the model outputs can be seen in figure 2.2).

6 Sensitivity tests

To test the effects of the surface wind stress, the bottom friction coefficient, the size of the cape, the grid resolution, the viscosity coefficient and the domain size on the recirculation process, several numerical experiments are conducted by varying one parameter at a time. Within realistic parameter values, the grid resolution and the viscosity do not have much effect on the cyclone generation. However, the nature of the cyclone does depend on bottom friction, wind and the size of the cape.

1 Influence of the wind stress

Using the same parameters as in the reference experiment, ten experiments are run with different along shore wind stress values, varying from $0.02 $ N.m$^{-2}$ to $0.2 $ N.m$^{-2}$. These wind stress values correspond to wind velocities at 10 m elevation of about $4$ m.s$^{-1}$ to $11$ m.s$^{-1}$ .

Figure 2.5: Barotropic velocities and sea surface elevation at day 100 for different values of the along shore wind stress. a) 0.05 $ N.m^{-2}$ (maximum velocity: 38 cm.s$^{-1}$, averaged velocity: 12 cm.s$^{-1}$), b) 0.15 N.m$^{-2}$ (maximum velocity: 100 cm.s$^{-1}$, averaged velocity: 36 cm.s$^{-1}$), c) 0.2 N.m$^{-2}$ (maximum velocity: 130 cm.s$^{-1}$, averaged velocity: 46 cm.s$^{-1}$). The horizontal coordinates are in kilometers and the greyscale range for the free surface elevation is in centimeters. The interval between the isolines is 2 cm for (a), 6 cm for (b), and 8 cm for (c).
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/sensibvent.eps,width=10cm}} \end{figure}

In all cases, the solution is steady after 50 days and the order of magnitude of the mean velocities is in agreement with the analytical solution (figure 2.5). The characteristic velocities range from $0.05$ m.s$^{-1}$ to $0.4$ m.s$^{-1}$. The associate scaling parameters are, using the same characteristic numbers as in section 2.4: This corresponds to the same regime as for the reference experiment: small Rossby and Ekman numbers, large viscous Reynolds number and friction Reynolds number $\sim O(1)$. The results of Becker [1991] tell us that if $Re_{friction}<1$ detachment should not occur, but the presence of bottom topography induces another detachment process, resulting from input of vorticity by vortex stretching, friction slope torque or wind stress slope torque. For example a detachment process can be explained by absolute vorticity conservation. If a water particle follows an isobath cyclonically bent with the coast on the left, like an equatorward current around Cape Columbine, it acquires cyclonic vorticity, $\xi < 0$. For the sake of absolute vorticity conservation ( $\frac{f+\xi}{H}=cst.$), the particle is displaced offshore onto a deeper isobath (because here, $f<0$). The particle needs a steeper turn to go back to the first isobath, hence stronger negative relative vorticity. This moves the particle into deeper waters and so on, and detachment occurs. This shows that the presence of bottom topography can favor detachment of the flow for an equatorward eastern boundary current past a cape. On the contrary, bottom topography can be a stabilization process for a poleward eastern boundary current. Hence, Becker's [1991] criterion does not apply in our case, and we still have detachment even for a small frictional Reynolds parameter (figure 2.5-a).

Figure: Along shore extension of the eddy at day 50 as a function of the wind stress. Comparison with the characteristic length scale: $l=\frac{H_0 \tau}{\rho r^2}$.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/eddysizewind.eps ,width=10cm}} \end{figure}

To characterize the size of the eddy as a function of wind stress, we choose to measure the distance between the tip of the cape and the place where the flow is the strongest towards the coast. The along shore eddy length given by this method is proportional to the wind stress (figure 2.6) and the trend is in agreement with the characteristic length scale found in section 6.2 (figure 2.5-a, 2.5-b, 2.5-c and 2.6). There is though an approximately constant discrepancy ($\sim 50$ km) between the numerical results and the characteristic length scale, that reveals that something is missing in deriving the analytical length scale. Within the eddy, the flow is weak and the along shore velocities are positive for $y<20$ km and $y>100$ km and negative around $y=60$ km. The constant nature of this pattern might be related to the slope of the shelf or to the size of the cape.

2 Influence of the linear bottom friction coefficient

Ten more numerical experiments with wind stress fixed at $0.05$ $ N.m^{-2}$ were conducted to explore the effects of the linear bottom friction coefficient. This parameter varies from $0.5 \times 10^{-4}$ m.s$^{-1}$ to $5 \times 10^{-4}$ m.s$^{-1}$. For low bottom friction values, the solution was not yet stationary at day 50, so day 100 is analyzed.

Figure 2.7: Barotropic velocities and sea surface elevation at day 100 forced by a constant wind stress (0.05 N.m$^{-2}$), for different values of the linear bottom coefficient r: a) $r=0.5 \times 10^{-4}$ m.s$^{-1}$ (maximum velocity: 129 cm.s$^{-1}$, averaged velocity: 49 cm.s$^{-1}$), b) $r=2.5 \times 10^{-4}$ m.s$^{-1}$ (maximum velocity: 35 cm.s$^{-1}$, averaged velocity: 14 cm.s$^{-1}$), c) $r=5.0 \times 10^{-4}$ m.s$^{-1}$ (maximum velocity: 19 cm.s$^{-1}$, averaged velocity: 8 cm.s$^{-1}$). The horizontal coordinates are in kilometers and the greyscale range for the free surface elevation is in centimeters. The interval between the isolines is 10 cm for (a), 4 cm for (b), and 2 cm for (c).
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/sensibfric.eps,width=10cm}} \end{figure}

The mean velocities are again in the same order of magnitude as predicted by the analytical calculations (figure 2.7). As in the previous section, the different characteristic velocities and characteristic parameters are derived. For increasing r, they follow the range: Again the Rossby and Ekman numbers are small and the viscous Reynolds number is very large. But for this set of experiments, the frictional Reynolds number which is the most important in the control of the detachment process, varies through two orders of magnitude. Because this number is equivalent to the island wake parameter of Wolanski et al. [1984], we can expect a domination of bottom friction for low values and instabilities for large values. For low bottom friction, the along shore size of the eddy does not match the characteristic length scale and three small eddy cells keep on moving inside a global structure(figure 2.7-a). The solution is not yet steady at day 100 and different dynamical balances should exist. It appears that in this experiment, we are in a eddy shedding regime.

Figure: Along shore extension of the eddy at day 100 as a function of the linear bottom friction parameter r. Comparison with the characteristic length scale: $l=\frac{H_0 \tau}{\rho r^2}$.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/eddysizefric.eps,width=10cm}} \end{figure}

For $r>10^{-4}$ m.s$^{-1}$ and corresponding $Re_{friction} < 7.5$, the eddy size does scale with $r^{-2}$ (figure 2.8), but as in the previous section this size is not in agreement with the characteristic length scale. This analysis confirms the key role of bottom friction in controlling the detachment processes and the importance of parameterizing it carefully.


3 Influence of the size of the cape

To explore the influence of the size of the cape on the detachment process, a set of experiments has been conducted with a size of the cape ranging from 25 km to 150 km and a wind stress ranging from 0.025 N.m$^{-2}$ to 0.2 N.m$^{-2}$. The cross-shelf width of the periodical channel has been extended up to 300 km in order to contain the biggest capes. The bathymetry has been computed in such a way that the distance between the tip of the capes and the shelf break remains identical between all the experiments. This implies that the width of the shelf in the bay varies with the size of the cape (figure 2.9). The other parameters remain identical to the reference experiment.

Figure 2.9: Bottom topography implemented in the model for (a) a cape of 150 km and (b) a cape of 25 km. The horizontal coordinates are in kilometers. The depths scale is in meters.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/topocapes.eps,width=15cm}} \end{figure}

In every case, the solution is stationary after 50 days and remains in the attached cyclonic eddy regime. The free surface elevation and the currents at day 100 for a wind stress of 0.1 N.m$^{-2}$ reveal large differences depending on the size of the cape (figure 2.10):
  1. In the bay, the drag induced by the presence of the cape produces an averaged weaker current as the size of the cape increases. This averaged velocity ranges from 32 cm.s$^{-1}$ for a cape of 25 km to 22 cm.s$^{-1}$ for a cape of 50 km (for a wind stress of 0.1 N.m$^{-2}$).
  2. The shelf break acting as an offshore barrier, mass conservation implies a stronger current at the tip of the cape for a larger cape than for smaller ones. The velocities at the tip of the cape range from 56 cm.s$^{-1}$ for a cape of 25 km to 80 cm.s$^{-1}$ for a cape of 150 km (for a wind stress of 0.1 N.m$^{-2}$).
  3. These large differences in velocity, imply that a scaling of the eddy length only dependent on the wind stress is no longer valid: large variations in the size of the eddy depends on the size of the cape (see figure 2.10 and figure 2.11).

Figure 2.10: Barotropic velocities and sea surface elevation at day 100 forced by a constant wind stress (0.1 N.m$^{-2}$), for different sizes of the cape. a) Cape of 25 km (maximum velocity: 56 cm.s$^{-1}$, averaged velocity: 32 cm.s$^{-1}$). b) Cape of 75 km (maximum velocity: 68 cm.s$^{-1}$, averaged velocity: 29 cm.s$^{-1}$). c) Cape of 150 km (maximum velocity: 80 cm.s$^{-1}$, averaged velocity: 22 cm.s$^{-1}$). The horizontal coordinates are in kilometers and the greyscale range for the free surface elevation is in centimeters. The interval between the isolines is 8 cm.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/sensibcapes.eps,width=10cm}} \end{figure}

The difference between the size of the eddy and the frictional characteristic length scale increases when the size of the cape decreases (figure 2.11). The balance advection / bottom friction is less valid for small capes. This can be explained by the reduction of the importance of the flat area in the lee of the cape for small capes (zero for a cape of 25 km, see figure 2.9). Because the sloping areas are relatively more important, the vortex stretching term gain importance relative to the bottom friction term when the size of the cape decreases. For a cape of 25 km, the size of the eddy as a function of the wind stress follows a totally different trend than for the other experiments. For large capes ($>$ 100 km), the characteristic length scale gives a good approximation for the size of the eddy as a function of the wind stress.

Figure 2.11: Size of the attached cyclonic eddy as a function of the wind stress and the size of the cape.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/tourbsize.eps,width=10cm}} \end{figure}

The size of the cape affects dramatically the recirculation patterns in the bay. The length scale derived in section (2.5.2) does not take the size of the cape into account. Hence, further investigations will be necessary to improve the understanding of the circulation in the bay.

7 An upwelling plume ?

In section 1.4.4, it has been emphasized that a characteristic pattern in St. Helena Bay is the development of an upwelling plume from the tip of Cape Columbine. Numerous studies have been conducted to explain the presence of this upwelling plume due to the cyclonic wind stress curl measured in the lee of the cape [Jury, 1985a,Jury, 1985b,Jury, 1988,Kamstra, 1985,Taunton-Clark, 1985]. Although cyclonic wind stress curl might be important in this area, and locally enhances upwelling, none of these studies were able to model or quantify the impact of wing stress curl on the upwelling structure. This has been done, using a one and a half layer reduced-gravity model for the upwelling of Point Arena on the US West Coast by Enriquez and Friehe [1995]. Although their model produced an enhanced upwelling due to the presence of cyclonic wind stress curl, it didn't generate a marked upwelling plume extending from the cape. In this section, I would like to propose the 'barotropic coastal flow detachment' as another possible process for the generation of the Cape Columbine upwelling plume. To illustrate this hypothesis, a 'barotropic' tracer has been introduced in the model to roughly simulate the sea surface temperature. It follows an advection equation:
\begin{displaymath} \frac{\partial T}{\partial t} + \bar{u}\frac{\partial T}{\partial x} + \bar{v}\frac{\partial T}{\partial y} = 0 \end{displaymath} (22)

To avoid the generation of numerical noise, biharmonic viscosity has been added. At the model initialization, the value of the tracer is 20 everywhere, except on a narrow band of 10 km at the tip of the Cape (x $<$ 30 km, y =110 km) where it has a value of 10, accounting for the coastal upwelling. During all the simulation, the tracer is nudged towards 10 in this band. Although the offshore Ekman drift is not taken into account, this experiment is performed to simulate the characteristic sea surface temperature pattern observed in St Helena Bay.

Figure 2.12: 'SST' tracer after 30 days in a experiment forced by a wind stress of 0.05 N.m$^{-2}$. The advection of low values of the tracer by the detached flow simulate the Cape Columbine upwelling plume. The interval between the isolines is 2.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/plume.eps,width=10cm}} \end{figure}

The tracer has been tested in a simulation forced by a constant wind stress of 0.05 N.m$^{-2}$, the other parameters remaining as in the reference experiment. After 30 days, even with this low wind forcing, a tongue of low 'SST' extends from the tip of the cape following the cyclonic eddy (figure 2.12). The shape of this tongue can be compared to a sea surface portrayal of the Cape Columbine upwelling plume (figure 1.10). Thus, the horizontal advection of water upwelled South of Cape Columbine around the attached cyclonic eddy can be an explanation of the development of the Cape Columbine upwelling plume. Wind stress curl is not necessary for the generation of this plume. Another interesting feature is the patch of low tracer value at the coast at 120 km downstream from the cape (figure 2.12). This seems to be related to wave like features visible in the currents downstream of the main perturbation. Perhaps, there is here a possible explanation for the presence of the Hondeklip Bay upwelling center.


8 Standing coastal trapped waves in the lee of Cape Columbine

The discrepancies between the characteristic length scale and the size of the eddy shown on figure (2.11) suggest that a balance between advection and bottom friction is not enough to describe the flow patterns in the Bay. Other processes may also contribute actively. Two indications reveal that the circulation observed in the lee of the cape might be related to standing coastal trapped waves. Firstly, a wave-like pattern can be seen along the coast, downstream of the cape (figure 2.2 for x $>$ 200 km). Secondly, if we reverse the forcing and the topography orientation (i.e. for a poleward eastern boundary current, figure 2.13), there is no more detachment of the flow. This dissymetry is characteristic of a poleward propagating process that can create a standing perturbation if it is advected by a mean equatorward flow. Two candidates correspond to the 'poleward propagation along a coastal eastern boundary' requirement: Kelvin waves and coastal trapped waves.

Figure 2.13: Barotropic velocities and free surface elevation at day 100 for a 'reversed' experiment where the wind stress of 0.05 N.m$^{-2}$ is poleward. The topography orientation is also reversed. The maximum velocity is 38.5 cm.s$^{-1}$ and the averaged velocity is 14 cm.s$^{-1}$. The horizontal coordinates are in kilometers and the greyscale range for the free surface elevation is in centimeters. The interval between the isolines is 2 cm.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/poleward.eps,width=10cm}} \end{figure}

1 Brief review on coastal trapped waves

For the last three decades, coastal trapped waves have been studied intensively. Buchwald and Adams [1968], derived the linear barotropic equation of motion on a f plane for along shore propagating waves. The variation of topography being of importance across the shelf, they kept the non-linear terms in the vertically integrated equation of continuity. Because they were looking for phenomena with spatial scales much smaller than the external radius of deformation, the rigid lid approximation was made. In this case, the cross-shelf structure of along shore propagating waves on an exponential topography follows a linear second order differential equation. In the Northern hemisphere, those free waves only propagate keeping the coast on their right (on their left in the southern hemisphere). The generation of these waves by atmospheric forcing has been analyzed by Gill and Schumann [1974] and they found that resonance can occur with propagating wind perturbations. By keeping the horizontal divergence, Huthnance [1975], derived 3 different kinds of barotropic trapped waves over the continental shelf: the Kelvin waves that propagate at the speed of long gravity waves, the edge waves with frequencies higher than f, and the coastal trapped waves of sub inertial frequencies. It happened that whereas edge waves may travel along the shelf in either direction, Kelvin and coastal trapped waves progress along the shelf in a cyclonic sense about the deep sea. There is a frequency at which the group velocity of coastal trapped waves vanishes. At this frequency, the wave energy cannot propagate along the shelf [Huthnance, 1975]. By including stratification, Huthnance [1978] showed analytically and numerically that the coastal trapped wave frequency increases with the measure of stratification. For weak stratification (small Burger number), these waves take the form of barotropic continental shelf waves, whereas for large stratification they take the form of internal Kelvin waves. For large along shore wavenumbers they take the form of bottom-trapped waves. The presence of a mean along shore flow can alter considerably the free wave properties by Doppler shifting and change in the background vorticity [Brink, 1991,Huthnance, 1981]. Martell and Allen [1979] have studied the generation of continental shelf waves by small along shore variations in bottom topography. They used a perturbation method and found that shelf waves can be forced by the interaction of a wind stress forced current with small topographic disturbances. The advective effects of the unperturbed velocities are of importance, and lee waves form when their phase velocity opposes the advective velocities. The generation of lee waves is accompanied by a drag on the topographic obstacle. The influence of the presence of a cape in scattering coastal trapped waves has been explored analytically in the barotropic case [Wilkin and Chapman, 1987] and numerically when stratification has been included [Wilkin and Chapman, 1990]. In the barotropic case, reflection occurs when a wave encounters a narrowing shelf. When the shelf widens, the energy is transfered into higher modes and a shadow zone of very weak currents is present in the vicinity of the cape. Its extent increases with the size of the cape and decreases as the inverse of the wave frequency [Wilkin and Chapman, 1987]. When stratification is added, scattering is found to be amplified and intensification of the flow occurs within the scattering region [Wilkin and Chapman, 1990]. The influence of the advection by a mean current has not been taken into account in these last studies. Coastal trapped waves have been observed along the coast of South Africa from tide gauges and current meters measurements. They propagate at a celerity ranging from 4.2 m.s$^{-1}$ to 6.7 m.s$^{-1}$. They induce variations in sea level up to 50 cm and strong current reversals along the South Coast. The Agulhas current damps dramatically wave propagation in the South-East [Schumann and Brink, 1990]. Large scale stationary features of the Benguela front observed from satellite imagery have been related to the existence of barotropic shelf waves; but their standing nature has not been explained [Shannon, 1985]. The ingredients for standing lee shelf waves generation are present in the barotropic numerical experiments conducted in this chapter. The wave guide is formed by the shelf and the shelf edge slopes, the interactions between the wind forced current and Cape Columbine can provide the forcing mechanism, and the mean equatorward current allows the presence of a standing solution. Hence, the recirculation process in the lee of Cape Columbine could be related to the presence of lee shelf waves.


2 Wave lengths selected by a mean along shore current

The problem can be summarized has follow: what are the effects of the interactions between a large cape like Cape Columbine and a mean wind forced along shore current, on the circulation on the shelf in the lee of the cape ? The interactions between a cape and a mean along shore current of constant velocity $U_0$ are approximately equivalent to the interactions between a cape that moves in the opposite direction (with a velocity $-U_0$) and a fluid at rest. If $(x,y,z,t)$ are the Eulerian coordinates in the fix referential and $(x_1,y_1,z_1,t_1)$ are the coordinates in the referential attached to the moving cape, the temporal derivations have to be rewritten in the moving frame:
\begin{displaymath} \frac{\partial}{\partial t}= \frac{\partial}{\partial x_1... ...rtial x_1}{\partial t}}_{U_0}+ \frac{\partial}{\partial t_1} \end{displaymath} (23)

For an observer moving with the cape, in the absence of wind stress, the vertically averaged momentum equation (2.15) and the continuity equation (2.4) take the form (removing the subscripts for the coordinates):
$\displaystyle \frac{\partial \vec{\bar{u}}}{\partial t} +U_0 \frac{\partial \ve... ...\partial x} +(\vec{\bar{u}}.\nabla)\vec{\bar{u}} +f\vec{k} \wedge \vec{\bar{u}}$ $\textstyle =$ $\displaystyle - g\nabla \zeta + \nu_4 \nabla^4 \vec{\bar{u}} - \frac{r\vec{\bar{u}}}{D}$  
$\displaystyle \nabla(D \vec{\bar{u}}) + \frac{\partial \zeta}{\partial t} +U_0 \frac{\partial \zeta}{\partial x}$ $\textstyle =$ $\displaystyle 0$ (24)

With the variables $\left\{ x , y , t, D, \vec{\bar{u}}, \zeta \right\}$ made dimensionless by the characteristic values defined in section (2.4) $\left\{ L, L, T, H_0, U, U, \frac{fUL}{g} \right\}$ ($T$ is a characteristic time scale, $\sim 50$ days), equations (2.24) leads to:
$\displaystyle \frac{1}{fT} \frac{\partial \vec{\bar{u}}}{\partial t} + \frac{U_... ...rtial x} +R_0 (\vec{\bar{u}}.\nabla)\vec{\bar{u}} +\vec{k} \wedge \vec{\bar{u}}$ $\textstyle =$ $\displaystyle - \nabla \zeta + E_H \nabla^4 \vec{\bar{u}} - \frac{E_v}{H+\frac{R_0 L^2}{{\cal R}^2}\zeta} \vec{\bar{u}}$  
$\displaystyle \nabla \left[ \left(H+\frac{R_0 L^2}{{\cal R}^2}\zeta \right) \ve... ...l t} +\frac{U_0 R_0}{U}\frac{L^2}{{\cal R}^2} \frac{\partial \zeta}{\partial x}$ $\textstyle =$ $\displaystyle 0$ (25)

where $R_0$ and $E_V$ are the Rossby and the vertical Ekman numbers defined in section (2.4), $E_H=\frac{\nu_4}{f L^4}$ is the horizontal Ekman number associated to the biharmonic operator, and ${\cal R}=\frac{\sqrt{gH}}{f}$ is the external Rossby radius of deformation. In the regime of parameters of the reference experiment (section 2.4), $fT$, $R_0$, $E_V$ and $E_H$ are small compared to 1. ${\cal R} \sim $ 400 km is greater than the characteristic length scale $O$(100 km) of our problem, allowing the use of the rigid lid approximation. Although no constraint is given on the value of $U_0$ in comparison to $U$, equations (2.25) are linearized keeping the term of advection of momentum by $U_0$. Hence the dimensional remaining equations of motion are:
$\displaystyle U_0 \frac{\partial \vec{\bar{u}}}{\partial x} + f \vec{k} \wedge \vec{\bar{u}}$ $\textstyle =$ $\displaystyle - g \nabla \zeta$  
$\displaystyle \nabla \left( H \vec{\bar{u}} \right)$ $\textstyle =$ $\displaystyle 0$ (26)

Equations (2.26) show that we are looking for a standing process looking from the point of view of a moving cape. The resulting vorticity equation takes the form:
\begin{displaymath} U_0 \frac{\partial \bar{\xi}}{\partial x} = f \vec{\bar{u}}.\frac{\nabla H}{H} \end{displaymath} (27)

where the vortex stretching balances the advection of vorticity due to the displacement of the frame. Because bottom topography is defined in the fixed referential, it does not make sense that variations of bottom topography move from the cape point of view. Hence, H must not be variable in the along shore direction. Equation (2.27) becomes:
\begin{displaymath} U_0 \frac{\partial \bar{\xi}}{\partial x} = f \bar{v}\frac{H'}{H} \end{displaymath} (28)

The continuity equation in (2.26) is not divergent for the transport and allows us to define a transport stream function of the form:
\begin{displaymath} H\bar{u}=-\frac{\partial \psi}{\partial y} \end{displaymath} (29)


\begin{displaymath} H\bar{v}=\frac{\partial \psi}{\partial x} \end{displaymath} (30)

Introducing the transport stream function into the vorticity equation (2.28) yields to:
\begin{displaymath} U_0 \frac{\partial}{\partial x} \left(\frac{\partial^2 \... ...x \partial y} +f \frac{\partial \psi}{\partial x} \right) =0 \end{displaymath} (31)

The vorticity equation (2.31) can be linearized as done by Wilkin and Chapman [1987] by using a bathymetry that follows an exponential function:
\begin{displaymath} H=H_{max} \,e^{-2\lambda(Y_{max}-y)} \end{displaymath} (32)

Where $Y_{max}$ is the position of the offshore boundary. This gives:
\begin{displaymath} U_0 \frac{\partial}{\partial x} \left(\frac{\partial^2 \... ...x \partial y} +f \frac{\partial \psi}{\partial x} \right) =0 \end{displaymath} (33)

Equation (2.33) accepts solutions in the form of standing shelf waves:
\begin{displaymath} \psi=\psi_0 e^{ikx}\phi(y) \end{displaymath} (34)

Introducing the wave solution (2.34) into (2.33) gives for $k \neq 0$:
\begin{displaymath} \phi''-2\lambda \phi' - \left(k^2 + \frac{2\lambda f}{U_0} \right) \phi =0 \end{displaymath} (35)

As in section (2.2) an the analysis of Wilkin and Chapman [1987], we will suppose that the offshore boundary is closed by a wall at $y=Y_{max}$. It has been demonstrated that the presence of a wall at the shelf edge does not affect dramatically the shelf wave structure and dispersion relation [Wilkin and Chapman, 1987]. For the sake of mass conservation, $\psi$ must be constant along the walls in y=0 and $y=Y_{max}$. For $k \neq 0$, this is only possible if $\phi(0)=\phi(Y_{max})=0$. This implies that the standing shelf waves do not produce any net transport. The solution for equation (2.35) that satisfies these boundary conditions is:
\begin{displaymath} \phi=e^{\lambda y}sin \left(\frac{n\pi y}{Y_{max}}\right) \, ;\;n=1,2,3,... \end{displaymath} (36)

Introducing the solution (2.36) into equation (2.35) selects an along shore wavenumber for each mode in the form:
\begin{displaymath} k_n^2=-\frac{n^2 \pi^2}{Y_{max}^2}-\lambda^2-\frac{2\lambda f}{U_0} \end{displaymath} (37)

The full solution of the vorticity equation (2.33) can be write in the form of a sum of standing waves:
\begin{displaymath} \psi_w(x,y)=\sum_{n=1}^{\infty}A_n e^{\lambda y}sin \left(\frac{n\pi y}{Y_{max}}\right)e^{ik_nx} \end{displaymath} (38)

In the Southern Hemisphere, the Coriolis parameter f is negative. Thus, equation (2.37) implies that lee shelf waves can form only for positive $U_0$, accounting for an equatorward eastern boundary current. A more general condition for the presence of a lee shelf wave of mode n is:
\begin{displaymath} 0<U_0<\frac{2\lambda \vert f\vert}{\frac{n^2 \pi^2}{Y_{max}^2}+\lambda^2} \end{displaymath} (39)

If the relation (2.39) is not satisfied, $k_n$ is imaginary and the wave is evanescent. For $n>\frac{Y_{max}}{\pi}\sqrt{-\lambda^2-\frac{2\lambda f}{U_0}}$ only evanescent waves can be generated. Following Lighthill [1966], lee waves produced by a moving perturbation have to propagate at a phase velocity equal to the speed of the perturbation. Taking the propagating wave solution of Wilkin and Chapman [1987], we obtain $-U_0 = c_\phi = \frac{2\lambda f}{k^2_n+\lambda^2+ \frac{n^2 \pi^2}{Y_{max}^2}}$, which is identical to equation (2.37). The condition (2.39) says that standing lee shelf wave can only exist if $U_0$ is opposite to the shelf wave propagation and smaller than the fastest shelf wave phase celerity. Following the same approach of the section (2.5.2) , the wind forced characteristic velocity defined by equation (2.9) can give a reasonable value for $U_0$ in equation (2.37). We obtain a standing shelf wave length $L_n$ for each mode as a function of the wind stress:
\begin{displaymath} L_n=\frac{2\pi}{\sqrt{-\frac{n^2 \pi^2}{Y_{max}^2}-\lambda^2-\frac{2\lambda f\rho r}{\tau}}} \, ;\;n=1,2,3,... \end{displaymath} (40)

To compare this result to the outputs from the previous experiments, the topographic parameters have been chosen: $\lambda =7 \times 10^{-6}$, $Y_{max}=180$ km and $H_{max}=500$ m, so that the exponential topography is relatively close to the bathymetry of the numerical model in St. Helena Bay (2.2) (figure 2.14).

Figure 2.14: Comparison between the topography in the bay described in section (2.2)(dotted line) and the exponential topography for $H_{max}=500$ m and $\lambda =7 \times 10^{-6}$ (dashed line).
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/anatopo.eps,width=10cm}} \end{figure}

The wavelengths of the first two modes are of the same order of magnitude as the size of the perturbations observed in the lee of the cape (for a cape of 100 km)(figure 2.15). Thus the detachment process is likely to be related to the presence of these standing shelf waves. Nevertheless, these wavelengths increase when the width of the shelf ($Y_{max}$) decreases. This is in disagreement with the pattern observed in section (2.6.3).

Figure 2.15: Standing wave lengths for first 6 modes as a function of wind stress for a bay of 180 km wide ($H_{max}=500$ m and $\lambda =7 \times 10^{-6}$). Comparison with the size of the eddy for a cape of 100 km.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/eddysizewave.eps ,width=10cm}} \end{figure}

3 Standing shelf waves excitation

In this section, I would like to propose and illustrate a possible process for standing waves excitation in the lee of a cape. To do so, the linear vorticity equation (2.27) is supposed valid even in the close vicinity of the cape. This linear problem implies that the moving cape does not carry in its movement any water in its wake, and thus does not generate any net transport. Because the cape moves in a channel bounded by a wall offshore, mass conservation implies that, looking from the frame related to the cape, the current in front of the tip of the cape is greater than the current ($U_0$) associated to the motion of the cape. The velocities at the tip of the cape are supposed constant and equal to $U_1$. One can imagine that the flow is redistributed inside a frictional boundary layer close to the cape. For example, the lateral boundary layer thickness defined by Becker [1991], dimensionalized by the size of the cape, is here around 5 kilometers. This is one order of magnitude smaller than a characteristic wave length. If we keep the topography defined in the previous section, global mass conservation for a cape of cross-shelf extension $Y_{cap}$ implies:
\begin{displaymath} \int_{Y_{cap}}^{Y_{max}} HU_1\, dy=\int_{0}^{Y_{max}}HU_0\, dy \end{displaymath} (41)

Giving $U_1$ as a function of $U_0$:
\begin{displaymath} U_1=U_0\frac{1-e^{-2\lambda Y_{max} }}{1-e^{-2\lambda (Y_{max}-Y_{cap})}} \end{displaymath} (42)

The velocity ($U_1$) offshore of the cape and the presence of the cape at x=0 imply that the along shore velocity, solution of the vorticity equation (2.27), must have equal values at x=0 (in the frame moving with the cape):
\begin{displaymath} U_{wave}=\left\{ \begin{array}{ll} -U_0 & \mbox{for }... ...) & \mbox{for } Y_{cap}< y < Y_{max} \end{array} \right. \end{displaymath} (43)

This implies a wave solution in the same order of magnitude of $U_0$. Hence, the term $\frac{U_0 R_0}{U} \frac{\partial \vec{\bar{u}}}{\partial x}$ has the same importance as the term $R_0 (\vec{\bar{u}}.\nabla)\vec{\bar{u}}$ in the addimentional momentum equation (2.25). Solving the linear vorticity equation (2.27), applying the boundary condition (2.43) for x=0, is equivalent asking if a set of linear propagating waves can account for the flow generated just at the back of a moving cape. The frictional boundary layer is no longer necessary to explain the redistribution of the flow behind the cape; this can be done by a set of linear standing waves. The boundary condition at x=0 for the transport stream function accounting for the lee shelf waves is:
\begin{displaymath} \psi_w(0,y)=\left\{ \begin{array}{ll} \frac{U_0H_{ma... ...ght) &\mbox{for } Y_{cap}< y<Y_{max} \end{array} \right. \end{displaymath} (44)

One can note that $\psi_w(0,y)$ is continuous and that $\psi_w(0,0)=\psi_w(0,L)=0$, in agreement with the properties of the standing waves. For the following we will define $\psi_0=\frac{U_0H_{max}e^{-2\lambda Y_{max}}}{2\lambda}$ and $\psi_1=\frac{U_0H_{max}e^{-2\lambda Y_{max}}}{2\lambda} \left(\frac{1-e^{2\lambda Y_{cap}}}{1-e^{-2\lambda(Y_{max}-Y_{cap})}}\right)$. For x=0, pressure and transport should be both continuous. In this case, although pressure can be obtained from the along shore momentum equation, there is no information for the pressure for $x \le 0 $. This information could be obtained by deriving also a wave solution upstream of the Cape (where the mean current is $U_1$). To stay simple, we will keep only local conservation of the transport for the matching conditions at x=0. This gives:
\begin{displaymath} \psi_w(0,y)=\sum_{n=1}^{\infty}A_n e^{\lambda y}sin \left(\frac{n\pi y}{Y_{max}}\right) \end{displaymath} (45)

Because $\frac{2}{Y_{max}}\int_0^{Y_{max}}sin \left(\frac{n\pi y}{Y_{max}}\right)sin \left(\frac{m\pi y}{Y_{max}}\right)dy=\delta_{mn}$ , we obtain:
\begin{displaymath} A_n= \frac{2}{Y_{max}}\int_0^{Y_{max}}sin \left(\frac{n\pi y}{Y_{max}}\right) e^{-\lambda y} \psi_{w}(0,y) \, dy \end{displaymath} (46)

Introducing the value of $\psi_{w}(0,y)$:
\begin{displaymath} A_n = \frac{2}{Y_{max}} \left(\psi_0 B_n - \psi_0 C_n + \psi_1e^{-2\lambda Y_{max}} D_n - \psi_1 E_n\right) \end{displaymath} (47)

With,
\begin{displaymath} \begin{array}{ccc} B_n & = & \int_0^{Y_{cap}} e^{\lambda... ...} sin \left(\frac{n\pi y}{Y_{max}}\right) \, dy \end{array} \end{displaymath} (48)

To confirm the validity of the assumptions made previously, a numerical experiment using SCRUM has been conducted. The configuration is the same as the reference experiment described in the section (2.4), except for the bottom topography and the value of the wind forcing. In this experiment, the bottom topography follows the equation (2.32) with $H_{max}=500$ m and $\lambda =7 \times 10^{-6}$ (see figure 2.14). The value of the constant wind stress is fixed at 0.05 N.m$^{-2}$. The solution is stationary after 50 days, thus a transport stream function can be extracted from the model outputs. The transport stream function and the barotropic velocities in the lee of the cape, for the model at day 100, are shown on figure (2.17-a). Using this topography, no recirculation is visible, the current follows the lee side of the cape to fill the bay. Nevertheless, large standing waves remain in the lee of the cape. They exhibit wave lengths ranging from roughly 50 km in the bay up to 100 km for the small oscillations visible near the offshore boundary. The averaged value of the along shore velocities is 0.1 m.s$^{-1}$, which is 40 % less than the wind forced velocities expected if there was no coastline variations (equation 2.9). This accounts for the drag induced by the cape on the wind forced circulation. The wave pattern is rapidly damped with increasing x, and oscillations are hardly visible after 3 wave lengths. The total transport is about 3.5 Sv. The transport isocontour closest to the coast is moving slightly offshore with increasing x, due to the presence of the downstream cape associated to the periodic channel.

Figure 2.16: Properties of the standing waves. a: Wavelengths in kilometers for the 20 first modes; the black dots accounts for the evanescent waves (in this case, it is the e-folding length that is represented). b: Amplitude relative to the first mode in percent of the 20 first modes; note that the seventh mode amplitude is already less than 1% than the first mode amplitude. c: Total difference in percent between the boundary condition and the wave solution in x=0; there is less than 10% difference after 17 modes added.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/diagwaves.eps,width=7cm}} \end{figure}

The mean along shore velocities of the numerical experiment (0.1 m.s$^{-1}$) gives the value for $U_0$ for the calculation of the standing wave solution. In this case, only the 5 first modes are non-evanescent. They correspond to wave lengths ranging from 61 km to 110 km (figure 2.16-a). Evanescent waves adjust the solution to the boundary condition close to the cape. Their amplitudes being small (figure 2.16-b), they do not strongly affect the solution in the bay. The computation of the relative difference between the solution at x=0 and the boundary condition (figure 2.16-c) shows that 17 modes are sufficient to have a solution close to the boundary condition at x=0. The graph (2.16-b) shows that most of the solution in the bay is picked up by the first mode. The wave length of this mode is 61 km, which is close to the value of the size of the eddy for this wind stress. The solution given by summing the 20 first shelf waves modes and the mean along shore current is represented on figure (2.17-b). The wave pattern close to the cape and the waves length are consistent with the numerical result (figure 2.17-a), but no damping of the waves occurs in the bay. On the contrary, variations happen to increase with increasing x in the first 300 km. This difference is due to the absence of bottom friction when deriving the wave solution. The effect of bottom friction can be roughly simulated by introducing an exponential damping for each wave mode. A scaling of the effects of bottom friction versus the advection by the mean current gives an e-folding length $X_0$:
\begin{displaymath} \frac{U_0^2}{X_0}\frac{\partial u'}{\partial x'}\sim -\frac{rU_0}{H_0}u' \Rightarrow X_0=\frac{rU0}{H_0} \end{displaymath} (49)

Where $H_0$ is the mean value of the bottom topography and r is the linear bottom friction parameter defined in section (2.2). This gives an e-folding length of 50 km. The new solution is shown on figure (2.17-c). It coincides relatively closely to the numerical solution (figure 2.17-a). This result confirms the assumptions made in deriving the standing waves equation and emphasizes the importance of bottom friction. Another discrepancy is the difference between the numerical and standing wave solutions for the incoming current at x=0. A remedy should be to resolve the standing wave dynamics upstream of the cape and to add a boundary condition on the free surface at x=0 as explained previously.

Figure 2.17: Transport stream function and barotropic velocities in the lee of a cape of 100 km width over an exponential bathymetry. a: Output at day 100 for a numerical experiment forced by an along shore wind stress of 0.05 N.m$^{-2}$. b: Standing waves solution for an along shore current $U_0=0.1$ m.s$^{-1}$. c: Same as b with the waves damped with an e-folding along shore length of 50 km. In each portrayal, the interval between the isolines is 0.5 Sv.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/standwaves_large.eps,width=9cm}} \end{figure}

This analysis shows that a local mass conservation condition is sufficient to generate standing waves forced by a mean current in the lee of a cape. Although stratification is not taken into account, this may be an explanation of the large scale stationary features observed in the upwelling front in the lee of Cape Columbine [Shannon, 1985].

4 Standing edge waves ?

Whereas the presence of standing shelf waves and a balance advection/bottom friction can explain the flow pattern in the lee of capes of an order of magnitude of 100 km, this is not satisfactory in the case of small capes. For small capes, perturbations develop in the bay close to shore, where the bottom topography follows a law corresponding more to:
\begin{displaymath} H=H_{max}\left(1-e^{-\lambda y}\right) \end{displaymath} (50)

accounting for a flat shelf rising at the coast. Ball [1967], derived the linear barotropic equations of motion over this topography for along shore propagating waves with no offshore limit. He found solutions in the form of edge waves. The same approach is applied in the case of standing waves. Therefore, as in section (2.8.2), we look for solutions in the standing wave form $Y e^{ikx}$, where $Y$ is a function of y only. Introducing this form for each variable in the linear equations of motion (2.26), we obtain the system:
\begin{displaymath} \left\{ \begin{array}{ccc} ikU_0 u - fv & = & - ikg\zet... ... \frac{\partial Hv}{\partial y} & = & 0 \end{array} \right. \end{displaymath} (51)

Where all the variables are now only y-dependent. The solution of the system (2.51) for $\zeta$ (the linear vorticity equation) is:
\begin{displaymath} H\frac{\partial^2 \zeta}{\partial y^2}+ \frac{\partial H... ...2+\frac{\frac{\partial H}{\partial y} f}{U_0}\right)\zeta =0 \end{displaymath} (52)

with $\zeta$ finite at the coast and zero at infinity. Introducing the value of H and applying the variable transformation:
\begin{displaymath} s=e^{-\lambda x} \end{displaymath} (53)

We obtain:
\begin{displaymath} s^2(1-s)\frac{\partial^2 \zeta}{\partial s^2}+ s(1-2s)\f... ...}{\lambda U_0}-\frac{k^2}{\lambda^2}\right)s\right]\zeta = 0 \end{displaymath} (54)

Following Ball [1967], this equation accepts solutions in the form:
\begin{displaymath} \zeta=s^p \sum_{j=0}^\infty A_j s^j \end{displaymath} (55)

where
\begin{displaymath} p^2=\frac{k^2}{\lambda^2} \end{displaymath} (56)

and
\begin{displaymath} \frac{A_{j+1}}{A_j}=\frac{(j+p+1)(j+p)+\frac{f}{\lambda U_0}-p^2}{(j+p+1)^2-p^2} \end{displaymath} (57)

In order to keep $\zeta$ finite for $y \rightarrow \infty$, we must have $p > 0$. The ratio (2.57) shows that the series (2.55) is divergent unless it terminates. To terminates the series, an integer n must exist such that:
\begin{displaymath} (n+p+1)(n+p)+\frac{f}{\lambda U_0}-p^2=0 \end{displaymath} (58)

Thus, selecting the wavelengths of the standing edge waves:
\begin{displaymath} L_n=-\frac{2\pi}{\lambda}\frac{1+2n}{n(n+1)+\frac{f}{\lambda U_0}} \, ;\;n=0,1,2,3,... \end{displaymath} (59)

Figure 2.18: Standing edge wave lengths for the 4 first orders has a function of the wind stress. Comparison with the size of the eddy for a cape of 25 km.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/edgewaves.eps,width=10cm}} \end{figure}

$\lambda$ as been chosen equal $6 \times 10^{-5}$ to compare the result with the size of the perturbation for a cape of 25 km. We can note on figure (2.18) that the zero order edge wave length coincides with the size of the eddy. This shows that smaller capes force this other kind of waves. It explains the differences for the eddy size found between the different capes in figure (2.11). To summarize, it has been demonstrated in this last section that although flat bottom dynamics control the recirculation in the bay, topographic waves are key phenomena in the barotropic detachment process on the shelf. Because they occur only in presence of an equatorward current (at the eastern boundary of an ocean), they can induce a dissymetry in the drag if the forcing oscillates. This can produce a poleward net current in some places.

9 Retention induced by the attached barotropic eddy

1 The "water age" tracer

To illustrate the effects of the development of the eddy on the coastal and offshore circulation, a tracer T has been introduced in the model. It follows a time forced advective equation:
\begin{displaymath} \frac{\partial T}{\partial t} + \bar{u}\frac{\partial T}{\partial x} + \bar{v}\frac{\partial T}{\partial y} = 1 \end{displaymath} (60)

The number 1 on the right of equation (2.60) is the time forcing term. To be consistent with the numerical SCRUM model, the equation (2.60) has been rewritten in the flux form and for the sake of numerical stability, biharmonic viscosity has been included.
\begin{displaymath} \frac{\partial}{\partial t} \left(DT \right) + \frac{\parti... ... \left(\frac{\hat{\bigtriangleup} \left(T \right)}{D} \right) \end{displaymath} (61)

$\nu_4$ has been kept as small as possible, and because in all experiments $Re_{biharm}$ is of the order of $O(10^{5})$, viscosity should not perturb the solution. T is kept at zero at the upwind boundary. Away from the boundary, T increments continuously. To test this, T has been introduced in an experiment with a rectilinear coastline and a flat bottom. Using the former analytical results, at day 50, when the solution is stationary (equation 2.9), the time since a water particle has left the upwind boundary is only x-dependent:
\begin{displaymath} T=\frac{\rho r x}{\tau} \end{displaymath} (62)

The rectilinear coastline numerical experiment is in agreement with this solution (figure not shown).

Figure 2.19: Spatial distribution of the water age tracer at day 100 for experiments similar to the reference experiment. The along shore wind stress is fixed during each experiment at: (a) 0.05 N.m$^{-2}$, (b) 0.1 N.m$^{-2}$, (c) 0.15 N.m$^{-2}$. The horizontal coordinates are in kilometers and the greyscale range for the water age tracer is in days.The interval between the isolines is 5 days in each portrayal.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/tracerage.eps,width=9cm}} \end{figure}

From the outputs at day 100 of the reference experiment, we can see that the distribution of T over the model domain is strongly influenced by the topography and the development of the eddy in the lee of the cape (figure 2.19-b). In the offshore area (y$>$200 km), T increases almost linearly towards the right side of the domain (the downwind and equatorward directions), this illustrates the advection of the water as it flows toward the equatorward side of the model domain. In the coastal area, T reaches a maximum (90 days) in the vicinity of the cape (x=40 km, y=100 km) and in the downstream near-shore area (figure 2.19-b, T=60 days, x=80 km, y=90 km). It reaches a local maximum inside the shelf break meander where the velocities are weak (figure 2.19-b, T=45 days, x=170 km, y=160 km). For a wind 50% stronger than the reference experiment, whereas the area of strong T values has increased with the size of the eddy, the maximal value of T does not vary significantly from the reference experiment (figure 2.19-c). In this experiment, the local maximum on the shelf edge has disappeared. For a wind 50% weaker, the eddy is small ($<$40 km) and its contribution does not affect dramatically the tracer distribution (figure 2.19-a). Even so, there is a local maximum at the tip of the cape (figure 2.19-a, T=40 days, x=45 km, y=95 km). In order to investigate the effects of the intensity of the wind forcing on the distribution of T, several successive runs of the model are performed using values of wind stress increasing from 0.02 N.m$^{-2}$ to 0.2 N.m$^{-2}$ by steps of 0.02 N.m$^{-2}$ (figure not shown). The spatial distribution of T is extracted at day 50. For low wind forcing (ranging from 0.02 to 0.06 N.m$^{-2}$) and weak flow detachment, there are small differences between the coast and the offshore areas; T increases almost steadily over the entire domain towards the right boundary. For moderate wind forcing ( ranging from 0.06 to 0.14 N.m$^{-2}$), the eddy starts to develop downwind of the cape. In the offshore area, the distribution of T is not affected by the eddy. In the coastal domain, the development of the eddy induces an increase of the age of the water retained in the eddy and on the coastal side of the eddy. T reaches a maximum value of about 30 days in the near-shore domain. For strong wind forcing ( greater than 0.14 N.m$^{-2}$), the size of the eddy growths as the wind but T does not increase in the eddy

Figure 2.20: Value of the water age tracer, $T_{off}$(a) and $T_{coast}$(b) at day 50 for values of wind stress forcing ranging from 0.02 N.m$^{-2}$ to 0.2 N.m$^{-2}$ by steps of 0.02 N.m$^{-2}$.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/tempsmoy.eps,width=10cm}} \end{figure}

These results are summarized on figure 2.20. In order to allow comparison between the offshore and coastal regions, T is averaged over two areas: a coastal/eddy area (T$_{coast}$: x=25 to 150 km and y=0 to 110 km) and an offshore area (T$_{off}$ : x=25 to 150 km and y=110 to 200 km). T$_{coast}$ and T$_{off}$ are calculated at day 50 for each of the successive runs performed with wind stress ranging from 0.02 N.m$^{-2}$ to 0.2 N.m$^{-2}$. In the offshore domain, T$_{off}$ decreases as an inverse function of the wind stress: the stronger is the wind, the stronger is the advection towards the right side of the model domain [e. g. equation (2.62)]. In the coastal domain, T$_{coast}$ decreases as the wind forcing increases from low to moderate (from 0.02 to 0.06 N.m$^{-2}$). In that range of wind forcing, the cape has little influence on the coastal circulation, the eddy size remains small compared to the size of the cape. For stronger wind forcing, the effect of the eddy on the coastal circulation is highly noticeable. For wind forcing ranging from 0.06 to 0.14 N.m$^{-2}$, the size of the eddy expands and T$_{coast}$ increases simultaneously with the wind. From 0.14 to 0.2 N.m$^{-2}$, the size of the eddy reaches or expands over the domain where T$_{coast}$ is calculated and T$_{coast}$ reaches a plateau at a value of 18 days. The variability of T$_{coast}$ and T$_{off}$ over a wide range of wind forcing illustrates the effects of the development of the eddy on the coastal and offshore circulation. When the wind forcing is strong enough ( greater than 0.06 N.m$^{-2}$), the eddy induces a pronounced recirculation in the coastal domain, but the offshore area is unaffected. Looking back at figure 2.2, one can see that while some water may enter the eddy as a slow equatorward flow near-shore, most of the water has circulated around the eddy and then entered from the bay side. The relative increase of T in the coastal eddy domain, when compared to the value in the offshore area, is an important consequence of the development of the eddy.

2 A retention index

The presence of the eddy contributes to the formation of two distinct patterns of circulation within the model domain: in the offshore area, the circulation is predominantly along shore; in the coastal area, the circulation is dominated by a cyclonic eddy and by the associated recirculation pattern. The size of the eddy is positively related to the intensity of wind forcing. The recirculation induced by the eddy tends to isolate the near-shore area from the offshore domain. It limits the cross-shelf exchange of water and retains water particles within the coastal domain, thus providing a mechanism for retention. Retention in this coastal domain is then closely related to the wind induced recirculation pattern. The next step in our analysis is to evaluate the strength of retention as a function of the strength of the wind forcing.

Figure 2.21: The relationship between the retention index $R_i$ and wind stress, ranging from 0.02 N.m$^{-2}$ to 0.2 N.m$^{-2}$ by steps of 0.02 N.m$^{-2}$.
\begin{figure} \centerline{\psfig{figure=Figures/Chap2/retentionindex.eps,width=10cm}} \end{figure}

A true retention index would have been given by the residence time computed from the advection fields provided by the model. However, the difference between the age of the water in the coastal domain (T$_{coast}$) and the offshore domain (T$_{off}$) can be used as a proxy to evaluate the intensity of the retention. The variability of this index (further noted as $R_i$) gives an indication on how the aging of the water differs between the two areas, a positive (negative) value indicates water particles being older (younger) in the coastal domain relative to the offshore area. From a biological point of view, this offshore-inshore gradient can be used to evaluate the potential advantage for larvae to be located in the coastal area. The variability of $R_i$ as a function of wind stress is presented on figure 2.21. $R_i$ increases as the wind forcing increases, it reaches a maximum (18 days) value for a wind stress of 1.8 N.m$^{-2}$ and then appears to level off or decrease. As mentioned before, $R_i$ represents the difference between the coastal and the offshore areas in the aging process of the water particles and $R_i$ should not be used as an indicator of residence times within the bight. This positive relationship between wind forcing and the retention index suggests that, in an upwelling system, circulation patterns such as a standing vortex may provide retention in the lee side of a cape. Further, stronger upwelling favorable winds can enhance retention within the near-shore area despite the intensification of the offshore flow related to the upwelling. Such structures, providing a positive coupling between upwelling and retention, have been documented in several upwelling systems [Graham and Largier, 1997,Roy, 1998]. In the Benguela upwelling system, most of the spawning occurs in late spring and summer, during the peak season of the upwelling. Fish eggs and larvae are transported from the spawning ground to the West Coast upwelling by a coastal jet. With the classical wind-driven coastal upwelling circulation structure of offshore transport of surface water, larvae will tend to be transported in the offshore domain and be lost. However, our modeling experiment shows that the Cape Columbine plume and the associated coastal recirculation in St Helena Bay provide a retention mechanism allowing the larvae to be retained within the productive coastal domain and thereby to avoid dispersion in the offshore area.

10 Summary

We used an idealized numerical model to understand the interactions between the wind-induced circulation on the shelf and topographical features such as Cape Columbine and St. Helena Bay. The setting of the model is based on the assumption that the circulation on the shelf follows barotropic dynamics. Hence, the barotropic equations of motion have been solved in a periodic channel and over an analytical bathymetry. They have been constrained by a constant upwelling favorable wind stress and a linear bottom friction. Despite these simplifications, it appears that the model is able to produce a cyclonic eddy in St. Helena Bay that corresponds to the observed circulation pattern [Holden, 1985,Shannon, 1985,Boyd and Oberholster, 1994]. The shape and size of this eddy are in relative agreement with the results of Oey [1996] and Boyer and Tao [1987a]. The eddy creates a dynamical barrier, limiting the exchanges between the near-shore area and the shelf edge domain. Diagnostic analysis shows that the recirculation process is controlled by a balance between the Coriolis acceleration and the pressure gradient. The velocities offshore are linked to a balance between wind stress and bottom friction and the external part of the eddy follows cyclo-geostrophic dynamics. When the numerical solution reaches a stationary state, a vorticity analysis reveals that a balance between the curl of advection and the curl of bottom friction controls the eddy dynamics. This balance gives a characteristic eddy length scale proportional to the wind stress and inversely proportional to the square of the linear bottom friction parameter. Several numerical experiments using a wide range of wind stress and bottom friction values, indicate that the along shore extension of the eddy follows the trends of this length scale when the dynamics are in the attached-cyclonic-eddy regime. The influence of the size of the cape has been tested, showing different regimes of perturbations. Different kinds of standing coastal trapped waves, in equilibrium with a mean along shore current have been proposed to explain the resulting flow patterns in the bay. An analytical solution has been derived to explain the standing shelf wave excitation by a mean along shore current past a cape. A barotropic tracer representing the sea surface temperature has been introduced into the model to illustrate the impact of an attached barotropic cyclonic eddy on the generation of upwelling plumes A tracer showing the age of the water is introduced to evaluate the intensity of recirculation generated by the eddy in the coastal domain. This tracer is used as a proxy for retention and to explore the relation between the intensity of retention and wind forcing. In accordance with previous observations, the model simulation indicates that the topography induces retention in the lee side of the cape. In regions where fish spawning occurs during the upwelling season, fish have successfully used such structures to reproduce [Roy et al., 1989,Bakun, 1996,Bakun, 1998]. In the Benguela upwelling system, St Helena Bay is recognized as the main nursery ground off the West coast where juvenile fish are concentrated [Hutchings, 1992], suggesting that this retention mechanism may be critical to recruitment success. This idealized work provides a conceptualized portrayal of some typical processes that we can encounter in St. Helena Bay.


3 A regional model of the South African West coast




Setting up a high resolution numerical model of the ocean circulation in the surroundings of the South and West coasts of southern Africa was the direction chosen by the VIBES project to explore the physical processes affecting fish, eggs, larvae and juveniles during the recruitment cycle. This chapter provides a description of the main characteristics of the numerical tools that were set up. A detailed analysis of the numerical model output is provided and is aimed at evaluating the validity of the modeling experiments. An illustration of the potential use of such numerical tools in fisheries oceanography is provided by a set of experiments designed to investigate the effect of transport on the fish spawning products using a passive tracer.


Afin d'explorer les processus physiques affectant poissons, \oeufs, larves et juvéniles durant le cycle de reproduction, la direction choisit par le projet VIBES consistait en la mise au point d'un modèle numérique à haute résolution de la circulation océanique autour des côtes Sud et Ouest de l'Afrique australe. Ce chapître apporte la description des caractéristiques principales des outils numériques développés. Une analyse détaillée des sorties du modèle numérique est conduite et est employée afin d'évaluer la validité des solutions. Une illustration de l'utilisation potentielle de ces outils numériques en Océanographie des pêches est fournie par une suite d'expériences utilisant un traceur passif, développées pour étudier les effets du transport sur les produits de la ponte des poissons.

1 Previous modeling work conducted in the Benguela

Few attempts have been undertaken to model the shelf and slope dynamics around the South and West coast of South Africa. Van Foreest and Brundrit [1982] developed a two-mode linear numerical model using a Galerkin procedure for the vertical and applied it to the South Benguela. The surface forcing was idealized (constant northward wind stress), and the time of integration was short (3 days). Nevertheless, the solution was showing the topographic steering of the baroclinic jet of Cape Peninsula. More recently, the Princeton Ocean Model (POM), a sigma coordinate model, has been applied to simulate the circulation around the Southern Africa from about 46$^o$ S to 12$^o$ S and 4$^o$ E to 30$^o$ E [Skogen, 1998]. 18 bottom following $\sigma$-layers are used in the vertical, and the horizontal resolution is 20 km. We can note that a condition of zero barotropic velocities is given for the initial and boundary conditions. The simulations are forced using data of a specific year. The results showed most of the characteristic features of the circulation of the Benguela, but the resolution was too coarse to resolve the mesoscale characteristics patterns that develops on the shelf in the West Coast area. Several basin-scale models had the Benguela included in their simulation domains [Barnier et al., 1998,Biastoch and Krauß, 1999,Marchesiello, 1995,de Miranda, 1996]. But their resolutions were also too coarse (1/3 of a degree for the bests) to provide a satisfactory circulation pattern for the transport of eggs and larvae around the South and West coasts of South Africa .

2 Model description

The first developed oceanic primitive equation models were using for the vertical, the geopotential coordinate z. In this case the bottom topography is represented by steps in the grid. This was not satisfactory for the simulation of bottom flows along the topography, of particular importance in coastal dynamics. Thus, traditionally in coastal modeling, it has been preferred to use models in which the vertical coordinate follows the topography. For this study, we use ROMS (the Regional Ocean Modeling Systems), a community code developed by the modeling groups at Rutgers University (USA) and at University of California Los Angeles (USA). It belongs to a hierarchy of terrain-following models originated with the semi-spectral primitive equation model (SPEM, [Haidvogel et al., 1991]), which solves the hydrostatic primitive equations with a rigid lid at the sea surface, coupled with advection/diffusion equations for potential temperature and salinity and the nonlinear UNESCO equation of state. SCRUM (S-coordinate Rutgers University Model), a free sea surface version of SPEM has been developed by Song and Haidvogel [1994]. It improves upon SPEM by allowing for a generalized nonlinear terrain-following coordinate, which provides enhanced resolution at either the sea surface or sea floor. This feature might be of importance for the biology, since biological activity occurs mostly in the surface oceanic layers. ROMS represents an evolution of SCRUM, including several substantial developments required to efficiently and robustly calculate the coastal circulations at high spatial resolution, such as subgrid scale parameterization, high order schemes, and high performance computing on SMP-class computer architecture (like the SGI Origin 2000 of the University of Cape Town).

1 Equations of motion

The equations of motion in Cartesian coordinates, following the Boussinesq and hydrostatic approximations, take the form:
$\displaystyle \frac{\partial u}{\partial t}$ $\textstyle =$ $\displaystyle -u\frac{\partial u}{\partial x} - v\frac{\partial u}{\partial y} ... ... u}{\partial z} + fv - \frac{\partial \phi}{\partial x}+ {\cal F}_u+ {\cal D}_u$ (63)
$\displaystyle \frac{\partial v}{\partial t}$ $\textstyle =$ $\displaystyle - u\frac{\partial v}{\partial x} - v\frac{\partial v}{\partial y}... ... v}{\partial z} - fu - \frac{\partial \phi}{\partial y}+ {\cal F}_v+ {\cal D}_v$ (64)
$\displaystyle \frac{\partial T}{\partial t}$ $\textstyle =$ $\displaystyle - u\frac{\partial T}{\partial x} - v\frac{\partial T}{\partial y} - w\frac{\partial T}{\partial z} + {\cal F}_T+ {\cal D}_T$ (65)
$\displaystyle \frac{\partial S}{\partial t}$ $\textstyle =$ $\displaystyle - u\frac{\partial S}{\partial x} - v\frac{\partial S}{\partial y} - w\frac{\partial S}{\partial z} + {\cal F}_S+ {\cal D}_S$ (66)
$\displaystyle \rho$ $\textstyle =$ $\displaystyle \rho(T,S,P)$ (67)
$\displaystyle \frac{\partial \phi}{\partial z}$ $\textstyle =$ $\displaystyle -\frac{\rho g}{\rho_0}$ (68)
$\displaystyle 0$ $\textstyle =$ $\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}$ (69)

With the vertical boundary conditions prescribed as follows:
$\displaystyle \hbox{at the top: }z=\zeta$ $\textstyle \kappa_M \frac{\partial u}{\partial z}$ $\displaystyle = \tau_{surf}^x$ (70)
  $\textstyle \kappa_M \frac{\partial v}{\partial z}$ $\displaystyle = \tau_{surf}^y$ (71)
  $\textstyle \kappa_T \frac{\partial T}{\partial z}$ $\displaystyle = \frac{Q_T}{\rho_0 C_p}$ (72)
  $\textstyle \kappa_S \frac{\partial S}{\partial z}$ $\displaystyle = \frac{(E-P)S}{\rho_0}$ (73)
  $\textstyle w$ $\displaystyle = \frac{\partial \zeta}{\partial t} + u\frac{\partial \zeta}{\partial x} + v\frac{\partial \zeta}{\partial y}$ (74)
$\displaystyle \hbox{at the bottom: }z=-h$ $\textstyle \kappa_M \frac{\partial u}{\partial z}$ $\displaystyle = \tau_{bot}^x$ (75)
  $\textstyle \kappa_M \frac{\partial v}{\partial z}$ $\displaystyle = \tau_{bot}^y$ (76)
  $\textstyle \kappa_T \frac{\partial T}{\partial z}$ $\displaystyle = 0$ (77)
  $\textstyle \kappa_S \frac{\partial S}{\partial z}$ $\displaystyle = 0$ (78)
  $\textstyle w$ $\displaystyle =- u\frac{\partial H}{\partial x} - v\frac{\partial H}{\partial y}$ (79)

Where, Equations (3.1) and (3.2) express the momentum balance in the x and y directions. Equations (3.3) and (3.4) express the time evolution of Temperature and Salinity. Equation (3.5) gives the non-linear equation of state. Under the hydrostatic approximation, the momentum balance in the vertical direction limits itself to a balance between the pressure gradient and the buoyancy forces (3.6). In the Boussinesq approximation, density variations are neglected in the momentum equations except in their contribution to the buoyancy forces in the vertical momentum equation (3.6). Equation (3.7) expresses the continuity equation for an incompressible fluid.

2 The s stretched vertical coordinate system and the horizontal curvilinear coordinates

With an active sea surface, a generalized topography-fitting coordinate takes the form:
$\displaystyle s=s\left(\frac{z-\zeta}{H+\zeta}\right)$ $\textstyle -1\leq s \leq 0$   (82)

As an extension to standard terrain-following transformations, a nonlinear stretching of the vertical coordinate is applied that depends on local water depth [Song and Haidvogel, 1994]. It can be used to generate a more uniform vertical resolution near the surface (or the bottom) and consequently a better representation of the mixed layer and the thermocline. The transformation used in SCRUM and ROMS is:
\begin{displaymath}z=h_s +(h-h_s)C(s) \end{displaymath} (83)

where $h_s$ is a constant to be chosen as a typical surface mixed layer depth, and
\begin{displaymath}C(s)=(1-\theta_b)\frac{\sinh(\theta s)}{\sinh(\theta)}+ \the... ...ft[\theta(s+ 1/2)\right]-\tanh(\theta/2)} {2 \tanh(\theta/2)} \end{displaymath} (84)

For large $\theta$, the coordinate lines are more tightly confined to the surface; additionally, if $\theta_b$ approaches 1, resolution at the bottom boundary is enhanced. These possibilities might be of great importance for the coupling with the biological components. The vertical velocity in the s coordinate is defined as:
\begin{displaymath} \Omega = \frac{\partial s}{\partial z}\left[ w- (1+s) \frac... ...t)_s - v \left(\frac{\partial z}{\partial y}\right)_s \right] \end{displaymath} (85)

In the transformed coordinate system, the kinematic boundary conditions (3.12) and (3.17), at the surface $(s=0)$ and at the bottom $(s=-1)$ simplify to:
\begin{displaymath}\Omega = 0 \end{displaymath} (86)

Figure 3.1: Curvilinear coordinates.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/curv.eps,width=6cm}} \end{figure}

In the horizontal, like SPEM and SCRUM, ROMS is written in horizontal curvilinear coordinates. This system can conform to irregular lateral boundaries or allows the placing of more computational resolution in regions of interest. These new coordinates are introduced by a transformation in the horizontal coordinate from (x,y) to $(\xi,\eta)$, where the relationship of horizontal arc length to the differential distance is given by:
$\displaystyle (ds)_{\xi}$ $\textstyle =$ $\displaystyle \left(\frac{1}{m}\right)d\xi$ (87)
$\displaystyle (ds)_{\eta}$ $\textstyle =$ $\displaystyle \left(\frac{1}{n}\right)d\eta$ (88)

Here, $m(\xi,\eta)$ and $n(\xi,\eta)$ are the scale factors which relate the differential distances $(\Delta \xi, \Delta \eta)$ to the physical arc lengths $\left((\Delta S)_\xi, (\Delta S)_\eta \right)$ (figure 3.1). Coastal boundaries can also be specified as a finite-discretized grid via land/sea masking.

3 Discretization

In the horizontal direction $(\xi,\eta)$, except where noted below, a centered second-order finite difference approximation is adopted on an Arakawa "C" grid (figure 3.2), which is well suited for problems with horizontal resolution that is fine compared to the first radius of deformation [Hedström, 1997].

Figure 3.2: Position of the variables on the Arakawa horizontal C-grid.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/cgrid.eps,width=6cm}} \end{figure}

The vertical also uses a second-order finite-difference approximation. Just as we use a staggered grid in the horizontal, the grid is staggered also in the vertical (figures 3.3 and 3.4). These choices are traditional for second-order, finite differences models, and provide for conservation of the first and second moments of momentum and tracers [Haidvogel et al., 2000].

Figure 3.3: Position of variables on the staggered vertical grid.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/staggrid.eps,width=3cm}}\end{figure}

Figure 3.4: Placement of variables on a ROMS cross section.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/vgrid.eps,width=12cm}} \end{figure}

For computational economy, the hydrostatic primitive equations for momentum are solved using a split-explicit time-stepping scheme which requires a special treatment and coupling between barotropic (fast) and baroclinic (slow) modes. A finite number of barotropic time steps, within each baroclinic step, are carried out to evolve the free-surface and vertically integrated momentum equations. In order to avoid the errors associated with the aliasing of frequencies resolved by the barotropic steps but unresolved by the baroclinic step, the barotropic fields are time averaged before they replace those values obtained with a longer baroclinic step. A cosine-shape time filter, centered at the new time level, is used for the averaging of the barotropic fields. In addition, the separated time-stepping is constrained to maintain exactly both volume conservation and constancy preservation properties which are needed for the tracer equations. Currently, all 2D and 3D equations are time discretized using a third-order accurate predictor (Leap-Frog) and corrector (Adams-Molton) time-stepping algorithm which is very robust and stable. The enhanced stability of the scheme allows larger time steps than in more traditional numerical scheme, by a factor of four, which more than offsets the increased cost of the predictor-corrector algorithm.


4 The pressure gradient scheme

The major advantage of sigma coordinate models is the transformation of the surface and sea bottom to coordinate surfaces. Unfortunately, this is also the source of their major disadvantage: the well known pressure gradient error. In linear $\sigma$ coordinate, the x-component of the pressure gradient force is determined by the sum of 2 terms:
\begin{displaymath} \left.\frac{\partial p}{\partial x}\right)_{z=cst}= \lef... ...rac{\partial p}{\partial \sigma}\frac{\partial h}{\partial x} \end{displaymath} (89)

The first term of the right involves the variation of pressure along a constant $\sigma$-surface and the second is the hydrostatic correction. Near steep topography, these 2 terms are large, comparable in magnitude and tend to cancel each other. A small error in computing either term can result to a relatively large error in the resulting horizontal pressure gradient [Song, 1998]. Another source of error, the hydrostatic inconsistency [Haney, 1991], can occur when using second order central differences in the discretization of equation (3.27). It can be demonstrated that the second order discretization of equation (3.27) is equivalent to interpolating pressure between the contiguous sigma layers to obtain the horizontal pressure gradient [Kliem and Pietrzak, 1999]. Hydrostatic inconsistency arises if the slope and horizontal resolution are such that extrapolation is performed instead of interpolation. Kliem and Pietrzak [1999] and Song [1998] proposed different ways to reduce errors: The scheme implemented in ROMS is the weighted Jacobian formulation proposed by Song [1998]. This formulation has been designed to minimize truncation errors and to retain integral properties. If $s$ is the generalized topographic following coordinate system, the horizontal pressure gradient can be rewrite in a Jacobian form:
\begin{displaymath} \left.\frac{\partial p}{\partial x}\right)_{z= cst}= \le... ...}{\partial x}\frac{\partial \rho}{\partial s'} \right\}\,ds' \end{displaymath} (90)

Hence, vertical variations in the horizontal pressure gradient are given by an integral of the Jacobian:
\begin{displaymath} J(z,\rho)=\frac{\partial z}{\partial s}\frac{\partial \rh... ...\frac{\partial z}{\partial x}\frac{\partial \rho}{\partial s} \end{displaymath} (91)

Song [1998] defined the standard Jacobian formulation as the second order central difference discretization of equation (3.29). He proposed another scheme, the weighted Jacobian, centered in z space rather than in s space, as for the standard Jacobian. In an idealized case, Song [1998] shows that whereas the standard Jacobian outperforms the weighted Jacobian when the hydrostatic consistency condition is satisfied, the weighted Jacobian gives superior results if the condition is violated (which is often the case in realistic configurations). The conservation of momentum and energy and the accurate representation of the bottom pressure torque has been validated [Song and Wright, 1998]. The conservation of these properties can constrain model errors.

5 The advection scheme

A classic approach in Ocean modeling for the treatment of advection is the use of second order central difference schemes. In conjunction, Laplacian, higher-order diffusion operators or low-pass spatial filters are employed to smooth the numerical noise induced by dispersive computational errors or turbulent cascades. The advection operator for the momentum and tracer variables in ROMS has been redesigned to reduce dispersive errors. It consists of an upstream-biased third order scheme. Using this scheme, explicit smoothing of the fields is no longer mandatory, enhancing the effective resolution of the solution for a given grid [Shchepetkin and McWilliams, 1998].

6 The turbulent closure scheme

The parameterization of the unresolved physical vertical mixing processes in ROMS is done via a non-local, K-profile planetary (KPP) boundary layer scheme [Large et al., 1994]. Two distinct parameterizations are conducted: one for the ocean interior and one for the oceanic surface boundary layer. The boundary layer depth (h) depends on the surface forcing, the buoyancy and the velocity profile and is determined by equating a bulk Richardson number relative to the surface to a critical value. Below the boundary layer, the vertical mixing is regarded as the superposition of 3 processes: vertical shear, internal wave breaking, and double diffusion. In the surface layer, the diffusivity is formulated to agree with a similarity theory of turbulence. At the base of the surface layer, both diffusivity and its gradient have to match the interior values. The KPP model has been shown to simulate accurately processes such as convective boundary layer deepening, diurnal cycling, and storm forcing.


3 Configuration


1 The grids

Several arguments constrain the choices for the computational domain:

Figure 3.5: The pie shaped grid for the low resolution experiment. The dark line represent the coastline and the 500 m isobath is an indicator of the position of the shelf break.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/grid_lowres.eps,width=10cm}}\end{figure}

To satisfies these criteria, the grid has been built following the arc of a circle. It follows roughly the South-west corner of Africa and the resolution is identical in the $\xi$ and $\eta$ directions. This way, the number of masked points, that are a dead weight in computing the solution, has been reduced. To facilitate the connections to the open Ocean, the Northern and Eastern open boundaries cut the shelf in its narrowest parts at about respectively $27.5 ^o$ S and $24.5^o$ E. Hence, the model domain includes all the South and South-West coasts of South Africa from Lüderitz to Cape St. Francis (figure 3.5). The Eastern boundary is not at a place where the currents are weak, since it cuts the strong Agulhas Current. We rely on a specific open boundary scheme to handle this feature properly. The offshore boundary is placed beynd the shelf break, on the ocean plane, roughly 300 kilometers away from the tip of the Agulhas Bank. This way, the domain is 1300 km long in its inner radius (at the coast), 2240 km long offshore, and 740 km wide. Two kinds of grid have been realized: Note that the grid is isotropic and thus doesn't produce a dissymetry in the dissipation of the turbulence. The resolution is higher at the coast where we need a more accurate solution. The first baroclinic Rossby radius of deformation is everywhere resolved in the high resolution experiment. Although the pressure gradient scheme should allow the use of a steeper slope, some degree of topographic smoothing may be necessary to ensure stable and accurate simulations using realistic bathymetry. A useful parameter is found to be:
\begin{displaymath} r=\frac{\Delta h}{2h}=\frac{h^{+1/2}-h^{-1/2}}{h^{+1/2}+h^{-1/2}} \end{displaymath} (92)

Empirical studies have shown that robust results are obtained if r does not significantly exceed a value of 0.2 [Haidvogel et al., 2000]. To filter the topography only where it is needed, we used a Shapiro filter weighted by the values of r: if r is smaller than a target value (here, 0.15), the weight is 0; if r is bigger, the weight is 1. The filter has been passed on the topography until $r<0.24$ for the low resolution experiments and until $r<0.15$ for the high resolution experiments. The low resolution experiment has a bigger $r$ value because it was not possible to filter further the topography at this resolution without removing all the shelf on the West Coast (see for example figure 3.8).

Figure 3.6: Raw ocean topography (greyscale range in meters) from the ETOPO2 dataset.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/topo-raw.eps,width=10cm}}\end{figure}

Figure 3.7: Smoothed ocean topography (greyscale range in meters) for the high resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/topo-higres.eps,width=10cm}} \end{figure}

Figure 3.8: Smoothed ocean topography (greyscale range in meters) for the low resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/topo-lowres.eps,width=10cm}} \end{figure}

Comparing figure (3.6) and figure (3.7), one can note that whereas some features like the Cape Canyon or the steep wall just South of the Agulhas Bank have been filtered out, the major topographical characteristics remain in the bathymetry used for the high resolution experiment. Although smaller, the Agulhas Bank is approximately well represented as well as the widening of the shelf in St. Helena Bay or the narrowing of the shelf South of Lüderitz. The filtering has removed the narrow shelf in the Cape Peninsula area. For the low resolution grid, most of the topographical features have been filtered out and there only remains a very smoothed shelf break that follows the coastline. Tests have been conducted for the high resolution configuration with a neutral stratification to validate the value of the smoothing on the pressure gradient error. On the vertical, we have chosen N=20 vertical levels. It would have been interesting to use more vertical levels to resolve more accurately the bottom layers, but tests along a vertical slice with an increased number of vertical levels did not produce a significant improvement of the solution. The minimum depth has been fixed at 30 meters and the maximum depth at 5000 meters. The number of levels were concentrated at the surface where most of the variability occurs and to keep a good resolution for the biological components (table 3.3.1 and figure 3.9). To do so, $\theta$ has been chosen equal 7, $\theta_b$ equal 0.3, and $h_s$ equal 20 meters. Note that with these parameter values, the resolution is less than 5 meters everywhere at the surface, but can be really coarse on the bottom of the deep ocean (more than 1000 m when h=5000 m) (table 3.3.1).

Table 3.1: Vertical s-coordinate system: depth in meters of the s levels for the w-points. The other variables are at intermediate levels.
       
Level at h$_{min}$ at h$_{moy}$ at h$_{max}$
       
20 0.0 0.0 0.0
19 -1.0 -2.8 -4.7
18 -2.0 -6.5 -11.0
17 -3.0 -11.9 -20.7
16 -4.1 -20.5 -36.9
15 -5.1 -35.2 -65.2
14 -6.2 -61.1 -116.0
13 -7.4 -106.4 -205.5
12 -8.7 -181.7 -354.8
11 -10.1 -294.2 -578.3
10 -11.7 -436.9 -862.2
9 -13.3 -586.2 -1159.0
8 -14.8 -719.0 -1423.2
7 -16.3 -831.1 -1645.8
6 -17.7 -934.0 -1850.4
5 -19.1 -1045.7 -2072.2
4 -20.7 -1184.8 -2348.9
3 -22.4 -1371.8 -2721.2
2 -24.5 -1631.7 -3238.9
1 -26.9 -1997.5 -3968.1
0 -30.0 -2515.0 -5000.0


Figure 3.9: Vertical s-coordinate system: vertical levels of the $\rho $ points for a section across the shelf North of St Helena Bay.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/vgridhighres.eps,width=7cm}} \end{figure}


2 Surface fluxes

The atmospheric forcing fields are based on monthly climatologies derived from the Comprehensive Ocean-Atmosphere Data Set (COADS) [Da Silva et al., 1994]. The momentum forcing is given by the longitudinal and latitudinal components of the wind stress. One can note on figure (3.10) that on the South Coast, the climatological wind field in never favorable to upwelling. North of $30^o$ S the upwelling favorable wind is perennial throughout the year, whereas it is seasonal South of $30^o$ S [Shillington, 1998]. A cyclonic wind stress curl is present on the shelf associated with the upwelling favorable wind.

Figure 3.10: Wind stress based on the COADS monthly climatology (1 vector is portrayed every 8 vectors). The greyscale range represent the wind stress curl ($\times 10^-7$) in N.m$^{-3}$.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/windstress.eps,width=12cm}}\end{figure}

Figure 3.11: Net heat flux in W.m$^{-2}$ based on the COADS monthly climatology.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/heatflux.eps,width=12cm}} \end{figure}

Setting up the thermohaline forcing of an ocean is less straightforward. The specification of the surface fluxes alone may lead to an undesirable drift of the model fields as it neglects the feed-back to the atmosphere entirely. Therefore, basin scale models are often forced by nudging towards monthly mean climatological surface fields. A method physically more satisfactory is to linearize the thermal forcing around climatological sea surface temperature $(T_{clim})$, in order to represent the model sea surface temperature $(T_{mod})$ feedback on the surface heat fluxes [Barnier et al., 1998], as
\begin{displaymath} Q_T=Q_{net}+\frac{\partial Q}{\partial T_{clim}}\left(T_{mod}-T_{clim}\right) \end{displaymath} (93)

The term $\frac{\partial Q}{\partial T_{clim}}$ is computed as $Q_{net}$ from the different contributions for the heat fluxes: infrared, sensible heat, and latent heat [Siefridt, 1994]:
\begin{displaymath} \frac{\partial Q}{\partial T_{clim}}= -4 \,\sigma \,T_{clim... ...atm} \,C_e\, L\, U \times 2353 \log(10) \,q_s\, T_{clim}^{-2} \end{displaymath} (94)

Where $\sigma$ is the constant of Stefan, $C_p$ is the specific heat of the atmosphere, $C_h$ is the sensible heat transfer coefficient, $U$ is the wind speed, $C_e$ is the latent heat transfer coefficient, $L$ is the latent heat of vaporization, and $q_s$ is the sea level specific humidity. The term $\frac{\partial Q}{\partial T_{clim}}$ is portrayed on figure 3.12. For a mixed layer depth of 50 m it corresponds to a nudging coefficient toward the climatology ranging from about 50 days in the South to a maximum 90 days in the West Coast. Figure (3.13) shows the sea surface temperature used for the correction of the surface heat flux, obtained from the COADS dataset. In this dataset, the Agulhas Current and the upwelling system are poorly represented.

Figure 3.12: Linear dependence of the net heat flux on the sea surface temperature in W.m$^{-2}$.C$^{-1}$ based on the COADS monthly climatology.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/dqdsst.eps,width=12cm}} \end{figure}

Figure 3.13: Sea surface temperature ($^o$C) of the COADS monthly climatology, used in the net heat flux correction term.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/sstclim.eps,width=12cm}}\end{figure}

Figure 3.14: Evaporation minus precipitation (cm.day$^{-1}$) of the COADS monthly climatology.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/e-p.eps,width=12cm}} \end{figure}

The fresh water fluxes are specified as salt fluxes based on the climatological precipitation (P) and evaporation (E) fields (figure 3.14). The use of E-P data independent of the model solution can induce a drift of the model solution [Haidvogel et al., 2000]. To counter this behavior a correction term for the surface salinity flux has been added to the formulation, identically to the heat flux correction term. We obtain:
\begin{displaymath} \hbox{Salt flux}=\frac{(E-P)S_{mod}}{\rho_0}+ \hbox{Cff} \left(S_{mod}-S_{clim}\right) \end{displaymath} (95)

This method has been chosen for our configuration, using a nudging coefficient, corresponding to 75 days, the same order of magnitude as the averaged value of the heat flux correction term. Unfortunately, a bug has crept into the model and the salinity flux remained equal to zero during all the simulations. Nevertheless, the cold water doesn't evaporate dramatically in an upwelling system and the precipitation in the Benguela is low (figure 3.14). This induces low values for the E-P field of the West Coast. The salinity fluxes coming from the lateral boundaries should overwhelm the surface flux. Hence, we expect the solution to not drift excessively from the climatology values.


3 Initial and boundary conditions

1 Open boundary scheme

Coastal modeling requires well behaved, long term solutions for configurations with open boundaries on up to 3 sides. A numerical boundary scheme should allow the inner solution to radiate through the boundary without reflection and information from the surrounding ocean to come into the model. The active open boundary scheme implemented in ROMS estimates the two dimensional horizontal phase velocities in the vicinity of the boundary [Marchesiello et al., 2000]. For each model variable $\phi$, following Raymond and Kuo [1984], the normal ($c_x$) and tangential($c_y$) phase velocities are:
$\displaystyle c_x$ $\textstyle =$ $\displaystyle -\frac{\frac{\partial \phi}{\partial t} \frac{\partial \phi}{\par... ...} {(\frac{\partial \phi}{\partial x})^2 + (\frac{\partial \phi}{\partial y})^2}$ (96)
$\displaystyle c_y$ $\textstyle =$ $\displaystyle -\frac{\frac{\partial \phi}{\partial t} \frac{\partial \phi}{\par... ...} {(\frac{\partial \phi}{\partial x})^2 + (\frac{\partial \phi}{\partial y})^2}$ (97)

If the propagation is towards the open ocean, the features produced inside the model are evacuated following the wave equation:
\begin{displaymath} \frac{\partial \phi}{\partial t} + c_x \frac{\partial \phi}... ...artial \phi}{\partial y} =\frac{\phi_{data}-\phi}{\tau_{out}} \end{displaymath} (98)

If the propagation is towards the interior, the value at the boundary is nudged towards data:
\begin{displaymath} \frac{\partial \phi}{\partial t} =\frac{\phi_{data}-\phi}{\tau_{in}} \end{displaymath} (99)

The tangential and normal propagations are discretized in ROMS in an upstream biased fashion, where the normal component is treated implicitly. Hence, this scheme allows large time steps without loss of stability. At the corners the averaged value of the two adjacent boundary points is taken. Mass conservation is enforced around the model domain [Marchesiello et al., 2000]. The inflow/outflow split scheme can cause problems in the sense that a grid point can be an inflow for one process and an outflow for another (for example the Eastern boundary is an inflow boundary for the Agulhas Current and the barotropic Rossby waves, but it is an outflow boundary for coastal trapped waves and Kelvin waves). A series of actions are taken to counter this problem in order to obtain a long term satisfactory behavior of the solution:
  1. The solution is weakly relaxed towards data in outflow conditions [right term of equation (3.36)].
  2. The solution is weakly nudged towards data in nudging bands close to the boundaries. The nudging coefficient is linearly decreasing in the 6 points near the boundaries (figure 3.15).
  3. Whereas no explicit mixing is mandatory in the model domain, a sponge layer with linearly increasing lateral mixing coefficient (figure 3.16) filters out the possible numerical noise or reflections produced by the open boundaries. Recent simulations of the US West Coast ocean model [Marchesiello et al., 2000] have shown that with the active boundary condition schemes, the sponge layers were no longer necessary.

Figure 3.15: Nudging time in days for the high resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/nudgetemp_hres.eps,width=9cm}} \end{figure}

Figure 3.16: Lateral Laplacian mixing coefficient (m.s$^{-2}$) for the high resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/sponge_hres.eps,width=9cm}}\end{figure}

This method filters the model solution to connect it to the surrounding smooth oceanic data, while allowing the development of a meaningful internal solution. The relaxation times are chosen empirically to avoid long-term drift and over-specification: Since the boundary values of the free surface elevation does not affect the inner solution on a C-grid, a simple non-gradient scheme is applied for the boundary condition of the free surface elevation. An important particularity of the regional model of the South African West Coast is the highly energetic Agulhas Current that is flowing into the domain via the Eastern boundary. In the presence of this meandering current, the solution happened to be instable after a year or two of simulations. To obtain a long term stable solution, a specific open boundary scheme has been implemented into the code for the barotropic component of the velocities perpendicular to the boundaries. This scheme was originally proposed by Flather [1976] for a tidal model, and combines a one dimensional radiation equation (3.38) with a one dimensional version of the linearized continuity equation (3.39):
$\displaystyle \frac{ \partial \bar{u}}{\partial t} \pm c \frac{ \partial \bar{u}}{\partial x}$ $\textstyle =$ $\displaystyle 0$ (100)
$\displaystyle \frac{ \partial \zeta}{\partial t} + H \frac{ \partial \bar{u}}{\partial x}$ $\textstyle =$ $\displaystyle 0$ (101)
$\displaystyle \Rightarrow \frac{ \partial \bar{u}}{\partial t}$ $\textstyle =$ $\displaystyle \pm \frac{c}{H} \frac{ \partial \zeta}{\partial t}$ (102)


\begin{displaymath} \Rightarrow \bar{u}_{bound}=\bar{u}_{data} \pm \frac{c}{H}(\zeta - \zeta_{data}) \end{displaymath} (103)

The sign in equation (3.41) depends on the position of the boundary. For the phase velocity, we assumes that the waves approaching the open boundary are mostly non dispersive surface gravity waves. Hence, c in equation (3.41) is fixed at $c=\sqrt{gH}$ (g being the gravity acceleration and H the water column depth). Equation (3.41) becomes;
\begin{displaymath} \bar{u}_{bound}=\bar{u}_{data} \pm \sqrt{\frac{g}{H}}(\zeta - \zeta_{data}) \end{displaymath} (104)

This open boundary scheme can be seen as a one way nesting scheme that conserves mass. The differences between the specified values ( $\bar{u}_{data}$ and $\zeta_{data}$), and those calculated by the model ($\bar{u}$ and $\zeta$) are forced to radiate at the speed of the external gravity waves [Palma and Matano, 1998]. Information is constantly provided by the boundary, that could over-determine the inner solution. Nevertheless, if consistent data were provided for the boundary, this scheme gave the best overall performance in test cases conducted using a shallow water model [Palma and Matano, 1998]. One should expect that for a realistic simulation, if data of good quality are given for the boundary, this scheme should stabilize the inner solution in comparison to a scheme that allows a more important degree of freedom. This is what it is observed during our simulation. The drawback is that an important control from the boundaries could limit the physical significance of the solution obtained inside the model domain.

2 Climatology fields

In a first attempt, the initial and boundary conditions have been derived from a global monthly climatology dataset [Levitus, 1982]. The sea surface elevation is computed as the dynamic height referenced to the 500 m level. The thermal wind relationship gave the baroclinic velocities. Several problems arose from this derivation:
  1. The solution was dependent of the arbitrary choice of the no-motion reference level.
  2. Extrapolation was necessary to obtain data on the shelf and gave spurious recirculation.
  3. The Agulhas Current was badly represented (see figure 3.13).
Whereas it was possible to obtain a solution by this method, it has been preferred to rely on a seasonal averaged data derived from a z-coordinate, 1/3 $^o$ resolution, rigid lid, basin scale model of the Indian and South-Atlantic Oceans [Biastoch and Krauß, 1999]. This model is currently implemented at the Oceanographic Department of the University of Cape Town. The rigid lid assumption made in this model forces us to diagnose a free surface elevation from the other variables. This seasonal data gives an annual cyclic information for each model variable (e.g. u, v, $\bar{u}$, $\bar{v}$, T, S, $\zeta$). To avoid discontinuities in the climatology (and forcing) fields, ROMS linearly interpolates in time the data to obtain a field at each model time. The summer values of the climatology (e.g. 15 February) are used for the initial conditions. For the model spin-up, it has been preferred to let the model adjust itself from rest on the density field associated with the temperature and salinity. The variables u, v, $\bar{u}$, $\bar{v}$ and $\zeta$ are set to zero a t=0. The circulation has to adjust to the stratification, the domain geometry, the surface and the lateral forcings. Because the domain is relatively small, this should occur relatively quickly, and we expect to obtain an equilibrium state after a few years of simulation.


4 Results for the low resolution model


1 Spin-up

Figure 3.17: Sea surface height and barotropic currents for the low resolution experiment. a: after 1 day of simulation. b: after 1 week. c: after 2 weeks. d: after 1 month. The greyscale range represents the sea surface elevation in centimeters. 1 current vector is portrayed every 4 vectors.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/spinup_lowres.eps,width=14cm}}\end{figure}

As explained by Penduff [1998], several mechanisms rapidly adjust the solution to the initial stratification, and the different forcing. After 1 day, the sea surface elevation shows the geostrophic adjustment to the initial stratification (figure 3.17-a), . From the Northern boundary, a signal of low elevation (associated with the upwelling) propagates along the West Coast to the South as a coastal trapped wave (figures 3.17-a, 3.17-b and 3.17-c), and connects with locally forced low sea surface elevation around Cape Peninsula. The adjustment processes induce relatively strong barotropic velocities on the Agulhas Bank (figures 3.17-a and 3.17-b). After 15 days, the velocities are weaker on the shelf and slope (figure 3.17-c), and a ring detaches from the Agulhas Current.

Figure 3.18: Sea surface height and barotropic currents for the low resolution experiment. a: 3 January of model year 2. b: 3 January of model year 5. c: 3 January of model year 10. d: 3 January of model year 15. The greyscale range represents the sea surface elevation in centimeters. 1 current vector is portrayed every 4 vectors.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/ssh_1jan_10years_lres.eps,width=14cm}} \end{figure}

Figure 3.19: Root mean square of the sea surface elevation over the domain The thick line represents the data after a low pass filter with a cutoff at 1 year.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/dist_zeta.eps,width=14cm}} \end{figure}

Though the model resolution is coarse (ranging from 18 km to 31 km), mesoscale activity is able to develop during the simulation. On figure (3.18), the sea surface elevation and the barotropic currents have been portrayed for 3 January of model year 2 (figure 3.18-a), model year 5 (figure 3.18-b), model year 10 (figure 3.18-c), and model year 15(figure 3.18-d). After 1 year of simulation (figure 3.18-a), the solution appears to be adjusted. Regardless of the mesoscale activity, very few differences occur in the following years. Cyclonic eddies appears to be shed from the Agulhas Bank and from the Cape Point / Cape Columbine area. Though very close to the open boundary, from time to time an anticyclonic eddy detaches from the Agulhas Current at the Agulhas retroflection (figure 3.18-b). The adjustment process can be seen on a graph of the domain averaged norm of $\zeta$: $ \sqrt{ \frac{1}{S} \int\!\!\!\int_S \zeta^2 ds} $ (figure: 3.19). The model "forgets" quasi immediately the shock of the start from rest. A low pass filtering with a cut-off at 1 year is applied to the data (thick line on figure 3.19) to show the seasonal and interannual variability. No significant long term trend is observed after the first adjustment of the sea surface elevation.

Figure 3.20: Volume averaged potential temperature ($^o$ C)for the low resolution experiment as a function of time.The thick line represents the data filtered by a low pass filter with a cut off at 4 years. The dotted line represents the trend of the data.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avgtemp_lres.eps,width=14cm}} \end{figure}

Figure 3.21: Volume averaged salinity (PSU) for the low resolution experiment as a function of time. The thick line represents the data filtered by a low pass filter with a cut off at 4 years. The dotted line represents the trend of the data..
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avgsalt_lres.eps,width=14cm}} \end{figure}

After the quick adjustment to the initial values, the slow response of the stratification occurs. These long term processes consist of the propagation of baroclinic waves, advection and mixing. The presence of the Agulhas Current associated with strong velocities accelerates the adjustment process. Because the domain is relatively small, it takes a relatively short time for a baroclinic wave or an advected water particle to cross it. An important measure of model fidelity is the degree to which it is able to preserve the globally averaged values of its tracer fields. The volume averaged temperature time series shows a strong seasonal pattern (figure 3.20). After 2-3 years it oscillates around a mean of approximately $4.65^o$ C. No significant trend is noticeable. But the solution exhibits variations on the interannual scale. This variability is quite surprising. It can be related to the presence of the Agulhas Current that produces, on an irregular basis, large eddies in comparison with the size of the domain. Interannual variability is also a characteristic pattern observed in the time series of the domain averaged salinity (figure 3.21). Freshening occurs during the 5 first years, in a range of 0.002 PSU. As the surface forcing didn't operate correctly during this simulation, a trend should have been noticed in the volume averaged salinity. The weakness of the observed trend emphasises an important constraint of the open boundaries on the solution. Tests should be conducted with less restrictive open boundary nesting values.

Figure 3.22: Top: Volume averaged kinetic energy (cm$^2$.s$^{-2}$) for the low resolution experiment. Bottom: Surface kinetic energy (cm$^2$.s$^{-2}$) for the low resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avg_ke_lres.eps,width=14cm}}\end{figure}

Time series of the volume averaged kinetic energy and of the averaged surface kinetic energy are shown on figure (3.22-top) and figure (3.22-bottom) respectively. During the first months of the simulation, the volume averaged kinetic energy rises rapidly to reach an averaged value of about 90 cm$^2$.s$^{-2}$. Little temporal trend is apparent in the time series. The seasonal pattern is not really dominant in the volume averaged kinetic energy. On the contrary, the surface kinetic energy shows the marked ocean response to the seasonal varying surface forcing. The absence of a significant trend is a sign of correct behavior of the solution. As expected, the surface averaged kinetic energy has a maximum during the upwelling season.


2 Time average

The outputs of the low resolution experiment have been time averaged from model year 3 to model year 10 in order to obtain a smooth 3-dimentional field of the mean circulation resolved by the model.

Figure 3.23: Annual surface climatology derived from the results of the low resolution experiment. a: sea surface temperature ($^o$C). b: surface currents (m.s$^{-1}$), Vmax is the maximum velocity, 1 vector is portrayed every 4 vectors. c: sea surface salinity (PSU). d: sea surface height (m). The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/surface_clim_lres.eps,width=14cm}} \end{figure}

For the low resolution experiment, the averaged sea surface temperature is close to the surface climatology employed for the heat forcing of the model (figure 3.23-a). Upwelling is relatively homogeneous with 14$^o$C waters all along the west coast from Cape Peninsula to the northern boundary. The isotherms tend to move away from the coast North of 30$^o$ S, but the northern boundary constrains them back towards the coast north of 29$^o$ S. The Agulhas current is characterized by a tongue of warm water ($> 20 ^o$C) that rounds the Agulhas Bank, enclosing colder surface water onto the bank. The presence of the Subtropical front is noticeable in the South Corner of the model domain. It shows a marked sea surface temperature gradient ranging form 13$^o$C to 17$^o$C in a short distance. The Agulhas Current does not retroflect properly in this experiment (figure 3.23-b). When it flows along the Agulhas Bank, it reaches on average a maximum velocity of 1.1 m.s$^{-1}$, which is a realistic value. Because the topography is smooth, the current does not detach from the southern tip of the Agulhas Bank, and rounds the bank to flow in a North-east direction. This behavior is not observed in nature. As a consequence, the retroflexion area is displaced westward close to the offshore boundary. On the Agulhas Bank, currents are weak as pointed by Boyd and Oberlholster [1994]. The Good Hope jet is not present on the western part of the bank. Along the west coast, the north westward surface current flows continuously towards the equator with characteristic velocities of about 30 cm.s$^{-1}$. Divergence is evident offshore of Cape Columbine (17$^o$ E - 33$^o$ S), in agreement with the schematic representations of Boyd and Shillington [1994] (figure 1.8) and Shannon [1985] (figure 1.13). The Subtropical front is well represented by the surface salinity (figure 3.23-c) with values ranging from 34.6 PSU to 35 PSU in the South corner of the model domain. The effect of the zero surface salinity flux is visible near the northern boundary where the solution has departed from the climatology forcing values and is restored back in the vicinity of the boundary. Nevertheless, the averaged surface salinity values remain close (in a range of $0.1\sim 0.2$ PSU) to the Levitus [1982] climatology data. The Agulhas Current does not have a marked signature in surface salinity. On the contrary, it dominates the averaged sea surface elevation signal 3.23-d). In the few kilometers between the Agulhas Bank and the southern part of the Agulhas Current, the averaged sea surface elevation ranges from -30 cm to 60 cm (figure 3.23-d). Along the west coast, the variations in sea surface height are very small ($\sim 10 $cm).

Figure 3.24: Averaged currents in m.s$^{-1}$ for the low resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The maximum velocity is given for each level. One vector is represented every 4 vectors. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/current_clim_lres.eps,width=14cm}} \end{figure}

The horizontal averaged currents have been portrayed for z = -100 m (figure 3.24-a), z = -500 m (figure 3.24-b), z = -1000 m (figure 3.24-c), and z = -2000 m (figure 3.24-d). The Agulhas current is present on the 3 first levels. Its magnitude diminishes smoothly with depth. For z=-100 m (figure 3.24-a), a narrow coastal jet with velocities of 30 cm.s$^{-1}$ flows past Cape Peninsula towards the equator. Between this jet and the coast, the equatorward flow is weak. For z=-500 m (figure 3.24-b), poleward motion is present along the topography North of 32$^o$ S. Between Cape Peninsula and 32$^o$ S equatorward motion is still present. Below 500 m weak poleward motion is present North of 36$^o$ S (figures 3.24-c and 3.24-d) as described by Nelson [1989].

Figure 3.25: Potential temperature annual climatology ($^o$C) for the low resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/temp_clim_lres.eps,width=14cm}} \end{figure}

Figure 3.26: Salinity annual climatology (PSU) for the low resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/salt_clim_lres.eps,width=14cm}} \end{figure}

The potential temperature and the salinity have been also portrayed for the 4 horizontal levels defined previously (figures 3.25 and 3.26). The upwelling signal all along the west coast is present for z=-100m with a minimum in temperature (12$^o$ C, figure 3.25-a) and a minimum in salinity (34.7 PSU, figure 3.26-a). This minimum in salinity is caused by the presence of the Antarctic Intermediate Water (AAIW), characteristic of the South-East Atlantic Ocean, with its core of minimum of salinity around 700-800 m (see section 1.2). This minimum of salinity can diminish the density gradient induced by the upwelling in comparison to the other upwelling systems. This is a particularity of the Benguela. The impact of this salinity minimum on the dynamics of the upwelling system still have to be quantified. On the eastern side of the Agulhas Bank, for z = -100 m, a local minimum in temperature stands at the offshore limit of the shelf. It can be the signature of the subsurface cool-water feature described by Boyd and Shillington [1994]. For z = -100 m, -500 m, and -1000 m, the Agulhas Current is recognizable by a maximum both in salinity and temperature. At these levels, a zonal tongue of local minimum of temperature and salinity extends from the Agulhas Bank at 35$^o$ S. An averaged cyclonic circulation of the surface currents (visible also in the sea surface elevation) is associated to this structure. It might be related to upwelled waters transported offshore by mesoscale eddies shed from the Agulhas Bank-Cape Peninsula area. For z = -2000 m, an equatorward increase of potential temperature of 0.4$^o$ C (figure 3.25-d) and for salinity of 0.15 PSU (figure 3.26-d) over the length of the domain is visible. This shows a transition from Antarctic Intermediate Water to North Atlantic Deep Waters (see the TS-diagram 1.4). Although the results of the low resolution experiment show interesting features that can be related to observations, some aspects of the solution, like the Agulhas Current retroflection or the coastal circulation, are not well represented at this level of resolution. The bottom topography appears to have an important control of the flow and need to be treated more accurately. Hence, a new set of experiments is conducted by doubling the horizontal number of grid points in each direction. Nevertheless, the low resolution configuration is a useful tool that provides a reference solution to compare with new results and that can be used to test new numerical schemes or parameterizations at a reasonable computer cpu time.


5 Results for the high resolution model

Keeping the same surface and lateral boundary forcing of the previous experiment, a new configuration has been set up using an horizontal grid with an accuracy ranging from 9 km at the coast to 16 km offshore. This configuration should resolve most of the mesoscale features observed along the southwestern coast of Africa. The criterion for the error in the pressure gradient allows the use of a less smoothed bottom topography (figure 3.7). Because the bathymetry is of primary importance in the control of the shelf and slopes dynamics, the behavior of the solution in this configuration should be more satisfactory.

Figure 3.27: Left panel: bottom layer currents (cm.s$^{-1}$) after 1 month for the sigma coordinate test case in high resolution. The greyscale range represents the total velocity in cm.s$^{-1}$. The isolines interval is 1 cm.s$^{-1}$.The horizontal coordinates account for the longitude and the latitude. Right panel: maximum current (cm.s$^{-1}$) over all the domain as a function of time in days.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/stratif_neutre.eps,width=14cm}} \end{figure}

In a first step, the effects of the s-coordinate related pressure gradient error are tested in an experiment with no forcing and no horizontal variations in density. Four walls are placed at the lateral boundaries of the domain. The vertical structures in temperature and salinity are defined analytically at the model initialization to account for the horizontally spatial average of the vertical density structure. During this test, no horizontal or vertical mixing is present, to concentrate on the effects of the pressure gradient. Because there is no forcing and no horizontal variations in density, the pressure gradient should remain equal zero during all the simulation. If currents are generated, they are spurious circulations induced by errors in computing the pressure gradient. The spurious circulation should be maximum in the bottom s-level, where the slopes in the s coordinates are the highest. As explained in section (3.2.4), the s-coordinate related pressure gradient error comes from the discretization of the pressure gradient in the s-coordinate system. If the density field and the s-coordinate system do not vary with time, the error in the pressure gradient should remain constant for this test case (the s-coordinate is dependent of the free surface elevation but this only significantly affects the surface layers). Because only the Coriolis and pressure gradient terms are solved during the test simulation, the spurious currents should reach rapidly a geostrophic equilibrium with the error in the pressure gradient. Hence, after a spin-up of a few days, the averaged spurious currents should not increase with time. It is then sufficient to run the test experiment for only a month to obtain a significant representation of the impact of the pressure gradient error on the circulation. During the test experiment, the maximum spurious currents reach a plateau of about 6 cm.s$^{-1}$ after 5 days (figure 3.27-right). It is a large value but looking at the horizontal currents of the s-bottom level after 1 month of simulation (figure 3.27-left), one might note that this largest currents are along the domain wall (that are open boundaries in the realistic experiment). Hence, these currents are not directly forced by errors in the pressure gradient, but are part of the general circulation generated to compensate the slow motion induced by the pressure gradient errors. At distance from the walls, the bottom spurious currents are of the order of 1 cm.s$^{-1}$ with a maximum of 2 cm.s$^{-1}$ South of the Agulhas Bank (figure 3.27-left). The pressure-gradient-error-related-currents are portrayed after 1 month of simulation at 4 horizontal levels: z = -100 m (figure 3.28-a), z = -500 m (figure 3.28-b), z = -1000 m (figure 3.28-c), and z = -2000 m (figure 3.28-d). At each level, the error results in a poleward along-slope current of around 1 to 2 cm.s$^{-1}$. The current is locally stronger ($\sim$ 5 cm.s$^{-1}$) at the southern tip of the Agulhas Bank for z = -1000 m. If we compare the error induced currents to the time averaged currents of the high resolution realistic experiment (figure 3.33), we can note that the relative error is of approximately 5 % for z = -100 m, 10 % for z = -500 m , and 20 % for z = -1000 m and z = -2000 m. These values are not small for the deep layers. A part of the solution can be induced by errors in the pressure gradient. This might be the case for the deep poleward motion observed in figure (3.33-d) and for the poleward undercurrent observed on figure (3.45-b). One can note firstly that the values obtained for these 2 currents corresponds to observation [Nelson, 1989]. Secondly the residual pressure gradient related current obtained in the test case are of the same order of magnitude of the error made in the measurements. Thirdly, the current pathway observed on figure (3.33-d) is not the same as the pathway followed by the error related current seen in figure (3.28-d).

Figure 3.28: Horizontal currents in cm.s$^{-1}$ after 1 month for the test of the pressure gradient error. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. One vector is represented every 4 vectors. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/test_pressureerror_lev.eps,width=14cm}} \end{figure}

The realistic simulation in high resolution has been conducted for 10 years. The data for the different modelized variables have been stored in an output file every 2 simulated days. This way, analysis of the model outputs are conducted with a high level of spatial and temporal accuracy. These data are also employed to force an individual based model developed at the University of Cape Town to study the behavior of the eggs, larvae, and juveniles of sardines and anchovies around the south and west coast of Africa. Like for the low resolution experiment, the spin up for the high resolution experiment occurs quickly. After 1 year of simulation, the solution appears to be already adjusted to the different forcing. A high level of mesoscale activity develops during the simulation, activity characterized by the generation of Agulhas rings from the Agulhas Current retroflection area, the shedding of cyclonic eddies form the southern tip of the Agulhas Bank, Cape Peninsula, and Cape Columbine. The upwelling front shows also an important variability, developing a series of meanders, plumes and filaments in a realistic manner. In the following sections, an analysis of the averaged behavior of the model solution and variability as well as comparisons with observations validate the simulation.

Figure 3.29: Volume averaged potential temperature ($^o$C) for the high resolution experiment as a function of time in years.The thick line represents the data filtered by a low pass filter with a cut off at 4 years. The dotted line represents the trend of the data.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avgtemp_hres.eps,width=14cm}} \end{figure}

Figure 3.30: Volume averaged salinity (PSU) for the high resolution experiment as a function of time in years. The thick line represents the data filtered by a low pass filter with a cut off at 4 years. The dotted line represents the trend of the data.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avgsalt_hres.eps,width=14cm}} \end{figure}

As for the low resolution experiment, time series are shown of the volume averaged potential temperature and salinity are portrayed respectively on figure (3.29) and on figure (3.30). The spin up appears to be as quick as for the low resolution experiment. The volume averaged temperature shows seasonal variations and a trend of -0.1$^o$C in 10 years (figure 3.29). A surprising result is the trend observed in the time series of the volume averaged salinity, 10 times greater than for the low resolution experiment (nevertheless it is still small). This exhibits the weaker influence of the open boundaries on the inner solution. Two important events are visible as minimums both in the temperature and salinity time series, at year 6 and after year 9. These minimums correspond to a bifurcation in the pathway of the Agulhas Current associated with a large cyclonic eddy perturbing the solution close to the Agulhas Bank. During the two events, the retroflection area has moved towards the eastern boundary of the domain. These variations in the position of the retroflection of the Agulhas Current have been observed in nature.

Figure 3.31: Top: Volume averaged kinetic energy (cm$^2$.s$^{-2}$) for the high resolution experiment. Bottom: Surface kinetic energy (cm$^2$.s$^{-2}$) for the high resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/avg_ke_hres.eps,width=14cm}}\end{figure}

Because smaller structures are resolved in the high resolution experiment, the volume averaged kinetic energy has increased from a mean value of 70 cm$^2$.s$^{-2}$ to 100 cm$^2$.s$^{-2}$ with increasing resolution (figure 3.31-top). The 2 events described previously are also visible as maximums in kinetic energy from year 5 to year 6 and from year 8 to year 9. The seasonal pattern in the mean surface kinetic energy is less marked than for the low resolution experiment. This reveals that more intrinsic instability processes, not directly forced by the wind, are now resolved during the high resolution experiment (figure 3.31-bottom). The mean level of surface kinetic energy is also greater (600 cm$^2$.s$^{-2}$) in the high resolution experiment than for the low resolution experiment (500 cm$^2$.s$^{-2}$).


1 Time averaged variables

By taking the average in time of the different model variables from year 3 to year 10, it has been possible to verify the improvement of the model solution when the resolution has been increased.

Figure 3.32: Annual surface climatology derived from the results of the high resolution experiment. a: sea surface temperature ($^o$C). b: surface currents (m.s$^{-1}$), Vmax is the maximum velocity, 1 vector is portrayed every 4 vectors. c: sea surface salinity (PSU). d: sea surface height (m). The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/surface_clim_hres.eps,width=14cm}} \end{figure}

For the sea surface temperature (figure 3.32-a), the pathway of the sea surface temperature signature Agulhas Current differs slightly from the low resolution experiment. It no longer rounds the Southern limit of the Agulhas Bank in an unrealistic way. The averaged position of the retroflection area is visible in the image of averaged surface current and sea surface elevation (figures 3.32-b and 3.32-d). Some wave like meanders in the upwelling front along the west coast are noticeable in figure (3.32-a), and a jet that follows the shelf edge is present between the southern tip of the Agulhas Bank and Cape Peninsula (figure 3.32-b) as observed in ADCP current measurements [Boyd and Oberholster, 1994]. These specific features of the coastal dynamics of the west coast are subject to a more detailed analysis further in the manuscript. Few differences are noticeable between the low and high resolution experiments for the averaged sea surface salinity (figure 3.32-c).

Figure 3.33: Averaged currents in m.s$^{-1}$ for the high resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The maximum velocity is given for each level. One vector is represented every 6 vectors. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/current_clim_hres.eps,width=14cm}} \end{figure}

The time averaged horizontal currents are portrayed for z = -100 m, -500 m, -1000 m, and -2000 m (figure 3.33). The Agulhas Current and the retroflection are present in the 3 first levels. At these 3 levels an anticyclonic recirculation is noticeable where the Agulhas Current encounter the western open boundary. This might be related to an average position for the Agulhas Rings that are shed from the Agulhas retroflection throughout the simulation. Coastal currents along the west coast are equatorward for z = -100 m. For the lowers levels, the currents along the topography appears to be very weak. In the wide area between the coast and the offshore open boundary the eastern limb of the South Atlantic subtropical gyre manifest itself by a meandering equatorward mean current visible for z = -100 m and z = -500 m. At deeper levels, current in this area appears to be more convoluted and a mean offshore poleward flow is present for z = -2000 m, as in the low resolution experiment.

Figure 3.34: Potential temperature annual climatology ($^o$C) for the high resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/temp_clim_hres.eps,width=14cm}} \end{figure}

Figure 3.35: Salinity annual climatology (PSU) for the high resolution experiment. a: for $z=-100$ m, b: for $z=-500$ m, c: for $z=-1000$ m, and d: for $z=-2000$ m. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/salt_clim_hres.eps,width=14cm}} \end{figure}

On the large scale, few differences appear in the comparison between the low and high resolution experiments for the average potential temperature (figures 3.25 and 3.34) and the average salinity (figures 3.26 and 3.35). For the high resolution experiment, the retroflection of the Agulhas Current is visible for temperature and salinity for z = -100 m, -500 m, and -1000 m. The low temperature and salinity tongue, described in section (3.4.2) extending from the Agulhas Bank is no longer present for z= - 500 m and -1000 m. This feature should be related to the smoothed topography of the low resolution experiment. For z = -100 m the subsurface cool water feature on the eastern part of the Agulhas Bank described previously is more marked in this simulation (figure 3.34-a) and can be seen also in salinity. For z = -500 m, a narrow band of warmer water of unknown origin is present along the shelf edge in the northern part of the domain.

Figure 3.36: Transport stream function (Sv) derived from annual climatology of the high resolution experiment. The isocontour interval is 10 Sv. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/streamfunc.eps,width=7cm}} \end{figure}

If we note $<\bar{u}>$, $<\bar{v}>$ and $<\zeta>$ the time averaged barotropic velocities and free surface elevation, and if we remark that the free surface elevation is small compare to the depth of the ocean depth $H$, the time averaged vertical integrated equation of continuity takes the form:
$\displaystyle \frac{\partial <\zeta>}{\partial t} + \frac{\partial}{\partial x} (H<\bar{u}>) + \frac{\partial}{\partial y} (H<\bar{v}>)$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle \rightarrow \frac{\partial}{\partial x} (H<\bar{u}>) + \frac{\partial}{\partial y} (H<\bar{v}>)$ $\textstyle =$ $\displaystyle 0$ (105)

Hence, the time averaged transport is non-divergent and a time averaged transport function can be extracted from the time averaged barotropic velocities. The resulting time averaged transport function for the high resolution experiment is portrayed on figure (3.36). It shows that the averaged transport associated with the Agulhas Current is around 65 Sv, and that the averaged transport of the simulated part of the Benguela ranges from 10 Sv to 20 Sv. These values can be compared to the measured transport of 75 Sv for the Agulhas Current and of 15 Sv for the a similar part of the Benguela current (figure 1.5) [Shannon and Nelson, 1996].


2 Comparison with temperature and salinity data

Figure 3.37: a: sea surface temperature for the high resolution experiment for the summer months. b: sea surface temperature for the high resolution experiment for the winter months. c: sea surface temperature from the pathfinder satellite data set for the summer months. d: sea surface temperature for the from the pathfinder satellite data set for the winter months. The interval between the isotherms is 1$^o$C.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/pathfinder_seasons.eps,width=14cm}} \end{figure}

Though the linear parameterization of the sea surface temperature feedback on the sea surface heat fluxes can be seen as a restoring term toward climatological sea surface temperature data, it makes sense to compare the averaged sea surface temperature generated by the model to more accurate data than the ones employed to force the model. In figure (3.37), the model sea surface temperature is compared to sea surface temperature measured from 1987 to 1999 by satellite Pathfinder with an accuracy of 9 km. To make the comparison possible, satellite and model data are averaged separately in the austral winter months (from April to September) and in the austral summer months (from October to March). It can firstly be noticed that the seasonal sea surface temperature patterns obtained from the high resolution simulation differ from the smooth climatology used to force the model (figure 3.13), whereas it compares remarkably to the Pathfinder averaged data (figure 3.37), denoting a good behavior of simulated smaller scale structures. For the summer months, one can note on the panel (3.37-a) that the upwelling starts from Cape Agulhas with identical values in the Pathfinder dataset (figure 3.37-c). The temperature in a very narrow band close to the coast around 32$^o$ S is 3$^o$C to 4$^o$C smaller in the simulated field (figure 3.37-a) than in the observation (figure 3.37-c). This can be related to the wind forcing which is too large close to the coast and/or to a small inaccuracy in the satellite measurement of the sea surface temperature close to the west coast. The summer pathway of the modeled 18$^o$C isotherm that detaches from Cape Columbine to come back to the coast 150 km downstream along the coast (figure 3.37-a) is identical, between 35$^o$ S and 31$^o$ S, to the pathway of the 17$^o$C isotherm in satellite data (figure 3.37-c). Further North, in summer, the modeled isotherm tends to move off-shore to come back towards the coast near the northern open boundary. This behavior is not observable on satellite data. A little bump in the 20$^o$C isotherm is present at the southern tip of the Agulhas Bank both in the model and satellite data for the summer months (figures 3.37-a and 3.37-c). It is the signature of the shelf edge equatorward current flowing from the southern tip of the Agulhas Bank towards Cape Peninsula. On the eastern part of the Agulhas Bank, a reverse plume is present in model data, whereas only a bend in the Agulhas Current is visible on satellite sea surface temperature data. For the winter months, the model solution (figure 3.37-b) is also close to the pathfinder sea surface temperature data (figure 3.37-d). Particularly, the cooling of surface waters on the inner Agulhas Bank and the standing wave pattern (with a wave length of about 150 km) just North of St Helena Bay can be observed in both portrayals (figures 3.37-b and 3.37-d).

Figure 3.38: a: average sea-bottom temperatures ($^o$C) over the continental margin off south-western Africa. b: average sea-bottom salinities (PSU) over the continental margin off south-western Africa. Adapted from Dingle and Nelson [1993].
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/dingle_all.eps,width=12cm}}\end{figure}

Figure 3.39: a: simulated climatological bottom temperatures ($^o$C) . b: simulated climatological bottom salinities (PSU).
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/bottom_clim.eps,width=14cm}}\end{figure}

Dingle and Nelson [1993] have extracted the temporal average of a large amount of temperature and salinity data for the bottom of the ocean along the west coast of South Africa (figure 3.38). The first level of the s-coordinate follows the bottom of the ocean. Hence, it is straightforward to extract averaged values for the simulated sea bottom temperature and salinity (figure 3.39) to compare to observations. The model bottom temperatures on the abyssal plain (2$^o$C) and on the slope along the west coast (figure 3.39-a) are identical to the observed ones (figure 3.38-a). On the shelf, the model bottom temperature is warmer ($\sim$ 11-12 $^o$C) than the observations ($\sim$ 9-10 $^o$C), this difference can be a consequence of the absence of a parameterization of the bottom mixed layer during the simulations. Because river runoffs are not taken into account during the simulation, the warm signature of the Orange River outflow, noticeable on figure (3.38-a), is not visible on the model outputs. The invasion of colder water on the shelf from the South, described by Dingle and Nelson [1993], is visible both in the model and the observations. The same remarks can be made for the salinity with bottom shelf waters saltier ($\sim$ 34.8-35 PSU, figure 3.39-b) than in the observations ($\sim$ 34.7-34.9 PSU, figure 3.38-b). The minimum of salinity due to the Antarctic Intermediate Water, observed along the slope (34.4 PSU, figure 3.38-b), is a bit smaller than the minimum visible in the outputs of the model (34.6 PSU, figure 3.39-b). This difference can be due to the data employed at the model initialization and in the forcing of the model open boundaries. These model-data comparisons show a correct behavior of the time averaged model solution. The analysis has now to be extended to check if the variability of the modelized circulation compares also to the observations. This is done in the next section.


3 Variability

The previous section was focusing on the time average model variables ($<X>$). In this section we will concentrate on the anomalies departing from the time averaged variables ($X'=X-<X>$). By definition, the time average of the anomalies is zero, thus we will concentrate on the variance of the anomalies ($<X'^2>$). The standard deviation, or root mean square, can be obtained by taking the square root of the variance ($\sqrt{<X'^2>}$). It is an indicator of the variability produced during the simulations.

Figure 3.40: Root mean square of the sea surface temperature anomaly ($^o$C). top: from the high resolution experiment. b: from monthly Pathfinder (1987-1999) satellite data. The interval between the isocontours is 0.1 $^o$C. The horizontal coordinates account for the longitude and the latitude
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/rms_temp.eps,width=8cm}} \centerline{\psfig{figure=Figures/Chap3/STDEV.ps,width=8cm}}\end{figure}

The root mean square of the sea surface temperature is portrayed on figure (3.40-top) for the model results from year 3 to year 10 and on figure (3.40-bottom) for monthly Pathfinder satellite data from 1987 to 1999. The model solution and the measured sea surface temperature variability compare fairly well. Quite surprisingly, there is a minimum of variability (1.5 $^o$C) in St. Helena Bay both for the model solution and for the observations. This can be due to the presence of the Cape Columbine upwelling plume that counters the seasonal variations of surface temperature. On the contrary, a maximum of variability is also present on the inner Agulhas Bank (2.5 $^o$C) both for the model solution and for the satellite measurements. It can be related to the strong seasonal variations associated with the local surface forcing of the area. Offshore of Cape Columbine, a maximum in sea surface temperature variability follows the pathway of the cyclonic eddies generated from Cape Columbine and the Cape Peninsula during the model simulation. Offshore of this maximum, a large area with low sea surface modelized temperature variability (1.4$^o$C) is in disagreement with satellite data (2.4 $^o$C) . This might be related to the absence of the Agulhas rings that induces important variations in sea surface temperature but which are immediately radiated by the offshore open boundary after their generation during the model simulation.

Figure 3.41: Root mean square of the sea surface elevation anomaly (cm). a: from the high resolution experiment. b: from satellite altimer data. The interval between the isocontours is 5 cm. The horizontal coordinates account for the longitude and the latitude.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/rms_zeta.eps,width=14cm}}\end{figure}

The root mean square of the sea surface elevation has been computed at high resolution (0.25$^o$) from combined Topex-Poseidon and ERS1 altimetry data from October 1992 to October 1997 [Ducet et al., 2000]. It shows a local maximum (50 cm) in the Agulhas retroflection area and minimums over the shelves of the South and West coasts (0-5 cm off the west coast and 5-10 cm of the Agulhas Bank, figure 3.41-b). Away from the open boundary sponge layers, this behavior is remarkably well simulated, qualitatively and quantitatively, in the high resolution experiment (figure 3.41-a). Difference between the model and the altimetry variance in sea surface elevation is present around 17.5$^o$ E and 37$^o$ S, in the pathway of the Agulhas rings. At this location, a maximum in sea surface elevation variance is noticeable in the altimetry data, but is not present in the outputs of the model simulation. This discrepancy can be related, as explained previously, to the vicinity of the open boundary not allowing the Agulhas rings to evolve properly.

Figure 3.42: Subsurface (-50 m depth) eddy kinetic energy (cm$^2$.s$^{-2}$). a: from the high resolution experiment. b: from satellite altimer data. The interval between the isocontours is 100 cm$^2$.s$^{-2}$. The horizontal coordinates account for the longitude and the latitude.
\begin{figure}\centerline{\psfig{figure=Figures/Chap3/eke.eps,width=14cm}} \end{figure}

The eddy kinetic energy per unit of mass $\left(\frac{1}{2}(u'^2+v'^2)\right)$ has been also derived from high resolution altimetry data [Ducet et al., 2000]. It also shows an important maximum in the Agulhas retroflection area (4000 cm$^2$.s$^{-2}$). The variations in the incoming Agulhas Current correspond to a narrow band of high eddy kinetic energy ($>$ 2000 cm$^2$.s$^{-2}$) (figure 3.42-b). North of 35$^o$ S and West of 20$^o$ E, the subsurface eddy kinetic energy (computed for z = -50 m) coincides qualitatively and quantitatively with the observed eddy kinetic energy (figure 3.42-a). This shows that the eddies formed from the Cape Peninsula and Cape Columbine have a correct behavior. Because the incoming Agulhas Current is forced at the eastern boundary from smooth climatology data the remote variations in the Agulhas Current, like Natal pulses and meanders, are not resolved by the model. Hence, very low eddy kinetic energy is present along the eastern part of the Agulhas Bank in comparison to the observations. As a consequence, the simulated eddy kinetic energy is 2 times smaller than the observations in the retroflection area.

Figure 3.43: Top: subsurface (-50 m depth) velocity-variance ellipses. One ellipse is portrayed every 6 grid points. Bottom: velocity-variance ellipses from the times series components derived from Geosat altimetry data, adapted from Morrow et al. [1994]. The scale is the same in each portrayal, represented by the ellipse in the bottom-left corner of the top panel.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/ellipses.eps,width=7cm}}... ...nterline{\psfig{figure=Figures/Chap3/ellipses_alti.eps,width=7cm}} \end{figure}

The eddy kinetic energy shows that important variability occurs near steep topography around the South West corner of Africa. Hence, we can expect the variability to be strongly anisotropic. This as been observed from satellite altimer data [Morrow et al., 1994]. An analysis is conducted to see if the model reveals this anisotropic aspect of the current variability. The magnitude and the direction of the eddy variability are represented as in Morrow et al. [1994], using variance ellipses. The current variations being $u'$ and $v'$, following Morrow et al. [1994], the direction $\theta$ of the major axis of variability, measured anti-clockwise from the direction of the u velocity component is:
\begin{displaymath} \tan{\theta}=\frac{\lambda_1-<u'^2>}{<u'v'>} \end{displaymath} (106)

where $\lambda_1$ is the variance along the major axis and is given by
\begin{displaymath} \lambda_1=\frac{1}{2}\left(<u'^2>+<v'^2>+ \sqrt{\left(<u'^2>-<v'^2>\right)^2+4<u'v'>^2} \right) \end{displaymath} (107)

and $\lambda_2$ is the variance along the minor axis given by
\begin{displaymath} \lambda_2=\left(<u'^2>+<v'^2> \right)-\lambda_1 \end{displaymath} (108)

Note that the eddy kinetic energy ($EKE$) is connected to the velocity variance by
\begin{displaymath} EKE=\frac{\lambda_1 +\lambda_2}{2} \end{displaymath} (109)

The subsurface variances ellipses are represented for the high resolution model for z = -50 m (figure 3.43-top). They show a high degree of similitude with the variances ellipses derived from Geosat satellite altimer data (figure 3.43-bottom). The remarks made for the eddy kinetic energy are also valid for the variances ellipses: good comparison along the west coast, whilst the ellipses are smaller for the model outputs in the Agulhas retroflection area. The incoming variability pattern of the Agulhas Current is illustrated by the two elongated ellipses on the Eastern part of the Agulhas Bank for the satellite analysis (figure 3.43-bottom). This shows important cross isobath variations (characteristic of the meanders and natal pulses) in the Agulhas Current around 23$^o$ E. As explained previously, this variability generated upstream of the area is not resolved by the model. As a consequence, the simulated variability in the Agulhas retroflection area is two times smaller than the observations. Both the model results and the observations show an important anisotropy in the variability, mostly near strong topographic gradients. This anisotropy is important along the southern tip of the Agulhas Bank. The variability appears to be more isotropic on the south west of the Agulhas Bank (where the Agulhas rings are shed) for the observations. This behavior is not noticeable in the model solution, the Agulhas rings being quasi-immediately in contact with the offshore open boundary. Along the West Coast, where altimeter data are available, the model and satellite variances ellipses compare fairly well. This is especially true near the Cape Peninsula, where the presence of the cape induces a strong anisotropy in the subsurface current variability. Animations of the model output reveals that cyclonic eddies are mostly generated from Cape Columbine, Cape Peninsula and the Southern tip of the Agulhas Bank. These areas of eddy generation coincides with areas of strong anisotropy of the current variability. Another interesting feature is the anisotropy of the variability close to the open boundaries. It happens that the open boundary inhibits the cross boundary velocity variance. This can be an effect of the Flather open boundary scheme that constrains the cross boundary barotropic velocities to remain close to smooth climatology data values. One can conclude from this section that though the strong variations in the Agulhas Current coming from the Indian Ocean are not taken into account in the model solution, and disregarding the Agulhas ring behavior when they have a size comparable to the model width, the variability developed in the model in the form of eddies, plumes,... compares remarkably well to observations. This is particularly true along the South African West Coast.


4 Along the West Coast

Because of its importance for the biological components, the model solution of the Benguela upwelling system along the South African West Coast is analyzed in more detail.

Figure 3.44: sea surface temperature along the west coast of South Africa the 30 November of model year 4.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/sst_westcoast.eps,width=8cm}}\end{figure}

The upwelling system is characterised by the presence of cold water at the coast induced by the Ekman divergence forced by the summer equatorward winds. As explained in the first chapter, the resulting upwelling front is highly convoluted and shows characteristic patterns such as the upwelling plumes extending from Cape Peninsula, from Cape Columbine and from the Namaqualand upwelling centers around 30.5$^o$ S. Other characteristic features are the cyclonic eddies that are shed from the capes. Because these patterns have been mostly related to the small scale spatial and temporal structure of the wind forcing [Jury, 1985a,Jury, 1985b,Jury et al., 1985,Jury, 1986,Jury and Taunton-Clark, 1986,Jury, 1988,Kamstra, 1985,Taunton-Clark, 1985], one might expect that a model solution only forced by a coarse smooth climatological wind field (see figure 3.10) should not generate these elements, characteristic of the Benguela upwelling system. This is not the case. Looking at the modelized sea surface temperature at the beginning of the upwelling season, the 30 November of model year 4 (figure 3.44), one can note that most of observed patterns of the upwelling front are present during the high resolution experiment: The shape and dimensions of these simulated features compare well with the observations. Hence, the mesoscale activity of the Benguela upwelling system is driven by intrinsic oceanic instability processes and is triggered by topography, as explained by Batten [1997]. The small scale wind structure doesn't appear to be the main driving forcing of this mesoscale activity.

Figure 3.45: a: the South and West Coast time averaged currents (cm.s$^{-1}$) at 30 m depth. One vector is portrayed every 3 vectors. b: along shore current (cm.s$^{-1}$, positive towards the equator) cross section North of Cape Columbine. The greyscale range represents the equatorward along shore current in cm.s$^{-1}$. The isocontour interval is 5 cm.s$^{-1}$. The position of the cross section is given by the dashed line on panel a
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/current_wcoast.eps,width=14cm}}\end{figure}

The time averaged subsurface modelized currents are highly controlled by the topography (figure 3.45-a). A large amount off Acoustic Doppler Current Profilers data has been collected by Boyd and Oberlholster [1994] to generate a time averaged map of the 30 m depth currents on the shelves of the South and West coasts of South Africa. Important quantitative similitude occurs in the comparison between the model outputs and the ADCP data: The principal discrepancy between the model and ADCP data is the current in St Helena Bay, in a narrow band just North of Cape Columbine. For the ADCP data, the flow in St Helena Bay is weak and show a cyclonic motion, as explained by Holden [1985] and simulated by the barotropic experiment in the second chapter. For the realistic simulation, an important flow of 20 cm.s$^{-1}$, follows the Northern part of Cape Columbine to fill St Helena Bay. The results of the second chapter might give an explanation of this phenomenon. In the barotropic experiment, the sluggish circulation is present inside the attached cyclonic eddy that develops on the flat portion of the shelf where the dynamics are controlled by a balance between advection of vorticity and bottom friction. Although weighted by $\frac{\nabla H}{H}$, the Shapiro filter employed to smooth the bottom topography for the realistic experiment removed the flat portion of the shelf in St Helena Bay. The resulting bottom topography corresponds more to the exponential topography employed for the analytical standing shelf wave solution (figure 2.14). In the case of the exponential topography, the vortex stretching term gains importance and the flow sticks to the downwind side of the cape (figure 2.17-a) as for the realistic experiment. A cross-shelf cross section of the time averaged along shore velocities North of Cape Columbine for the upper 500 m depth is portrayed on figure (3.45-b). It shows an equatorward baroclinic jet of 20 cm.s$^{-1}$ above the shelf break. A net poleward motion with velocities of 5 cm.s$^{-1}$ is present on the shelf break in agreement with the current meter measurements of Nelson [1989]. A subsurface poleward current is also present at the coast with velocities of 5 cm.s$^{-1}$. This is also in agreement with observations [Boyd and Oberholster, 1994,Nelson, 1989].

Figure 3.46: a:the South and West Coast time averaged barotropic currents (cm.s$^{-1}$). All velocity vectors are represented. b: annual averaged free surface elevation. The greyscale range represents the free surface elevation in m.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/baro_wcoast.eps,width=12cm}} \end{figure}

In order to compare the output of the realistic simulation to the results of the second chapter, the time averaged barotropic velocities and free surface elevation are portrayed for a small area centered on St Helena Bay in figure (3.46). Note that in presence of a baroclinic pressure gradient induced by the stratification, the baroclinic velocities are no longer in geostrophic equilibrium due solely to the pressure gradient induced by free surface elevation. The annual averaged along shore wind stress utilized to force the model in the area limited by the bounds of figure (3.46) is 0.5 N.m$^{-2}$. The characteristic barotropic velocities on the shelf in figure (3.46-a) are of about 7.5 cm.s$^{-1}$ which is slightly smaller than the values obtained in section (2.8) for the case of the exponential shelf for this wind forcing. For both the barotropic velocities and the free surface elevation, a standing wave pattern is noticeable in the lee of Cape Columbine (figure 3.46-a). It has a wave length of approximately 100 km. The wave length for the first barotropic standing wave mode derived in section (2.8), using an exponential topography and an along shore mean current of 7.5 cm.s$^{-1}$ is 60 km. This is 1.6 times smaller than the wave length observed in the realistic experiment. Different arguments can be advanced to explain this difference: firstly, one can note that the topography of the realistic experiment is not constant along West Coast, with the 200 m isobath moving towards the coast around 31.5$^o$ S. This can alter the vorticity equation employed to derived the standing wave equation. Secondly, the presence of stratification can alter the behavior of the standing coastal trapped waves. Looking at the dispersion relation for coastal trapped waves for different stratification [Huthnance, 1978], one can note that the wave length decreases for a given phase velocity (hence for a fixed advection by a mean along shore current in the standing wave case) with increasing stratification, invalidating this second argument. Thirdly, the hypothesis made in section (2.8) that the along shore mean current is barotropic is surely not valid in the realistic case where the along shore currents associated with the upwelling are a surface baroclinic jet. The process of a standing barotropic shelf wave excitation by a baroclinic surface jet has not been treated in section (2.8). In this case, the model of a cape moving in still water is invalid. Surprisingly, if we introduce the subsurface velocity observed on figure (3.45) as the mean along shore current (20 cm.s$^{-1}$) in the standing wave equation, we obtain exactly 100 km for the wave length of the first mode. This can not be taken as a proof. Nevertheless, we still can note that the time averaged circulation simulated on the shelf in St. Helena Bay for the realistic experiment is qualitatively in agreement with the results of the standing shelf wave analytical study of section (2.8).

Figure 3.47: a: south and west coast averaged barotropic currents (cm.s$^{-1}$) for February. All velocity vectors are represented. b: averaged free surface elevation for February. The greyscale range represents the free surface elevation in m.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/baro_wcoast_feb.eps,width=12cm}}\end{figure}

The same analysis is made for the middle of the upwelling season: for the month of February (figure 3.47). For this particular month, the along shore wind stress on the west coast has an averaged value of 0.075 N.m$^{-2}$. The averaged barotropic currents portrayed in figure (3.47-a) have a characteristic velocity of about 10 cm.s$^{-1}$. An interesting feature in figure (3.47) is the separation of the flow into 2 branches past Cape Columbine: one branch filling St Helena Bay, the other one flowing north westward parallel to the coastline to come back to the coast 150 km North of Cape Columbine. This is characteristic of the Columbine divide described by Shannon [1985] (figure 1.13). Whereas the inner branch shows a standing wave pattern comparable of the annual averaged circulation, the behavior of the outer branch resembles the attached cyclonic eddies modelized in the second chapter. For this value of wind stress, the size of the eddy described in chapter 2 is 60 km (figure 2.6), which is smaller than the size of the detachment pattern observed in figure (3.47). On the contrary the length scale predicted by the balance between advection and bottom friction (equation 2.21) is 125 km, which is relatively close to the size of the pattern observed in figure (3.47). This last result should be taken cautiously, since stratification or variations in bottom topography can seriously alter the circulation within St. Helena Bay.


5 Generation of cyclonic eddies by the Agulhas Current in the lee of the Agulhas Bank

Though the realistic configurations have been principally set up to study the shelf dynamics around the South-west corner of the African continent, and particularly the Benguela upwelling system, the results of model regarding the Agulhas Current has motivated the redaction of two papers. They focus attention on specific mesoscale processes produced by the model, observed in nature, and not simulated before in this area. The present section describes the generation of cyclonic eddies by the Agulhas Current in the lee of the Agulhas Bank [Penven et al., accepted]. The second paper, still in preparation, forms the next section. The inter-ocean exchanges brought about by the shedding of Agulhas rings south-west of Africa has been much studied because of their global climatic implications [ Lutjeharms, 1996; De Ruijter et al., 1999]. Recent models have shown that this movement of warm water from the Indian to the Atlantic may in fact control the rate of thermohaline overturning of the whole Atlantic [Weijer et al., 1999]. There are also other, smaller sources of exchange including Agulhas filaments [Lutjeharms and Cooper, 1996] and intrusions of cold subantactic water [Shannon et al., 1989]. In addition, a number of cold, cyclonic eddies have been observed in the South-east Atlantic [e.g. Duncombe Rae et al., 1996; Gründlingh, 1995]. For a proper quantification of the inter-ocean exchanges the characteristics of all these components need to be known. The origin, behavior and potential role in inter-ocean exchanges of the cyclonic eddies have not been established. We verify if the high resolution model might simulate these cyclonic eddies, and we investigate the origin of such cyclonic eddies.

Figure 3.48: (a) Model simulation of the sea surface height and barotropic velocities (1 vector every 4 grid points) on 27 April of model year 2. (b) Naval Research Laboratory, MODAS 2.1, sea surface height derived from in-situ and satellite altimetry data on 16 January 1993.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/PENVENFIG1.eps,width=14cm}}\end{figure}

The model gives a realistic portrayal of the known circulation in this ocean region (figures 3.48-a and 3.50-b), including mesoscale details such as the Agulhas Current, shear edge features on the landward border of the current, Agulhas rings and the wind-driven coastal upwelling along the west coast. The Agulhas Current overshoots the concave eastern part of the Agulhas Bank and has its maximum surface velocities of about 1 m.s$^{-1}$ at the southernmost tip of the shelf. These simulations are all in very good agreement with observation [e.g. Lutjeharms et al., 1989]. The features evident in the modeled sea surface height are also well represented in the sea surface height observed by altimetry (figures 3.48-a and 3.48-b). These consist of an anti-cyclonic Agulhas ring as well as some cyclonic eddies of unknown origin. One of these cyclonic eddies lies between the Agulhas Current and the western side of the Agulhas Bank in both portrayals. Cyclonic eddies are prevalent at this particular location in the model and this portrayal is therefore quite characteristic in this respect.

Figure 3.49: The drift patterns for a few characteristic cyclonic eddies west of the continental shelf. The bold tracks are for model eddies shed during model year 3; the dotted tracks for cyclones observed in Naval Research Laboratory, MODAS 2.1 sea surface height analysis for 1993.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/PENVENFIG2.eps,width=7cm}}\end{figure}

An analysis of the generation, drift and persistence of cyclonic eddies at this location as observed in satellite altimetry is highly consistent with the model simulations (figure 3.49). They are formed at roughly the same location, drift off in the same westerly direction, are produced with the same frequency, and can dissipate rapidly or grow in intensity. These results give us confidence that these features are not model artifacts. It would be important to establish the depth and other hydrographic characteristics of these particular cyclonic eddies.

Figure 3.50: A comparison between a characteristic cyclonic eddy generated off the Agulhas Bank in the model (b and d) and one observed at sea (a and c). The panel b shows the simulated surface currents and sea surface temperature on 21 November of model year 4. It also shows the line along which a vertical temperature section (d) has been extracted from the model at the same date. The sea surface temperature generated from NOAA 11 AVHRR data (a) is for 27 November 1992. The straight broken line shows the cruise track of the research vessel "Polarstern" that left Cape Town on 3 December 1992. The dotted line shows the location of the warm filament on 4 December 1992 when the vessel would have been in the center of this feature. The vertical temperature section (c) was measured by XBT (expendable bathythermograph) from the ship. The locations of the 200 m, 500 m and the 1000 m isobaths are shown in panels a and b.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/PENVENFIG3.eps,width=14cm}}\end{figure}

The sea surface temperature and current portrayals in the upper panels of figure (3.50) exhibits substantial agreement. The model shows a plume of warm surface water that encircles a cyclonic eddy in the lee of the continental shelf (figure 3.50-b). To the North, there is faint indication of an anti-cyclonic eddy in the current vectors, encircled by the extension of the warm filament. The satellite thermal infrared image shows much the same (figure 3.50-a), except that the locations of the two features are slightly displaced. A vertical temperature section through the two eddies in the model shows the warm surface filaments as well as an anti-cyclonic eddy centered at 35.3$^o$ S and a cyclonic eddy at 36.7$^o$ S (figure 3.50-d). In the model, the eddy is still noticeable at 1500 m depth. An XBT section was undertaken across this region in a very similar location [Bathmann et al., 1994]. It is shown in the left-hand panels of figure (3.50). Regrettably, the sea surface temperature image was largely obscured by cloud on the day of the hydrographic section. Nonetheless, the position of the feature could be located and is shown relative to that of the 27 November 1992 by the dotted line in figure (3.50-a). The vertical temperature section for this line resembles closely that of the model, both in the surface filaments as well as in the presence of both a cyclonic as well as an anti-cyclonic eddy (figures 3.50-c and 3.50-d). The strong resemblance of the hydrographic and model results shows that the cyclonic eddies generated off the Agulhas Bank in the model are not inconsistent with observations and extend to substantial depths. In the model, the coastal part of the Agulhas Current tends to follow the topography at the southern tip of the Agulhas Bank, carrying warm surface filaments spreading northward towards Cape Peninsula. At times, the southern part of this flow detaches from the topography and a cyclonic meander developes. After approximately 1 month, it detaches from the Agulhas Bank, taking away the warm filament (see Figures 3-a and 3-b). Occasionally, a cyclonic perturbation may follow the shelf break from the eastern Agulhas Bank and trigger the cyclonic eddy generation process. The averaged values of the Rossby ($R_o=0.04$) and Burger ($S=3.8$) numbers have been computed as described in Boyer and Tao [1987], taking the width of the Agulhas Current as a characteristic length scale (100 km). They lie in the cyclonic eddy shedding regime found in rotating tank experiments, when a linearly stratified flow encounters an obstacle [Boyer and Tao, 1987]. Thus, the generation of cyclones past the Agulhas Bank can be explained by a flow detachment process. Their characteristic diameters range from 50 km to 200 km. Although they are generated irregularely, the number of eddy shed each year is relatively constant, respectively from year 1 to year 10: {3,3,4,4,4,5,4,5,4,5}. Because the model domain has a limited size, most of the cyclonic eddies are radiated through the offshore boundary before decaying (figure 3.49). The first simulations of the southern Agulhas Current by a model with high spatial resolution suggests that the disposition of the current relative to the southernmost part of the African continental shelf is such that cyclonic lee eddies may be generated. This may be an intermittent process since the Agulhas Current does not follow the most southerly part of the shelf consistently [Lutjeharms et al., 1989]. The model indicates that these eddies may be shed at irregular intervals. They may subsequently move off in a number of directions in the South-West Indian Ocean [Gründlingh, 1995]. Their further dispositions, their contribution to the melange of water masses in this ocean region as well as their potential role in stabilizing Agulhas rings (Kizner, personal communication) all need to be determined. If all cyclonic rings in the south-western Atlantic have this suggested origin, they will not contribute to inter-ocean exchanges since they are generated in situ.


6 Shear edge eddies of the southern Agulhas Current

The southern and the northern parts of the Agulhas Current consistently exhibit differences in their path characteristics. Whereas the trajectory of the northern part is unusually stable for a western boundary current [Gründlingh, 1983], the southern part has large meanders [Harris et al., 1978,Lutjeharms, 1981]. This contrasting kinematic behavior is a consequence of the steepness of the continental shelf slope [De Ruijter et al., 1999a]. A very steep slope and narrow continental slope constrains the northern part of the current whereas a more gradual slope along the wider continental shelf south of Africa (the Agulhas Bank) allows more side-ways meandering. The exception is a small region of weaker slope along the northern Agulhas Current, at the Natal Bight, where small meanders and accompanying eddies have also been observed [Gründlingh and Pearce, 1990] along the edge of the current. Cyclonic eddies are embedded in the landward border of current meanders in the southern Agulhas Current and seem to gain energy from the lateral shear due to the juxtaposition of fast moving water in the current and more sluggish water over the shelf. These cyclonic shear eddies may have a substantial influence on adjacent shelf waters of the Agulhas Bank. They are usually attended by warm plumes drawn from the surface waters of the current [Harris et al., 1978,Lutjeharms, 1981,Schumann and van Heerden, 1988,Lutjeharms et al., 1989,Goschen and Schumann, 1990]. These plumes have in fact been the prime manifestation of shear edge eddies that has been used for their study by thermal infrared sensing from satellite. Depending on the reigning wind conditions, this anomalously warm surface water may then spread over substantial parts of the adjacent shelf [Lutjeharms et al., 1989,Goschen and Schumann, 1994]. It has even been surmised [Swart and Largier, 1987] that this warm surface water may thus make an important contribution to the intense seasonal thermocline over the eastern part of the Agulhas Bank [Schumann and Beekman, 1984]. This hypothesized contribution has as yet not been quantified. Regrettably, few of these shear edge eddies have to date been hydrographically surveyed. A study by Lutjeharms et al. [1989] has used two sets of hydrographic observations as well as satellite imagery to give a general description of these features. More eddies are found the further downstream one measures. In general their lateral dimensions also increase downstream. They seem to be particularly prevalent in the shelf bight on the eastern side of the Agulhas Bank. Judged by the trailing plumes of warm water, they all move downstream, but an insufficient number of time series in the satellite imagery has prevented a statistically reliable propagation speed to be derived. Once past the southern tip of the Agulhas Bank, warm plumes may be advected equatorward into the southern Atlantic Ocean, making a modest contribution to the inter-ocean leakage of heat and salt [Lutjeharms and Cooper, 1996]. The fate of the eddies themselves remains unknown. Shear edge eddies my have a diameter of about 50-100 km and are represented by a well-defined thermal dome (figure 3.53-b) inshore of the landward border of the Agulhas Current. This portrayal of these eddies is complicated by the intermittent passage of a Natal Pulse through the region. Natal Pulses are substantial, single meanders in the trajectory of the Agulhas Current [Lutjeharms and Roberts, 1988] that are generated by current instability at the Natal Bight [De Ruijter et al., 1999a] and that propagate downstream at a rate of about 20 to 30 km.day$^{-1}$. When they move past the Agulhas Bank, they seem to slow down to roughly 5 km.day$^{-1}$ [Lutjeharms and Roberts, 1988,Van Leeuwen et al., 2000] and become difficult to distinguish from the other meanders normally found here. Like these other meanders, Natal Pulses also incorporate cyclonic eddies [Gründlingh, 1979] that move downstream with them. Once past the southernmost tip of the Agulhas Bank, Natal Pulses may trigger the shedding of an Agulhas ring [Van Leeuwen et al., 2000] by loop occlusion at the Agulhas retroflection [Lutjeharms and van Ballegooyen, 1988]. The role this process plays in interocean exchanges South of Africa has been extensively reviewed by De Ruijter et al. [1999b] and Lutjeharms [1996]. It is not known what happens to the cyclone that is carried in a Natal Pulse once the meander has passed the tip of the Agulhas Bank. There is evidence that a cyclonic eddy may on occasion be generated in the lee (western side) of the Agulhas Bank by the passing Agulhas Current [Penven et al., accepted] and it has been surmised that the cyclonic eddy that comes with a Natal Pulse may be absorbed by such a lee eddy. Similarly, it has been hypothesized that shear edge eddies may be absorbed by cyclonic eddies in a passing Natal Pulse, thus enhancing it and even contributing to the efficaciousness with which it influences Agulhas ring shedding downstream. The outputs of the high resolution experiment are employed in order to understand the processes involved in the generation and subsequent behavior of shear edge eddies

1 Simulation of the shear edge features

Figure 3.51: Sea surface temperature ($^o$C) and surface currents (m.s$^{-1}$) over the Agulhas Bank for the 10 April of model year 9. The greyscale range represents the sea surface temperature in $^o$C. The interval between the isocontours is 0.5 $^o$C.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge10april9.eps,width=8cm}} \end{figure}

The sea surface temperature over the Agulhas Bank for the 10 April of model year 9 is given in figure (3.51). Simulated surface currents are also shown. Temperatures in the core of the Agulhas Current are 23$^o$C and 24 $^o$C and decrease downstream. Temperatures on the shelf lie between 18 $^o$C and 20$^o$C . These values are all very representative for these features for this time of year [Schumann and Beekman, 1984,Lutjeharms, 1996]. The core of the Agulhas Current does not follow the shelf edge very rigorously, but overshoots the bight in the Agulhas Bank at about 23$^o$ E, and eventually crosses the 1000 m isobath at the southernmost tip of the shelf (figure 3.51). Velocities in the current are up to 1 m.s$^{-1}$; over the Agulhas Bank in general much lower ($<$ 20 cm.s$^{-1}$). Both these values are very realistic [Schumann and Perrins, 1982]. Two surface plumes of warm water at the inshore edge of the Agulhas Current are evident in figure (3.51). A small plume lies at 20$^o$30' E, the larger one starts at 22$^o$ E. The latter represents a fully developed shear edge feature around the concave part of the shelf break with surface counter currents of up to 40 cm.s$^{-1}$. The implied shear edge eddy has dimensions of approximately 80 by 150 km. These simulations of the surface characteristics of this region show a high verisimilitude to those evident in figure (3.53-a) and give a most realistic portrayal of the known conditions for the region, i.e. for this time of the year.

Figure 3.52: The simulated sea surface temperature ($^o$C) for the Agulhas Bank and direct vicinity for a: 2 June, b: 2 August, c: 2 October and d: 2 December of model year 7. The greyscale range represents the sea surface temperature in $^o$C. The interval between the isocontours is 1 $^o$C. The dashed lines represent the 200 m, 500 m and 1000 m isobaths.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_y7.eps,width=14cm}}\end{figure}

A portrayal of model results for all seasons of model year 7 is given in figure (3.52). Two aspects of the inter-seasonal variability for the sea surface is immediately apparent. The first is the expected seasonal temperature variations. Both the temperatures in the core of the Agulhas Current and those of the shelf waters exhibit a seasonal cycle of at least 2$^o$C. These values also are realistic [Christensen, 1980]. The second thing that is clear is that the location of the simulated Agulhas Current is not entirely stable. This is particularly noticeable downstream of the Agulhas Bank - from June to October the current shifted by at least 240 km. Along the eastern edge of the Agulhas Bank, the meandering of the Agulhas Current is less noticeable, but inspection of its location at about 23$^o$ E in June (figure 3.52-a) and in October (figure 3.52-c) shows a noticeable southward shift. The main shear edge eddy is a recurrent feature of the Agulhas Bank bight, but may lie farther upstream (figure 3.52-a) or downstream (figure 3.52-d). Figures (3.51) and (3.52) therefore demonstrate that the model simulates the known surface conditions very well.

Figure 3.53: a: distribution of temperature ($^o$C) at 10 m over the Agulhas Bank in March 1968. b: Temperature cross section across a border eddy south of Mossel Bay. Adapted from Lutjeharms et al. [1989].
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge1.eps,width=14cm}}\end{figure}

Figure 3.54: a: This panel is identical to that in Figure (3.51). The broken line indicates the location of a simulated vertical section from the land across the shelf and shelf edge an crossing the edge of the Agulhas Current that is given in the right panel. b: A vertical temperature section to 200 m depth along the broken line in the upper panel. Note the warm water of the Agulhas Current, the dome of cold water forming a shear eddy and the plume of warm water extending to a depth of 20 m.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_section1.eps,width=14cm}} \end{figure}

Figure 3.55: a: distribution of temperature ($^o$C) at 10 m over the Agulhas Bank in September 1968. b: Temperature cross section across a border eddy south of Mossel Bay. Adapted from Lutjeharms et al. [1989].
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge2.eps,width=10cm}}\end{figure}

Figure 3.56: a:The sea surface temperature ($^o$C) simulated for 15 September of model year 7. Otherwise as in Figure (3.51). The broken line indicates the location of a simulated vertical section from the land across the shelf and shelf edge an crossing the edge of the Agulhas Current that is given in the right pane. b: potential temperature ($^o$C) vertical section to 200 m along the line.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_section2.eps,width=14cm}} \end{figure}

To judge the full effectiveness of the model, one would want to test this in the vertical as well. A vertical section is shown in figure (3.54). The simulated section in this figure is taken across the eastern Agulhas Bank, across a warm plume, a shear edge eddy as well as the landward edge of the Agulhas Current. The location as well as the time of year has been selected to make comparison easier with a hydrographic section taken at this location and portrayed in figure (3.53-b). The main features are represented well, including the thermal gradient of the edge of the Agulhas Current, the shape and size of the cold dome representing the eddy as well as the depth and shape of the warm surface plume heading upstream. The plume overlies the landward edge of the eddy, as represented by the strongest horizontal, sub-surface temperature gradient. This corresponds to what has been observed in nature [Lutjeharms et al., 1989]. Even the vertical gradient of the seasonal thermocline over the Agulhas Bank is entirely representative for this time of year [Lutjeharms et al., 1996]. Shear edge eddies are also present in winter as can be seen in figure (3.56) for the model and in figure (3.55) for hydrographic data. For this time of the year, the thermic vertical structure is also remarkably well represented by the model, showing a relatively homogeneous temperature (15$^o$C) onto the Agulhas Bank (figures 3.56-b and 3.55-b). The sea surface temperature shows the warm plume that rounds the shear edge eddy both in the model (figure 3.56-a) and in the hydrographic data (figure 3.56-a). The reverse plume appears to be further downstream ($\sim$ 100 km) in the hydrographic data than in the model results. At this time of the year, the cold dome happens to be more marked than in summer. Regrettably there are insufficient data on shear edge features of the Agulhas Current to afford more inter-comparisons between the model simulations and actual observations. Nonetheless, those that have been presented here are such that they give us considerable confidence in the reliability of the model.

2 Evolution of shear edge eddies

Figure 3.57: The simulated evolution of shear edge eddies along the eastern side of the Agulhas Bank. The broken lines denote the 200 m, 500 m and 1000 m isobaths. Shown is the sea surface height (contour lines from -60 to 20 cm, the latter being the value south of the Agulhas Current). The interval between the isocontours is 5 cm. The dates are a: 7 December of model year 7, b: 15 December 7, c: 23 December 7, d: 1 January 8, e: 7 January 8, f: 15 January 8, g: 23 January 8 and h: 1 February 8.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_serie.eps,width=14cm}} \end{figure}

A time series for sea surface elevation off the Agulhas Bank and general vicinity is portrayed in figure (3.57). It is given with 8 day intervals, starting on 7 December of model year 7. In the first image (figure 3.57-a) there is a well-developed Agulhas Current with a large cyclonic shear eddy in the eastern bight of the edge of the Agulhas Bank. West of the Agulhas Bank there is a weakly developed anti- cyclone about 250 km in diameter. The presence of an anti-cyclone in this region has been shown to occur at least 12 % of the time [Lutjeharms and Valentine, 1988]. The model also shows a small anti-cyclone in the north-easterly corner of the modeled field over the Agulhas Bank. It remains largely unchanged throughout the sequence. We are unsure if it is an artifact of the boundary conditions. A week later (figure 3.57-b), the shear edge eddy has shrunk somewhat in size, having split off a very small eddy that is now moving downstream at about the rate such eddies are known to propagate [Lutjeharms et al., 1989]. The anti-cyclonic eddy had grown somewhat in strength. On the next panel these developments continue (figure 3.57-c). By 1 January (figure 3.57-d) the smaller shear edge eddy has grown in size and has left the eastern shelf edge. The large shear eddy in the bight has again spawned a protrusion, this time considerably bigger. A week later this protrusion has split of as a slightly larger eddy and is moving downstream as did its predecessor. Its predecessor has by now developed into a strong Agulhas Bank lee eddy, characterized by a sea surface height 25 cm lower than its South Atlantic environment. Two weeks later (figure 3.57-g) the second shear edge eddy had been absorbed into this lee eddy, had strengthened it to a 30 cm anomaly, and the lee eddy can be seen moving away into the South Atlantic Ocean [Penven et al., accepted]. Such cyclones have been observed in this region in altimetry [Gründlingh, 1995] and from hydrographical measurements [C. Veth, S. Drijfhout, personal communication]. They have on occasion been traced in altimetry to this location that is associated with a lee eddy. On the last panel of this sequence (figure 3.57-h) the lee eddy has moved totally eastward and the main shear edge eddy is producing another downstream protrusion. Note that the downstream leakage from this main shear eddy is always associated with a slight offshore meander in the Agulhas Current trajectory.

Figure 3.58: Tracking the westward leakage of cyclonic vorticity in the form of a bud-off eddy. The left panel (a) depicts the sea surface height (5 cm between the isocontours). Note the cores of the shear edge eddy between 23$^o$E and 24$^o$E and the Agulhas Bank lee eddy between 18$^o$E and 19$^o$E. The broken line is used to track the movement of water with a cyclonic motion (low sea surface height). The Hovmüller diagram on right panel (b) shows the movement in time (increasing upward on the ordinate). The distance on the abscissa is from the easternmost point on the solid line in the upper panel, i.e. x=0 is at 24.25 E and 35 S; the distances on the abscissa are positive traveling westward. The interval between the isocontours is 3 cm.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_fig6.eps,width=14cm}} \end{figure}

These images are a piecemeal selection from a more detailed series. This process can therefore be shown with considerably better temporal resolution (figure 3.58). The left panel shows the setting for this region with the thick broken line indicating the path along which the sea surface elevation will be followed (figure 3.58-a). In figure(3.58-b), clear greyscale shows low sea surface deepening in comparison to the surroundings, thus cyclonic motion. Hence, the displacement of cyclonic shear edge eddy and the lee eddy can be followed along the broken line by looking at low sea surface height values. The Hovmüller diagram in the right panel (figure 3.58-b) shows the development over time, starting at the bottom and moving upward. Note that the direction is not the same as in the left panel (figure 3.58-a), but has been inverted. Starting from 1 December of model year 7 (see figure 3.57-a) there is a well-developed shear edge eddy at 100 km and an anticyclone at 500 km, i.e. in the lee of the Agulhas Bank. The subsequent leakage of cyclonic eddies from the shear edge eddy to the lee of the bank is shown most clearly by the clear track from the left-hand side of the panel to the right-hand side starting on 1 January and ending at 450 km distance on 20 January. On the first part of its trip, the shear edge eddy travels along the line at approximately 6 km.day$^{-1}$. On the 15 January, at the Southern tip of the Agulhas Bank (x = 350 km), it accelerates up to 32 km.day$^{-1}$. On the 20 January, it is absorbed into by a larger cyclonic eddy developing in the lee of the Agulhas Bank [Penven et al., accepted] The replacement of an anti-cyclone by a cyclone in the lee of the Agulhas Bank is represented by the change from dark to clear at 500 km distance along the line on about 1 January of model year 8. These simulations leave one with the impression that it is the Agulhas Current that sequesters the shear edge eddy in the eastern bight of the Agulhas Bank and, when it meanders, allows a certain part of the eddy to escape. The cyclonic vorticity of this propagating eddy may then be taken up by a lee eddy. How does this scenario compare to what has been observed at sea? Satellite imagery for the region shows that warm plumes, assumed to be associated with shear edge eddies, consistently exhibit a downstream motion. A plume might of course be associated either with an independent, separated eddy or with the leading edge of a protrusion from a shear eddy trapped in the eastern bight. Therefore both possibilities remain open: there may be distinct eddies moving downstream, or occasional protrusions still attached to a trapped eddy. Statistically, the bight is known for the prevalence of shear edge eddies [Lutjeharms et al., 1989] so that the concept of a near-stationary, but leaky, eddy at this location is not inconsistent with the data. However, with very limited data it cannot be confirmed in a rigorous manner at present. If this single simulation of the evolution of shear edge eddies at the Agulhas Current edge seems reasonably sound, how representative is it of other such sequences in the model?

3 Interannual variation in eddy evolution

Figure 3.59: Hovmüller diagrams of simulations of the temporal behavior of circulatory features along the Agulhas Bank. The sea surface elevation along the line defined on Figure (3.58-a). The interval between the isocontours is 3 cm. The left panel (a) represents model years 3 to 4, the central panel (b) years 5 to 6 and the right hand panel (c) years 7 to 8. Otherwise the images are identical to that of the lower panel of Figure 7.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/trackshearedge.eps,width=14cm}} \end{figure}

The simulated behavior of shear edge eddies along the edge of the Agulhas Bank for a 6-year period in the model run is presented in figure (3.59). Except for the extended periods the portrayal is identical to that of figure (3.58-b). The sea surface height is tracked along the line used in figure (3.58-b). A number of notable features are immediately apparent. First, their is considerable inter-annual and intra-annual variation, generated by the internal physics of the model itself. In this depiction, the center of the trapped shear eddy is located at a distance of 100 km in figure (3.59). This location seems to be relatively invariant, but the dimensions and intensity of the eddy varies considerably with time. There are events lasting about 6 weeks when it is particularly well-developed (e.g. November year January year 6, figure 3.59-b), and longer periods when is seems very weak or totally absent. The absence of a trapped shear eddy seems to coincide roughly with the presence of the Agulhas Current closer inshore. This movement of the current is reflected by the appearance of sea surface height elevations at a distance of about 300 km along the track and a simultaneous sea level enhancement in the far eastern part of the domain (left side of columns in Figure 8). On June of model year 7 (figure 3.59-c) these two regions of higher sea level nearly touched, showing strong perturbations in the large scale currents. At the westernmost part of the field (right side of columns in figure 3.59) marked positive and negative departures from the mean sea surface height show the spasmodic exchange of a cyclonic and an anticyclonic motion. This may be interpreted as the replacement of a northward meander in the simulated Agulhas Current with a cyclonic lee eddy (e.g. May year 3, figure 3.59-a). Perhaps of most interest is the clear signs of intermittent but frequent leakage of cyclonic motion (figure 3.59-b) from the upstream region to the downstream region. These are shown as feint clear lines from left to right in the columns of figure 3.59. More marked eddy leakage can be observed on Mars 3, September 3, June 4, September 5, December 5, September 6, May 7, December 7, September 8 (figure 3.59). They indicate that these eddies that carry cyclonic motion downstream propagate at roughly 8 km.day$^{-1}$, very close to estimates made from satellite imagery [Harris et al., 1978,Lutjeharms et al., 1989] for shear edge eddy progression and even for the movement of Natal Pulses along the Agulhas Bank [Lutjeharms and Roberts, 1988,Van Leeuwen et al., 2000]. The frequency with which there is a movement of from the trapped shear eddy downstream seems to have some regularity if not affected by the large scale movement of the Agulhas Current (e.g. March to November of year 4; figure 3.59-a). The spawning of a traveling eddy seems to occur at intervals of roughly 20 days in the model. Observations to date are insufficient to validate this, mostly due to interference to the satellite thermal infrared measurements by cloud cover. However, indications are of a considerably more irregular process.

4 Shear edge eddy trapping by topography

The strong indication by the model that a shear eddy will get trapped in the bight of the shelf and that this is largely determined by the interaction of the Agulhas Current with the topography of the bight is tested in this section by a set of idealized numerical experiments. The bight that lies at the eastern side of the Agulhas Bank around 23$^o$ E, is characterized by the bend toward shore of the 500 m isobath, the isobaths deeper than 1500 m running quasi rectilinearly South-westward. This creates an embayment of about 200 km long and 100 km wide. The goal of this section is to show that the interaction between a density driven current with a topographic accident such as the Agulhas Bight is able to generate a standing recirculation process identical to the large shear edge eddy observed in the realistic experiments.

Figure 3.60: a: bathymetry (m) employed for an idealized model of the circulation along an indentation in a shelf slope. b: The vertical potential temperature ($^o$C) distribution used for initial and boundary conditions, on a section from the coast to the deep ocean.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_fig8.eps,width=14cm}}\end{figure}

The ROMS model has been configured as follows:

Figure 3.61: Sea surface height (5 cm interval) after 60 model days. The currents have been adjusted with the density structure. In the upper panel (a) advective terms and bottom friction have been retained, in b there is no advection of momentum and in c both the advection of momentum and the bottom friction have been left out.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/shearedge_fig9.eps,width=7cm}} \end{figure}

After 60 days of simulation, the model has reached a steady state where the along shore velocities have been adjusted to the stratification. The resulting surface current attains values of 90 cm.s$^{-1}$ above the 1000 m isobath. The width of the current ranges between 50 and 100 km. We are in the parameter range of the Agulhas Current. The resulting free surface elevation is portrayed on figure (3.61-a). Its slope represents the along shore geostrophic surface currents. Although the temperature is constrained to remain close to its initial value, the free surface elevation reveals that a cyclonic recirculation stands in the bight. Hence, this simulation shows that the interaction between the Agulhas Current and the topography of the bight is likely to generate a cyclonic eddy. To test if the recirculation might be induced by an advection process, the same experiment has been conducted, removing the advective terms from the different prognostic equations. After 60 days, the figure (3.61-b) represents the resulting sea surface elevation. The along shore currents have roughly the same characteristics as the previous experiment. Surprisingly, the cyclonic recirculation has grown in size and intensity. It tends to propagate towards x negative, as a coastal trapped wave, but the nudging of temperature retains it close to the bight. Hence, the resulting shape of the recirculation is distorted. This reveals that whereas the advection doesn't seems to be the driving mechanism of the recirculation in the bight, its alongshore component can help to obtain a standing process. Bottom friction can be a serious candidate for driving this process. The model of the arrested topographic wave [Csanady, 1978], can explain the crossing of the isobaths by a barotropic current. A last experiment has been conducted, removing this time the advection terms and the bottom friction. Without the damping effect of the bottom friction, the solution was not stationary after 60 days. The resulting free surface elevation (figure 3.61-c) is less steep than for the previous experiments. Nevertheless, the cyclonic pattern is larger and more intense than for the first case. This implies that whereas bottom friction plays an important contribution during the adjustment of the current to the density field in this simulation, it is not responsible of the cyclonic recirculation.

5 Summary

The realistic model simulates the creation of cyclonic shear edge eddies and their attendant surface plumes of warm water with a high degree of verisimilitude. The dimensions of the features, their hydrographic structure and geostrophic velocities in the model bear a strong resemblance to what has been observed. The model shows that a shear edge eddy remains trapped in the bight on the eastern side of the shelf break of the Agulhas Bank and that eddies that travel downstream represent leakages from this resident shear eddy. This is not inconsistent with observations to date, but cannot be unambiguously verified. Although the current axis shows some degree of meandering in the model, this is much less than in nature, suggesting that much larger parts - or even the full - trapped eddy might occasionally escape and move downstream. Once they reach the southernmost tip of the Agulhas Bank, the traveling eddies may be absorbed by a cyclonic eddy in the western lee of the bank or may start such an eddy. The model does not indicate unequivocally whether such absorption has an effect on the timing of the eastward departure of the lee eddy. An idealized model of a baroclinic current along an indented shelf edge shows the development of a trapped shear edge eddy. If the advection of momentum is excluded, the cyclonic eddy tends to propagates along shore as a coastal trapped wave. If bottom friction is removed, the solution is no longer stationary. This eliminates two candidates for the generation process of a standing cyclonic eddy in the bight.


7 Transport patterns from the Agulhas Bank to the South African West Coast

The previous sections validate the results of the realistic simulation of the circulation around the South-western corner of Africa. These results can now be utilized to study the physical processes affecting the recruitment of pelagics in the Benguela upwelling system. The spawning of sardines and anchovies occurs essentially during the austral spring and fall. The wind induced mixing and the Ekman transport of the coastal waters towards the deep Ocean are unfavorable for the reproduction along the West coast. To reproduce, sardines and anchovies spawn onto the Agulhas Bank. After a few days, eggs and larvae reach the upwelling system of the South African West Coast. A few months after spawning, an important biomass of young recruits concentrate in St Helena Bay. Hutchings [1992] identified the key processes for the recruitment of sardines and anchovies in the Southern Benguela:
  1. Feeding of the adults during the spawning on the Agulhas Bank.
  2. Feeding of the larvae.
  3. Transport of the eggs and larvae from the Agulhas Bank to the Benguela upwelling system.
  4. Migration and retention of the larvae in the Coastal productive area.
  5. Feeding of young recruits.
If we consider that the swimming capabilities of the larvae are limited in comparison to the magnitude of the surrounding currents, the points 3 and 4 involve principally physical processes of transport from the Agulhas Bank to the South African West Coast, dispersion, offshore Ekman transport and retention at the coast.

Figure 3.62: Initial condition for the tracer. a: horizontal repartition. b: vertical repartition.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/ini_eggs.eps,width=14cm}}\end{figure}

In order to explore these physical processes of transport and dispersion, short term simulations has been conducted, starting from different moments of the long term simulation, in which a passive tracer, representing the probability of presence of an egg, has been introduced into the model. At the beginning of each short term simulation, the initial value of the tracer 'egg' is defined as a tridimentional Gaussian patch (figure 3.62):
\begin{displaymath}T_{egg}=C_0\,e^{-\left(\frac{R_{Earth}}{40} \frac{\pi}{180}(... ...hi}{180}}(G-G_0)\right)^2} \,e^{-\left(\frac{z}{20}\right)^2} \end{displaymath} (110)

Where $R_{Earth}$ is the radius of the Earth in km, $\Phi$ is the latitude, $\Phi_0$ is the latitude of the center of the patch, $G$ is the longitude, $G_0$ is the longitude of the center of the patch, $z$ is the vertical coordinate in m, and $C_0$ is a coefficient set up to obtain $\int\!\!\!\int\!\!\!\int_V T_{egg}\, dv=1$. Hence, only 1 egg is injected into the model at the beginning of each simulation, with a Gaussian probability of presence to be in a circle of 40 km radius centered at $\Phi_0$ and $G_0$, in the 20 first meters from the surface. For these simulations, the initial patch is centered at $\Phi_0=35^o$ S and $G_0=20.5^o$ E accounting for an egg spawned on the spawning ground of the western Agulhas Bank (see Hutchings [1992]). This initial patch is portrayed on figure (3.62). During the simulation the tracer 'egg' is forced by the advective and the vertical mixing schemes of the model, employing the coefficient of vertical mixing of momentum. The open boundaries for this tracer are passive and radiative to allow its evacuation from the model domain. After the tracer release, the short term experiments are conducted for 3 months to include all the whole egg and larvae states for sardines and anchovies. After the larvae state, the swimming capabilities of the fishes invalidate the use of a passive tracer to simulate their behavior.

Figure 3.63: Time series of the tracer surface concentration for an egg released the 1 September of model year 4. a: after 5 days. b: after 35 days. c: after 65 days. d: after 90 days.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/tracer_sep.eps,width=14cm}}\end{figure}

The impact of the wind forcing of the transport patterns from the Agulhas Bank to the Benguela upwelling system is studied by releasing the tracer 'egg' each month of the summer period, for the low and high resolution experiments, from September of model year 4 to April of model year 5. A time series of the surface tracer behavior for a release just before the upwelling season, the 1 September of model year 4 for the high resolution experiment is portrayed on figure (3.63). Because of the important vertical mixing present in the surface mixed layer, the surface distribution of the tracer is representative of the horizontal repartition for all the mixed layer. The averaged wind stress for this period of 3 months is around 0.075 N.m$^{-2}$, starting at 0.049 N.m$^{-2}$ for September and finishing at 0.088 N.m$^{-2}$ for the end of November. After 5 days, the tracer didn't moved considerably, sticking to the coast near Cape Agulhas (figure 3.63-a). After 1 month, a portion of the tracer reaches Cape Columbine, but the core of the patch stays near Cape Peninsula. For this period, the patch didn't disperse considerably (figure 3.63-b). At the beginning of November (figure 3.63-c), the patch has separated into 3 tongues: 1 extending from Cape Peninsula due to the presence of an upwelling filament, 1 following the shelf edge jet from Cape Columbine, and 1 filling St Helena Bay. At the end of November, the probability of presence of the egg is approximately uniformly distributed on the shelf along the west coast (figure 3.63-d).

Figure 3.64: Time series of the tracer surface concentration for an egg released the 1 December of model year 4. a: after 5 days. b: after 35 days. c: after 65 days. d: after 90 days.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/tracer_dec.eps,width=14cm}} \end{figure}

Figure (3.64) represents the time series of the tracer evolution for an egg spawned in the middle of the upwelling season, the 1 December of model year 4. For this period, the averaged wind stress is 0.085 N.m$^{-2}$, starting in the beginning of December at 0.088 N.m$^{-2}$ and ending at the end of February at 0.075 N.m$^{-2}$. For this simulation, things are going quickly. After 5 days, the Northern part of the patch has already reached Cape Peninsula (figure 3.64-a). After 1 month, the all of the tracer patch has passed Cape Peninsula, the effect of the offshore Ekman transport is important with the major part of tracer heading north-westward, away from the West Coast. A secondary maximum of tracer concentration is trapped around 34$^o$ S and 17$^o$ E, inside a cyclonic eddy shed from Cape Peninsula (figure 3.64-b). The effect of the offshore transport is dramatic after 2 months, for the beginning of February (figure 3.64-c), the largest concentration of tracer is now situated more than 400 km offshore from the coast. After 3 months, there is almost no chance for a passive larvae issued from an egg released in December to remain on the shelf along the West Coast (figure 3.64-d).

Figure 3.65: Time series of the tracer surface concentration for an egg released the 1 March of model year 5. a: after 5 days. b: after 35 days. c: after 65 days. d: after 90 days.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/tracer_mar.eps,width=14cm}}\end{figure}

For the last time series, the egg is released at the end of the upwelling season, the 1 March of model year 5. The averaged wind stress along the 3 months of simulation is 0.045 N.m$^{-2}$, starting at 0.075 N.m$^{-2}$ for the beginning of March and ending at 0.028 N.m$^{-2}$ for the end of May. The displacement start quickly. After 5 days (figure 3.65-a), the pattern is similar as the previous time series (figure 3.64-a). After 1 month (figure 3.65-b), an important portion of the tracer is already present in St Helena Bay. There are also 2 patches of tracer that are trapped into cyclonic eddies generated from the Cape Peninsula-Cape Columbine area. For the beginning of May, the dispersion has slowed down with the diminishing wind stress and few variations are present between the last 2 months (figures 3.65-c and 3.65-d). After 3 months, the tracer distribution is approximately equivalent to the end of the case for the release in September (figure 3.63-d).

Figure 3.66: Time series of the tracer concentration integrated on the volume of the target zone, in percentage of the total amount of tracer released. Top: for the low resolution experiment. Bottom: for the high resolution experiment.
\begin{figure} \centerline{\psfig{figure=Figures/Chap3/tracer_concentr.eps,width=14cm}} \end{figure}

To quantify the probability for an egg spawned on the Western Agulhas Bank to reach the nursery ground of St Helena Bay, time series of the tracer integrated on the volume limited by the dashed line on figure (3.65) has been portrayed on figure (3.66). This integrated value accounts for the probability of the presence in the nursery ground of St Helena Bay of an egg spawned on the Western Agulhas Bank. For the low resolution experiments, the pattern is clear (figure 3.66-top). The probability of an egg to reach St Helena Bay is maximum (almost 25 % chance) if it is released before the upwelling season. This maximum occurs after around 45 days. The more we advance in the upwelling season, the more the offshore Ekman transport is active, and the less the eggs have a chance (only 5 % for January) to reach the nursery ground. The chance increases again at the end of the upwelling season. The time series for the high resolution experiment follows a more complex pattern (figure 3.66-bottom). For a release in September the percentage of tracer in the target area follows approximately the same curve as for the low resolution experiment. For October and November, the maximum is the same as for September, but the tracer arrives more rapidly in St. Helena Bay. The tracer is also washed out more quickly from the bay, as we advance in the upwelling season. For the release in December, the curve differs dramatically from the low resolution experiment, showing an important amount of tracer entering the zone in less than a month and leaving the area in also less than a month. Figure (3.64-b) reveals that this narrow maximum of tracer in St Helena Bay can be associated with the crossing of the area by a patch of tracer trapped in a small eddy. This shows that the mesoscale activity alters considerably the transport pattern from the Agulhas Bank to St Helena Bay. For the releases of January, February and March, the amount of tracer in St. Helena Bay follows a relatively constant pattern: rapid increase to reach a maximum of 15 % after less than 1 month and slow decrease. The impact of the interannual variability observed in the outputs of the high resolution experiment is tested for the transport of eggs by making the comparison between an experiment for an egg released the 1 September of model year 5 and the experiment for an egged release the 1 September of model year 4. The dashed line on figure (3.66) reveals that although a certain degree of interannual variability is present in the model outputs, the global transport patterns do not differ dramatically from one year to the other. This allows sardines and anchovies to adapt their reproductive strategies to these relatively stable global transport patterns. One can note that differences in the transport behavior between the low resolution and the high resolution experiments occur principally during the upwelling season, when the mesoscale activity along the west coast is the strongest. This result connects to the conclusion of the second chapter that shows that during the upwelling season, although the Ekman transport can carry the material away from the coast, eddies can trap it for a certain amount of time in the coastal domain.

6 Summary

In this chapter, a primitive equation model, the Regional Oceanic Model Systems, has been configured to reproduce the characteristic patterns of the circulation around the South-western corner of Africa. The model resolves the free surface, hydrostatic, tri-dimensional primitive equations featuring high order schemes and new parameterizations over variable topography using stretched, terrain following coordinates. The horizontal, orthogonal, curvilinear grid has been designed in a pie shape to fit the curve of the South African coastline. Specific open boundary conditions involving a radiative one way nesting of seasonal time averaged data from a large scale oceanic model allow long term stable simulations. Two configurations are set up: a low resolution one with a grid space ranging from 18 km to 31 km and a high resolution one with a grid space ranging from 9 km to 16 km. The surface fluxes are derived from the coarse monthly climatology of the COADS dataset. After a spin-up of 2 years, the model has reached a statistical equilibrium. The high resolution experiment produces important intrinsic instabilities associated with the upwelling process and the Agulhas Current dynamics, with formation of plumes, filaments and eddies. For the low resolution experiment, whereas on a large scale the time averaged circulation, temperature and salinity compares to the observations, some aspects of the solution such as the Agulhas Current retroflection or the coastal circulation are not correctly represented. In the high resolution case, the solution is much more satisfactory and compares quantitatively to the known circulation. The volume averaged level of kinetic energy is higher than for the low resolution experiment due to the better resolved mesoscale processes. The averaged position of the retroflection of the Agulhas Current now corresponds to the observations, and important aspects of the coastal circulation such as the coastal jets are now properly resolved. The signature of the Agulhas Current is still present at 1000 m depth. At this depth and below, the circulation in the Benguela is slow, meandering and principally poleward. The depth integrated transport for the Benguela is ranging from 10 to 20 Sv and is more than 60 Sv for the Agulhas Current, in agreement with previous studies. The time averaged sea surface temperature compares quantitatively to satellite data. The principal difference is the stronger upwelling observed in the model outputs. This can be related to the low resolution dataset employed for the wind forcing. The bottom temperature and salinity values stay in reasonable agreement with long term observations. The length of the experiment allows a statistic analysis of the variability. The sea surface temperature variance shows a maximum on the Agulhas Bank where the currents are weak and a minimum in St Helena Bay where the presence of the upwelling plume seems to balance the seasonal variations. This pattern is quantitatively confirmed by sea surface satellite data. The root mean square of sea surface height is compared to altimeter data. It reveals both in model and data a maximum in variability in the Agulhas retroflection area and minimums on the shelves along the South and the West coasts. The same pattern is observed from the subsurface eddy kinetic energy. Although the modelized EKE compares quantitatively to data north of 35$^o$ S, where the variability is produced in-situ, an important difference is noticeable in the area influenced by the Agulhas Current. This discrepancy can be explained by the smooth time averaged data used to generate the Agulhas current at the Eastern boundary and by the constrain generated by the western boundary on the spawning off the Agulhas rings. This difference is confirmed by the ellipses of subsurface velocity variance. They also reveals an important anisotropy of the variability near Cape Columbine, near Cape Peninsula and South of the Agulhas Bank. These area of anisotropy in variability are also the preferred area for eddy generation. Important anisotropy in variability is also present along the open boundaries which tend to minimize the cross boundary variability. Although the wind forcing is coarse in space and time, the modelized upwelling system of the west coast shows important features such as the upwelling plume of Cape Columbine, the Namaqualand upwelling center, the upwelling eddies generated by the major capes, and the filaments spreading from the main upwelling front. This shows that the temporal and spatial small scale structures of the wind field are not the primary generators of the mesoscale activity of the upwelling system. It is rather induced by intrinsic instability processes. On the shelves, the time averaged circulation is highly controlled by the topography. A part of the Agulhas Current tends to follow the western part of the Agulhas Bank to feed the Good Hope jet. A surface equatorward baroclinic jet follows the shelf edge and a poleward undercurrent flows along the shelf break. The subsurface coastal counter current is present in the high resolution simulation. Standing coastal wave patterns can be noticed in St Helena Bay, in agreement with the analysis of the second chapter, but the treatment of the topography induces a steepening of the shelf, inhibiting the generation of a cyclonic mean circulation in the bay. Mesoscale processes in association to the Agulhas Current are also simulated by the model. They include: This analysis of the model outputs gives confidence in the validity of the simulation. A set of experiments has been set up to study the transport patterns of sardines and anchovies from the spawning area of the western Agulhas Bank to the nursery ground of St Helena Bay. A passive tracer has been released at different times of the simulation. Each `tracer simulation' has been conducted for 3 months. The amount of tracer that manages to reach St Helena Bay is highly dependent on the date of the release. If the tracer release occurs before the upwelling season, it fills slowly St. Helena Bay. An important portion of tracer is still along the West Coast after 3 months. On the contrary, if the tracer release is done during the upwelling season, though the tracer arrives very quickly in the West Coast, it is also rapidly carried away by the Ekman transport. If the release occurs after the upwelling season the amount of tracer in St. Helena Bay increases again. During the upwelling season, the loss of tracer is more important for the low resolution experiment than for the high resolution experiment. This reveals that the mesoscale activity is able to counter the effects of the offshore Ekman transport, or that the along shore jets responsible of transport are not well reproduced for the low resolution experiment.

Conclusion

Here is the end of the manuscript, the time to take the stock of what has been achieved during this project. The goals of the PhD were
  1. to study the dynamics of mesoscale processes related to retention within the St Helena Bay nursery ground
  2. to develop a high resolution regional model to simulate the shelf circulation along the South and West coasts of South Africa.
The work could have been focused on the design of a 3D realistic model or on the study of a particular process of the system. The step by step approach chosen was more general: starting with a description of the system that has led to the selection of a few processes, concentrating on a specific process in St Helena Bay using both numerical and analytical approaches, and ending by a realistic simulation of the circulation of the Southern Benguela.

The barotropic study reveals that a wind driven equatorward current can generate a standing process in the lee of a cape. In the case of St Helena Bay, the numerical study shows that this process should take the form of a standing cyclonic eddy, controlled by a balance between advection and bottom friction. A length scale derived from this balance, which can be considered as an e-folding length scale, can then predict the size of the eddy as a function of the wind stress and the bottom friction parameter. Sensitivity tests reveal some discrepancies between this length scale and the size of the eddy, especially for smaller capes where the slope in the bay has gained importance. In the presence of bottom slope, standing shelf waves are more likely to develop. An analytical study shows that standing shelf waves can be excited by a mean current past a cape. The tracer, representing the age of the water, has been introduced in the model. It exhibits the retention induced by the standing recirculation process. Upwelling favorable winds can generate currents responsible for eggs and larvae dispersion. But, at the same time, these currents can induce recirculation in the lee of capes like Cape Columbine that can trap eggs and larvae in a favorable environment.

The particularity of the Benguela system, with the Agulhas Current retroflecting just South of the area of interest did not facilitate the implementation of a realist 3-D model of the region. Tests have been conducted using a low resolution configuration to set-up the open boundary conditions. The treatment of the bottom topography for this simulation gave an incorrect representation of the shelf circulation and the detachment of the Agulhas Current. For the high resolution experiment the solution is more satisfactory, and compares with observations for most of the processes. They include:

The major ingredients of Southern Benguela are present in the simulation. It is the first time that these local processes, described in numerous publications, have been modelized with this level of accuracy. Although the surface forcing is derived from a smooth monthly climatology, the mesoscale activity generates a significant high frequency variability. Its amplitude along the West Coast of South Africa compares quantitatively with observations. This result shows that the variability along the West Coast is more a consequence of intrinsic instability of the coastal ocean rather than a direct forcing from small scale wind variations. Some differences have also been noticed when comparing model results with data:

A tracer, representing the probability of presence of an egg spawned on the western Agulhas Bank has been introduced to simulate the transport patterns between the Agulhas Bank and the West Coast of South Africa. It shows the negative effect of the upwelling favorable wind for the transfer from the Agulhas Bank to the West Coast and the positive effect of the mesoscale eddies and jet on the retention of the biological materials in the favorable areas. This is in agreement with the last result of the second chapter.




Prospects

Many other improvements can be added to the actual model to produce a better representation of the Benguela ecosystem, such as the assimilation of data, the introduction of tides, the modeling of the river outflows, or the modeling of the primary production... Each increase of the complexity of the modelized system, will result in an increase of the complexity of the solution. New tools should then be designed to help the understanding of the key processes that are structuring these solutions. Recently, an important effort has been conducted to understand the dynamics of coastal ecosystems, an example is the work conducted for the West Coast of the United States [Miller et al., 1999]. The experience gained and the tools designed for the study of the Benguela ecosystem could be applied to other coastal domains of the world. Important fundamental insight could be obtained by comparing the dynamics of the different domains.

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A numerical study of the Southern Benguela circulation with an application to fish recruitment.

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