Comparative study of different advective schemes:Application to the MECCA model
H. SMAOUI∗ and B. RADI∗∗E.N.S.A.M.-Meknès, Marjane II, B.P. 4024 Béni M’hamed, Meknès, Morocco
Received 15 February 2001; accepted in revised form 23 January 2002
Abstract. A 3-D version of the MECCA model (Model of Estuarine and Coastal Circulation As-sessment) is used to simulate the dynamics of the Eastern part of the English Channel. This areais characterized by a strong tidal turbulent regime and a frontal zone identified near the Frenchcoast by a low-salinity band. The model uses the upwind scheme to approximate the advectiveterms. Results show that the model overestimates the band width of the frontal zone and that thisanomaly is definitely caused by the numerical diffusion introduced by the so-called upwind scheme(first-order approximation). In this paper, we study the flux-limiter schemes as an alternative tothe upwind method in order to reduce this non-physical diffusion. To illustrate the improvementsprovided by this type of schemes, a comparison in 2-D schematic cases is made between the upwindand centered scheme with more recent higher-order schemes combined with limiter namely Minmod,Superbee, Van Leer and Monotonized Centered (MC) (called also MUSCL scheme). Respecting theCFL condition, our numerical simulations show that the flux-limiter schemes reduce the numericaldiffusion and eliminate the oscillations caused by the non-limited higher-order schemes. For theschematic and realistic cases, the Superbee limiter is a good compromise between shape preservationand computational cost.
Many phenomena can be modeled by a single advection-diffusion equation: waterpollution, contaminant diffusion, etc. So, the numerical study of this equation isvery important. We present here some flux-limiter schemes, which satisfy manyof the requirements of a good advective scheme. They are Total Variation Dimin-ishing (TVD) schemes, which are mass conserving and less diffusive than simplerschemes.
It is known that the scalar transport is the combination of different physicalprocesses: (1) advection by the statistical mean of the velocity which transports thescalar quantity in the direction of the flow without enlarging its initial distribution,(2) diffusion due to the turbulent velocity fluctuation which enlarges and smoothesthe scalar distribution. The numerical approximation of these processes requiresimportant choices and compromises.∗Corresponding author, E-mail: [email protected]∗∗Associating address: LME, Universite Chouaib Doukali, Eljadida, Morocco.
362 H. SMAOUI AND B. RADI
One example of such compromise is the numerical treatment of the advectiveterms in the transport equation. These compromises are necessary, to minimizeboth artificial numerical diffusion and dispersion. The numerical diffusion mayseverely damp the flow and produce inaccurate results, whereas artificial numericaldispersion may introduce non-physical oscillations called wiggles.
Traditionally, large-scale oceanographic and meteorological models adopt theupwind central-difference scheme to discretize the advective terms present in theflow equation and passive scalar equations, as done for example in the PrincetonOcean Model (POM) [1] and the Modular Ocean Model (MOM) [2]. However,these schemes introduce numerical instability or hyper-numerical diffusion. Toovercome these problems, Gerdes [3] suggested to use the flux corrected transport(FCT) method [4] in the MOM code. This method uses a low-order monotonicscheme in the regions with strong gradients and an higher-order scheme elsewhere.In addition, Hecht et al. [5] have carried out a detailed comparison of the advectionschemes in the study of a two dimensional passive tracer within the Stommel gyre.The main purpose of this paper is to improve the advection of the variables withsharp gradients in numerical ocean modeling. Such situation occurs in frontal re-gions, especially in estuarine flows situations, such as in the Eastern part of theEnglish Channel [6, 7].
The outline of this paper is as follows. In Section 2, we describe succinctly theflux-limiter techniques for one and two dimensional purely advective equationsand briefly review some notions of TVD schemes. In Section 3, the compari-son between first-order (upwind) and second-order (centered) schemes with fourflux-limiter schemes is presented for 2-D analytical and schematic cases. The im-plementation of the flux-limiter schemes is performed for realistic simulationsof the English Channel general circulation. The relevance of these results to thecomparison procedures is discussed. The numerical results are summarized anddiscussed in Section 4.
2. Transport Equation and their Discretization
We treat the transport equation with a purely advected concentration C. This equa-tion is given by
∂C
∂t+ ∇.(�VC) = 0, (1)
where �V is the velocity vector. In the 2D case, Equation (1) becomes:
∂C
∂t+ ∂(uC)
∂x+ ∂(vC)
∂y= 0, (2)
where u and v are the velocity components in x– and y–directions, respectively.The time stepping of Equation (2) in spatially discretized form gives:
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 363
Cn+1i,j = Cn
i,j − αni− 1
2 ,j
(Cni,j − Cn
i−1,j
) + βn
i+ 12 ,j
(Cni+1,j − Cn
i,j
)−λn
i,j− 12
(Cni,j − Cn
i,j−1
) + µn
i,j+ 12
(Cni,j+1 − Cn
i,j
), (3)
where
αni− 1
2 ,j= α
(uni−2,j , u
ni−1,j , u
ni,j , u
ni+1,j
)βn
i+ 12 ,j
= β(uni−1,j , u
ni,j , u
ni+1,j , u
ni+2,j
)λni,j− 1
2= λ
(vni,j−2, v
ni,j−1, v
ni,j , v
ni,j+1
)µn
i,j+ 12
= µ(vni,j−1, v
ni,j , v
ni,j+1, v
ni,j+2
). (4)
Scheme (3) is monotonic if [8]:
αni− 1
2 ,j≥ 0, βn
i+ 12 ,j
≥ 0, λni,j− 1
2≥ 0, µn
i,j+ 12
≥ 0. (5)
Spekreijse [8] has also shown that a 2D explicit monotonic scheme is not neces-sarily TVD. However, numerical experience [8] has shown that 2D schemes usingthe splitting technique (1D second-order accurate TVD scheme in each directionperpendicular to the cell’s face) give oscillation-free accurate results.
For algorithmic simplicity, we use the 2D extension of flux-limiting schemes assplitting mode in each direction. The application of this technique to Equation (2)with the cell-centered finite volume discretization gives the semi-discrete equation:
dCi
dt�x�y = − [uC]
i+ 12 ,j
i− 12 ,j
− [vC]i,j+ 1
2
i,j− 12. (6)
The interface values Cn
i+ 12 ,j
and Cn
i,j+ 12
for an arbitrary velocity field are given
as follows:
Cn
i+ 12 ,j
=
Cni,j + 1
2
(Cni+1,j − Cn
i,j
)�
(θ+i+ 1
2 ,j
)if ui+ 1
2 ,j≥ 0
Cni+1,j − 1
2
(Cni+1,j − Cn
i,j
)�
(θ−i+ 1
2 ,j
)if ui+ 1
2 ,j< 0
(7)
Cn
i,j+ 12
=
Cni,j + 1
2
(Cni,j+1 − Cn
i,j
)�
(θ+i,j+ 1
2
)if vi,j+ 1
2≥ 0
Cni,j+1 − 1
2
(Cni,j+1 − Cn
i,j
)�
(θ−i,j+ 1
2
)if vi,j+ 1
2< 0
(8)
364 H. SMAOUI AND B. RADI
Figure 1. Total Variation Diminishing region and the centered differences flux.
with
θ+i+ 1
2 ,j= Cn
i,j − Cni−1,j
Cni+1,j − Cn
i,j
, θ−i+ 1
2 ,j= Cn
i+2,j − Cni+1,j
Cni+1,j − Cn
i,j
θ+i,j+ 1
2= Cn
i,j − Cni,j−1
Cni,j+1 − Cn
i,j
, θ−i,j+ 1
2= Cn
i,j+2 − Cni,j+1
Cni,j+1 − Cn
i,j
,
where � is the limiter function.The expressions for Cn
i− 12 ,j
and Cn
i,j− 12
are be obtained simply by substituating
respectively the index i by i− 1 in formula (7) and the index j by j − 1 in formula(8).
If the limiter function � satisfies the following conditions:
0 ≤ �(r)
r≤ 2 and 0 ≤ �(r) ≤ 2 (9)
then the scheme, given by (7) and (8), is monotonic.The region defined by (9) is shown in Figure 1 along with the limiter cor-
responding to the centered differences. Since this scheme is known to producespurious wiggles in the solution with strong gradient variations, it is not surprisingthat this scheme is not uniform within the TVD region. Among the proposals,which have been discussed by Sweby [9] and Leveque [10], the limiters used inour study satisfy the (9) constraints:
We consider the following limiter functions:
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 365
One can find more details about these advection schemes in the book by Durran[11]. In our case, the limiter function � verifies �(1) = 1.
3. Numerical Tests and Results
3.1. TEST PROBLEM
The test problem concerns the advection of a cone profile in a uniform velocityfield. This problem is governed by the advective transport equation:
∂C
∂t+ u
∂C
∂x+ v
∂C
∂y= 0, (x, y) ∈ [0, 1] × [0, 1] (10)
in which we take u = v = 0.5.The following initial conditions are selected:
C(x, y, 0) = C0(x, y), (11)
where C0 is given by:
C0(x, y) ={
4(R−rR) if r ≤ R
0 otherwise(12)
and r = √(x − xc)
2 + (y − yc)2 with (xc, yc) and R being respectively the cen-
ter’s coordinates and the radius of the cone base.The analytic solution of this problem is well known: As time progresses, the ini-
tial cone distribution is transported in the direction of the flow without deformation.So, the exact solution of this problem is given by:
C(x, y, t) = C0(x − ut, y − vt), (13)
where (x, y) ∈ [0, 1]×[0, 1] and t ∈ [0, T ] with T is the duration of the simulation.To quantify the order of each method, computations have been performed on
25 × 25, 50 × 50 and 100 × 100 grids. In order to estimate the error solution of thedifferent schemes under investigation (upwind, centred, Minmod, Superbee, VanLeer and MC), we define the discrete norms L1 and L2 as:
‖�C‖1 =∑Nx
i=1
∑Ny
j=1 | Ci,j,exact − Ci,j,numerical |N
(14)
‖�C‖2 =√∑Nx
i=1
∑Ny
j=1(Ci,j,exact − Ci,j,numerical)2
N, (15)
366 H. SMAOUI AND B. RADI
where Nx and Ny are the number of grid points in x– and y–direction, respectively,and N = Nx × Ny is thus the total number of grid points in the computationaldomain.
Furthermore, monotonicity and conservation of the schemes are examined re-spectively by:
Cmin = min1≤i,j≤N
Ci,j,numerical, Cmax = max1≤i,j≤N
Ci,j,numerical (16)
and
|1 − rmass| =∣∣∣∣∣1 −
∑Nx
i=1
∑Ny
j=1 Ci,j,numerical∑Nx
i=1
∑Ny
j=1 Ci,j,exact
∣∣∣∣∣ . (17)
More details on these quantities and the motivation for their use can be found inVreugdenhil and Koren [12]. The measurements of these quantities for all investi-gated schemes are summarized in Table I.
Defining h as the maximal grid width, one can remark that the solution errorsbehave O(h2) with respect to ‖�C‖2 and O(h) with respect to ‖�C‖1, except forthe centered scheme. Among the monotonic schemes, MUSCL yields the smallesterrors on all three grids, according to both L2 and L1 norms. But, Superbee givesthe smallest mass errors, on three grids and converges faster than the Van Leer andMUSCL schemes (see Table II).
3.2. SCHEMATIC CASE
In Smaoui [13], numerical results were presented for one and two spatial dimen-sions using a non-limited Lax-Wendroff scheme (second-order in both time andspace) to evaluate the numerical flux denoted Fn
i± 12. In one spatial dimension,
three different types of initial conditions were used (results not presented in thisstudy): (1) steep gradient, (2) triangle form and (3) smooth gaussian form. Thesesimulation types permit performance comparison among schemes (upwind, Lax-Wendroff, Minmod, Superbee, Van Leer and MC). It is seen that Superbee givesremarkable solutions, followed by MC, Van Leer and Minmod limiters.
Our second test examines the schematic case of a tidal sea with an entry offresh water through the left boundary as shown on Figure 2. The choice of thissituation is motivated by a similar phenomenon observed along the French coast inthe eastern part of the English Channel [6]. The flux-limiter scheme of this studyis implemented in our 3D version of the MECCA model.
For the schematic case, the flow is computed by solving the linearized shallow-water equation coupled with a purely advective tracer equation (ex. salinity) inorder to isolate numerical diffusion. These equations are solved numerically byfinite differences in a semi-enclosed flat-bottom basin with a depth of 10 m (Fig-ure 2). The discretization is �x = �y = 250 m in space and �t = 60 s in time, in
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 367
Table I. Numerical results for different schemes on different grids.
25 × 25 50 × 50 100 × 100
Cmin 0 0 0
Cmax 1.8421E+00 2.4460E+00 2.9142E+00
Upwind | 1 − r | 1.1080E-01 5.2682E-02 2.5455E-02
‖.‖2 1.3359E-02 4.2801E-03 1.2301E-03
‖.‖1 1.8280E-01 1.1331E-01 6.2031E-02
Cmin -1.6760E-01 -1.6109E-01 -4.0701E+01
Cmax 3.3695E+00 3.3208E+00 4.2310E+01
Centred | 1 − r | 3.3683E-03 2.1533E-03 1.6001E-03
‖.‖2 4.0937E-03 1.3965E-03 6.2200E-02
‖.‖1 5.8254E-02 3.9094E-02 2.4417E+00
Cmin 0 0 0
Cmax 2.5264E+00 3.1382E+00 3.5204E+00
Minmod | 1 − r | 1.7228E-02 1.0084E-02 4.3235E-03
‖.‖2 6.3314E-03 1.3582E-03 3.3944E-04
‖.‖1 8.3041E-02 3.3083E-02 1.4355E-02
Cmin 0 0 0
Cmax 2.9083E+00 3.4474E+00 3.7422E+00
Superbee | 1 − r | 2.3211E-02 2.3853E-03 1.0479E-03
‖.‖2 3.6500E-03 1.0938E-03 5.2414E-04
‖.‖1 4.2918E-02 1.9454E-02 1.6188E-02
Cmin 0 0 0
Cmax 2.7268E+00 3.3080E+00 3.6581E+00
VanLeer | 1 − r | 1.5285E-03 4.5751E-03 2.8459E-03
‖.‖2 4.8020E-03 9.2212E-04 3.1270E-04
‖.‖1 5.8497E-02 2.0041E-02 1.0671E-02
Cmin 0 0 0
Cmax 2.8270E+00 3.3895E+00 3.7148E+00
MUSCL | 1 − r | 4.9484E-03 3.6225E-03 2.6142E-03
‖.‖2 4.3090E-03 7.9762E-04 3.1903E-04
‖.‖1 5.2877E-02 1.6631E-02 1.0083E-02
368 H. SMAOUI AND B. RADI
Table II. CPU time for different schemes.
upwind centered Minmod Superbee Van Leer MUSCL
25 × 25 0.86 0.98 1.33 1.29 1.37 1.3
50 × 50 3.38 3.97 5.58 5.21 5.73 5.23
100 × 100 13.6 16.38 23.52 21.23 24.05 21.3
Figure 2. Basin geometry for the schematic case.
order to ensure stability of the calculations. Along the open boundary (x = 15 km),we impose a 12-h tidal wave of 1 m amplitude and with 625 m3/s of flow rate atthe river mouth (x = 0). The salinity concentration is assumed to be uniformlydistributed (at 35 ppt) except at the inflow region where it is lower (Sin = 10 ppt).The goal of this schematic case is only the study of the influence and sensitivity ofthe flux-limiter scheme on a dynamical frontal zone.
Figure 3 depicts the simulated field velocity at 27 hours representing the mini-mum velocity at the entrance of basin. This instantantenous result output is chosenin order to examine the influence of the flow-rate intensity on the advection of freshwater. Results using the various discretization schemes are presented in Figure 4,which shows the horizontal salinity distribution, and Figure 5, which shows salinitytransects.
Figures 4a and 5a show the computed solution at 27 h (maximum velocity atx = 15 km) using the upwind scheme. This solution exhibits a monotonic solutionand damps the fresh water and salt water interface. This result is shown clearlyin Figure 5a which exhibits one transect (y = 3.5�y) of the numerical solution.Compared to the upwind scheme, the centered-difference scheme (Figures 4b and5b) limits the numerical diffusion but exhibits some undershoots, so that the gra-dient is sharper but monotonicity is not preserved. The limiter schemes eliminatethese undershoots completely and simultaneously limit the numerical diffusion in
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 369
Figure 3. Velocity field at 27 h.
comparison with the upwind scheme. Figure pairs 4e–5e and 4f–5f show that theMC and Van Leer limiters achieve the same level of performance. These schemesproduce a negligible numerical diffusion as compared with the Minmod limiter(see Figures 4c and 5c).
The Superbee limiter (Figures 4d and 5d) gives the best results among all:limited numerical diffusion and excellent mass conservation (see also [13] for theone-dimensional case). Among the higher-order schemes and for this schematiccase, the Superbee limiter seems to be a good compromise between accuracy andcomputational efficiency.
3.3. REALISTIC CASE
Smaoui [13] remarked that the numerical diffusion (implicit mixing) of certainadvection schemes depends strongly on the velocity field. This is most pronouncedwith the upwind scheme. In the General Ocean Circulation Model (GOCM), thevelocity field depends on the individual configuration of the model, and we mayexpect large differences between a flat-bottom box ocean model (schematic case)and one with realistic bottom topography. In this subsection, we implement theflux-limiter scheme in a GOCM and we apply our numerical model to the easternpart of the English Channel.
3.3.1. Description of the MECCA Model
The MECCA model, initially developed by Hess [14], uses finite-difference ap-proximations to solve the discretized 3D equations governing conservation of mo-mentum, mass, heat and salt on a beta plane, subject to the hydrostatic and Boussi-nesq approximations. It is able to simulate time-varying water currents, salini-ties and temperatures in hallow-water domains at time scales ranging from a fewminutes to several months, and space scales stretching from a few kilometers to a
370 H. SMAOUI AND B. RADI
Figure 4. Salinity distribution at 27 h.
few hundred kilometers. The model is designed to simulate circulation driven bytides, wind, water density gradients as well as atmospheric pressure gradients.
3.3.2. Governing Equations
In the orthogonal cartesian coordinate system with horizontal coordinates (x, y)oriented eastward and northward, respectively, and the vertical coordinate z in-
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 371
Figure 5. Salinity transect at y = 0.8 km at 27 h.
creasing upward from the sea surface, the expressions of the linearized equationsof motion are: in the x− direction
∂u
∂t+ ∂uu
∂x+ ∂uv
∂y+ ∂uw
∂z− f v = − 1
ρ0
∂p
∂x+ ∂
∂z
(Av
∂u
∂z
)+ Fu (18)
372 H. SMAOUI AND B. RADI
in the y− direction
∂v
∂t+ ∂vu
∂x+ ∂vv
∂y+ ∂vw
∂z+ f u = − 1
ρ0
∂p
∂y+ ∂
∂z
(Av
∂v
∂z
)+ Fv (19)
and in the z− direction, the hydrostatic balance is
ρg = −∂p
∂z. (20)
The conservation equation for temperature T is:
∂T
∂t+ �V.∇T = ∂
∂z
(Dv
∂T
∂z
)+DT + RT , (21)
where R represents sources and sinks of heat, and a similar equation holds for theconservation of salt S, with diffusion term DS and source/sink term RS . The fluiddensity ρ is determined by a general equation of state:
ρ = ρ0[1 + Fρ(S, T )], (22)
where Fρ is an empirical function used to calculate the perturbation density fromthe temperature and salinity. Finally, the conservation of mass is expressed by theincompressible continuity equation:
∂u
∂x+ ∂v
∂y+ ∂w
∂z= 0. (23)
In these equations, (u, v, w) are the components of the three-dimensional velocityvector �V with (x ,y, z) components in the east, north and upward directions; f isthe Coriolis parameter (f = 2ω sin θ , where ω is the angular rotation rate of theearth and θ is the latitude.
3.3.3. Mixing and Diffusion
In the preceding momentum, temperature and salinity conservation equations, theterms Fu, Fv , DT,S represent turbulent horizontal mixing and diffusion. In theMECCA model, these terms are computed as in the POM model [1]:
Fu = ∂
∂x
[2Ah
∂u
∂x
]+ ∂
∂y
[2Ah
(∂u
∂y+ ∂v
∂x
)](24)
Fv = ∂
∂y
[2Ah
∂v
∂y
]+ ∂
∂x
[2Ah
(∂u
∂y+ ∂v
∂x
)](25)
DT,S = ∂
∂x
[Dh
∂(T , S)
∂x
]+ ∂
∂y
[Dh
∂(T , S)
∂y
], (26)
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 373
Figure 6. Bottom topography of the eastern portion of the English Channel.
in which the horizontal turbulent viscosity Ah and diffusivity Dh depend on thelocal velocity shear, deformation rate and the horizontal grid spacing � accordingto the Smagorinsky [15] formula:
Ah = Ah0 + β�2
[2
(∂u
∂x
)2
+ 2
(∂v
∂y
)2
+(∂u
∂y+ ∂v
∂x
)2] 1
2
(27)
Dh = Ah, (28)
where β is a constant with assigned value of 10−2 and Ah0 is a small backgroundhorizontal viscosity of 10 m2/s.The instantaneous vertical turbulent viscosity Av and diffusion coefficients Dv arecalculated using a one-equation turbulence model given in the next subsection.
3.3.4. Previous Results from the MECCA Model
The eastern part of the English channel is characterized by an irregular bathymetry(Figure 6) and a strong tidal turbulent regime (Figure 7). Over the years, manystudies (observations, simulations, etc.) of this region have shown that the tidalcurrents are greatly influenced by the bottom irregularities and are not symmetricaldue to the presence of a strong advective term, which generates the M4 harmonic.More detailed studies of the hydrodynamics of the English Channel can be foundin Le Provost and Fornerino [16], Smaoui [7] and Nguyen and Ouahsine [17].
Figure 8 shows the field velocity every two hours during one tidal cycle whenthe forcing consists mainly in M2 and S2 waves and without wind forcing. This
374 H. SMAOUI AND B. RADI
Figure 7. Streamlines of the tidal residual transport.
figure gives some general idea of the turbulent regime in the region. The MECCAmodel gives good results concerning the wave propagation (mean velocity andsea elevation) compared to the available data or other existing models. However,simulations of the dispersion of the fresh water plume show some difference be-tween the available observations [6]: MECCA overestimates the width of the freshwater plume. This difference has been attributed to the turbulence-closure schemeinitially used in MECCA model (zero-equation model), and to the implicit mixingcaused by the spatial discretization of the advective term.
The comparison of the range of turbulence-closure schemes used in three-dimen-sional tidal models by Davies et al. [18] has revealed that a fine grid in the verticalis absolutely necessary to ensure that the additional physics accompanying thehigher-order turbulence-closure schemes (the q2 − q2l model, for example) arenot spoiled by artificial smoothing or spurious grid-scale oscillations due to coarsegrids or poor numerics. Naturally, fine grids and high-order turbulence closure willgive rise to significant computational effort in three-dimensional calculations. Analternative to the use of two-equation closure schemes is a prognostic equation forturbulence, with the mixing length determined from the turbulence or specified inan adiagnostic manner.
For the turbulence-closure scheme, we have introduced a one-equation modelthat consists in solving an equation for the turbulent kinetic energy, K, and a semi-empirical expression for the mixing length. The selection of this turbulent schemeis justified by (1) the shallow depth in the simulated region, which makes theboundary layer thickness occupy nearly the totality of the water column and (2)our modest means of computation.
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 375
This kind of turbulence parametrization is formulated as follows:
∂K
∂t+ �V.∇K = ∂
∂z
(Av
σK
∂K
∂z
)+ P +G+DK − ε, (29)
where P is the shear production of turbulent kinetic energy, G is the rate of con-version of turbulent kinetic energy into potential energy, which appears as thedissipation of turbulent kinetic energy by buoyancy, DK is the horizontal diffusionof turbulent kinetic energy and ε is the turbulent energy dissipation rate.
The general expression for the mixing length 3 is a function of the turbulentkinetic energy and depends on the mixing lengths 30 and 3m [19], where the mixinglengths 30 and 3m are given by:
30 = κz
1 + κz/3m(30)
and
3m = γ0
∫ surfacebed z
√Kdz∫ surface
bed
√Kdz
(31)
in which κ = 0.4 is the Von Kárman constant and γ0 is a constant in the range 0.1to 0.4.
In stratified flow configurations, the MECCA model uses the corrected mixinglength 3 proposed by Munk and Anderson [20]:
3 = 30 (1 + 10Ri)−0.5 , (32)
where Ri denotes the Richardson number defined as:
Ri = −g
ρ
∂ρ
∂z
[(∂u
∂z
)2
+(∂v
∂z
)2]−1
. (33)
The eddy viscosity Av and diffusivity Dv are computed as functions of theturbulent kinetic energy and the mixing length according to:
Av = CµK1.53 , Dv = Av
σs, (34)
where σs is the Prandtl number.Simulations with this one-equation closure scheme show some improvement by
reducing the width of the fresh water plume, but not sufficiently to reproduce theobservations [6]. The remaining cause behind the disagreement, namely implicitmixing, is the motivation of this study.
376 H. SMAOUI AND B. RADI
Figure 8. Velocity field during one tidal cycle.
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 377
3.3.5. Implementation in the MECCA Model
The MECCA code solves the vertically averaged momentum equations to obtainthe barotropic velocity and sea level height, using an alternating-direction im-plicit (ADI) scheme [21]. The residual momentum equations are then solved forthe internal mode, using an implicit scheme to determine the horizontal velocitycomponents at each level of the grid cell. The vertical velocity at each level isdetermined explicitly from the continuity equation. A staggered placement of vari-ables within the grid cell is used, with the horizontal velocity components specifiedon the orthogonal sides of the cell and the sea level height and vertical velocityspecified at the cell center. The mode-splitting technique allows the internal modeto be calculated on a time step longer than that used for the free surface deformationand the vertically integrated velocities.
Initially, in the MECCA model, scalars and momentum diffusivities are func-tions of local velocity gradients and may therefore vary over time and space. (Suchturbulence-closure scheme is known as a zero-equation model.) In addition, ver-tical diffusivities are reduced in the presence of vertical density gradients as afunction of the local Richardson number, in order to take into account the possiblestratification effects.
The MECCA model uses the upwind scheme to solve the advection part of theequations for any scalar (temperature and salinity). This scheme conserves totalmass and has first-order accuracy. It is free from dispersive effects (oscillations).Its main disadvantage is the large amount of implicit diffusion (Figure 9e). We haveimplemented the flux-limiter schemes discussed above in the MECCA model andnow present the findings of our simulations.
In Figure 9e, we present the salinity concentrations computed by the variousschemes at 3 hours after high water at Calais. In parallel, we recall the mainobservational results [6], namely:
(1) the fresh water plume remains near the French coast from the Bay of Sommeto Cap Gris-Nez,
(2) the North Sea water is not affected by the fresh water plume from the SommeRiver,
(3) A discontinuity exists between the Seine River waters and those of the Sommeand a local discontinuity near Authie Bay.
Examination of Figure 9 reveals that the upwind scheme (Figure 9e) violatesthe properties (1) and (2), as well as (3) in part. The computations obtained bythe characteristic method (Figure 9a) shows an important implicit diffusion nearCap d’Antifer, a region characterized by strong velocities (cape effects), but theaforementioned properties are respected for the most part.
The Minmod limiter scheme (Figure 9b) yields hybrid results situated betweenthose obtained with the characteristic method and the upwind scheme: It has a
378 H. SMAOUI AND B. RADI
lesser numerical diffusion than the upwind scheme at Cap Gris-Nez and Caped’Antifer. The remaining limiter schemes (Figures 9c, 9d and 9f) resproduce re-markably well the three properties found from the mobservations. Performance ofthe MUSCL limiter (Figure 9f) could be situated between those of the Van Leer(Figure 9c) and the Superbee (Figure 9d) limiters. Concerning the width of thefresh water plume along the French coast, the Superbee scheme gives the best resultand holding its place of superior performance that we noted in the schematic case.The fresh water plume can be very well seen in the numerical simulation aroundCap d’Antifer (Figure 9d).
4. Conclusion and Discussion
Several alternative advection schemes were investigated and tested in idealizedtwo-dimensional flow situations. The second-order scheme was compared withthe upwind scheme and several higher-order TDV flux-limiter schemes. Theseschemes have the ability to limit the numerical diffusion and reduces the numericaldispersion introduced by the third-derivative term due to truncation errors. TheLax–Wendroff or leap-frog scheme has become a trusted scheme in numericalocean modeling because it is easy to implement, has no numerical damping andconserves quadratic quantities, but it is dispersive. The usual technique to suppressthe under- and over-shooting of the central scheme is to use higher-values of theSmagorinski coefficient in the formulation of the horizontal diffusivity. However,this leads to large damping of the solution and smearing of interfaces. It alsohighlights the subjective choice of the coefficients, which are justified more bythe need for numerical stability than by the physical processes. In response tothese shortcomings, a large number of advective schemes have been developedover the past few decades, including the flux-corrected transport (FCT) scheme,the essentially non-oscillatory (ENO) scheme and various anti-diffusion schemes.
The classical tests illustrate clearly the problem encountered during the numer-ical modeling of sharp discontinuities. This is a particular problem when dealingwith sharp gradients and it highlights the need for both high resolution and higher-order schemes. The tests presented here demonstrate that the higher-order TVDlimited schemes can easily be implemented. And, the one recommended here is theSuperbee limiter [22]. The Superbee scheme is selected because the detailed one-and two-dimensional cases in which it was tested show its superior performanceover a number of other limiters commonly referenced in the literature.
The one-dimensional TVD schemes are monotonic if the CFL criterion is sat-isfied. Multidimensional flux-limiters schemes have been formulated based on theone-dimensional flux limiters by using the splitting technique. They were foundto be suitable for ocean modeling. They are easy to implement and computation-ally efficient, resolving well discontinuities with a minimal amount of numericaldiffusion. Our ongoing research involves the application of these schemes to the
COMPARATIVE STUDY OF DIFFERENT ADVECTIVE SCHEMES 379
Figure 9. Salinity field at 3 h after high water at Calais.
380 H. SMAOUI AND B. RADI
advection of momentum with a semi-implicit time scheme in order to reduce thecomputational cost.
Acknowledgements
The work presented herein was performed in the framework of the PROMISEprogramme (PRe-Operational Model In Seas of Europe) funded by the EuropeanCommunity under a MAST III contract (MAS3–CT950025). Thanks to Dr K. W.Hess of N.O.A.A. for a helpful discussion about the MECCA model.
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