Diagnoses of vertical transport in a three-dimensional finite element

Estuarine, Coastal and Shelf Science 74 (2007) 655e669www.elsevier.com/locate/ecss

Diagnoses of vertical transport in a three-dimensional finiteelement model of the tidal circulation around an island

Laurent White a,b,*, Eric Deleersnijder a,b

a Centre for Systems Engineering and Applied Mechanics (CESAME), Universite catholique de Louvain, 4, Avenue G. Lemaıtre,B-1348 Louvain-la-Neuve, Belgium

b G. Lemaıtre Institute of Astronomy and Geophysics (ASTR), Universite catholique de Louvain, 2, chemin du Cyclotron,

B-1348 Louvain-la-Neuve, Belgium

Received 10 May 2006; accepted 28 July 2006

Available online 14 September 2006

Abstract

A three-dimensional finite element model is used to investigate the formation of shallow-water eddies in the wake of Rattray Island (GreatBarrier Reef, Australia). Field measurements and visual observations show that stable eddies develop in the lee of the island at rising and fallingtides. The water turbidity downstream of the island suggests the existence of strong upwelling that would be responsible for carrying bed sed-iments up to the sea surface. We first propose to look at the upwelling velocity and then use the theory of the age to diagnose vertical transport.The water age is defined as the time elapsed since particles of water left the sea bottom, where the age is prescribed to be zero. Two versions ofthis diagnosis are considered. Although the model predicts upwelling within the eddies, it is not sufficiently intense to account for vertical trans-port throughout the water column during the life span of the eddies. As mesh resolution increases, this upwelling does not intensify. However,strong upwelling is then resolved off the island’s tips, which is confirmed by the results obtained with the age. This study also shows that thefinite element method, together with unstructured meshes, performs well for representing three-dimensional flow past an island.� 2006 Elsevier Ltd. All rights reserved.

Keywords: age; upwelling; unstructured mesh; finite element method; island wake; Rattray Island

1. Introduction

In shallow coastal regions, flow disturbances caused by to-pographical features, such as islands, headlands, reefs and nar-row passages, can have strong effects on marine ecosystems.Topographically generated circulation affects the distributionof sediments and can significantly influence the local dispersalof pelagic organisms (Hamner and Hauri, 1981; Wolanski andHamner, 1988; Wolanski et al., 1988; Wolanski, 1994; Coutisand Middleton, 1999, 2002). As pointed out by Wolanski et al.(1984), this has important implications in the location of fish-eries and waste outfalls. Due to the presence of islands and

* Corresponding author. Centre for Systems Engineering and Applied Me-

chanics (CESAME), Universite catholique de Louvain, 4, Avenue G. Lemaıtre,

B-1348 Louvain-la-Neuve, Belgium.

E-mail address: [email protected] (L. White).

0272-7714/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ecss.2006.07.014

reefs, oncoming currents separate and, as water is strippedaway at the surface, it is replaced by upwelled water (Hamnerand Hauri, 1981). Upwelled water is generally enriched withnutrients, which may locally alter the biotic diversity (Wolan-ski et al., 1988). Depending on flow characteristics and islandgeometry, stable or unstable eddies may develop in the islandwake (Wolanski et al., 1984; Pattiaratchi et al., 1986; Ingramand Chu, 1987; Wolanski et al., 1996). These eddies may havea local impact on the ecosystem because of secondary circula-tion, enhanced turbulence and upwelling (Hamner and Hauri,1981; Wolanski and Hamner, 1988; Wolanski et al., 1988).

Of particular concern are the shallow-water eddies gener-ated in the wakes of islands by oscillating tidal flows. By shal-low water, it is meant here that the ratio of the water depth tothe island width (facing the current) is much less than one.Shallow-water flows are characterized by dominant bottomfriction, which has important consequences onto their

mailto:[email protected]

http://www.elsevier.com/locate/ecss

656 L. White, E. Deleersnijder / Estuarine, Coastal and Shelf Science 74 (2007) 655e669

dynamics (Ingram and Chu, 1987; Tomczak, 1988). Unstead-iness of tidal flows was shown to play a crucial role in the for-mation of eddies (Black and Gay, 1987). To take into accountbottom friction in the description of the island wake dynamics,Wolanski et al. (1984) suggested to use the ‘‘island wake pa-rameter’’ P rather than the usual Reynolds number prevailingin the description of two-dimensional wake flows. Pattiaratchiet al. (1986) confirmed this concept with laboratory and fieldexperiments. The island wake parameter P is defined as the ra-tio of nonlinear acceleration to bottom friction. For P< 1, nowake is present. For Px1, two stable eddies form in the lee ofthe island and remain attached while for increasing values ofP, instabilities occur in the wake, following by eddy shedding.In this paper, we will focus on the case Px1 for which eddiesessentially result from a balance between radial pressure gra-dient and centrifugal effect due to flow curvature. Near thebottom, the azimuthal velocity decreases because of bottomfriction. However, a constant pressure gradient is maintainedthroughout the water column. Therefore, the balance breaksdown near the seabed, leading to flow convergence towardsthe center of the eddy and upwelling within the water column.

Among shallow-water islands for which stable tidal eddiesare observed, Rattray Island (Great Barrier Reef, NortheastAustralia) has been the focus of many studies in the pasttwo decades (Wolanski et al., 1984; Falconer et al., 1986;Black and Gay, 1987; Wolanski and Hamner, 1988; Deleer-snijder et al., 1992; Wolanski et al., 1996, 2003). RattrayIsland is 1.5 km long and lies in well-mixed water approxi-mately 25 m deep. The currents are dominated by the tides,whose ellipses are strongly polarized and essentially orientedfrom northwest to southeast. The island was subject to an ex-tensive field survey in 1982 (Wolanski et al., 1984). Twenty-six current meters were deployed in four transects in thewake of the island during flood tide and made clear the exis-tence of a clockwise-rotating eddy extending across the wake.No measurements were made at falling tide but aerial photo-graphs suggested the presence of two counter-rotating eddies.Aerial photographs also showed turbid water in the wake bothat rising and falling tides, suggesting upwelling capable of car-rying bed sediments upwards. While two-dimensional numer-ical models are able to faithfully represent depth-averagedfeatures (e.g., Falconer et al., 1986; Black and Gay, 1987),only three-dimensional models can account for vertical mo-tion. There have been only a few attempts in the past at ana-lyzing vertical motion in shallow-water island wakes.Deleersnijder et al. (1992) utilized a 200-m horizontal resolu-tion finite-difference model to compute the three-dimensionalvelocity field in the vicinity of Rattray Island. Although up-welling was predicted within the bulk of the eddies, its inten-sity was too low to account for vertical transport across theentire water column during the lifetime of the eddies. It wasput forward that the model used a resolution that was toolow to accurately represent velocity gradients (and hence di-vergence). Wolanski et al. (1996) used the same model asthat by Deleersnijder et al. (1992) with an additional parame-terization to account for the impact of the free shear layer ex-tending downstream from the tips of the island. Later on,

Alaee et al. (2004) also used a three-dimensional finite-differencemodel but they focused on upwelling at the tips of an idealizedelliptic island. The main difference between the work byAlaee et al. (2004) and the previous one is that Alaee et al.(2004) worked with flow regimes for which no eddies weregenerated.

In this paper, we wish to continue along the path set out byDeleersnijder et al. (1992) with two crucial improvements.The first one is concerned with the numerical method, as weuse a three-dimensional finite element model. The finite ele-ment method (FEM) allows for conveniently using unstruc-tured meshes, whose resolution may vary and increasewithin regions of interest to attain higher accuracy. In addition,the ability of unstructured meshes to conform to complexcoastlines is attractive. The FEM has been successfully uti-lized in the past for the modeling of coastal flows and oceanicprocesses (e.g., Walters, 1992; Lynch et al., 1996; Le Rouxet al., 2000; Hanert et al., 2005b; Pietrzak et al., 2005; Whiteet al., 2006) and its popularity within the ocean modeling com-munity is likely to grow in the future. With the FEM, we willbe able to increase the mesh resolution around the island andin the wake, and analyze the sensitivity of upwelling intensityon resolution. The second improvement is achieved by usinga more sophisticated diagnosis of vertical transport. By look-ing at the upwelling velocity, Deleersnijder et al. (1992) con-sidered vertical transport due to advection only. However,because vertical transport is a combination of advective anddiffusive effects, we need a diagnosis that is able to accountfor both. The concept of the age may be used for that purpose.It is presented in detail by Delhez et al. (1999), Deleersnijderet al. (2001) and Delhez and Deleersnijder (2002). It is a com-ponent of CART (Constituent-oriented Age and Residencetime Theory, http://www.climate.be/CART). The age of a par-ticle of seawater is defined as the time elapsed since the par-ticle under consideration left the region in which the age isprescribed to be zero. In the theory of the age, all classical ad-vectionediffusion operators are accounted for. By prescribingthe age of all particles lying on the seabed to be zero, we maytrack the time needed for those particles to reach the sea sur-face. The main goal of this paper is to determine whether theeddies are partly or fully responsible for the vertical transportof bed sediments. Thus, we would like to answer the followingquestion: How much time is required for particles of a non-buoyant, passive tracer to travel from the seabed to the seasurface?

This paper is organized as follows. Section 2 describes thethree-dimensional hydrodynamic model together with its im-plementation with the FEM. In Section 3, two different diag-noses of vertical transport using the age are laid out. Resultsare presented in Section 4 and we conclude with Section 5.

2. Model description

In this section, the underlying physical assumptions are out-lined. The equations, the domain of interest and the boundaryconditions are presented. The finite element method is brieflyexplained. Because of the limited extent of the region of

http://www.climate.be/CART

657L. White, E. Deleersnijder / Estuarine, Coastal and Shelf Science 74 (2007) 655e669

interest, the f-plane approximation is made. As pointed out byWolanski et al. (1984), Rattray Island lies in well-mixed waterwith very little contrast of salinity and temperature. Therefore,a constant density is assumed throughout the domain. Finally,we work within the scope of the hydrostatic approximation.The currents around Rattray Island being dominated by tidalforcing, we neglect wind stress. All these assumptions weremade by Deleersnijder et al. (1992) and we deliberately decideto work within the same framework to be able to assesswhether or not coarse mesh resolution was responsible forunderestimating the upwelling velocity.

Let v ¼ ðu; v;wÞ be the velocity, u ¼ ðu; vÞ be the horizontalcomponents of v and h be the free-surface elevation with re-spect to the constant reference height z¼ 0 taken to be themean sea level. We will distinguish between the three-dimen-sional gradient operator V and the horizontal gradient operatorVh, affecting only the horizontal components of a vector. Un-der the aforementioned assumptions, the horizontal momen-tum equation is:

vu

vtþ ðv$VÞuþ fbez^u¼�gVhhþ v

vz

�nz

vu

vz

�þD; ð1Þ

where f is the constant Coriolis parameter taken at a latitude of�20�, bez is the upward-pointing unit vector, g is the gravita-tional acceleration and nz is the vertical eddy viscosity coeffi-cient. Horizontal momentum diffusion is parameterized by D.Eq. (1) is complemented with the usual continuity and free-surface equations, which are not reproduced here. As wasdone by Deleersnijder et al. (1992) and directly inspired byFischer et al. (1979), a simple turbulence closure is consideredfor which the vertical eddy viscosity nz is given by:

nz ¼ ku�ðhþ zÞ�

1� dhþ z

H

�; ð2Þ

where k is the von Karman constant, u� is the bottom frictionvelocity (the square root of the norm of the bottom stress di-vided by the water density) and d is an adjustable parameter,taken to be 0.6 in all numerical experiments. The horizontaleddy viscosity term D is:

D¼ v

vx

�nh

vu

vx

�þ v

vy

�nh

vu

vy

�;

where nh is computed via a Smagorinsky scheme (Smagorin-sky, 1963):

nh ¼ csD2

�vu

vx

vu

vxþ 0:5

�vu

vyþ vv

vx

�2

þvv

vy

vv

vy

�1=2

; ð3Þ

in which cs is a constant and D is a measure of the local meshsize. For triangular meshes, D2 is taken to be the surface areaof the local triangle (Akin et al., 2003). In expression (3), theoverbar denotes depth-averaged quantities, as suggested byLynch et al. (1996).

Because the tidal ellipses are strongly polarized (Wolanskiet al., 1984), we take the y-axis of the domain to be parallel to

the major axis of the tidal ellipses, namely oriented fromsoutheast to northwest as depicted in Fig. 1. In so doing, wemay assume the side boundaries to be impermeable to theflow so that only the southeast and northwest boundaries ehereafter referred to as lower and upper boundaries, respec-tively e remain open. Using available field measurements,the depth-averaged normal velocity and the elevation are im-posed at both the lower and upper boundaries by prescribingthe incoming characteristic variable un � h

ffiffiffiffiffiffiffiffig=h

p, where un

is the depth-averaged normal velocity. The phase lag betweenboth boundaries is less than 20 min and is neglected in themodel. At the bottom, a slip condition is imposed on the hor-izontal velocity and the bottom stress t is parameterized bythe following logarithmic law:

tðx; y;xbÞ ¼�

nz

vu

vz

�z¼�h

¼�

k

lnðxb=x0Þ

�2

kubðx; y;xbÞkubðx; y; xbÞ; ð4Þ

in which xb is the distance to the seabed where the appropriatebottom velocity ub is defined and x0¼ 5� 10�3 m is theroughness length (Black and Gay, 1987).

The three-dimensional finite element model SLIM (Sec-ond-generation Louvain-la-Neuve Ice-ocean Model, http://www.climate.be/SLIM) is used. The detailed implementationis described by White et al. (in preparation). The model isessentially built upon the work by Hanert et al. (2005a) andHanert et al. (2005b) for its two-dimensional structure. Thethree-dimensional space discretization is based upon down-ward extrusion of two-dimensional triangular meshes and col-umn splitting into prisms. The location of horizontal andvertical velocity nodes is shown in Fig. 1. All nodes are freeto move in the vertical. This allows for tracking the free sur-face and for dynamical mesh redistribution and refinementin the vertical following a scheme based e.g., on a posteriorierror estimates, as proposed by Hanert et al. (2006). The latter,however, is not yet implemented. The solution technique is se-quential. We first solve the equations for the depth-averagedvelocity and the elevation. All linear terms are semi-implicitin time, which permits to circumvent the stability constraintdue to the propagation of inertia-gravity waves. The mesh ge-ometry is updated with the new elevation field. We then solvefor the full three-dimensional horizontal velocity by alternat-ing at each time step between u and v so that the Coriolis forcedoes not produce nor dissipate energy. The vertical velocity iscomputed after correction of the full horizontal velocity by thedepth-averaged velocity. The same time step is used for bothmodes.

Because the last velocity node lies on the seabed, the bot-tom stress (4) is computed by using the mean value of thelast two velocity nodes. The distance to the seabed xb is calcu-lated accordingly. The constant cs used in the parameterizationof the horizontal momentum diffusion coefficient, Eq. (3), typ-ically lies in the range 0.1e0.3. It is slightly larger than thevalue recommended by Smagorinsky (1963) but on the same

http://www.climate.be/SLIM



Horizontal velocity

Vertical velocity

15

15

20

20

20

20

20

2525

25

25

2525

25

25

25

25

30

3030

30

30

30

30

35

35

35

35

40

N

1 km

x

y

Fig. 1. Top left: domain of interest with bathymetry in meters (Rattray Island is the black area at the center). Top right: basic prismatic element used to interpolate

both components of the horizontal velocity (non-conforming linear in the horizontal and linear in the vertical) and the vertical velocity (constant in the horizontal

and linear in the vertical). Bottom: unstructured meshes used to simulate the flow around Rattray Island. The mesh on the left contains 3024 triangles and has

a resolution of 140 m around the island. The mesh on the right contains 6096 triangles and has a resolution of 85 m in the vicinity of the island. This is to be

compared with the 200-m resolution used by Deleersnijder et al. (1992) and Wolanski et al. (2003). Note that uniform refinement over the whole domain would

quadruple the number of triangles. Local refinement avoids this drawback.

order as that used in usual finite element models such as that ofLynch et al. (1996). In our model, this choice leads to peakvalues in the horizontal eddy viscosity nh of less than 1 m2/swhen the flow is the swiftest in the wake of the island. Thisis in agreement with a value of 0.5 m2/s estimated by Wolanskiet al. (1984) in the field. The value of nh attains 2e3 m2/s offthe island’s tips due to flow curvature and acceleration. How-ever, this effect is strongly localized. To enhance robustness,advection terms are computed with an upwind-biased scheme,following the method by Hanert et al. (2005a). Although the

scheme is somewhat numerically dissipative, the amount ofdissipation only damps out unresolved (or poorly resolved)scales with little effect on the main solution. Examples of un-structured meshes used for numerical experiments are shownin Fig. 1.

3. Diagnoses of vertical transport

As model resolution increases over time, so does theamount of output data and the difficulty of having a relevant


and unbiased view of the results. A few cuts through a three-dimensional domain at some selected timesteps to display onevariable gives quite a poor rendition of an otherwise N-dimen-sional problem. In this paper, we focus on vertical transportand three relevant diagnoses are presented.

3.1. The upwelling velocity

Deleersnijder et al. (1992) looked at the advective verticaltransport by resorting to the upwelling velocity (Deleersnijder,1989, 1994). The latter is devoid of topographic effects and isentirely due to intrinsic upwelling mechanisms. Placing our-selves in a sigma-coordinate system (terrain-following coordi-nates), the upwelling velocity is that component of the verticalvelocity that causes a relative change in the vertical positionwithin the water column. This should be differentiated fromthe topography-induced component that is only nonzero be-cause of the nonflat bathymetry and free surface. The valueof the upwelling velocities modeled by Deleersnijder et al.(1992) peaked at 10�3 m/s at mid-depth within the eddies,which is more than twice smaller than the value needed fora water particle having this velocity to travel across the entirewater column during the life span of the eddies. This argumentrests on the assumption that the intensity of the eddies isstrong enough during about 2e3 h. The reason put forwardfor the underestimation was the grid coarseness.

3.2. The age

Despite its simplicity e this is a postprocessing step and nocomputation is involved e the main drawback of the upwell-ing velocity is that it does not account for all processes respon-sible for vertical transport and it gives an instantaneoussnapshot of the dynamics. The theory of the age is a holisticapproach (Delhez et al., 1999; Deleersnijder et al., 2001; Del-hez and Deleersnijder, 2002). That is, the age is a diagnosisthat accounts for advective and diffusive transport and dependson all model results. The question that we want to answer is:

How much time is required for particles of a non-buoyant, pas-sive tracer to travel from the seabed to the sea surface? Thetheory presented by Deleersnijder et al. (2001) allows for com-puting the mean age of particles since they left the regionwhere the age is set to zero. It is not uncommon to regard wa-ter masses as passive tracers (Cox, 1989; Hirst, 1999; Goosseet al., 2001). In that respect, the concentration of the passivetracer is interpreted as the concentration of the water mass un-der consideration. In our situation, we trace the bottom water,that is the water that has touched the seabed.

We shall define two types of age. The first age is defined asthe arithmetic average of the times that have elapsed since theparticles left the seabed for the last time. The age of a waterparticle keeps increasing as long as it does not touch the sea-bed again. Once a water particle touches the seabed, its age isreset to zero. The second age is defined to be the time neededto travel from the seabed to the sea surface. Similarly to thefirst type of age, the age of a water particle is reset to zerowhen it touches the seabed. However, once the water particletouches the sea surface, it is disregarded until it touches theseabed again. This key difference between both types of ageis illustrated in Fig. 2. From now on, all variables associatedwith the type of age i will have a subscript i. The ages willbe referred to as age 1 and age 2.

To compute the age, we have to solve advectionediffusionequations for the bottom water concentration Ci and the ageconcentration ai. Once those two variables are known, theage ai is given as the ratio of the age concentration to the waterconcentration:

ai ¼ai

Ci

ði¼ 1;2Þ: ð5Þ

The water concentration Ci is solution to (Delhez et al., 1999):

vCi

vtþV$ðvCiÞ ¼

v

vz

�Kz

vCi

vz

�þD ði¼ 1;2Þ; ð6Þ

t

t1t2

t3

Sea surface

Fig. 2. A water sample containing three water particles is taken at the sea surface at time t. Each particle has a different history, as illustrated by their trajectories.

For the first type of age, the age keeps increasing as long as the particle does not touch the seabed again. Therefore, the first age is

a1¼ [(t� t1)þ (t� t2)þ (t� t3)]/3. For the second age, the particle is disregarded when it touches the sea surface. Therefore, only the first particle need be ac-

counted for: a2¼ (t� t1).


where Kz is the vertical eddy diffusivity coefficient and D pa-rameterizes turbulent horizontal diffusivity with a Smagorinskyscheme similar to Eq. (3) The age concentration ai obeys thefollowing equation (Delhez et al., 1999):

vai

vtþV$ðvaiÞ ¼ Ciþ

v

vz

�Kz

vai

vz

�þD ði¼ 1;2Þ; ð7Þ

where the water concentration Ci is the so-called aging term.Note that the water concentration varies between 0 and 1.Boundary conditions are yet to be prescribed to close thesystem. Because the vertical eddy diffusivity at the bottomis zero, it is convenient to introduce a bottom roughnesslength xb. This is similar to the roughness length introducedfor the computation of the bottom stress, see Eq. (4). Wemay similarly define a surface roughness length xs. Thebottom and surface boundary conditions will be enforcedat zb¼�hþ xb and zs¼ h� xs. At sea bottom, the waterconcentration is 1 and the age concentration is 0 for bothages:

½Ci�z¼zb¼ 1 and ½ai�z¼zb

¼ 0 ði¼ 1;2Þ; ð8Þ

which translates the fact that we want to track water particlesthat leave the seabed with the age reset to 0. At the sea surface,we have to distinguish between age 1 and age 2. Since the freesurface is impermeable to the first age, we have:�

Kz

vC1

vz

�z¼zs

¼ 0 and

�Kz

va1

vz

�z¼zs

¼ 0; ð9Þ

which means that age 1 keeps increasing even when water par-ticles touch the free surface. For the second age, we have toimplement the fact that once a particle touches the surface,it is disregarded. In other terms, the water concentration iszero at the surface:

½C2�z¼zs¼ 0: ð10Þ

Finally, in order to avoid infinite values of age 2 at the surface,Eq. (5) requires to have:

½a2�z¼zs¼ 0: ð11Þ

The conditions applied on lateral boundaries depend onwhether they are closed or open. Along closed boundaries,a no-flux condition is enforced on the water and age concen-trations. At outflow open boundaries, both the water and ageconcentrations are advected out of the domain. At inflowopen boundaries, incoming water and age concentrationsmust be prescribed to compute the advective flux. The incom-ing water (age) concentration is taken to be the mean outgoingwater (age) concentration. In other terms, the incoming age isprescribed to be the mean outgoing age, leading to periodicboundary conditions on the age in the mean sense. This isbased on the hypothesis that horizontal age contrasts appearclose to the island and that homogeneity prevails far awayfrom it.

At this point, two remarks can be formulated:

(1) we will take the vertical eddy diffusivity coefficient Kz to beequal to the vertical eddy viscosity coefficient nz. In otherwords, the Prandtl number e which is the ratio of viscosityto diffusivity e is assumed to be equal to one. This hypoth-esis is generally accepted for unstratified fluids and it there-fore applies to Rattray Island (Munk and Anderson, 1948);

(2) the computation of age 2 at the surface implies to evaluatean indeterminate limit of type 0/0. We have to make surethat the limit exists.

3.3. A one-dimensional water-column model

A simplified one-dimensional water-column model is veryuseful to gain additional insight into both types of age. It willalso give indications as to the difficulties lying ahead of us inthe numerical resolution, and in particular those pertaining tothe boundary conditions. Steady state and horizontal uniformityare assumed for all variables. Since the horizontal componentsof the velocity field are deemed horizontally homogeneous, thevertical component must be constant everywhere in the watercolumn to satisfy the continuity equation. Because of the im-permeability of the seabed, the vertical component of the veloc-ity must therefore vanish throughout the water column. Onlyvertical diffusivity remains. To simplify calculations, we as-sume that the domain extends from z¼ 0 to z¼H. The originof the z-axis coincides with the seabed and no longer with themean sea level. The vertical eddy diffusivity is given by:

Kz ¼ ku�z�

1� dz

H

�: ð12Þ

It follows that the bottom and surface boundary conditions areenforced at zb¼ xb and zs¼H� xs. The differential problem(6)e(11) simplifies to:

d

dz

�Kz

dCi

dz

�¼ 0 and

d

dz

�Kz

dai

dz

�þCi ¼ 0 ði¼ 1;2Þ; ð13Þ

subject to the boundary conditions:

C1ðzbÞ ¼ 1;

�Kz

vC1vz

�z¼zs

¼ 0; a1ðzbÞ ¼ 0;

�Kz

va1vz

�z¼zs

¼ 0;

C2ðzbÞ ¼ 1; C2ðzsÞ ¼ 0; a2ðzbÞ ¼ 0; a2ðzsÞ ¼ 0: ð14Þ

The problem (13) and (14) can be nondimensionalized. Di-mensionless variables e denoted by primes e are defined by:z0; z0b; z

0s;x0b;x0s

¼ ðz; zb; zs;xb;xsÞ=H; K0z ¼ Kz=Kz;

C0i ¼ Ci; a0i ¼ aiKz=H2;

where z0s ¼ 1� x0s and Kz is the depth-average of Kz defined byEq. (12). The dimensionless diffusivity coefficient is:

K0z ¼6

3� 2dz0ð1� dz0Þ: ð15Þ


Henceforth, only dimensionless variables will be used and wemay omit the primes. All notations remain the same and theproblem to solve is still referenced by Eqs. (13) and (14).The analytical solution to Eqs. (13) and (14) e whose devel-opment is given in the Appendix e is shown in Fig. 3. Theseexact solutions feature three important aspects:

(1) age 1 is about seven times larger than age 2 at the surface.This suggests that age 2 will not be influenced by the lat-eral boundaries as much as age 1 in the vicinity of the is-land. Assuming a depth-average eddy diffusivity of0.05 m2/s and a typical depth of 20 m around the island,age 1 is on the order of 5 h while age 2 is on the orderof 45 min. Since tidal flow is reversing approximately ev-ery 6 h, age 1 has barely the time to fully develop duringa half tidal cycle while this is not a concern for age 2. Wehave to keep that in mind when designing open boundaryconditions on the age;

(2) both ages feature a logarithmic bottom layer whose lengthscale is on the order of the bottom roughness length. Un-less we choose vertical increments to resolve the bottomlayers e which is infeasible in practice e somethingwill have to be done to account for them;

(3) the definition of age 2 at the surface gave rise to an inde-terminate limit of type 0/0. This limit exists, as is shown inthe Appendix.

3.4. Numerical resolution of the water-column model

The presence of a logarithmic bottom layer in the age pro-file makes it more challenging to solve numerically. We pro-pose here to compare three finite element discretizations inthe vertical for the resolution of the age concentration. Wewill focus on age 1. In this paper, we do not aim at giving a de-tailed overview of the finite element method. Therefore, eachdiscretization method will be covered conceptually withoutdelving into technical details.

The first discretization is based on approximating the exactsolution by a piecewise linear field. That is, the water columnis divided into N elements with Nþ 1 nodes, the bottommostlying at z¼ zb and the topmost coinciding with the free surfaceat z¼ zs. The field is interpolated with linear polynomials be-tween nodes. The second discretization is based upon the ex-tended finite element method (Moes et al., 1999). The latterconsists in enriching a few bottom nodes with a global loga-rithmic function ln(z/zb). In so doing, we actually increasethe number of degrees of freedom. Within the bottom elementsadjacent to the selected enriched nodes, we add an enrichedfield to the piecewise linear field. The enriched field is madeup of a piecewise linear field modulated by the global logarith-mic function. The third discretization consists in substitutingthe linear shape functions for logarithmic shape functionswithin the bottommost element (Hanert, personal communica-tion). Instead of approximating the exact field by a linear poly-nomial, we approximate it with a linear combination oflogarithmic functions.

A comparison between all three discretizations is shownin Fig. 4. Clearly, the first discretization e the usual piece-wise linear interpolation e fails at decently representingthe exact solution. The second discretization e enrichinga few bottom nodes e yields a very accurate approximation.Unfortunately, the tridiagonal structure of the linear system islost and a very high-order quadrature rule is required toachieve this level of accuracy (typically more than 20 Gausspoints). This proved to be too expensive in three dimensionseven though more Gauss points could be used near the bot-tom by dividing out the bottommost element into non-uniform subelements in which a low-order quadrature rulecould be used. With the third discretization, an accuratesolution is obtained while the tridiagonal structure of thematrix is preserved and a low-order (5 Gauss points)quadrature rule suffices. This is the employed method tocompute the solutions to all age-related advectionediffusionequations.

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Exact solutions to nondimensionalized problem Eqs. (13) and (14) with d¼ 0.6, H¼ 20 m and xb¼ xs¼ 5� 10�3 m. At the surface (z¼ zs), age 1 is about

seven times larger than age 2. For age 1, since the water concentration is 1 across the entire domain, the age concentration is equal to the age. Notice also the

logarithmic bottom layers and the finite value of age 2 at the surface (the limit was indeterminate).


0 0.5 1 1.5 2 2.50

0.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.50

0.10.20.30.40.50.60.70.80.9

1A B C

Fig. 4. Numerical resolution of the water-column model for age 1 with 10 elements. In all panels, the solid line is the exact solution while the numerical approx-

imation is represented by the circles. (A) Piecewise linear approximation. (B) Piecewise linear approximation with enrichment of the two bottom nodes with

a global logarithmic function. (C) Discretization for which the linear shape functions are substituted for logarithmic shape functions within the bottommost

element. In all other elements, linear shape functions are used. The third discretization yields the best quality-price ratio.

4. Results and discussion

A 5-day run was carried out on each mesh displayed inFig. 1. In the vertical, 10 and 16 layers were used for thecoarse and the fine mesh, respectively. The Smagorinsky con-stant cs is 0.1 for the coarse mesh and 0.3 for the fine mesh.These values were chosen to remain as close as possible tothe estimate of the horizontal eddy viscosity coefficient byWolanski et al. (1984). However, the total amount of dissipa-tion (physical and numerical) is larger for the coarse mesh.This is due to the artificial dissipation introduced by the up-wind-biased advection scheme. This artificial dissipation is de-pendent on the mesh size. Although not required, the sametime step of 15 s was used for both meshes so that the sole ef-fect of spatial discretization was taken into account. To ensurethat a regime solution is attained, the results are presented af-ter 3 days. We first show the results on the upwelling velocityand then proceed with the age. In the subsequent discussion,the northern (southern) island’s tip will be referred to as theright (left) tip.

4.1. The upwelling velocity

The upwelling velocity on each mesh is shown at two dif-ferent times during falling tide in Fig. 5. Note that the upwell-ing velocity is taken at mid-depth, as was done byDeleersnijder et al. (1992). Panels BeC in Fig. 5 show the up-welling velocity during falling tide when the free-stream speedis maximum at about 0.5 m/s. Though the distribution of up-welling and downwelling zones is quite similar for bothmeshes, two important differences are observed. First, forthe coarse mesh (panel B), the upwelling velocity field is quitenoisy, contrary to that obtained with the fine mesh. Second, forthe latter (panel C), a small-scale intense upwelling zone(4 mm/s) is resolved off the right tip of the island (see thewhite patch). This upwelling zone remains unresolved withthe coarse mesh. Snapshots EeF are taken at the end of fallingtide, shortly before tide reversal. The free-stream speed isabout 0.1 m/s. The results for both meshes are very similar be-cause only larger-scale features persist. Aside from these

detailed differences and similarities, a few general remarksmay be formulated regarding the upwelling velocity:

(1) in all snapshots, the upwelling velocity within the eddies isabout 1 mm/s. This is particularly clear on the right panelsin Fig. 5EeF, where the eddies are well formed and stillintense shortly before tide reversal. Because those eddiestypically start to develop when the tidal velocity is maxi-mum and die off after tide reversal, their life span isroughly 3 h. Therefore, a distance of 10 m could at bestbe traveled by water particles motioned by the upwellingvelocity and leaving the seabed at the onset of the eddyformation. Since the depth around the island is about20 m, the predicted upwelling is not sufficiently intenseto carry bottom water up to the sea surface during theeddies lifetime.

(2) the mesh resolution does not significantly modify the up-welling intensity within the eddies. The orders of magni-tude obtained in this study are in line with thosepredicted by Deleersnijder et al. (1992).

(3) some smaller-scale features, such as the intense upwellingzone off the right island’s tip shown in Fig. 5C, are not re-solved by the coarse mesh. Note that increasing the meshresolution up to 65 m confirms the presence of thosesmall-scale features (not shown). Upwelling is also pre-dicted off the left tip of the island for the fine mesh. Del-eersnijder et al. (1992) also reported upwelling off theright island’s tip but the intensity was much smaller (about1 mm/s) than our predictions.

4.2. The age

The surface age for age 1 and age 2 obtained with the finemesh is shown in Fig. 6 at two different times during risingtide and in Fig. 7 at two different times during falling tide. Ex-periments on extended meshes (in the free-stream direction)have been made to assess the sensitivity of the predicted ageon the distance at which boundary conditions are imposed. Inall cases, the predicted age remains within a few percent ofthat computed with the original mesh (not shown). Hence, the


A

C

D

E

F

B

Fig. 5. Upwelling velocity at mid-depth (mm/s) during falling tide. Each column shows (from top to bottom): the depth-averaged velocity field on the fine mesh, the

upwelling velocity for the coarse mesh and the upwelling velocity for the fine mesh. Snapshot times correspond to: (AeC) December 2 at 15h00 at peak ebb

velocity. (DeF) December 2 at 18h20 at the end of falling tide, shortly before tide reversal.

boundaries are located far enough from the island. At the onsetof the simulation, the water concentration is prescribed to be thesolution of the water-column model presented in the previoussection. The initial age concentration is zero. For age 1, the waterconcentration is never computed and remains equal to onethroughout the domain and at all time. For age 2, both the waterand age concentrations are computed.

Let us first concentrate on age 1, which is shown on panelsB and E of each figure. See also Movie 1 (available online)showing age 1 at the surface during a full tidal cycle. A strik-ing feature is visible in Figs. 6B and 7B, where bottom water(the darker elongated patches) emanates off the island’s tips.This water is less than 1 h old at both tips. The fact that theage of the bottom water is roughly the same off both island’s


A

B

C

D

E

F

Fig. 6. Surface age in hours for the fine mesh during rising tide. Each column shows (from top to bottom): depth-averaged velocity field, surface age 1 and surface

age 2. Snapshot times correspond to: AeC. December 1 at 8h00 at peak flood velocity. DeF. December 1 at 10h40 at the end of rising tide, shortly before tide

reversal.

tips can be explained by the depth difference. Off the right tip,the depth is about 30 m while off the left tip, it is about 15 m.However, upwelling on the right is found to be at least twice asintense as that on the left, which could explain the resultingsymmetry. When the tide keeps rising, some of the bottom wa-ter recirculates within the island’s wake while the rest is ad-vected downstream. At the end of rising tide, age 1 is about2e3 h downstream of the island, as shown in Fig. 6E. Finally,

at the end of falling tide, the age varies between 3 and 4 h witha few exceptions of younger water located near the centers ofboth eddies. The age of these patches is about 2 h. This isshown in Fig. 7E. Since the age of bottom water within theisland’s wake shortly before tide reversal is roughly 3 h, itcould be hypothesized that this water mainly originates fromthe island’s tips when the free-stream speed is large enoughto initiate upwelling.


A

B

C

D

E

F

Fig. 7. Surface age in hours for the fine mesh during falling tide. Each column shows (from top to bottom): depth-averaged velocity field, surface age 1 and surface

age 2. Snapshot times correspond to: AeC. December 2 at 15h00 at peak flood velocity. DeF. December 2 at 18h20 at the end of rising tide, very shortly before

tide reversal.

The interpretation of age 2 at the surface is more delicate.Although not as definite as for age 1, we may also discernyoung patches originating from the island’s tips in Figs. 6Cand 7C. See also Movie 2 (available online) showing age 2at the surface during a full tidal cycle. Along the downstreamedge of the island, a patch of older water is visible (lighterpatch). These patches coincide with regions of very low hori-zontal velocity, as can be seen on the right panels showing the

depth-averaged velocity field. Because of less intense circula-tion, vertical diffusion decreases, which has a direct impact onthe age. Finally, shortly before tide reversal, age 2 behavesrather counter-intuitively. The surface age pattern seems tobe opposite to that for age 1 (see Figs. 6F and 7F). The largestvalues of the surface age is found around the centers of theeddies where the upwelling velocity is the largest. This isvery clear for the left-hand side eddy in Fig. 6F and the


right-hand side eddy in Fig. 7F. This behavior motivates thefollowing investigation regarding the effect of vertical advec-tion on the age within eddies.

4.3. The effect of vertical advection

The water-column model previously presented includedvertical diffusion only. Although incomplete, it allowed forgrasping important features in the age such as the logarithmicbottom layer and the quantitative differences between bothtypes of age. We now wish to go one step further by addingthe effect of vertical advection. In this case, it is incorrect toassume horizontal homogeneity in the horizontal velocity,for otherwise the vertical velocity would be zero. Since weare primarily interested in the vertical transport within eddies,we will instead formulate the problem in cylindrical coordi-nates (r, q, z). The eddy is rotating about the z-axis and is sup-posed to be axisymmetrical. The corresponding velocitycomponents are (vr, vq, w). The domain of interest is a thin cyl-inder extending from the bottom to the top and whose radius r0

is such that 0< r0< e, with e� 1. In cylindrical coordinates,the advection term of the advectionediffusion equation fora scalar f is:

V$ðvfÞ ¼ vr

vf

vrþ vq

1

r

vf

vqþw

vf

vz; ð16Þ

using the fact that V$v ¼ 0. Because the problem is axisym-metrical, we have vf/vr¼ 0 at r¼ 0. It can also be contendedthat vf/vr is arbitrarily small as e approaches 0. Therefore, thefirst term in the right-hand side of Eq. (16) is neglected every-where in the water column, provided that vr remains finite. Forsmall values of r0, the azimuthal component, vq, can be linear-ized to vqxkr (for some constant k). Hence, the second termof the right-hand side of Eq. (16) reduces to k vf/vq, whichvanishes because f is independent of q. Thus, only vertical ad-vection remains, that is V$ðvfÞ ¼ w vf=vz. Although less le-gitimate, we will also neglect the effects of horizontaldiffusion. Note that, for the continuity equation to be satisfied,vr must depend on z. This is so because the continuity equationreduces to V$v ¼ vr=r þ vw=vz within the thin cylinder. Withvertical advection, the differential problem (13) becomes:

wðzÞdCi

dz¼ d

dz

�Kz

dCi

dz

�and

wðzÞdai

dz¼ d

dz

�Kz

dai

dz

�þCi ði¼ 1;2Þ; ð17Þ

subject to the same boundary conditions (14). The vertical ve-locity must vanish at z¼ 0 and z¼H. We will assume a para-bolic profile for which the maximum speed w0 is reached atmid-depth:

wðzÞ ¼ 4w0

H2zðH� zÞ: ð18Þ

This expression is inspired by the fact that the upwelling ve-locity typically vanishes at the bottom and the surface while

it reaches its maximum somewhere near mid-depth. The non-dimensional version of Eq. (18) is:

w0ðz0Þ ¼ 4z0ð1� z0Þ;

so that nondimensionalization of Eq. (17) gives rise to:

Pe wðzÞdCi

dz¼ d

dz

�Kz

dCi

dz

�and

Pe wðzÞdai

dz¼ d

dz

�Kz

dai

dz

�þCi; ði¼ 1;2Þ; ð19Þ

where all variables are dimensionless and primes are againomitted. The Peclet number Pe ¼ w0H=Kz measures the im-portance of advective transport relative to diffusive transport.The differential problem (19), subject to boundary conditions(14), is not amenable to analytical solutions when Pe s 0. Nu-merical solutions are therefore sought and for each value ofPe s 0, the surface age e that is, the age at z¼ zs e is com-pared with the surface age computed with Pe¼ 0. The objec-tive is to investigate the relative effect that advection has onthe surface age. The results are reported in Fig. 8 for age 2only.

The behavior pertaining to the latter is counter-intuitive.When there is downward advection (Pe< 0), the surface agedecreases. For a range of positive Peclet numbers(0< Pe< Pe*), the surface age increases while for Pe> Pe*,the surface age decreases. To shed light on this behavior, let ustake the situation in which we have a moderately intense up-ward advective transport (say Pe¼ 1). In this case, the waterconcentration C2 reaches higher values within the water col-umn due to advection. This, in turn, brings about an increasein the age concentration a2 because the aging term in the equa-tion for a2 increases. There is, however, a counter-balancingeffect due to the upward advection of smaller values of a2

coming from the seabed. The net effect is an increase in a2.When computing the ratio a2/C2, both the numerator and thedenominator have increased but the former has done so bya larger factor, leading to an increase in the surface age. Thesame reasoning applies to the situation in which Pe< 0. Fi-nally, for values of Pe larger than Pe*, advective transportof low values of a2 overcomes the aging process, caused bythe fact that saturation takes place for C2, which gets closeto 1 and does not increase any more.

Within the eddies, the numerical model predicts upwellingat an intensity of 0.001 m/s and an average vertical eddy dif-fusivity coefficient of about 0.02 m2/s. Taking a depth of20 m leads to a Peclet number of 1 while the critical Pecletnumber for this situation is around 6.5. We thus fall in therange for which positive vertical advection contributes to in-crease the surface age at the center of the eddies.

Finally, it should be stressed that the threshold Peclet numberPe* depends itself on the dimensionless number xb/H, namelythe ratio of the bottom roughness length to the water-columnheight. This dependence is shown for age 2 in Fig. 8 for three dif-ferent values of xb/H, where it can be seen that the threshold Pec-let number Pe* increases as xb/H decreases. The chosen values


−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

Pe

10−4

10−3

b/H = 10−2

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

Pe

Pe*

ξ

Fig. 8. Left: relative deviation of surface age 2 from the reference surface age (obtained for Pe¼ 0) as a function of the Peclet number for xb¼ xs¼ 5� 10�3 m,

H¼ 20 m and d¼ 0.6. Negative (positive) Peclet numbers indicate downward (upward) advection. Age 2 features a counter-intuitive behavior by showing a de-

crease when Pe< 0 and an increase when 0< Pe< Pe*, where Pe* is a threshold Peclet number. For Pe> Pe*, a decrease in the surface age 2 is observed. Right:

relative deviation of surface age 2 from the reference surface age (obtained for Pe¼ 0), for three different values of xb/H. As xb/H decreases, the critical Peclet

number Pe*> 0 at which the surface age starts decreasing reaches higher values.

of xb/H have a negligible impact on the deviations of the surfaceage 1, which are not shown here.

5. Conclusions

The use of unstructured meshes together with the finite el-ement method proves to be effective at representing the three-dimensional tidal flow past an island. In particular, highermesh resolution in the vicinity of the island allows for captur-ing smaller-scale processes, such as intense upwelling off theisland’s tips, which cannot be resolved with coarse meshes.

The theory of the age was used to design two sophisticateddiagnoses of vertical transport. Age 1 was defined as the timeelapsed since water particles left the seabed while age 2 wasdefined as the time required for water particles to travelfrom the seabed to the sea surface. A simple water-columnmodel was employed to compare both ages and to investigatethe boundary conditions. The logarithmic boundary layer,present for both ages, was successfully resolved numericallywith a modified finite element method.

The pattern of age 1 at the surface confirms the presence of in-tense upwelling off the island’s tips. Most importantly, the valueof age 1 at the surface, downstream of the island and shortly be-fore tide reversal, suggests that the water at the surface originatefrom the tips of the island and recirculate within the wake. Therole of the age in explaining this circulation pattern is crucialas the latter could not readily be proposed by a simple look atthe upwelling velocity. Furthermore, this flow description issomewhat in contradiction with the sketch proposed by Wolanskiand Hamner (1988), in which upwelling only takes place withinthe eddies. The results presented in this work motivate further re-search toward a better understanding of the three-dimensionalflow circulation around shallow-water islands.

The pattern pertaining to age 2 also exhibits upwelling off theisland’s tips. However, within the eddies where upwelling

velocity is the largest, the surface age increases. This counter-in-tuitive behavior was validated by a simplified water-columnmodel including both advection and diffusion. Nevertheless, atthis stage, the effect of advection upon age 2 remains physicallynot well understood. Drawing conclusions based on age 2 is notstraightfoward and, undoubtedly, requires additional effort.

The age-based diagnoses considered in this work rely upontracking water masses. In so doing, we do not take into ac-count the weight and buoyancy of the sediments. The nextversion of the model should also include a higher-order turbu-lence closure such as MelloreYamada level 2.5 and an ade-quate parameterization of the free shear layer. Anotherimprovement would consist in relaxing the slip condition onthe horizontal velocity at the bottom. As was done to computethe age, we could impose the horizontal velocity to vanish atthe bottom and use logarithmic shape functions within the bot-tommost element.

Finally, as we tend to use higher-resolution meshes, wecould resolve smaller-scale processes for which the hydro-static approximation might be invalidated. In particular, witha local mesh size of roughly 5 m, the free shear layer couldbe resolved with a non-hydrostatic model. Therefore, a morefaithful rendition of the physical processes around shallow-water islands could be achieved by using a non-hydrostaticmodel. Nevertheless, the size of the eddies and the fact thatthey are well represented by two-dimensional models indicatethat for those, the hydrostatic approximation remains valid.

Acknowledgements

Laurent White and Eric Deleersnijder are a Research fellowand a Research associate, respectively, with the Belgian Na-tional Fund for Scientific Research (FNRS). The present studywas carried out within the scope of the project ‘‘A second-gen-eration model of the ocean system’’, which is funded by the


Communaute Francaise de Belgique, as Actions de RechercheConcertees, under contract ARC 04/09-316. This work isa contribution to the construction of SLIM, the Second-gener-ation Louvain-la-Neuve Ice-ocean Model (http://www.clima-te.be/SLIM). The authors are indebted to Eric Wolanski forvery useful comments he provided during the preparation ofthis manuscript.

Appendix A. Exact solution to the water-column model

The exact solution to problem (13) and (14) is developedherein. The water concentration C1 for the first type of ageobeys the following nondimensional differential problem:8><>:

v

vz

�Kz

vC1

vz

�¼ 0

C1ðzbÞ ¼ 1 and

�Kz

vC1

vz

�z¼zs

¼ 0:

Note that Kz has the following form:

Kz ¼6

3� 2dzð1� dzÞ:

It is readily seen that the solution is C1(z)¼ 1. It follows thatthe age concentration a1 satisfies the problem:8><>:

v

vz

�Kz

va1

vz

�þ 1¼ 0

a1ðzbÞ ¼ 0 and

�Kz

va1

vz

�z¼zs

¼ 0:

After twice integrating and using the boundary conditions, weobtain:

a1ðzÞ ¼3� 2d

6

�ln

�z

zb

�þ�

1� d

d

�ln

�1� dz

1� dzb

�� ð1� zsÞln

�zð1� dzbÞzbð1� dzÞ

��;

which, using the fact that 1� zs � 1, can be approximatedby:

a1ðzÞx3� 2d

6

�ln

�z

zb

�þ�

1� d

d

�ln

�1� dz

1� dzb

��:

The water concentration C2 for the second type of ageobeys the following differential problem:(

v

vz

�Kz

vC2

vz

�¼ 0

C2ðzbÞ ¼ 1 and C2ðzsÞ ¼ 0:

Twice integrating and using the boundary conditions yields:

C2ðzÞ ¼lðzsÞ � lðzÞ

lðzsÞ;

where:

lðzÞ ¼Z z

zb

1

KzðtÞdt ¼ ð3� 2dÞ

6ln

�zð1� dzbÞzbð1� dzÞ

�:

Finally, the age concentration a2 is solution to:(v

vz

�Kz

va2

vz

�þC2ðzÞ ¼ 0

a2ðzbÞ ¼ 0 and a2ðzsÞ ¼ 0;

which yields:

a2ðzÞ ¼mðzsÞlðzÞ � lðzsÞmðzÞ

lðzsÞ;

where:

mðzÞ ¼Z z

zb

1

KzðxÞ

�Z x

zb

lðzsÞ � lðtÞlðzsÞ

dt

�dx: ðA1Þ

An expression for age 2 can be arrived at by computing the ra-tio of a2(z) to C2(z), namely:

a2ðzÞ ¼mðzsÞlðzÞ � lðzsÞmðzÞ

lðzsÞ � lðzÞ :

At z¼ zs, the above expression equals an indeterminate ratioof type 0/0. The limit when z goes to zs exists and can be com-puted by using L’Hospital’s rule. We obtain:

a2ðzsÞ¼ limz/zs

a2ðzÞ

¼ lðzsÞðzs� zbÞ � mðzsÞ �Z zs

zb

lðxÞdx:

A closed form may be obtained for m(z) by evaluating Eq.(A1):

mðzÞ¼ð3�2dÞ6

�zbln

�zbð1�dzÞzð1�dzbÞ

��1

dln

�1�dz

1�dzb

��ð3�2dÞ

6d

�

8>><>>:2Li2ð1�dzÞ�2Li2ð1�dzbÞþln

d2zzb

ln

�1�dz

1�dzb

�ln

�zsð1�dzbÞzbð1�dzsÞ

�9>>=>>;;

where Li2 is the dilogarithm function defined as:

Li2ðzÞ ¼ �Z z

0

lnð1� xÞx

dx¼XNk¼1

zk

k2:

Assuming that zb � 1 and zsx1, the above expression for m

can be simplified to:

mðzÞxð3� 2dÞ6d

26642Li2ð1� dzÞ �p2

3þ lnð1� dzÞln

�d2z

1� dzs

�lnðzbð1� dzsÞÞ

3775




where the following approximation was used: Li2ð1� dzbÞxLi2ð1Þ ¼ p2=6.

Appendix B. Supplementary data

Supplementary data associated with this article can befound, in the online version, at doi:10.1016/j.ecss.2006.07.014.

References

Akin, J.E., Tezduyar, T., Ungor, M., Mittal, S., 2003. Stabilization parameters

and Smagorinsky turbulence model. Journal of Applied Mechanics 70 (1),

1e9.

Alaee, M.J., Ivey, G., Pattiaratchi, C., 2004. Secondary circulation induced by

flow curvature and Coriolis effects around headlands and islands. Ocean

Dynamics 54, 27e38.

Black, K.P., Gay, S.L., 1987. Eddy formation in unsteady flows. Journal of

Geophysical Research 92 (C9), 9514e9522.

Coutis, P.F., Middleton, J.H., 1999. Flow-topography interaction in the vicinity

of an isolated, deep ocean island. Deep-Sea Research Part I Oceanographic

Research Papers 46, 1633e1652.

Coutis, P.F., Middleton, J.H., 2002. The physical and biological impact of

a small island wake in the deep ocean. Deep-Sea Research Part I Ocean-

ographic Research Papers 49, 1341e1361.

Cox, M.D., 1989. An idealized model of the world ocean. Part I: the global-

scale water masses. Journal of Physical Oceanography 19, 1730e1752.

Deleersnijder, E., 1989. Upwelling and upsloping in three-dimensional marine

models. Applied Mathematical Modelling 13, 462e467.

Deleersnijder, E., 1994. An analysis of the vertical velocity field computed by

a three-dimensional model in the region of the Bering Strait. Tellus 46A,

134e148.

Deleersnijder, E., Norro, A., Wolanski, E., 1992. A three-dimensional model

of the water circulation around an island in shallow water. Continental

Shelf Research 12 (7/8), 891e906.

Deleersnijder, E., Campin, J.-M., Delhez, E.J.M., 2001. The concept of age in

marine modelling: I. Theory and preliminary results. Journal of Marine

Systems 28, 229e267.

Delhez, E.J.M., Deleersnijder, E., 2002. The concept of age in marine model-

ling: II. Concentration distribution function in the English Channel and the

North Sea. Journal of Marine Systems 31, 279e297.

Delhez, E.J.M., Campin, J.-M., Hirst, A.C., Deleersnijder, E., 1999. Toward

a general theory of the age in ocean modelling. Ocean Modelling 1,

17e27.

Falconer, R.A., Wolanski, E., Mardapitta-Hadjipandeli, L., 1986. Modeling

tidal circulation in an island’s wake. Journal of Waterway, Port, Coastal

and Ocean Engineering 112, 234e254.

Fischer, H.B., List, E.Y., Koh, R.C.Y., Imberger, J., Brooks, N.H., 1979. Mix-

ing in Inland and Coastal Waters. Academic Press, New York.

Goosse, H., Campin, J.-M., Tartinville, B., 2001. The sources of Antarctic bot-

tom water in a global ice-ocean model. Ocean Modelling 3, 51e65.

Hamner, W.M., Hauri, I.R., 1981. Effects of island mass: water flow and

plankton pattern around a reef in the Great Barrier Reef lagoon, Australia.

Limnology and Oceanography 26 (6), 1084e1102.

Hanert, E., Le Roux, D.Y., Legat, V., Deleersnijder, E., 2005a. Advection

schemes for unstructured grid ocean modelling. Ocean Modelling 7,

39e58.

Hanert, E., Le Roux, D.Y., Legat, V., Deleersnijder, E., 2005b. An efficient Eu-

lerian finite element method for the shallow water equations. Ocean Mod-

elling 10, 115e136.

Hanert, E., Deleersnijder, E., Legat, V., 2006. An adaptive finite element water

column model using the MelloreYamada level 2.5 turbulence closure

scheme. Ocean Modelling 12, 205e223.

Hirst, A.C., 1999. Determination of water component age in ocean models: ap-

plication to the fate of North Atlantic Deep Water. Ocean Modelling 1,

81e94.

Ingram, R.G., Chu, V.H., 1987. Flow around islands in Ruppert Bay: an inves-

tigation of the bottom friction effect. Journal of Geophysical Research 92

(C13), 14,521e14,533.

Le Roux, D.Y., Lin, C.A., Staniforth, A., 2000. A semi-implicit semi-lagrangian

finite element shallow-water ocean model. Monthly Weather Review 128,

1384e1401.

Lynch, D.R., Ip, J.T.C., Naimie, C.E., Werner, F.E., 1996. Comprehensive

coastal circulation model with application to the Gulf of Maine. Continen-

tal Shelf Research 16, 875e906.

Moes, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack

growth without remeshing. International Journal for Numerical Methods in

Engineering 46, 131e150.

Munk, W.H., Anderson, E.R., 1948. Notes on a theory of the thermocline.

Journal of Marine Research 3, 276e295.

Pattiaratchi, C., James, A., Collins, M., 1986. Island wakes and headland

eddies: a comparison between remotely sensed data and laboratory exper-

iments. Journal of Geophysical Research 92 (C1), 783e794.

Pietrzak, J., Deleersnijder, E., Schroter, J., 2005. The second international

workshop on unstructured mesh numerical modelling of coastal, shelf

and ocean flows. Ocean Modelling 10, 1e252 (Special issue).

Smagorinsky, J., 1963. General circulation experiments with the primitive

equations. 1 The basic experiment. Monthly Weather Review 91 (5),

99e165.

Tomczak, M., 1988. Island wakes in deep and shallow water. Journal of Geo-

physical Research 93 (C5), 5153e5154.

Walters, R.A., 1992. A three-dimensional, finite element model for coastal and

estuarine circulation. Continental Shelf Research 25, 83e102.

White, L., Beckers, J.-M., Deleernijder, E., Legat, V., 2006. Comparison be-

tween free-surface and rigid-lid finite element models of barotropic insta-

bilities. Ocean Dynamics 56, 86e103. doi:10.1007/s10236-006-0059-0.

White, L., Deleernijder, E., Legat, V. Three-dimensional finite element marine

modeling, in preparation.

Wolanski, E., 1994. Physical Oceanographic Processes of the Great Barrier

Reef. CRC Marine Science Series. CRC Press.

Wolanski, E., Hamner, W.M., 1988. Topographically controlled fronts in the

ocean and their biological influence. Science 241 (4862), 177e181.

Wolanski, E., Imberger, J., Heron, M.L., 1984. Island wakes in shallow coastal

waters. Journal of Geophysical Research 89 (C6), 10,553e10,569.

Wolanski, E., Drew, E., Abel, K.M., O’Brien, J., 1988. Tidal jets, nutrient up-

welling and their influence on the productivity of the alga Halimeda in the

Ribbon Reefs, Great Barrier Reef. Estuarine, Coastal and Shelf Science

26 (2), 169e201.

Wolanski, E., Asaeda, T., Tanaka, A., Deleersnijder, E., 1996. Three-dimensional

island wakes in the field, laboratory experiments and numerical models.

Continental Shelf Research 16 (11), 1437e1452.

Wolanski, E., Brinkman, R., Spagnol, S., McAllister, F., Steinberg, C.,

Skirving, W., Deleersnijder, E., 2003. Merging scales in models of water

circulation: perspectives from the Great Barrier Reef. In: Lakhan, V.C.

(Ed.), Advances in Coastal Modeling. Elsevier Oceanography Series,

vol. 67. Elsevier Science, pp. 411e429 (Chapter 15).

http://dx.doi.org/doi:10.1016/j.ecss.2006.07.014

http://dx.doi.org/doi:10.1016/j.ecss.2006.07.014

Documents

Diagnoses of vertical transport in a three-dimensional finite element