Ocean Dynamics (2010) 60:65–79DOI 10.1007/s10236-009-0243-0

On the parameters of absorbing layersfor shallow water models

Axel Modave · Éric Deleersnijder · Éric J. M. Delhez

Received: 17 June 2009 / Accepted: 5 November 2009 / Published online: 18 November 2009© Springer-Verlag 2009

Abstract Absorbing/sponge layers used as boundaryconditions for ocean/marine models are examined inthe context of the shallow water equations with theaim to minimize the reflection of outgoing waves atthe boundary of the computational domain. The op-timization of the absorption coefficient is not an is-sue in continuous models, for the reflection coefficientof outgoing waves can then be made as small as weplease by increasing the absorption coefficient. Theoptimization of the parameters of absorbing layers istherefore a purely discrete problem. A balance mustbe found between the efficient damping of outgoingwaves and the limited spatial resolution with which theresulting spatial gradients must be described. Using aone-dimensional model as a test case, the performancesof various spatial distributions of the absorption coef-ficient are compared. Two shifted hyperbolic distrib-utions of the absorption coefficient are derived fromtheoretical considerations for a pure propagative and apure advective problems. These distribution show goodperformances. Their free parameter has a well-definedinterpretation and can therefore be determined on aphysical basis. The properties of the two shifted hyper-bolas are illustrated using the classical two-dimensional

Responsible Editor: John Wilkin

A. Modave (B) · É. J. M. DelhezMARE - Modélisation et Méthodes Mathématiques,Université de Liège, Sart-Tilman B37, 4000 Liège, Belgiume-mail: A.Modave@ulg.ac.be

É. DeleersnijderCenter for Systems Engineering and Applied Mechanics(CESAME), Université catholique de Louvain, AvenueGeorges Lemaître 4, 1348 Louvain-la-Neuve, Belgium

problems of the collapse of a Gaussian-shaped moundof water and of its advection by a mean current. Thegood behavior of the resulting boundary scheme re-mains when a full non-linear dynamics is taken intoaccount.

Keywords Boundary condition · Absorbing layer ·Sponge layer · Shallow water model

1 Introduction

Open boundary conditions (OBC) are often seen as amajor source of uncertainty or even error in numeri-cal model simulations. There are two reasons for this.First, the conditions are supposed to account for whathappens outside the model domain, which is usuallypoorly known. Unless the model is embedded in alarger scale model or high-resolution observations areavailable, climatological mean data are usually usedto force models along their open boundaries. Second,the open boundary conditions are supposed to describeaccurately the outward propagation of signals and per-turbations of all kinds generated in the model interioras if the model domain was unbounded.

While the former problem cannot be solved with-out appropriate data being available, the latter reliesentirely on the mathematical and numerical formula-tions of the differential problem. It should therefore besolved by the implementation of appropriate boundaryconditions. A large number of numerical treatmentsof open boundary conditions have been proposed inthe literature and systematically compared in varioussituations (e.g., Raymond and Kuo 1984; Chapman

66 Ocean Dynamics (2010) 60:65–79

1985; Røed and Cooper 1986; Jensen 1998; Palma andMatano 1998, 2001; Marchesiello et al. 2001; Treguieret al. 2001; Nycander and Döös 2003; Gan and Allen2005; Cailleau et al. 2008; Herzfeld 2009). Because ofthe multitude of dynamical phenomena interacting inrealistic models, the issue has, however, not yet re-ceived a general answer.

Many OBC schemes are derived from theradiation condition first introduced by Sommerfeld(1949) and later modified by various authors (e.g.,Pearson 1974; Flather 1976; Engquist and Majda 1977;Orlanski 1976; Camerlengo and O’Brien 1980; Israeliand Orszag 1981; Miller and Thorpe 1981; Raymondand Kuo 1984; Palma and Matano 1998). The basic ideabehind such schemes is that disturbances propagateacross the boundary as waves that can be describedby simplifications of the full model dynamics. Similarassumptions (e.g., linearization) are also often madeto identify the Riemann invariants and characteristicvariables associated with the hyperbolic part of thedynamics (e.g., Hedström 1979; Røed and Cooper1986; Ruddick et al. 1994, 1995; Nycander et al. 2008).Besides, Blayo and Debreu (2005) show how theradiation methods can be revisited from the point ofview of characteristic variables.

Relaxation methods form a second class of widelyused open boundary conditions that are particularlyappealing by both their simplicity and their efficiency(e.g., Røed and Cooper 1986; Palma and Matano 1998;Nycander and Döös 2003). The idea is to extend thecomputational domain with an absorbing layer (orsponge layer) (Fig. 1), where the numerical solutionis progressively relaxed towards the external solution(Davies 1976). This flow relaxation scheme (FRS) can

Fig. 1 FRS in a two-dimensional domain. The solution is nudgedtowards the exterior solution in an absorbing layer that is addedto the domain of interest

be interpreted as adding a nudging term to each originalmodel equation

∂φ

∂t+ F(φ) = 0, (1)

which becomes

∂φ

∂t+ F(φ) = −σ

(φ − φext), (2)

where φ is any model variable, φext is the correspondingexternal solution, and σ is a positive coefficient thatcan be taken as a constant or allowed to vary withinthe absorbing layer (Martinsen and Engedahl 1987).The full dynamics of the model is therefore takeninto account in a natural way. The implementation isalso straightforward even in complex geometries, andthe method is therefore applicable to a wide range ofproblems.

The FRS has received new attention with the in-troduction of the concept of the perfectly matchedlayer (PML), first in the context of electromagnetism(Berenger 1994, 2007) and then in computational fluiddynamics (Hu 1996, 2001, 2008). The scheme hasbeen introduced in the oceanographic community byDarblade et al. (1997) and Navon et al. (2004) andmodified recently by Lavelle and Thacker (2008). Allthese PML approaches are equivalent to the FRS inone-dimensional problems but differ in two and threedimensions by the introduction of a directional splittingin PML schemes.

When the external solution φext is zero, both theFRS and PML amount to the introduction of a lineardamping term in the absorbing or PML that dependson the coefficient σ in Eq. 2. The idea is to ensurea sufficiently smooth variation of σ to avoid spuriousreflections at the interface between the model interiorand the absorbing layer and, at the same time, intro-duce enough damping of incoming disturbances in theabsorbing layer. To this end, various spatial distribu-tions of σ have been used in previous studies, but thechoice of these distributions and of the thickness of thesponge layer were largely empirical. In this paper, wepropose, therefore, to carry out a detailed study of theinfluence of these parameters in order to optimize theperformance of the boundary scheme.

2 Optimum absorbing layer for linear gravitywaves: a discrete problem

As a first attempt to identify the optimum parame-ters of an absorbing layer, we consider the prop-agation of linear gravity waves in the semi-infinite

Ocean Dynamics (2010) 60:65–79 67

one-dimensional domain x ∈] − ∞, 0] (Fig. 2a). Theelevation η(t, x) and the velocity u(t, x) evolve ac-cording to

∂η

∂t+ h

∂u∂x

= 0 (3)

∂u∂t

+ g∂η

∂x= 0, (4)

where h is the local depth (assumed to be constant) andg is the gravitational acceleration.

Instead of prescribing local open boundary condi-tions at x = 0 as would be done with radiation condi-tions, we extend the domain by adding an absorbinglayer of thickness δ to the right of the domain of inter-est. The aim of this layer is to absorb and damp wavesas they leave the domain of interest. Assuming theexternal solution to be zero, the differential equationsfor the elevation η�(t, x) and the velocity u�(t, x) in theabsorbing layer are therefore

∂η�

∂t+ h

∂u�

∂x= −σ(x) η� (5)

∂u�

∂t+ g

∂η�

∂x= −σ(x) u�, (6)

where σ(x) ≥ 0 is the absorption coefficient.The above two equations (Eqs. 5 and 6) are valid

in the absorbing layer, i.e., for x ∈ ]0, δ], while Eqs. 3and 4 must be solved in the domain of interest, i.e., forx < 0. Alternatively, Eqs. 5 and 6 can be considered tobe applicable in the whole extended domain but withσ(x) = 0 for x < 0.

Classical arguments of the theory of differentialequations (integrating the differential equations on avanishingly small control domain extending on bothsides of the interface, see, for instance, Morse andFeshbach 1999) can be used to show that the fields mustbe continuous at the interface between the domain andthe absorbing layer (at x = 0), i.e.,

η(t, 0) = η�(t, 0) (7)

u(t, 0) = u�(t, 0) (8)

In both the domain of interest and the absorbinglayer, the propagation of a signal can be described asthe superposition of an incident wave and a reflectedwave. Thanks to the continuity of the fields at theinterface between the domain and the absorbing layer,the frequency ω and the wavenumber k are identicalin the two parts. The elevation and velocity may thenbe expressed as (see, for instance, LeBlond and Mysak1978):

η(x, t) = � {Z+ei(ωt−kx) + Z−ei(ωt+kx)

}(9)

u(x, t) = � {U+ei(ωt−kx) + U−ei(ωt+kx)

}(10)

η�(x, t) = � {Z+ei(ωt−kx)e−γ (x) + Z−ei(ωt+kx)eγ (x)

}(11)

u�(x, t) = � {U+ei(ωt−kx)e−γ (x) + U−ei(ωt+kx)eγ (x)

}, (12)

where � takes the real part of its complex variable andwhere

k = ω√

gh(13)

U± = ±√

gh

Z± (14)

γ (x) =∫ x

0

σ(x′)√

ghdx′ (15)

Incidently, it is worth noting that the use of the sameabsorption coefficient σ in both Eqs. 5 and 6 ensuresthat the ratio η/u for a single wave component takesthe same value in the domain of interest and in theabsorbing layer. Therefore, while there is no physicalinterpretation for the absorbing term in Eq. 5 (theabsorbing term can be interpreted as a linear frictionterm in the momentum equation), its introduction isnecessary to avoid reflections of incident waves on theinterface x = 0 between the domain of interest and theabsorbing layer (Berenger 2007).

At the outer boundary of the absorbing layer, i.e.,at x = δ, it is assumed that outgoing waves have beendamped while travelling across the absorbing layerand that the solution reduces to the external solution,which is zero in the experimental set-up considered

Fig. 2 Geometry (a) andspatial discretization (b) ofthe one-dimensional linearwave problem

68 Ocean Dynamics (2010) 60:65–79

here. Of course, less reflective boundary conditions,like Neumann conditions or even a radiation condi-tion, could be implemented at x = δ to allow (damped)outgoing waves to leave the domain. As confirmed bynumerical experiments (not shown), the exact natureof the boundary condition at x = δ has, however, littleinfluence on the solution (in both the model domainand the sponge layer) if outgoing waves are efficientlydamped in the absorbing layer. Since the absorbinglayer must not only absorb outgoing waves but mustalso allow forcing the model with an external solution,and since this external solution is assumed to be zerohere, both the free surface elevation and the velocityare prescribed to vanish at x = δ.

The Dirichlet boundary condition at x = δ inducesdefinite relationships between the complex amplitudesZ+, Z−, U+, and U− of the outgoing and reflected wavecomponents. The efficiency of the absorbing layer canthen be quantified using the reflection coefficient

r =∣∣∣∣

Z−Z+

∣∣∣∣ =

∣∣∣∣U−U+

∣∣∣∣ (16)

that measures the ratio of the amplitude of the reflectedand incident waves in the domain of interest. Thesquare of the reflection coefficient gives the ratio of themean total energy (e.g., Holthuijsen 2007)

Etot = 1

T

∫ T

0

(1

2gη2(t, x) + 1

2hu2(t, x)

)dt with x ≤ 0

(17)

(where T = 2π/ω is the wave period) associated withthe reflected and incident wave components. Usingthe boundary condition at x = δ and the continuityof the solutions at x = 0, the reflection coefficient canbe expressed as

r = exp

[

− 2√

gh

∫ δ

0σ(x) dx

]

(18)

It can therefore be made as small as we please by usingan arbitrarily large absorption coefficient. In particular,reflection of the incident wave can be totally avoided ifthe absorption coefficient is unbounded and if∫ δ

0σ(x) dx = +∞ (19)

A similar conclusion was reached by Collino andMonk (1998) in the context of electromagnetism, andby Bermúdez et al. (2007) in the context of acoustics.Therefore, it makes no sense to look for an optimumabsorption coefficient or an optimum thickness of theabsorbing layer in the continuous case.

Consider now the discrete version of the linearwave problem (Eqs. 3 and 4) with an absorbing layer.Using a classical FBTCS scheme (e.g., Beckers andDeleersnijder 1993) on a staggered grid (Fig. 2b), thenumerical integration scheme can be written as

ηn+1j−1/2 − ηn

j−1/2

�t+ h

unj − un

j−1

�x= −σ j−1/2η

n+1j−1/2 (20)

un+1j − un

j

�t+ g

ηn+1j+1/2 − ηn+1

j−1/2

�x= −σ ju

n+1j , (21)

where �t and �x denote, respectively, the integrationtime step and the grid size, where

unj = u(n�t, j�x) (22)

and

ηnj−1/2 = η(n�t, j�x − �x/2) (23)

are the numerical approximations of the velocity andfree surface elevation and where σ j is the spatially vary-ing absorption coefficient. The absorption coefficient isset to zero in the domain interior and differs from zeroin an absorbing layer made up of N = δ/�x grid cells.The velocity is assumed to vanish at the outer boundaryof the absorbing layer.

When the discrete model is triggered from the in-terior with a sine wave, a reflected wave is created inthe absorbing layer. The resulting reflection coefficientin the model interior depends on the level of lineardamping of both the incident and reflected wave in theabsorbing layer. This reflection coefficient is plottedin Fig. 3 as a function of the absorption coefficientwhen a uniform value σ is used for both variablesin the absorbing layer. The corresponding reflection

10−6

10−4

10−2

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ [1/s]

Ref

lect

ion

coef

ficie

nt

Continuous problemDiscrete problem

Fig. 3 Reflection coefficient as a function of the uniform absorp-tion coefficient σ for the continuous case (solid curves) and thediscrete case (dashed curves). The solution is computed with g =9.85 m2/s, h = 100 m, �x = 10 km and �t = 150 s. The period ofthe outgoing wave is 20�t and the thickness δ of the absorbinglayer is 30 km (three grid cells)

Ocean Dynamics (2010) 60:65–79 69

coefficient computed with the continuous model is plot-ted for comparison. Contrary to the continuous casewhere the reflection coefficient can be made as smallas we please by increasing σ , an optimum value of theabsorption coefficient is apparent in the discrete case.

For small values of σ , the reflection coefficientsfor the continuous and discrete models are both closeto unity because the linear damping in the absorbinglayer then has nearly no influence and the outgoingwave undergoes a nearly perfect reflection at the outerboundary of the layer (at x = δ). As σ increases, theabsorbing layer succeeds in reducing the amplitude ofthe reflected wave in both the continuous and discretemodels. At some point, however, the exponential spa-tial decrease of the wave amplitude that is apparentin the analytical continuous solution in Eqs. 11–12 inthe absorbing layer cannot be captured by the discretemodel because of its limited spatial resolution. Foreven larger values of the absorption coefficient, theperformance of the discrete boundary scheme furtherdeteriorates. The outgoing wave is reflected at the in-terface between the model domain and the absorbinglayer that then behaves like a wall.

In a discrete model, an optimum value of σ is ex-pected to exist that achieves a trade-off between theaim to damp outgoing waves as efficiently as possi-ble and the limited spatial resolution of the modelthat cannot describe very steep spatial variations ofthe amplitude. Using Eq. 15, a characteristic lengthscale

�σ =√

ghσ

(24)

can be associated with the damping of waves in theabsorbing layer. Features with such a characteristiclength scale will be captured by the numerical grid onlyif �σ is of the order of magnitude as �x. One expects,therefore, the optimum absorption coefficient to scalelike√

gh�x

(25)

In addition, in order to get a significant damping ofthe outgoing waves in the absorbing layer, the width ofthe absorbing layer δ should also be sufficiently longerthan �σ .

3 Non-uniform absorption coefficient

The low value of the reflection coefficient achieved inFig. 3 with the optimum uniform absorption coefficientcan be further improved by using spatially varying

absorption coefficients. Such spatial distributions wereproposed by Davies (1976), Martinsen and Engedahl(1987), Berenger (1994), and Jensen (1998). Thesedistributions are purely heuristic, and it is not clearwhether they correspond to any kind of optimum.

A frequently used spatial distribution is the polyno-mial profile

σ(x) = σm

( xδ

)α

, (26)

with α > 0 and where σm is the value of the absorptioncoefficient at the outer side of the absorbing layer.Note that such a spatial distribution would not providea perfect absorption in the continuous case since thecondition in Eq. 19 is not met.

As an alternative, the hyperbolic distribution ini-tially proposed by Bermúdez et al. (2007) in the contextof electromagnetism can be adapted to shallow watermodels as

σ(x) =√

ghδ − x

(27)

This profile satisfies the condition in Eq. 19 for a perfectabsorption in the continuous case.

Using the continuous solution in Eqs. 11–12 and 15,it is easy to show that the hyperbola in Eq. 27 replacesthe exponential decrease of the linear waves (for aconstant absorption coefficient) in the absorbing layerby a linear one. Since the development of a boundarylayer is avoided, the good behavior of Eq. 27 in thecontinuous case should also transfer to discrete models.As shown in the Appendix, a linear decrease of theamplitude of a linear wave with wave number k canindeed be produced in the semi-discrete case when theabsorption coefficient varies according to

σ j =√

ghδ − j�x

cos

(k�x

2

)(28)

With such a distribution of the absorption coefficient,a sine wave can be completely damped, without re-flection, in an absorbing layer of thickness δ. Contraryto the continuous case, these coefficients depend onthe wave number k. For well resolved waves, i.e., fork�x � 1, however, the distribution is close to the con-tinuous one.

Unfortunately, Eq. 28 cannot be used as such todesign a perfect absorbing layer because the corre-sponding (semi-)discrete solutions in the model do-main, without damping, and in the absorbing layer,with linear damping, cannot be matched perfectly. To

70 Ocean Dynamics (2010) 60:65–79

Fig. 4 Geometry of theexperimental set-up. AGaussian pulse propagates tothe right and is partlyreflected by the absorbinglayer. The initial Gaussianpulse is described by η(0, x) =exp

[−(x + L/4)2/(3�x)2]

and u(0, x) = η(0, x)√

g/h

ensure a smooth transition from the model domain tothe absorbing layer, it is advisable that σ vanishes attheir interface. This suggests to shift the hyperbolicdistribution in Eq. 27 according to

σ0(x) = σ(x) − σ(0) =√

ghδ

xδ − x

(29)

With this modification, the linear damping of the solu-tion will be lost, but it is expected that large gradientscan be avoided.

To compare the performances of the different distri-butions of the absorption coefficient, the experimentalset-up of the previous section (Fig. 2) is used again.However, since the aim of the procedure is to devisean optimum distribution of the absorption coefficientsthat works for a wide range of frequencies, a Gaussianpulse is considered instead of a sine wave. Initially,the Gaussian pulse is located in the second half of afinite domain (x ∈ [−L/2, 0]), as shown on Fig. 4. Astime goes by, this pulse propagates to the right and ispartly reflected by the absorbing layer. The final timet f = L/2

√gh of the numerical experiment and the size

of the computational domain are such that the reflectedsignal ends in [−L/2, 0] and any influence from the leftboundary (x = −L) can be avoided in [−L/2, 0] at theend of the simulation.

A reflection ratio for the Gaussian pulse is thencomputed as

Reflection ratio

=√

Error at t f with the absorbing layerError at t f with the wall condition u = 0

, (30)

where the error gauge used is based on energetic con-siderations and is given by

Error(t) =∫ 0

−L/2

(1

2g [η(t, x) − ηref(t, x)]2

+1

2h [u(t, x) − uref(t, x)]2

)dx (31)

The fields ηref(t, x) and uref(t, x) in this expression areobtained as the solution of a reference simulation in

an extended domain, i.e., without any reflection at theopen boundary.

The various spatial distributions of the absorptioncoefficient are shown in Fig. 5 in the set-up consideredin Fig. 3 in the particular case when the absorbing layeris made of four grid cells. The corresponding reflectionratios are shown in Fig. 6 for various widths of theabsorbing layer.

The polynomial (and constant) distributions (Eq. 26)contain a free parameter σm. In Fig. 5 and in Fig. 6,this was optimized numerically in order to minimizethe reflection ratio. The hyperbola and the shiftedhyperbola can also be optimized by using a multiplica-tive factor. But the preliminary results show that thebest multiplicative factor is equal to unity for each case,and thus, the profiles are already optimal.

The determination of the spatial distribution of theabsorption coefficient can also be approached as a fulloptimization problem, with the absorption coefficients

0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

x [m]

Abs

orpt

ion

coef

ficie

nt σ

(x)

[1/s

]

Optimum uniformOptimum polynomial (α=1)Optimum polynomial (α=2)Optimum polynomial (α=3)HyperbolaShifted hyperbolaOptimum non–uniform

Fig. 5 Spatial distributions of the absorption coefficient using thesame parameters as in Fig. 3 when the absorbing layer containsfour grid points. The parameters σm used in the polynomial distri-butions are set by numerical optimization. The parameters of thepolynomial and uniform distributions are optimized numerically

Ocean Dynamics (2010) 60:65–79 71

2 4 6 8 10 12

x 104

10–5

10– 4

10– 3

10– 2

10– 1

100

Absorbing layer length δ [m]

Ref

lect

ion

ratio

Optimum uniform

Optimum polynomial (α=1)

Optimum polynomial (α=2)

Optimum polynomial (α=3)HyperbolaShifted hyperbolaOptimum non–uniform

Fig. 6 Reflection ratio as a function of the thickness of theabsorbing layer for different distributions of the absorption co-efficient. The parameters are identical to those used in Fig. 3.The optimum parameters are used for each distribution (uniform,polynomial, and non-uniform) and each absorbing layer length

σ j at the various grid points as control variables. Thecorresponding optimization problem is then

min Reflection ratio(σ1/2, σ1, σ3/2, ..., σN−1, σN−1/2)

s.t. 0 ≤ σ1/2 ≤ σ1 ≤ σ3/2 ≤ · · · ≤ σN−1 ≤ σN−1/2,

(32)

where it is also requested that the absorption coeffi-cient is a positive and non-decreasing function of thex coordinate in the absorbing layer. Such an additionalconstrain helps to improve the convergence of theprocedure.

The numerical resolution of Eq. 32 using the opti-mization toolbox of Matlab� suffers from an irregularconvergence when the absorbing layer is broad or whenthe model is triggered with a sine wave instead of aGaussian signal. In both cases, a nearly perfect behaviorof the absorbing layer can be achieved with slightlydifferent distributions of the absorption coefficients,so that the optimum distribution is poorly defined.When the system is triggered with a sine wave, inparticular, a nearly perfect behavior can be obtainedeven for thin absorbing layers, but the optimum dis-tribution (not shown) depends strongly on the wavenumber/frequency.

Figure 5 shows the optimum spatial distribution ofthe absorption coefficient obtained through numericaloptimization in the set-up considered. The gradual in-

crease of the absorption coefficient in the absorbinglayer introduces a progressive damping of the incomingwave and avoids, therefore, the development of largegradients in the solution. The polynomial profiles andthe shifted hyperbola exhibit qualitatively similar be-haviors but with some quantitative differences.

The performances of the different distributions arecompared in Fig. 6 when varying the thickness of theabsorbing layer. The unshifted hyperbola in Eq. 27 doesnot perform much better than the (optimum) uniformdistribution. The reflection ratio is larger than 10−1

for relatively thin absorbing layers and only slightlydecreases for larger absorbing layers. This poor behav-ior of these distribution can therefore be ascribed tothe jump of the absorption coefficient at the interfacebetween the model domain and the absorbing layer.

Values of the reflection ratio below 10−2 can beobtained with all other profiles for moderately thin ab-sorbing layers. While providing reflection ratios one ortwo orders of magnitude larger than the fully optimizedsolution, both polynomial distributions and Eq. 29 do agood job and are relatively equivalent for thin absorb-ing layers. When the thickness of the layer increases,the parabolic and cubic distributions provide very ef-ficient absorption properties, while the reflection ratioassociated with shifted hyperbola (Eq. 29) does notdecrease to values much lower than 10−3. Polynomialdistributions, however, contain a free parameter σm

that has been optimized to produce the correspondingcurves in Fig. 6. On the contrary, the distribution Eq. 29does not contain any free parameters but provides rea-sonable performances without having recourse to anykind of optimization.

4 Collapse of a Gaussian-shaped mound of water

In order to test the performances of the different distri-butions in a more realistic context, we consider the two-dimensional case of the collapse of a Gaussian-shapedmound of water. This test case was previously used byMcDonald (2002), Navon et al. (2004), and Lavelle andThacker (2008) to compare different kinds of boundaryconditions.

We consider a rectangular domain (x, y) ∈[−Lx, Lx] × [−Ly, Ly] with absorbing layers ofthickness δ added along each side (Fig. 7). Contraryto most previous authors who considered a squaredomain, a rectangular domain is used here to break thesymmetry associated with circular perturbations in asquare domain and to emphasize the skew propagationof these perturbations with respect to the boundary

72 Ocean Dynamics (2010) 60:65–79

Fig. 7 a Geometry of thetwo-dimensional problem.b–c Absorbing layers wherethe absorption coefficientsσx(x, y) and σy(x, y) differfrom zero (in gray)

(see below). A Gaussian-shaped mound of water isprescribed as initial condition

η(0, x, y) = η0 exp

(− x2 + y2

R2

)(33)

and the fluid is at rest in both the domain of interest andthe absorbing layers.

Using the “pretty good sponge” approach developedby Lavelle and Thacker (2008), the linearized shallowwater equations modified to account for the absorbinglayers are

∂η

∂t+ h

(∂u∂x

+ ∂v

∂y

)= −(σx + σy)η (34)

∂u∂t

+ g∂η

∂x− fv = −σxu (35)

∂v

∂t+ g

∂η

∂y+ f u = −σyv, (36)

where u and v are the components of the velocity inthe x and y directions and where σx and σy are theabsorption coefficients. These coefficients vanish in theinterior of the domain. In the two absorbing layers thatare normal to the x-direction (resp. y-direction), σy

(resp. σx) is zero (except in the corner) and σx (resp.σy) varies according to the 1D spatial distributionsconsidered in the previous section (Fig. 7). The right-hand sides of these equations introduce, therefore, adamping of the wave components perpendicular to theboundary while the along boundary components areunaffected.

The equations are integrated forward in time withthe FBTCS scheme (e.g., Beckers and Deleersnijder1993). The elevation and the transports are discretizedon an Arakawa C-grid. The parameters used for thistest case are given in Table 1. They are similar tothose used by Lavelle and Thacker (2008) (but with arectangular domain).

Figure 8a shows some snapshots of the referencesolution for the collapse of the initial mound of waterand the associated progressive geostrophic adjustment.This reference solution is computed using an extendedcomputational domain in order to avoid any reflection

at the open boundaries. During the collapse, circularwaves are created and propagate outwards. After about5 h, the main wavefront is expected to reach the upperand lower boundaries of the model domain and to hitthe absorbing layers at normal incidence. As time goesby, the wavefront will propagate along the boundariesand approach the absorbing layers with a decreasingangle of incidence.

Snapshots of the error are shown in Fig. 8b for theparticular case of an absorbing layer with δ = 13 �xwith the absorption coefficient prescribed according toEq. 29. When the wavefront hits the boundary, theerror is small until time t = 6 h. This small error is as-sociated with a weak reflection of the waves with quasi-normal incidence. This is in agreement with the resultsof the previous section where normal waves have beenshown to be properly handled by the one-dimensionalboundary scheme. The snapshots taken at later timesshow that the error increases as the incidence of thewavefront approaching the boundary decreases and thewaves tend to propagate along the boundary. The initialwavefront is partly reflected at the open boundary andthe reflected (and damped) wave propagates inside thedomain.

The numerical experiment of the collapse of themound of water and its geostrophic adjustment canbe repeated using the different distributions of theabsorption coefficient discussed in the previous section.In Fig. 9, the error at the end of the simulations (t = 9 h)is shown for two different widths of the absorbing layer.The free parameters of the constant and parabolicdistributions are optimized numerically to reduce theassociated error for that particular simulation. As inthe previous test case, an optimization of the hyper-bola or of the shifted hyperbola by adjustment of a

Table 1 Parameters used for the two-dimensional test case

Physical parameters f = 1.028 10−4 s−1, g = 9.85 m/s2

and h = 100 mGeometrical parameters Lx = 1,100 km and Ly = 510 kmInitial conditions η0 = 1 m and R = 50 kmNumerical parameters �x = 10 km, �y = 10 km

and �t = 150 s

Ocean Dynamics (2010) 60:65–79 73

Fig. 8 Linear simulation ofthe collapse of aGaussian-shaped mound ofwater. Snapshots of theelevation in the domain forthe reference solution (a) andof the error on the elevation(b) at time t = 4 h, 5 h, ... and9 h. The width of theabsorbing layer length δ is13 �x and the absorptioncoefficient is prescribedaccording to Eq. 29. Theother parameters are givenin Table 1

(b)

(a)

multiplicative factor is useless: the parameters of theseprofiles are already optimum. The results obtained witha parabolic distribution and the value σM = 0.9/�t ad-vocated by Lavelle and Thacker (2008) are also shown.The case where the waves are perfectly reflected iscomputed using a wall condition (i.e., u · n = 0, wheren is the external normal vector) at x = ±Lx andy = ±Ly.

The error patterns obtained with the parabolic dis-tributions are similar to the ones shown in Fig. 8b forthe distribution of the absorption coefficient Eq. 29.

The different solutions are qualitatively similar withvery good properties for normal waves and a re-duced efficiency for tangential waves. The optimumuniform coefficient also gives better results for nor-mal waves than for tangential waves, but they areglobally worse than with the non-uniform distributionsconsidered.

As expected, the best results are obtained with thebroadest absorbing layer (δ = 13�x in Fig. 9). For athin absorbing layer (δ = 5�x in Fig. 9), only smalldifferences can be seen between the results obtained

74 Ocean Dynamics (2010) 60:65–79

Fig. 9 Linear simulation ofthe collapse of aGaussian-shaped mound ofwater. Error on the elevationat the end of the simulation(t = 9 h) with differentabsorption coefficients after9 h. The width of theabsorbing layer δ is 5 �x (left)or 13 �x (right)

with the three spatially varying absorption coefficients.Slightly larger differences between the different resultsappear when a broader absorbing layer is used.

In order to compare the performances of the differ-ent absorption coefficients in a quantitative way, theenergy associated with the error fields is computedaccording to

Error(t) =∫ Ly

−Ly

∫ Lx

−Lx

(1

2g

[η(t, x, y) − ηref(t, x, y)

]2

+ 1

2h

[u(t, x, y) − uref(t, x, y)

]2

+1

2h

[v(t, x, y) − vref(t, x, y)

]2)

dx dy

(37)

The fields ηref(t, x, y), uref(t, x, y) and vref(t, x, y) cor-respond to the final solution of a reference run inwhich reflections are avoided by using a much largercomputational domain. As in the previous section, thereflection ratio Eq. 30 is used to measure the reflectedpart of the solution at the final time t f = 9h.

The reflection ratio values listed in Table 2 confirmthe visual impression from Fig. 9. The results obtainedwith the shifted hyperbola are slightly better than thoseproduced with the optimum parabolic distribution, es-pecially with the 13�x absorbing layer where the re-flection ratio associated with the shifted hyperbola issmaller than with the optimum parabolic distributionby about 20%.

The fact that the shifted hyperbolic profile performsbetter than the parabolic profile seems to contradictthe results plotted in Fig. 6 for the 1D numerical

Ocean Dynamics (2010) 60:65–79 75

Table 2 Reflection ratio for different absorption coefficients andfor two values of the width of the absorbing layer (δ = 5 �x andδ = 13 �x) using both the linear and non-linear shallow water

equations. The Coriolis and advection terms are neglected inthe absorbing layer A of the non linear case, but are taken intoaccount in the absorbing layer B

δ = 5�x δ = 13�x

Linear Non linear Linear Non linear

Layer A Layer B Layer A Layer B

Optimum uniform profile 0.181591 0.183183 0.181718 0.111063 0.119862 0.111301Parabola with σM = 0.9/�t 0.120980 0.125272 0.121175 0.093762 0.105474 0.094015Optimum parabola 0.114715 0.119776 0.114877 0.070595 0.092772 0.070830Shifted hyperbola (Eq. 29) 0.114302 0.120193 0.114538 0.059168 0.087185 0.059372

experiment. This difference is associated with the betterrepresentation of the skew propagation of the signal inthe 2D simulation.

Increasing the width of the absorbing layer allows fora more gradual damping of outgoing waves obtainedusing smaller absorption coefficients. For the 13�xabsorbing layer, the optimum value of the parameterσM of the parabolic distribution is about 1.5 × 10−3 s−1,while it reaches 3 × 10−3 s−1 for the 5�x absorbinglayer. Relating σM to the time step �t and not tothe width of the absorbing layer is therefore not thebest solution. The value of σM = 0.9/�t = 6 × 10−3 s−1

estimated using the formula of Lavelle and Thacker(2008) is about twice the optimum value for the 5�xabsorbing layer. This has, however, little effect on thereflection ratio in this numerical experiment. With thebroader 13�x absorbing layer, there is, however, a 25%difference between the reflection ratios obtained withthe two parabolic distributions.

As claimed in the introduction, one of the advan-tages of relaxation methods is that the full nonlineardynamics can be taken into account. In particular,absorption coefficients can be easily included in thenon-linear shallow water equations. When written inconservative form, the resulting equations are

∂ H∂t

+ ∂(Hu)

∂x+ ∂(Hv)

∂y= −(σx + σy)H (38)

∂(Hu)

∂t+ ∂(gH2/2)

∂x+ ∂(Hu2)

∂x+ ∂(Huv)

∂y− f Hv

= −σx Hu (39)

∂(Hv)

∂t+ ∂(gH2/2)

∂y+ ∂(Huv)

∂x+ ∂(Hv2)

∂y+ f Hu

= −σy Hv, (40)

where H(x, y, t) = h + η(x, y, t) is the total depth.These equations, which are in a suitable form for theirnumerical discretization using the finite volume ap-proach, include the advection term for momentum andaccount for the variability of the water depth.

Table 2 shows the reflection ratio when simulatingthe collapse of a Gaussian mound of water with thenon-linear shallow water equations. Two options areconsidered for the absorbing layer. In the first imple-mentation, labeled “layer A”, the Coriolis and advec-tion terms are neglected in the absorbing layers. Thesetwo terms are taken into account in the variant labeled“layer B”.

The performances of the different absorption co-efficients are very similar to those of the linear caseprovided that the full dynamics (layer B) is taken intoaccount in the absorbing layers. The error increasesthen by less than 1% with respect to the linear case andthe conclusions remain valid. In particular, the shiftedhyperbola performs slightly better than the parabolicprofile, especially with the 13�x absorbing layer, with-out requiring any tuning.

A quick comparison of the A and B variants listedin Table 2 shows that the simplification of the dynamicsin the absorbing layer can lead to a significant increaseof the reflection ratio, especially for the most efficientboundary conditions. In order to get the most out ofthe absorbing layer, it is therefore important to takeadvantage of the ability of the absorbing layer to ac-commodate the full dynamics of the system—which is,of course, also a sensible thing to do from a physicalperspective—in order to avoid spoiling the benefit ofusing a more appropriate spatial distribution of theabsorption coefficient.

5 Moving eddy: an advection dominated test case

The test cases considered so far are dominated by thepropagation of surface gravity waves. While the shiftedhyperbola Eq. 29 gives good results in such numericalexperiments, one might wonder if the same distribu-tion can be used in advection-dominated problems. Inthis section, we address therefore the problem of amoving eddy, i.e., the advection on an eddy by a meanflow and its own velocity field. This test case is also

76 Ocean Dynamics (2010) 60:65–79

used, in slightly different forms, by Navon et al. (2004),McDonald (2002), and Lavelle and Thacker (2008).

The non-linear shallow water equations (Eqs. 38–40)are used to describe the movement of an eddy that isinitially in geostrophic balance with the same Gaussian-shaped mound of water as used in the previous testcase. The model parameters and the basing geometryare also identical. To move the eddy, a uniform currentof 5 m/s is added in both horizontal directions in theinitial velocity fields and is supported by a geostrophictilt of the surface elevation (see Lavelle and Thacker2008). In the absorbing layer, the fields are relaxed to-wards the external solution, including only the uniformcurrent and the associated tilt of the surface elevation.As time goes by, the eddy is swept out of the opendomain by the advecting flow.

The time evolution of the solution is described andillustrated by Lavelle and Thacker (2008). During thefirst hours of the simulation, the amplitude of theGaussian eddy decreases through the generation ofadjustment waves, since the initial fields do not achievea nonlinear balance. Afterwards, the eddy is advectedtowards the open boundary while the height of therotating mound undergoes a small decrease because ofnumerical diffusion.

The reflection of the outgoing signal on the openboundary can be quantified by computing the energyassociated with the error fields Eq. 37. This is done herefor the different profiles of the absorption coefficientconsidered above and for two thicknesses of the absorb-ing layers Fig. 10. The uniform profile σ , the parameterσM of the parabola, and a multiplicative factor of theshifted hyperbola are tuned to minimize the area underthe curves of Fig. 10. In all the simulations, the errorincreases during the first hours due to the partial re-flection of the eddy on the absorbing layer. After sometime, the errors are simply transported by the uniformcurrent and leave the domain, so that the global errordecreases.

Increasing the thickness of the absorbing layer from5�x to 13�x leads to a significant reduction of theerror. In the best cases, the error decreases by a factorof 5. Note that the maximum error in Fig. 10 appearslater when the layer length is increased because theeddy then covers a larger distance in the absorbinglayer before being reflected.

For both thicknesses of the absorbing layer, theshifted hyperbolic profiles performs significantly betterthan the parabolic distribution used by Lavelle andThacker (2008). The error is, however, much largerthan with numerically optimized distributions (uni-form, parabolic, and shifted hyperbolic) of the absorp-tion coefficient.

(a) = 5km

0 5 10 15 20 25 30 35 40 45 500

0.005

0.01

0.015

Time [t]

Err

or(t

) / I

nitia

l ene

rgy

Optimum uniform profilOptimum parabolaOptimum shifted hyperbola

Parabola : σ(x) = 0.9/ x (x/δ)2

Shifted hyperbola : σ(x) = (gh)1/2/δ x/(x )Shifted hyperbola : σ(x) = U/δ x/(x−δ

−δ)

δ

(b) = 13kmδ

0 5 10 15 20 25 30 35 40 45 500

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Time [t]

Err

or(t

) / I

nitia

l ene

rgy

Optimum uniform profilOptimum parabolaOptimum shifted hyperbola

Parabola : σ(x) = 0.9/ x (x/δ)2

Shifted hyperbola : σ(x) = (gh)1/2/δ x/(x )Shifted hyperbola : σ(x) = U/δ x/(x−δ

−δ)

Fig. 10 Moving eddy: evolution of the error in the domain for dif-ferent profiles of the absorption coefficient and two thicknesses ofthe absorbing layer

The reasons for the bad result obtained with theshifted hyperbolic distribution are associated with thephysics of the problem. With the shifted hyperbolic dis-tribution Eq. 29, it is implicitly assumed that the signalpropagates as surface gravity waves while advection isdominant. The speed is therefore incorrectly estimated,and this obviously has an adverse effect on the per-formance of the absorbing layer. The problem can beeasily addressed, however, by using the normal velocityU in the shifted hyperbolic distribution Eq. 29 insteadof

√gH. As shown in Fig. 10, the corresponding distrib-

ution leads to optimum results that are comparable withthose obtained with numerically tuned distributions.

The use of the normal velocity in Eq. 29 can alsobe justified on theoretical grounds using the same ap-proach as in Section 3. Indeed, if advection dominates,the dynamics in the absorbing layer can be approachedby

∂φ

∂t+ U

∂φ

∂x= −σ(x) φ, (41)

where U > 0 is the advection velocity and σ(x) is theabsorption coefficient, which vanishes in the domain

Ocean Dynamics (2010) 60:65–79 77

interior. For a constant value of the absorption co-efficient, φ undergoes an exponential damping whileadvected across the sponge layer. Such an exponentialdamping can be made very efficient by using largevalues of σ , but it is then difficult to describe using anumerical model with limited spatial resolution. As inSection 3, however, a linear decrease of the amplitudeof outgoing waves can be achieved in the absorbinglayer with the hyperbolic profile

σ(x) = Uδ − x

(42)

Removing any discontinuity of the distribution at theinterface with the model interior leads then to themodified shifted hyperbolic distribution

σ0(x) = σ(x) − σ(0) = Uδ

xδ − x

(43)

The results obtained with this modified distributionare clearly comparable to (and even slightly betterthan) those obtained with the numerically optimizeddistributions. The great advantage of the shifted hy-perbolic distribution is, however, that no numericaloptimization nor trial-and-error procedure is requiredbut that the free parameter has a clear physical inter-pretation and can be adjusted from the knowledge ofthe dynamics of the problem.

6 Conclusion

Because of their easy implementation and their goodproperties (e.g., Røed and Cooper 1986; Palma andMatano 1998; Nycander and Döös 2003), absorb-ing/sponge layers are very attractive boundary condi-tions for numerical ocean models. The 1D and 2Dnumerical examples discussed above demonstrate,however, that the actual performance depends on thewidth of the absorbing layer and on the spatial variationof the absorption coefficient in the absorbing layer.

Obviously, the reflection coefficient decreases whenthe width of the absorbing layer increases. The as-sociated numerical cost of such an extension of themodel domain must, however, be taken into accountand a balance must be found between the additionalcomputer load and the performances of the absorbinglayer.

The numerical experiments carried out in this papershow that the reflection ratio can also be decreasedat no additional computer cost by adjusting the ab-sorption coefficient in the absorbing layer. When tryingto improve the performance of the boundary scheme,

this option should therefore be considered before anyfurther extension of the model domain.

The 1D test case discussed in Section 2 shows thatthe discrete character of the numerical problem leadsto bad performances of absorbing layers with too largevalues of the absorption coefficient. When a uniformcoefficient is used, an optimum value emerges as atradeoff between the increased damping produced byincreasing the absorption coefficient and the deterio-ration of the solution because of the resulting poorlyresolved spatial gradients. The quest for the maximumvalues of the absorption coefficient that are compat-ible with the numerical stability should therefore beavoided.

The best results are obtained with absorption coeffi-cients that vary within the absorbing layer. The resultsof the 2D simulation of the collapse of a mound ofwater (Table 2) show, for instance, that a spatiallyvarying absorption coefficient used in a 5�x absorbinglayer gives about the same error as a 13�x absorbinglayer with the optimum uniform constant coefficient.

For a relatively thin absorbing layer, all the spatiallyvarying absorption coefficients considered above leadto comparable results. In this case, the precise valuesand profiles of the absorption coefficient seem ratherunimportant. What matters most is that the absorptioncoefficient increases gently from a small value at theinner side of the absorbing layer to a larger value at theouter boundary.

In order to get the maximum benefit out of alarger absorbing layer, particular attention must be paidto the precise profile of the absorption coefficient. Inthe 2D numerical experiment considered in this paper,the shifted hyperbolic profile (Eq. 29) introduced inthis paper performs significantly better than the widelyused parabolic profile, with both the linear and non-linear shallow water equations. The shifted hyperbola(Eq. 29) offers, therefore, a valuable alternative, whichis nearly optimum without artificial tuning. The shiftedhyperbolic distribution does contain a free parameter,but this can be clearly associated with the propagationspeed of the signal. Good performances can there-fore be achieved by adjusting the distribution of theabsorption coefficient according to the physics of theproblem.

Acknowledgements EJMD and ED are, respectively, Hon-orary Research Associate and Research Associate at the Na-tional Fund for Scientific Research (Belgium). This work wassupported by the French Community of Belgium (RACE, ARC-05/10-333) and by the Interuniversity Attraction Poles Pro-gramme TIMOTHY-P6/13 (Belgian Science Policy). This paperis MARE publication n◦181.

78 Ocean Dynamics (2010) 60:65–79

Appendix: Semi-discrete model with lineardecrease of linear waves

Consider the semi-discrete linear wave equations on astaggered grid

∂η j−1/2

∂t+ h

u j − u j−1

�x= −σ j−1/2 η j−1/2 (44)

∂u j

∂t+ g

η j+1/2 − η j−1/2

�x= −σ j u j, (45)

where �x is the grid size and σ j−1/2 and σ j are so farundetermined absorption coefficients.

We request that the semi-discrete solution decreaseslinearly on a distance δ = N�x, i.e.,

η j−1/2 = δ − x j−1/2

δcos

(ωt − kx j−1/2

)√

hg

(46)

u j = δ − x j

δcos

(ωt − kx j

), (47)

where x j−1/2 = ( j − 1/2)�x and x j = j�x.Substituting Eq. 46 into Eq. 44 leads to the dispersion

relation

ω = 2√

gh�x

sin

(k�x

2

)(48)

and the value of the absorption coefficient needed tohave a solution with a linear decrease

σ j =√

ghδ − x j

cos

(k�x

2

)(49)

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