A circulation model for the European ocean margin

A circulation model for the European ocean margin

A. Santos a,*, H. Martins a, H. Coelho b, P. Leit~aao a, R. Neves a

a Instituto Superior T�eecnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugalb Universidade do Algarve, Campus de Gambelas, 8000 Faro, Portugal

Received 15 February 2000; received in revised form 10 May 2001; accepted 2 July 2001

Abstract

This paper presents a new numerical model for the study of circulation in coastal and deep ocean re-gions. Sigma coordinate models are appropriate where currents follow topography, but there are problemsof numerical diffusion in stratified areas. Some of these problems can be partly overcome by applyinginstead a cartesian model, whose outcome, however, is dependent on variable ocean depth. As a com-promise between these different models with their varying advantages and disadvantages, a double sigmatransformation model offers perhaps the best solution. Accordingly, in the simulations presented in thispaper, the double sigma coordinate model was applied as a diagnostic tool to calculate features of oceancirculation off the coast of Western Europe, particularly in the area between La Chapelle Bank and Por-cupine Bank. Calculations derived from the model indicated density-driven poleward currents continuousfrom the Portuguese to the Irish shelves in accordance with actual observations and numerical resultsavailable. The model also indicates that seasonal winds strongly modify flow over the shelf and upperslope. � 2002 Elsevier Science Inc. All rights reserved.

Keywords: Sigma model; Slope current; JEBAR; Numerical diffusion

1. Introduction

From the several ocean circulation models presented in the literature, three main model cate-gories can be distinguished, corresponding to different concepts for the discretization of thevertical coordinate. The most familiar examples from each category are: the isopycnal-coordinatemodel based on the Miami Isopycnic Coordinate Ocean Model – MICOM – code [1]; the car-tesian model based on the Modular Ocean Model – MOM – code [2]; and the sigma coordinatemodel based on the Semi-spectral Primitive Equation ocean circulation Model – SPEM – code [3].

Applied Mathematical Modelling 26 (2002) 563–582

www.elsevier.com/locate/apm

*Corresponding author. Tel.: +351-218417986; fax: +351-218417398.

E-mail address: [email protected] (A. Santos).

0307-904X/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

PII: S0307-904X(01)00069-5

That there is no ideal solution is clear from looking at the results of DYnamics of North AtlanticMOdels – DYNAMO – project [4]. The most satisfactory grid should always be oriented with theflow and so must be a compromise between the various available possibilities, depending on thephysical processes that determine the flow in any particular region. Sigma models reveal strongtopographically determined currents, making these models the best choice whenever flow is con-strained by depth contours. However, if the flow follows surfaces of constant density, as may bethe case near the seasonal thermocline in periods of low turbulence intensity, sigma models can,numerically speaking, erode these surfaces, and in such instances isopycnal models are the betterchoice, in spite of the numerical difficulties associated with them. The shortcomings of sigmamodels in stratified regions can be reduced by a slight change in the conceptual formulation. Thecomputational model may be divided vertically into two sigma models, separated by an interfaceplaced at a level of nearly horizontal motion [5,6]. This is a compromise between cartesian andsigma coordinates, often called a double sigma coordinate model. In fact, at the limit where thenumber of sigma domains is equal to the number of layers, a cartesian model is achieved with avertical grid resolution dependent on the local depth. This paper presents a fine resolution doublesigma coordinate model for shelf and slope regions developed during the first phase of OceanMargin Exchange (OMEX) project. The model can be used to obtain diagnostics for both density-and wind-driven circulations. The study region is the Northern Gulf of Biscay, particularly thearea between La Chapelle Bank and Porcupine Bank (Fig. 1). Special attention has been paid tothe so-called shelf edge current (or slope current). In fact, slope current constitutes the mostprominent pattern of the flow and is presumably responsible for the transport of material from themore active tidal shelf to the southeast of the OMEX cross-section (located at the Goban Spur).Any kind of prognostic calculation is beyond the scope of this paper. Further analysis of ex-changes between continental and deep ocean will be the subject of future work. However someinsight on physical processes occurring on the ocean margin can be found in [34].

1.1. Poleward flow at eastern ocean boundaries

Many authors have provided evidence for a poleward flow along European slopes [7–13]. Verysimilar poleward flows have been described in other eastern boundary regions such as the Cali-fornia Current System [14] and the Leeuwin current at the West Coast of Australia. These flows,mainly concentrated along the upper continental slope and outer continental shelf, appear asundercurrents in the upwelling season and sometimes as surface currents in the non-upwellingseason. Barton [15] suggested that the poleward flow is continuous along the entire easternboundary and attributed to the Iberian poleward flow a key role in the transport of Mediterra-nean water ultimately into the Norwegian sea. Frouin et al. [11] described a flow 200 m deep withgeostrophic velocities ranging from 0.2 to 0:3 ms�1 and associated transports varying from300� 103 m3 s�1 at about 38 N to 500–700� 103 m3 s�1 at about 41 N. They concluded that thepoleward current off the Iberian Peninsula runs for about 1500 km along the upper continentalslope of western Portugal, northwest Spain, northern France and southwest France and that it isabout 25–40 km wide. Other reports such as Ambar [7] suggest that the current extend from1600 m deep to the bottom of the surface layer during the upwelling season and to the surfaceduring the non-upwelling season. Pingree and LeCann [13,16] summarised current meter datacollected from the Bay of Biscay and presented a residual circulation pattern. Further mention of

564 A. Santos et al. / Appl. Math. Modelling 26 (2002) 563–582

their work can be found in the section where the model results are discussed. During the lasttwenty years, several driving models to explain slope currents have been put forward. Of these, themost-frequently studied have been wind-stress [9], wind-stress curl [17] and thermohaline forcing[11,12,14,18–20]. In fact, off the Iberian coast, onshore Ekman convergence induced by south–south-westerly winds forces a poleward surface flow. The shelfward transport induced by thesewinds causes a rising of sea level near the coast. The geostrophic adjustment to this sea leveldistribution will then generate a poleward current. In this case, the longshore acceleration is givenby oV =ot ¼ sy=qH , where H is the depth of the frictional layer, V the longshore velocity averaged

Fig. 1. Topography of the area of interest for OMEX.

A. Santos et al. / Appl. Math. Modelling 26 (2002) 563–582 565

over the depth H, sy the meridional component of wind-stress and q the seawater density. Usings ¼ 0:03 Nm�2, q ¼ 1027 kgm�3 and H ¼ 200 m, Frouin [11] found a longshore acceleration of0:013 ms�1 d�1, which gives V ¼ 0:4 ms�1 after 30 days. However, the authors argued that othereffects, particularly friction, retard the flow. Assuming that a steady-state reached when thebottom stress balances the wind stress (CdV 2 ¼ sy=q, being Cd the bottom drag coefficient takenequal to 0.001), they obtained V ffi 0:17 ms�1 which is in agreement with observations. Thiscurrent should decay seaward from the shelf break and the associated spatial scale is the internalradius of deformation (ffi 15 km off Iberia). This is what is generally observed both from satelliteimages and from in situ observations. Evaluation of the Ekman volume transports based onwindstress measured at Cabo Carvoeiro revealed that only 1/5 of the estimated transport could beexplained by the wind, which could not therefore be regarded as the main mechanism driving thepoleward current. On the other hand, estimates of large-scale geostrophic eastward transport give1:0 m2 s�1 per meter of meridional coastline. A value of the same order as that estimated fromhydrographic sections can be calculated by integrating along the western Iberian coast and addingwind-driven transport. The poleward cooling of the sea surface leads to a meridional increase ofsurface density causing the dynamic height to drop towards the pole. The large-scale eastwardflow is generated by this meridional pressure gradient and occurs in the upper 200–300 m. Nearthe eastern ocean boundary, this flow forces coastal downwelling and a surface poleward current,as confirmed by model results obtained by McCreary et al. [19] and Weaver and Middleton [20]for the Leeuwin Current. Huthnance [18] showed that a combination of shelf-slope bathymetrywith a northward density gradient provides a local mechanism that can drive a current towardsthe pole, as can be expressed by the relation qog=oy ¼ �hoq=oy, where g is the sea surface ele-vation and h the water depth (see also [13]). This relation states that sea level decline is propor-tional to depth h. Therefore sea level declines faster in deep water than in shallow water, soimplying a cross-slope sea level gradient. The existence of this gradient leads to a poleward flowover the slope. The cross-slope sea level gradient increases northward and so consequently doesthe along-slope transport, but this is not a situation that can continue, since friction acts tobalance the forcing mechanism [16]. Huthnance also showed that if the cross-shelf density dif-fusion is large, the along-slope current is given by

m ¼ 1

2

gqoqoy

Hkh 1

�� hH

�;

where H is the oceanic thermal depth and k the bottom friction coefficient. According to thisequation maximum velocity must be expected over the slope.

2. The mathematical model

The model assumes the Boussinesq approximation and hydrostatic pressure and solves twoequations for horizontal momentum, the continuity equation, two transport equations for heatand salt and an equation of state [21], relating density ðkg m�3Þ, salinity (P.S.U.) and temperature(�C). In the following equations p stands for pressure, patm for the atmospheric pressure, ðu; v;wÞthe components of velocity, g the surface level, h the local depth, H the total depth ðH ¼ hþ gÞ, qthe density, qr a constant reference value for density, q0 a perturbation of density ðq0 ¼ q � qrÞ, S


the salinity, T the temperature, AH and AV the horizontal and vertical turbulent viscosities, KH andKV the horizontal and vertical turbulent diffusivities, f the Coriolis frequency, g the acceleration ofgravity and FS and FT the source/sink terms for salt and temperature

ovot

þ uovox

þ vovoy

þ wovoz

þ fu ¼ � 1

qr

opoy

þ o

oxAH

ovox

� �þ o

oyAH

ovoy

� �þ o

ozAV

ovoz

� �; ð1Þ

ouot

þ uouox

þ vouoy

þ wouoz

þ fv ¼ � 1

qr

opox

þ o

oxAH

ouox

� �þ o

oyAH

ouoy

� �þ o

ozAV

ouoz

� �; ð2Þ

opoz

þ qg ¼ 0; ð3Þ

ouox

þ ovoy

þ owoz

¼ 0; ð4Þ

oðSÞot

þ oðuSÞox

þ oðvSÞoy

þ oðwSÞoz

¼ o

oxKH

oSox

� �þ o

oyKH

oSoy

� �þ o

ozKV

oSoz

� �þ FS; ð5Þ

oðT Þot

þ oðuT Þox

þ oðvT Þoy

þ oðwT Þoz

¼ o

oxKH

oTox

� �þ o

oyKH

oToy

� �þ o

ozKV

oToz

� �þ FT; ð6Þ

q ¼ ð5890þ 38T � 0:375T 2 þ 3SÞ=ð1779:5þ 11:25T � 0:0745T 2 � ð3:8þ 0:01T ÞSþ 0:698ð5890þ 38T � 0:375T 2 þ 3SÞÞ: ð7Þ

The integration of the continuity equation (4) over the entire water column yields the well-knownprognostic equation for the sea surface elevation:

ogot

¼ � o

ox

Z n

�hudz� o

oy

Z n

�hvdz ð8Þ

and the vertical integration of the hydrostatic pressure equation gives

pðzÞ ¼ patm þ gqrðg � zÞ þ gZ n

zq0 dz; ð9Þ

which relates the pressure at any depth to the atmospheric pressure, the surface level and thevertical integral of density anomaly between that level and free surface.

2.1. The vertical coordinate

The choice of the vertical coordinate in a circulation model is still a matter of discussionamongst marine modelling specialists. Ideally, the mesh should always be oriented with the flow tominimise numerical diffusion. The sigma ðrÞ-type coordinates [22] transform the model domaininto a constant depth domain, that is to say, smaller computation cells located in shallow areas areexpanded to the same proportions as the cells located in the deeper areas. The r-coordinate allowsthe same number of grid points whatever the local depth and thus is the most adequate to solve


problems where topography plays a major role in the determination of the flow. In stratified flowsisopycnals are nearly horizontal and thus, along the shelf break in the r-domain, they are rep-resented by sloping lines that can cross several layers. Also, stratified flows require a verticalresolution related to the density gradient, which is not the case in r-coordinates. A double sigmacoordinate does not completely solve these problems but can minimise them. As stated above, inthis type of coordinate the water column is divided into two domains and in each domain ar-transformation is applied. In this way, the grid used in the upper ocean is not linked to the localdepth unless the ocean bottom is above the interface dividing the two domains. If the splittingplan is located under the thermocline the layers in the upper domain are nearly horizontal and canrepresent correctly the flow along isopycnals, thus minimising numerical diffusion. There existalternative isopycnic and cartesian coordinates. In the former, grid lines are coincident with is-opycnals and in the latter they are horizontal. Isopycnic coordinates are suitable to simulate flowswhere density plays the major role, while cartesian coordinates are a compromise between thethree types, since they are not optimised for any process existing in the ocean. In general theydemand a large number of vertical layers and so in computational terms may become expensive. Adisadvantage common to every coordinate transformation is that in different regions of the oceanthe relative importance of each process is different, a problem that none of them can completelydeal with. Martins et al. [23] proposed a numerical method based on finite-volumes allowing thesame computational code to accommodate any of the described coordinate transformations andused it successfully in tidal estuaries. In the present application the major concern is the flow alongand across the ocean margin, where both topographic and density effects are important. A double-sigma coordinate was used as a compromise. The splitting plan was located 100 m deep toguarantee that areas of strong temperature gradients are confined to the upper domain and thusdiscretised independently of the local depth. In each vertical sub-domain the equations have beentransformed according to the r-transformation:

~xx ¼ x; ~yy ¼ y; ~zzU ¼ LU

HU

ðzþ hUÞ; ~zzL ¼ LL

HL

ðzþ hÞ; ð10Þ

where

HU ¼ hU þ g; HL ¼ maxð0; h� hintÞ; hU ¼ minðh; hintÞ: ð11ÞIn these equations, hint is the depth of the interface between the two sigma domains, HU and HL arethe total real depths of the upper and lower sigma domains, hU is the local depth of the uppersigma domain and LU, LL are respectively the sigma depths of the upper and lower domains. Thehorizontal velocities are equal in the real and transformed domains ð~uu ¼ u; ~vv ¼ vÞ and theequation of continuity, for the upper sigma domain, can be written:

oHU

otþ oðHU~uuÞ

o~xxþ oðHU~vvÞ

o~yyþ oðHU ~wwUÞ

o~zzU¼ 0: ð12Þ

For the lower sigma domain the water depth is constant in time and so the continuity equationbecomes:

oðHL~uuÞo~xx

þ oðHL~vvÞo~yy

þ oðHL ~wwLÞo~zzL

¼ 0: ð13Þ


From the vertical integration of these equations, an equation for the sea surface level can bederived as follows:

ogot

¼ � o

o~xx

Z LL

0

HL

LL

� �~uud~zzL �

o

o~yy

Z LL

0

HL

LL

� �~vvd~zzL �

o

o~xx

Z LU

0

HU

LU

� �~uud~zzU

� o

o~yy

Z LU

0

HU

LU

� �~vvd~zzU: ð14Þ

The vertical integration of the continuity equations also gives the vertical sigma velocity for bothdomains:

ðHU ~wwUÞ~zzU ¼ o

o~xx

Z LU

~zzU

ðHU~uuÞd~zzU þ o

o~yy

Z LU

~zzU

ðHU~vvÞd~zzU þ ðLU � ~zzUÞogot

; ð15Þ

ðHL ~wwLÞ~zzL ¼ o

o~xx

Z LL

~zzL

ðHL~uuÞd~zzL þo

o~yy

Z LL

~zzL

ðHL~vvÞd~zzL þ ðHL ~wwLÞ~zzL¼LL: ð16Þ

In the last equation the lower sigma vertical velocity appears at the interface. Taking into accountthat at the interface only one real vertical velocity exists and assuming that the interface does notchange position in time, the two sigma vertical velocities at the interface are easily related with thereal vertical velocity by

wint ¼HL

LL

� �ð~wwLÞ~zzL¼LL

¼ HU

LU

� �ð~wwUÞ~zzU¼0: ð17Þ

For the horizontal velocity component ~uu (similar for ~vv) and salinity S (similar for the temperatureT), the equations are the same for both sigma domains. Dropping the indexes L (lower sigmadomain) and U (upper sigma domain) for simplicity, evolution equations can be written as:

o~uuot

þ ~uuo~uuo~xx

þ ~vvo~uuo~yy

þ ~wwo~uuo~zz

� f ~vv ¼ � 1

qr

opox

þ o

o~xxAH

o~uuo~xx

!þ o

o~yyAH

o~uuo~yy

!þ L

H

� �2o

o~zzAV

o~uuo~zz

!;

ð18Þ

oðHSÞot

þ oðH ~uuSÞo~xx

þ oðH~vvSÞo~yy

þ oðH ~wwSÞo~zz

¼ Ho

o~xxKH

oSo~xx

� �þ H

o

o~yyKH

oSo~yy

� �þ L

H

� �2o

o~zzKV

oðHSÞo~zz

� �þ FS:

ð19Þ

Advection and diffusion being among the most important terms, conservative form is preferredfor scalars. For the horizontal diffusion, Mellor [24] proved that these terms should be computedalong the sigma isolines, because the sigma transformation equations for the horizontal diffusionterms, although mathematically correct, give unrealistic solutions and are therefore best avoided.In the momentum equations another term that warrants special attention is the baroclinic com-ponent of the pressure gradient. Referring back to Eq. (9), taking the derivative along the hori-zontal direction and applying Leibnitz’s rule for the derivative of an integral, we arrive at theformula:


opðzÞox

¼ opatmox

þ gqg

ogox

þ gZ n

z

oq0

oxdz: ð20Þ

Baroclinic pressure can be computed only if the density is known at the same level of the hori-zontal velocity. In a cartesian domain this is straightforward because the grid cells are not de-formed by the depth profile, but such is not the case in a sigma domain. It is necessary to makesome assumption for the density profile between two adjacent sigma cells and in the present modelwe assumed a linear evolution.

2.2. The numerical algorithm

The temporal discretization uses a time-splitting algorithm. Time-splitting methods are widelyused in hydrodynamic modelling to separate the calculation of processes with different propa-gation properties, and according to these methods, in order to complete the calculation in a time-step, one or more intermediate time levels have to be considered. A set of processes (e.g.advection/diffusion) modify the property values known at the beginning of the time-step and thenother processes, present in the equation, correct these estimated values to conclude time iteration.These methods are generally more stable than the explicit methods and allow the calculation ofthe different terms of the equation using different numerical schemes. Benqu�ee et al. [25] uses timesplitting to allow the calculation of advection by a method of characteristics and of the pressureby a classical Eulerian implicit algorithm. In our model a major goal of the splitting method is toobtain the solution using only tridiagonal matrixes with a time-centred Coriolis term, in order toincrease the precision of its calculation. Several methods are used in vertical integrated modelsthat can be easily extended to a 3D calculation [26–28]. One method [26] uses six finite-differenceequations in each time-step, the others use four. The first can be better suited to simulate intertidalareas using 2D models, but the latter are more efficient in deeper zones if the Coriolis term is time-centred. In our model, we applied a method using four finite-difference equations [28]. For the freesurface finite-difference equation the unknown velocities are eliminated using the correspondingmomentum equation, leading to tridiagonal matrices. Knowing the new elevation in each 1=2time-step the corresponding momentum equation can be resolved with an implicit calculation forvertical transport, inverting again a tridiagonal matrix. In this way, the most limiting stabilityfactors (gravity wave propagation and vertical diffusion) are resolved implicitly, and in the in-terests of numerical accuracy, Courant numbers of 5 ðCourant ¼ Dt

ffiffiffiffiffiffiffigH

p=Dx ¼ 5Þ are generally

used. For a horizontal grid size of Dx ¼ 10 km and water mean depth equal to H ¼ 4000 m, thisgives a numerical time-step roughly equal to Dt ¼ 250 s. As stated above, for the spatial dis-cretization a staggered Arakawa C grid was applied. With the exception of advective terms, all thespecial derivatives are centred. The ideal advection scheme must combine the transportiveproperty (this means that a perturbation is numerically advected only in the direction of thevelocity) with low numerical diffusion. The upwind scheme satisfies the first requirement but it canproduce substantial numerical diffusion, as do all first-order methods. Central differences are lessprone to diffusion but pay no regard to the transportive property of advection and so may causenegative concentrations; they are second-order methods, which usually produce numerical dis-persion errors. Most of the numerical algorithms are in some way combinations of both types.For the present model and for the advection of momentum a linear combination was applied:


Cb ¼ aðCbÞupwind þ ð1� aÞðCbÞcentered; ð21Þ

where Cb means a boundary value of any advected property and a is the weighting factor. In thismodel a constant value a ¼ 0:5 was considered for the whole domain, following Silva [29] whosuccessfully used a similar algorithm for wind wave simulation in a Boussinesq model. Localvalues based on the Peclet number (Spalding scheme for advection), or on the density gradient[30], could as well be considered. For the advection of temperature and salinity a TVD scheme [31]was applied, which combines the advantages of the upwind and Lax–Wendroff schemes.

2.3. The turbulent closure

The vertical turbulent diffusion coefficient depends on the vertical shear and density stratifi-cation. The present model includes turbulence closures based on empirical formulae [32], thePrandtl mixing length formulation and a kinetic energy model [33,34]. The results presented heretake into consideration a mixing length approach corrected according to the local Richardsonnumber:

AV ¼ l2mdj~VV jdz

with lm ¼ HnzH

� �/ðRiÞ: ð22Þ

There are several equations for the function nðz=HÞ. The equation used in this paper [21] is

nzH

� �¼ v

zþ hH

1

�� zþ h

H

�1=2

; ð23Þ

where v ¼ 0:4 is the Von Karman constant. For the function /ðRiÞ, most of the authors agree onthe equation /ðRiÞ ¼ expð�aiRiÞ with ai � 0:8. The Richardson number is also applied in therelation between vertical viscosity and vertical diffusivities. The sub-grid turbulence is parame-terised with the horizontal viscosity. The model assumes that turbulence is isotropic in the hor-izontal direction, which means that horizontal viscosity can be computed with the 4/3Kolmogorov’s law:

AH � e1=3Dx4=3; ð24Þ

where the length scale is the mesh size and e is the turbulent kinetic energy dissipation rate.According to Ozmidov, for a mesoscale model e ¼ 10�8 m2 s�3 [35]. Assuming a mesh sizeDx � 104 m, the last equation produces AH � 500 m2 s�1. The diffusion rate of mass and heat issimilar to this value.

2.4. The simulation domain and the grid

OMEX aims to study exchanges along the Ocean Margin. In OMEX I the main study area wasthe Goban Spur. The model grid was chosen to cover a geographical area sufficiently large as to:(1) take into account the evolution scale of the slope current which is hundreds of kilometres inlength [13]; and (2) keep open boundaries away from the area of interest, thus minimising errors


due to lack of information needed to fully specify the boundary conditions. The simulation do-main area is bounded within the limits 30�N–60�N and 25�W–8�E. The numerical model usesvariable step both in vertical and horizontal directions. The horizontal grid (Fig. 2) has a spatialstep varying between a maximum of 40 km and a minimum of 8 km (over the slope and shelfareas). The vertical grid is of the type represented in Fig. 3. The maximum layer thickness is 2700m in the bottom layer of deeper areas and 10 m for the upper layers. Bottom topography is basedon ETOPO5 database.

Fig. 2. Model domain and bottom topography used.


2.5. Initial and boundary conditions

For temperature and salinity, initial conditions are derived from seasonal climatological datareported by Levitus [36]. Prognostic equations for salinity and temperature are computed for aperiod of two weeks with very high horizontal viscosity and diffusivity ðffi 104 m2 s�1Þ in order toallow further smoothing of the initial conditions. After that period, temperature and salinity arefrozen and the model runs for a period of 60 days. The currents computed represent the oceanadjustment to initial conditions. At the open boundaries, if the water flows into the domainboundary, values are relaxed to climatological conditions with a prescribed relaxation timescale,and if there is an outflow, boundary values are computed from inner values with an advectionscheme. To solve the momentum equations, surface boundary values must be specified. In ourmodel, surface level is computed with a radiation condition [37], assuming a propagation velocityequal to c ¼

ffiffiffiffiffiffiffigH

p, valid for long gravity waves. The aim of this condition is to prevent artificial

reflections of surface waves leaving the domain. Besides the radiation condition, this model canalso handle a boundary condition based on the Flow Relaxation Scheme [38], whenever a localsolution is known. At the free surface the model inputs are fluxes of momentum (wind stress),energy (sensible heat, latent heat, long wave and short wave radiation) and mass (evaporation,

Fig. 3. Schematic representation of the vertical layers in a double sigma coordinate model. The upper figure is an ex-

panded plot of the surface domain. The lower figure represents both domains.


precipitation), but for the present simulations, and when the model is only driven by density, nofluxes need be imposed. On the other hand, if the model also considers wind forces, measures ofmonthly mean wind stresses have to be introduced. At the ocean bottom, a logarithmic velocityprofile is assumed to compute the bottom stress. Following Stanev [39], climatological tempera-ture and salinity data were assimilated in order to avoid unrealistic results.

3. Results and discussion

Three different simulations were carried out using the ocean circulation model previously de-scribed. To investigate a possible cause for and some of the characteristics of slope current alongthe western European margin, two simulations were undertaken where the currents are driven bythe initial density fields. These simulations were made for Spring and Autumn conditions, that isto say, the model was initialised with climatological Spring and Autumn density fields. Thesesimulations were named DS and DA, respectively. In order to illustrate the role played by wind inthe circulation in the Northern Bay of Biscay another simulation was made in which the model isdriven as well by monthly mean climatological windstress for April [40]. This simulation wasnamed WS. Results and discussion are presented in the following two sub-sections. Althoughdetailed comparisons with actual measurements are not meaningful, some quantitative compar-isons of flow strength and seasonality have been undertaken with currentmeter data presented in[13,34]. The flow pattern obtained was also compared with DYNAMO results [4].

3.1. Density-driven simulations

The major objective of this numerical experiment was to show that a realistic slope current canbe driven by seawater density distribution in the North Atlantic. The idea is not a new one, since asimilar experiment was performed by Pingree and LeCann [13], but the present simulations werealso useful to test the three-dimensional model developed during OMEX I. Furthermore, themodel used in this experiment is a three-dimensional one allowing a vertical structure for the slopecurrent, and in this sense, the present work represents an advance on previous studies. Residualcurrents after two months resulting from simulations DS and DA are presented in Figs. 4 and 5,respectively. Figs. 4(a) and 5(a) show results for the surface layer, while Figs. 4(b) and 5(b) showresults for the bottom layer. It should be noted that the bottom layer encompasses a wide range ofdepths, since it is a sigma layer (see also Fig. 3 for a better understanding of this point). In bothsimulations (DS and DA) there is a slope current resulting from pressure gradient JEBAR [18]that is associated with the density field in the Atlantic Ocean. Slope current is mostly confined tothe upper slope and to the outer shelf as it is shown in a cross-section for DS simulation at theGoban Spur (Fig. 6). In simulations DS and DA the slope current appears to be continuous (inthe sense that there is along-slope flow throughout) from the ocean margins of western Portugal tothose of the west of Ireland. The same results were obtained in simulations carried out with all themodels used in DYNAMO project. That currents are generally broader near the surface than atdeeper levels can perhaps be accounted for by topographical constraints. However, it should benoted that the width of slope current near the surface seems to be exaggerated. Speeds are gen-erally larger than reported in the literature [13,34]. A similar discrepancy was found by


Fig. 4. (a) Residual currents after 2 months of simulation with Spring climatological conditions, in the surface layer.

(b) Residual currents after 2 months of simulation with Spring climatological conditions in the bottom layer.


Fig. 5. (a) Residual currents after 2 months of simulation with Autumn climatological conditions in the surface layer.

(b) Residual currents after 2 months of simulation with Autumn climatological conditions in the bottom layer.


DYNAMO investigators with SPEM, suggesting that this is a common feature of sigma models,that is that these kinds of models tend to produce strong topographically-determined flows. Inshort, there is a transport of 4 Sv at the Goban Spur – between 100 and 1000 m contours – that isprobably 3–4 times greater than in reality. In their analyses of historical currentmeter data,Pingree and LeCann [13] found three residual flows that cannot be accounted for by reference tolocal winds: first, the coastal flow to the northwest along the Armorican shelf; secondly, that tothe SW of Ireland; and thirdly the counter-flow to the southeast along the outer Celtic shelf. Theyargued that these coastal flows might result from pressure gradients set up in response to densitygradients or non-local wind stress. In fact, the flow directed to the northwest along the Armoricanshelf appear in model results both in Spring and Fall, although it is not a coastal flow but a weakand broad current that follows roughly the 100 m contour. The model also predicts a counter flowin the outer Celtic shelf, particularly clear in Figs. 4(b) and 5(b). Similar flows were predicted bytwo of the three models used in the DYNAMO Project (SPEM and MICOM), the predictionsfrom SPEM (a sigma coordinate model) being in particular remarkably close to our own. Thenorthwest flow SW of Ireland reported by Pingree and LeCann is not present in our results and

Fig. 6. Cross-section of meridional residual velocity after two months of simulation in Spring at Goban Spur. Interval

contour is 0:04 ms�1. Dashed lines refers to southward velocity.


it is also not evident in any of the models used in DYNAMO. As previously stated, comparisonswith currentmeter data are not meaningful. It is, however, possible to compare strengths atspecific locations in order to evaluate the capability of the model. Huthnance [34] found that meancurrent near the surface at the Goban Spur in October – when the slope current is more intenseand persistent – is about 10 cms�1 (Fig. 6). Maximum speed in the model at the same location isof the same order of magnitude in DS simulation (Spring) but slightly larger in DA simulation(Autumn). The current decreases with depth and vanishes between 1500 and 2000 m. At LaChapelle Bank, DS simulation currents are generally weaker near the surface (about 5 cms�1)and, as shown by Huthnance [34], directed offslope at the 2000 m contour in April. Near thebottom, the current reaches 15 cms�1, a value much larger than that found by Pingree andLeCann [13]. In DA simulation, currents are stronger through the water column, reaching10 cms�1 near the surface and more than 20 cms�1 near the bottom. October averages presentedby Huthnance [34]) also show stronger currents aligned along-slope, reaching more than10 cms�1 near the surface. In summary, we can say that the present model has a certain ability toproduce a flow in the right direction but with speeds that are generally larger than observed. Therewere differences between Spring and Autumn that must arise from large-scale seawater distri-bution in the North Atlantic. Slope current is stronger in DA simulation, a result which findsconfirmation from observations that reveal that slope current is more consistently northwards(along-slope) in October than in April. However, it should be kept in mind that we have not takeninto account the wind stress that might possibly affect the weakening of the flow during Spring.

3.2. Wind-driven simulation

Typical winds over the Celtic Sea tend to be predominantly southwesterly in winter and westnorthwesterly in summer. The winds are much stronger in winter than in summer and thewindstress is generally 2–3 times greater. Over the Armorican shelf, the winds blow from the westtending west-northwest in the southern Bay of Biscay. Barotropic responses to idealised windswere studied in a previous paper by Pingree and LeCann [13]. Our purpose here is to illustrateresponse to a steady wind in the presence of pressure gradients associated with density gradients.An experiment was conducted for the month of April (Fig. 7 data from [40]), when at some lo-cations over the slope we would expect to find reversals in the poleward flow. The consequentresults are shown in Fig. 8. The first conclusion is that although the flow is strongly modified overthe shelf, over the slope, nearer the ocean bottom (not shown here), the currents are very similarto those obtained in DS simulation. This conclusion is supported by currentmeter analyses madeby Huthnance et al. [34], which indicated a downward decreasing of current variance. Over theshelf and upper slopes large responses to northwesterly/southwesterly winds are expected.However in April the winds are from the northwest in the southern Bay of Biscay and from thewest in Celtic Sea. Results from the vertically integrated model presented by Pingree and LeCann[13] show that under these conditions barotropic response should result in a flow to the southeastover the Celtic and Armorican shelves. Our results indicate a much more complex flow. Veryinteresting is the response on the Armorican shelf. The flow continues to the northwest, as in DSand DA simulations, although very much weakened (less than 5 cms�1). From actual observa-tions made in April 1987, Pingree and LeCann [13] reported something very similar. Results fromour model clearly suggest that this flow is driven by pressure gradients associated with seawater


density distribution, since similar results are obtained without wind forcing. The results from WSexperiment show a weaker flow to the northwest but one still to some extent resistant to the lowwinds typical for April. Another interesting feature shown by the present model is the counterflow on the outer Celtic Sea. Again, this flow was present in DS and DA simulation, althoughseemingly more realistically in WS simulation, thereby suggesting that this counter current is atleast partially induced by the wind. Pingree and LeCann [16] found that the flow in a mooringlocated at 48�42; 90N, 8�58; 30W was correlated with local winds, so confirming the same hy-pothesis. To the southwest of Ireland, the picture changes. There, the flow is to the northwest,something which does not appear in DS and DA simulations. Pingree and LeCann [13] foundlittle correlation between this flow and local winds. In model results this flow seems to be part ofthe current that runs clockwise around southern Ireland. Further investigation is needed for abetter understanding of the mechanism governing it. In summary, the present model (under theparticular conditions mentioned) shows a circulation in the area around the Goban Spur thatmatches very reasonably the diagram depicted by Pingree and LeCann [13] in Fig. 15 of theirpaper. The model also fits some of the observations made in the course of the OMEX Project andreported by Huthnance et al. [34]. Reversals of the near surface flow are to be seen, especially tothe southeast of La Chapelle Bank.

Fig. 7. Climatological monthly mean windstress from [40] for April. Minimum vector is 0:03 Nm�2, maximum vector

is 0:07 Nm�2.


4. Conclusions

From the described above some major conclusions arise:

• The model shows to be potentially useful for the description of shelf-slope circulation. Doublesigma coordinate allows the use of more realistic bottom topographies than classical sigmamodels.

• It was possible to show that a realistic slope current can be driven by seawater density distribu-tions in the North Atlantic. The slope current is mostly confined to the upper slopes and outershelf. The vertical extent of the current is 1500–2000 m. It was also shown that the current ismore intense in autumn than in spring. This autumn intensification must be related with sea-water density distribution in the North Atlantic.

• The slope current is continuous from the ocean margins of western Portugal to those of westIreland. Models used in DYNAMO also support this idea.

• Speeds and width of the current seems to be exaggerated, resulting in transports larger than theexpected. This appears to be an intrinsic problem of sigma models, since DYNAMO investiga-tors using SPEM found similar discrepancies.

• The model was able to produce some patterns of the flow in the Celtic and Armorican shelvesthat are not related with local winds.

• In the more realistic experiment, WS (spring density and wind forcing), the model reproducedmost of the circulation patterns of the residual flow described by Pingree and LeCann [13]. Re-versals of the slope current in spring were also predicted.

Fig. 8. Residual currents in the surface layer after 2 months of simulation with a Spring density field as initial con-

ditions and winds from April.


In the future, tides will be included in the model allowing a more realistic simulation of the shelf-slope circulation. Prognostic simulations have been carried out using realistic atmospheric forcingthat includes heat exchanges across the air–sea interface. Results are under analyses and will besubject of a future paper. A vertical generalised coordinate model is under development [23] al-lowing the use of several vertical coordinates, since it uses the finite volume concept. One of themost attractive solutions allowed by this model, is the extension of the double sigma concept. Bythis way it will be possible to use a vertical coordinate that is a mixture between sigma andcartesian coordinates.

Acknowledgements

This work was supported by the EU through the MAST programme, contract MAS3-CT96-0056 (Ocean Margin Exchange – OMEX). H. Coelho, R. Miranda and P. Chambel were fundedby the Portuguese Ministry of Science and Technology through PRAXIS XXI Programme.

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A circulation model for the European ocean margin

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