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Ocean Dynamics (2012) 62:1201–1215DOI 10.1007/s10236-012-0555-3
Adaptive volume penalization for ocean modeling
Shanon M. Reckinger · Oleg V. Vasilyev ·Baylor Fox-Kemper
Received: 15 December 2011 / Accepted: 18 May 2012 / Published online: 10 June 2012© Springer-Verlag 2012
Abstract The development of various volume penal-ization techniques for use in modeling topographicalfeatures in the ocean is the focus of this paper. Dueto the complicated geometry inherent in ocean bound-aries, the stair-step representation used in the majorityof current global ocean circulation models causes ac-curacy and numerical stability problems. Brinkman pe-nalization is the basis for the methods developed hereand is a numerical technique used to enforce no-slipboundary conditions through the addition of a term tothe governing equations. The second aspect to this pro-posed approach is that all governing equations aresolved on a nonuniform, adaptive grid through theuse of the adaptive wavelet collocation method. This
Responsible Editor: Pierre Lermusiaux
This article is part of the Topical Collection on Multi-scalemodelling of coastal, shelf and global ocean dynamics
S. M. Reckinger (B)School of Engineering, Department of MechanicalEngineering, Fairfield University, 1073 North Benson Rd.,Fairfield, CT 06824, USAe-mail: [email protected]
O. V. VasilyevDepartment of Mechanical Engineering, UCB 427,University of Colorado at Boulder,Boulder, CO 80309, USAe-mail: [email protected]
B. Fox-KemperDepartment of Cooperative Institute for Researchin Environmental Sciences (CIRES), and Departmentof Atmospheric and Oceanic Sciences, UCB 216,University of Colorado at Boulder,Boulder, CO 80309, USAe-mail: [email protected]
method solves the governing equations on tempo-rally and spatially varying meshes, which allows highereffective resolution to be obtained with less compu-tational cost. When penalization methods are coupledwith the adaptive wavelet collocation method, the flownear the boundary can be well-resolved. It is espe-cially useful for simulations of boundary currents andtsunamis, where flow near the boundary is important.This paper will give a thorough analysis of these meth-ods applied to the shallow water equations, as wellas some preliminary work applying these methods tovolume penalization for bathymetry representation foruse in either the nonhydrostatic or hydrostatic primitiveequations.
Keywords Immersed boundary methods ·Shallow water equations · Adaptive mesh refinement ·Wavelet collocation · Complex geometry
1 Introduction
In recent years, substantial progress has been madein the development of numerical methods used inocean modeling. The newly developed ocean modelshave started incorporating more sophisticated numer-ical methods, such as finite element methods, spectralmethods and finite volume methods, which are solvedon adaptive and/or unstructured grids (Chen et al. 2003;Danilov et al. 2004; Fringer et al. 2006; Iskandaraniet al. 2003; Lynch et al. 1996; Marshall et al. 1997;Herrnstein et al. 2005; Pain et al. 2005; Popinet andRickard 2007; White 2007). Structured grid modelshave been the mainstay of ocean modeling for over50 years with compromises for complex bathymetry
1202 Ocean Dynamics (2012) 62:1201–1215
(Adcroft et al. 1997) and unstructured grid models arebeginning to allow for much more complex bathymetry.However, Brinkman penalization is an approach thatcan be used for complex bathymetry regardless of gridstructure, and it can even be used for bathymetry thatevolves in time (e.g., sedimentation, turbines, seafloorearthquakes). Since the ocean is a strongly coupledmultiscale system, accurate representation of the entirerange of scales calls for state of the art numerical meth-ods and techniques. In handling this immense rangeof spatial and temporal scales, there is a need to dy-namically resolve significant structures. The proposedapproach not only solves the ocean-governing equa-tions on an on-the-fly adaptive grid but also providesa computationally efficient technique for representingcomplex boundaries.
The focus of this paper is to present a completeapproach for modeling topographical features in theocean using various forms of volume penalization.Modeling complex boundaries is a pressing issue inthe field of ocean modeling. The majority of currentocean models use body-fitted meshes, which are ex-pensive and often have stability issues when repre-senting boundaries with complicated geometry (Collinset al. 2006). Immersed boundary methods are wellknown for the their efficient implementation of solidboundaries of arbitrary complexity on fixed nonbodyconformal Cartesian grids (Mittal and Iaccarino 2005;Peskin 2002). Immersed boundary techniques are rarelyused in ocean modeling (Tseng and Fersiger 2003).Brinkman penalization, a type of immersed boundarymethod, has been used in many engineering problemsto simulate the presence of arbitrarily complex solid ob-stacles and boundaries (Arquis and Caltagirone 1984).This volume penalization technique is a way to enforceboundary conditions to a specified precision withoutchanging the numerical method or grid used in solvingthe equations. Its main idea is to model arbitrarilycomplex solid obstacles as porous media with poros-ity, φ, and permeability approaching zero. The mainadvantage of Brinkman penalization, when comparedto other penalization methods, is that the error can beestimated rigorously and controlled via a penalizationparameter (Angot et al. 1999). This allows for com-plete control of the accuracy of the boundary condi-tions. Additionally, it can be shown that the penalizedequations converge to the exact solution in the limitas the penalization parameter tends to zero (Angot1999). The work of Adcroft and Marshall (1998) showsthat implementing no-slip boundary conditions can beerror-prone for traditional approaches, while Adcroftet al. (1997) demonstrate the importance of capturing
topographic features that are not neatly captured bythe edges of finite-volume cells. Brinkman penalizationautomatically handles both cases, for arbitrary bottomslope and shelf orientation.
Immersed boundary methods have been developedfor incompressible flows and more recently have beenextended to compressible flows (Liu and Vasilyev2007). Both of these formulations have been adaptedto be used on the ocean-governing equations (includingthe shallow water equations and the hydrostatic andnonhydrostatic primitive equations). This paper will beprimarily focusing on the extension of these methodsto the shallow water equations but will present prelim-inary results of applying Brinkman penalization to theprimitive equations.
In order to model complex geometries, a nonuni-form, adaptive mesh is ideal. For many adaptive mod-els, the main challenge is grid generation. Not only isgrid generation difficult, but the process used is oftena trial and error. It is also computationally expensive.Additionally, many other adaptive grid models requiregrid generation at every time step. Ideally, the gridshould follow the structures in the flow, in additionto adapting to the complicated curves of the bathym-etry. In this work, this is done by the combination oftwo mathematical approaches: Brinkman penalization(Arquis and Caltagirone 1984) and the adaptive wave-let collocation method (Vasilyev 2003; Vasilyev andBowman 2000; Vasilyev and Kevlahan 2002). The adap-tive wavelet collocation method efficiently resolves lo-calized flow structures in complicated geometries, whilethe Brinkman penalization efficiently implements arbi-trarily complex solid boundaries. Brinkman penaliza-tion is a natural technique to use on problems withadaptive methods because the adaptive meshes willensure adequate resolution at the boundary. This isespecially important for problems where the physicsnear the boundaries play a considerable role in theoverall features of the flow.
The hybrid wavelet collocation–Brinkman penaliza-tion method has been previously investigated for threecases: incompressible Navier–Stokes equations both invorticity (Vasilyev and Kevlahan 2002) and primitivevariable formulations (Kevlahan et al. 2000; Kevlahanand Vasilyev 2005), and compressible Navier–Stokesequations in primitive variable formulation (Liu andVasilyev 2007). High Reynolds number flows can besimulated while greatly reducing the number of wave-number modes and controlling the error. The compu-tational cost of the wavelet method is independent ofthe dimensionality of the problem. It is O(N ), whereN is the total number of wavelets actually used. The
Ocean Dynamics (2012) 62:1201–1215 1203
adaptive wavelet collocation method uses second gen-eration wavelets, which allows the order of the methodto be variable. Also, the method is easily applied in bothtwo and three dimensions.
In the following sections, the adaptive waveletcollocation method will be discussed, followed by anexplanation of the extension of the compressible for-mulation of Brinkman penalization to be used on theshallow water equations. It was found that the appli-cation of the existing Brinkman penalization (devel-oped for the compressible equations) on the shallowwater equations was not sufficient and therefore severalmodifications of the Brinkman penalization methodwere made and fully tested. Finally, some preliminarywork is presented on the development of a volumepenalization method to be used on the hydrostatic andnonhydrostatic primitive equations for both no-slip andslip boundary conditions.
2 Adaptive wavelet collocation method
The adaptive wavelet collocation method is a generalmethod for the solution of a large class of linear andnonlinear partial differential equations (Vasilyev andBowman 2000; Vasilyev 2003; Vasilyev and Kevlahan2005; Regele and Vasilyev 2009). The method has al-ready been successfully applied in wide range of fluidmechanics problems, e.g., that of Vasilyev et al. (1997),Vasilyev and Kevlahan (2002), Kevlahan et al. (2007),Reckinger et al. (2010), and Schneider and Vasilyev(2010). In this section, the methodology is briefly re-viewed.
The benefit of using wavelets is that they are local-ized in both space and time. They are ideal for use incomplex flows where localized structures exist in thesolution. The wavelet collocation method takes advan-tage of wavelet compression properties. Functions with
localized structures or regions with sharp transitionsare well compressed using wavelet decomposition. Thiscompression is achieved by keeping only the waveletswith coefficients that are greater than an a priori thresh-old parameter. This allows high resolution computa-tions to be carried out only in the regions where itis necessary. It also allows a solution to be obtainedon a near optimal grid for a given accuracy. Figure 1shows an example of a simulation of the 2010 Chiletsunami using the adaptive wavelet collocation methodto solve the shallow water equations with real variablebathymetry. The right side of the figure shows thatthe grid is localized near the tsunami and near all thecontinental boundaries.
Any function u(x) in an n-dimensional space can bedecomposed as that of Chui (1997), Daubechies (1992),and Mallat (1998)
u(x) =∑
k∈K0
c0kφ
0k(x) +
+∞∑
j=0
2n−1∑
μ=1
∑
l∈Lμ, j
dμ, jl ψ
μ, jl (x), (1)
where φ0k(x) are scaling functions on the lowest level
of resolution and ψμ, jl (x) are the wavelet basis func-
tions. Also, c0k and dμ, j
l are the scaling and waveletcoefficients, respectively. The wavelet coefficients, dμ, j
l ,are small except near areas with large gradients.Equation 1 can be decomposed into two terms whosewavelet coefficients are above and below a chosenthreshold parameter ε,
u(x) = u≥(x) + u<(x), (2)
where
u≥(x) =∑
k∈K0
c0kφ
0k(x)+
+∞∑
j=0
2n−1∑
μ=1
∑
l ∈ Lμ, j
|dμ, jl | ≥ ε‖u‖
dμ, jl ψ
μ, jl (x),
(3)
Fig. 1 Preliminary resultsfrom 2010 Chile tsunamisimulation, with sea surfaceheight on the left and theadaptive grid (colored bylevel) on the right using theadaptive wavelet collocationmethod
1204 Ocean Dynamics (2012) 62:1201–1215
u<(x) =+∞∑
j=0
2n−1∑
μ=1
∑
l ∈ Lμ, j
|dμ, jl | < ε‖u‖
dμ, jl ψ
μ, jl (x), (4)
Donoho (1992) was able to show that for a regularfunction, the error is bounded as
‖u(x) − u≥(x)‖ ≤ C1ε‖u‖, (5)
which means that the number of grid points needed tosolve a numerical problem can be significantly reducedwhile still retaining a prescribed level of accuracy de-termined by the threshold parameter ε.
In the wavelet collocation method, there is a one-to-one correspondence between grid points and wavelets.This makes calculation of nonlinear terms simple andallows the grid to adapt automatically to the solution ateach time step by adding or removing wavelets. In ad-dition to the points with significant wavelet coefficients,several other checks are performed to ensure the reso-lution is sufficient for the given simulation. The way themethod works is, at the beginning of each time step,the wavelet coefficients are calculated. Wavelets withsignificant coefficients are identified. Next, in order toaccount for the evolution of the solution over time, thenearest neighbor wavelet coefficients in position andscale are also added (Liandrat and Tchamitchian 1990).After these significant and adjacent points are kept,the wavelets that are below the threshold ε and arenot in the adjacent zone are removed. It can be shown
that the L∞ error for this approximation is boundedby ε. This allows the grid to automatically follow theevolution of the solution. Then, reconstruction pointsare added, which are points needed to compute thewavelet transforms. Lastly, ghost points are added;these are points needed to calculate spatial deriva-tives. The spatial derivatives are calculated using finitedifferences. Since this method uses second generationwavelets (Sweldens 1998), the order of the wavelet (andalso finite difference) can be easily varied.
Figure 2 shows a one-dimensional example of a solu-tion (top) and its adaptive grid (bottom). The verticallines show the magnitude of the wavelet coefficientsat each location in space for each level of resolution.It is clear that at the location in the center of the x-axis where the solution has a sharp gradient, there islocalized refinement on the grid.
There are some additional computational costs as-sociated with the use of the adaptive multiresolutionwavelet method. Currently, the cost per grid point isapproximately three to five times greater than the costof a standard nonadaptive method. However, in casesof large compressions (Kevlahan and Vasilyev 2005)(up to 103), the compression greatly outweighs this cost.There is also some memory savings associated withusing adaptive methods, which allows higher resolutionsimulations with the same computational resources.
In summary, the dynamically adaptive wavelet col-location method is an adaptive, variable order methodfor solving partial differential equations with localized
0 0.2 0.4 0.6 0.8 1
0
0.5
1
x
f(x)
Original Function
0
1
2
3
4
5
6
xkj
d kj
Interpolating Wavelet Transform
Fig. 2 A one-dimensional example of grid adaptation using the adaptive wavelet collocation method
Ocean Dynamics (2012) 62:1201–1215 1205
structures that change their location and scale in spaceand time. Since the computational grid automaticallyadapts to the solution (in position and scale), we donot have to know a priori where the regions of highgradients or structures exist.
3 Brinkman penalization for shallow water model
The shallow water equations are mathematically similarto the compressible Euler equations. A compressibleformulation of Brinkman penalization has been devel-oped by Liu and Vasilyev (2007). This compressibleform is extended to the shallow water equations,
∂η
∂t= −
[1 +
(1
φ− 1
)χ
]∇ · (ηu), (6)
∂u∂t
+ u · ∇u + 1
Rof k̂ × u = − 1
Fr2∇η − χ
ηpenu, (7)
where the Rossby number, Ro = U/Lf , the Froudenumber, Fr = U/
√gH, the Brinkman penalization pa-
rameter, ηpen � 1, the porosity parameter, φ � 1, and,
χ(x, t) ={
1 if x ∈ Oi(x),
0 otherwise,(8)
which is called a masking function. Oi(x) is any obsta-cle, or, in the case of shallow water ocean simulations,is the continental boundaries.
Analysis of the equations and numerical testing showthat there are three main differences between the shal-low water equations and the compressible equations.These differences result in a different treatment ofthe penalization parameters and the numerical set-upcompared to the compressible form.
The following analysis describes these differences,while also considering amplitude and phase error analy-sis for the case of gravity wave propagation in the smallamplitude limit.
3.1 Wave speeds
One of the assumptions that is made when performingthe error analysis for compressible Brinkman penal-ization is that the speed of sound is the same in thefluid region and porous media region (Liu and Vasilyev2007). For the penalized shallow water equations, thegravity wave speed is different in the fluid versus theporous media. Consider the one-dimensional shallowwater equations,
∂u∂t
+ A∂u∂x
= 0, (9)
where u = (η, ηu)T is the vector of conservative, dimen-sional variables, and the Jacobian matrix is
A =
⎡
⎢⎢⎣
01
φ
gη − (ηu)2
η2
2(ηu)
η
⎤
⎥⎥⎦ . (10)
Note that when φ = 1, the equations reduce to thetraditional shallow water equations. Therefore, φ = 1represents the fluid case, while all other cases are forthe porous media. The eigenvalues of A are
λ = u ±√
u2 − 1
φ(u2 − gη). (11)
Assuming that u/√
gH = O(ε) and η/H = 1 + O(ε),where H is the mean depth of the ocean and ε is somesmall perturbation (ε � 1), the eigenvalues become
λ = u ±√
gHφ
. (12)
When φ = 1, the eigenvalues are u ± √gH as expected
for the shallow water gravity wave speed. However,for any other φ value, the eigenvalues are different.Inside the porous media, where φ � 1, the gravity wavespeed is much larger than in the fluid region. Theimplementation issues related to these difficulties arediscussed in Section 4.2.
3.2 Impedance
Some of the properties associated with the classical the-ory of acoustics (Blackstock 2000) are used in the devel-opment of the compressible formulation of Brinkmanpenalization (Liu and Vasilyev 2007). Consider theplane wave reflection and transmission at the inter-face between two different media. To model the one-dimensional problem of wave propagation from a fluidinto a porous media, it can be thought of as a suddenchange in cross-sectional area (Liu and Vasilyev 2007).In acoustics theory, the acoustic impedance at a givensurface is the ratio of the surface-averaged acousticpressure to the fluid volume velocity,
Z = ρcS
, (13)
where ρ is density, c is gravity wave speed in thefluid (c = √
gH), and S is the cross-sectional area. Inorder to have most of the wave reflected, the obsta-cle’s acoustic impedance needs to be sufficiently large,which is the basis for the impedance mismatch method(Chung 1995). For the shallow water equations, since
1206 Ocean Dynamics (2012) 62:1201–1215
the gravity wave speed is also a function of φ, theimpedance becomes
Z = ρcφ3/2
, (14)
which is a higher impedance than what was found forthe compressible Brinkman penalization formulation,which was Z = ρc/φ (Liu and Vasilyev 2007). Thisallows for a slightly lower range of reasonable φ valuesfor negligible wave transmission in the shallow watercase compared to the compressible case. This is one ad-vantage of the shallow water formulation of Brinkmanpenalization.
3.3 Analogy to Euler equations
The compressible equations in the isothermal limityield similar results as the shallow water equations. The1-D Euler equations with either isothermal or adiabaticconditions (i.e., ∂ P/∂x = c2∂ρ/∂x), reduce to
∂ρ
∂t+ 1
φ
∂ρu∂x
= 0, (15)
∂ρu∂t
+ ∂
∂x(ρuu) + c2 ∂ρ
∂x= − 1
ηpenu. (16)
These assumptions eliminate the energy equation. Un-like the full compressible form of Brinkman penaliza-tion, this form (like the shallow water equation form)only has two penalization parameters, φ and ηpen.
Similar analysis as in Section 3.1 yields the followingeigenvalues:
λ = u ± c√φ
. (17)
This analysis highlights the similarities between theshallow water and compressible equations. Eliminationof the energy equation effectively removes entropywaves leaving only two acoustic waves in the equations,with a subsequent change of wave speed inside theBrinkman zone.
3.4 Amplitude and phase errors by asymptotic analysis
The use of asymptotic analysis provides a way to esti-mate the amplitude and phase errors associated withthe penalized shallow water equations. In addition,by looking at the equations in different asymptoticlimits, information about the behavior of the systemis obtained from a rigorous mathematical viewpoint.The “ocean region” simply refers to the part of thenumerical domain where the shallow water equationsare being solved. The “continental region” refers to the
part of the numerical domain where the penalized shal-low water equations are solved. The following analysisassumes small amplitude waves in the ocean region andis similar to the analysis performed for the penalizedcompressible equations (Liu and Vasilyev 2007).
3.4.1 Asymptotic analysis for the ocean region
The ocean region variables are written as
ηo = 1 + εη′o, (18)
uo = εu′o, (19)
where ε � 1, small perturbations from the mean. Seasurface height is the mean ocean depth plus the seasurface height perturbations, where the velocity is onlya function of the velocity perturbations. If Eqs. 18and 19 are substituted into Eqs. 6 and 7 in the limit oflarge Rossby and Froude numbers, and only the leadingperturbation terms are retained, the result is a waveequation,
∂2η′o
∂t2= ∂2η′
o
∂x2, (20)
∂2u′o
∂t2= ∂2u′
o
∂x2. (21)
Thus, in the fluid or ocean region, the equations reduceto a wave equation.
3.4.2 Asymptotic analysis for continental region
For the continental region, the variables can be writtenas follows:
ηc = 1 + εη′c, (22)
uc = εηpenu′c. (23)
The leading perturbation terms in this case are differentfrom the ocean region because of the strong Brinkmandamping term in the momentum equation. If Eqs. 22and 23 are substituted into Eqs. 6 and 7 and only theleading order terms are retained, the result is a diffusionequation,
∂η′c
∂t= α
∂2η′c
∂x2(24)
∂u′c
∂t= α
∂2u′c
∂x2, (25)
where α = ηpen/φ. Thus, in the porous media or con-tinental region, the equations reduce to a diffusionequation.
Ocean Dynamics (2012) 62:1201–1215 1207
3.4.3 Asymptotic analysis for boundary layer
The asymptotic analysis for the ocean and continentalregion is valid only away from the interface because ofthe length and magnitude scales of the perturbationsused. Therefore, the boundary layer region needs to beanalyzed separately. The boundary layer variables canbe written as follows:
ηbl = 1 + εη′bl, (26)
ubl = εηpenu′bl. (27)
If Eqs. 26 and 27 are substituted into Eqs. 6 and 7 andonly the leading order terms are kept, the result is adiffusion equation,
∂η′bl
∂t= α
∂2η′bl
∂x2, (28)
∂u′bl
∂t= α
∂2u′bl
∂x2. (29)
This is where the penalized shallow water equationsdiffer greatly from the penalized compressible equa-tions. For the compressible equations, there is a bound-ary layer region that provides a natural transition be-tween the two other asymptotic solutions (Liu andVasilyev 2007). That is, in addition to the fluid regionbeing governed by the acoustic wave equation and theporous media region being governed by a diffusionequation, there exists a boundary layer region in be-tween that which is governed by a diffusive wave equa-tion. This boundary layer region does not exist in thepenalized shallow water equations. As demonstratedin the above analysis, the boundary layer region isthe same as the continental region, leaving no natural,mathematical transition. As a result, in order to en-sure numerical stability, the shallow water formulation
requires a numerical boundary layer, which needs to beresolved. This makes it slightly more computationallyexpensive compared to the compressible formulation.Note that the no-slip boundary conditions along thecoastal line are already an approximation, since thedepth of the ocean floor gradually decreases, thus nu-merically smears no-slip boundary conditions, whichmay even result in a more accurate representation ofthe coastal boundary conditions.
4 Numerical experiments and validation of method
4.1 Test setup
A one-dimensional test case is used to verify conver-gence of this new Brinkman penalization technique forthe shallow water equations. A 1-D normal wave isinitialized for the sea surface height. The velocity isinitialized to zero, as shown in Fig. 3. The wave splitsand propagates to the east and west. On the east side, ithits a Brinkman zone; and on the west side, it hits a con-ventional boundary wall, where no-penetration bound-ary conditions are enforced. This allows for comparisonbetween the two methods. After complete reflection,the solution is shown in Fig. 4.
4.2 Parameter study
While there are two parameters specific to the com-pressible Brinkman penalization formulation, φ andηpen, there are two additional parameters to considerwith the shallow water formulation. The first additionalparameter is δpen, which controls how sharp the transi-tion from land to water is at the Brinkman boundary.This parameter is required in the definition of the
η
η
x0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5x 10
−4 Ux
Ux
x0 0.2 0.4 0.6 0.8 1 1.2
−8
−6
−4
−2
0
2
4
6
8x 10
−5
Fig. 3 Example of initial conditions for a 1-D wave
1208 Ocean Dynamics (2012) 62:1201–1215
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
7
8x 10
ηη
x0 0.2 0.4 0.6 0.8 1 1.2
0
2
4
6
8x 10 Ux
Ux
x
Fig. 4 Example of a 1-D wave after complete reflection
masking function, χ , which was stated in general for anysolid boundary in Eq. 8. For a domain with solid straightboundaries on either side of the horizontal domain,a hyperbolic tangent function can be used to definemasking function,
χ = 1
4
[tanh
(x − xO1
δpen
)+ tanh
(x − xO2
δpen
)]2
, (30)
where xO1 and xO2 are the locations of the Brinkmanzone boundaries (or where χ = 1). Therefore, whenxO2 > x > xO1, then χ = 0 and the fluid equations aresolved. When xO1 > x or xO2 < x, then χ = 1 and thepenalized equations are solved. The parameter, δpen,controls how sharp the transition between 0 and 1 is.
The second additional parameter is the length scaleassociated with the initial conditions. This could bedefined in many ways; but for the 1-D normal waveused for this convergence study, it is simply the thick-ness of the wave. The initial conditions for this case areof the form
η(x, 0) = ηamp exp
(− (x − x0)√
δη
2)
, (31)
where ηamp is the amplitude of the sea surface heightwave, x0 is the location of the center of the wave, and√
δη is the thickness of the wave (the length scale ofinterest). If the wave approaching the Brinkman zoneis smoother or sharper, it is going to affect the lengthscale and the error convergence of the problem. Ifthe thickness of the wave approaching the boundary issmaller, the error will be bigger. If the thickness of thewave approaching the boundary is larger, the error willbe smaller.
Both length scales are important to consider whensetting the parameters and reproducing the results
presented here. However, since√
δη and δpen are bothsimply length scales of the problem, it is their ratio thatis important. For all work presented here, these twoparameters will be combined into one, γ = δpen/
√δη.
This means a sharp wave approaching a smooth bound-ary has a similar error to a smooth wave approachinga sharp boundary. For example, in practice, it meansthat if the shallow water formulation of Brinkmanpenalization is being used for wind-driven gyres in abasin, there will be a large length scale associated withthat phenomenon. If it were being used to representbathymetry in tsunami simulations, the tsunami wavelengths are a much shorter length scale. Thus, one canget away with a much smoother mask in a wind-drivengyre problem to maintain the same error as a tsunamisimulation with a sharper mask.
The parameter relationships are studied in order tobe able to control these new and old parameters fordifferent cases. To start, the additional shallow waterequation Brinkman parameters (δpen,
√δη, and the sea
surface height initial conditions) are set to the same val-ues used in the compressible Brinkman formulations inorder to do a direct comparison. Therefore, Fig. 5 showsthe convergence for exactly the same conditions thatare used in the compressible Brinkman formulation.
Comparing the shallow water formulation to thecompressible formulation from Liu and Vasilyev(2007), it is clear that the order of convergence is betterfor the shallow water case for the more limited rangeof parameters. For the compressible formulation, theorder of convergence is O(φ3/4), where for the shallowwater formulation (for the same initial conditions), theorder of convergence is approximately O(φ).
The improved convergence is likely due to the stron-ger impedance inherent in the shallow water Brinkmanformulation. Since a larger porosity parameter results
Ocean Dynamics (2012) 62:1201–1215 1209
10−2
10−1
100
10−3
10−2
10−1
100
Porosity Parameter, φ
Max
imum
Err
or, L
°
O(φ)
Fig. 5 Plot of max error versus porosity parameter, φ, whichdemonstrates convergence of approximately O(φ) (dotted lineshows O(φ) convergence). This convergence uses the same seasurface height incoming wave as that in Liu and Vasilyev (2007)
in more reflection and less wave transmission, the shal-low water formulation converges faster than the com-pressible formulation. The larger porosity requirementis due to the wave speed change inside the Brinkmanregion. Smaller values of porosity depend strongly onthe resolution and on the thickness of the transition be-tween the fluid and the solid, namely δpen. When poros-ity becomes too small, the increase and jump in thewave speed at the Brinkman boundary cause numericalinstabilities and stiffness.
Figure 6 shows the reflected wave with variousporosity parameters after complete reflection for thecompressible formulation comparison case. Figure 4shows the sea surface height and velocity solutions aftercomplete reflection for the entire domain, includingthe Brinkman region. This not only demonstrates how
0.5 0.6 0.7 0.8 0.9 10.9999
1
1.0001
1.0002
1.0003
1.0004
1.0005
x
Sea
Su
rfac
e H
eig
ht,
η
Sea Surface Height varying φ
Exactφ = 2.0e−01φ = 1.0e−01φ = 1.0e−02
Fig. 6 Plot of sea surface height for various porosity parameters
the no-penetration boundary conditions are satisfied inthe Brinkman zone but also shows the exponentiallydecaying solution for the sea surface height in theBrinkman zone. The small amount of wave transmis-sion is negligible because it is actually multiplied by φ
in the solution. These results match the compressibleBrinkman formulation results.
In addition to a direct comparison with the com-pressible Brinkman tests, the following tests are con-ducted to further understand the influence of the pa-rameters, especially with the γ parameter. For thesetests, δpen = 1 × 10−4, 5 × 10−4, and 1 × 10−3, ηamp =2.5 × 10−4, and δη = 5.0 × 10−4. This is equivalent toγ = 4.46 × 10−3, 2.23 × 10−2, and 4.46 × 10−2. Theseparameter values are chosen to give a large range offeasible φ values to test. However, even the largest δpen
parameter tested is smaller than what would be used forpractical applications for the current state of the code.Figure 7 shows convergence results for α = ηpen/φ =10−2.
It is clear from this convergence study that, givena γ ratio, as the porosity parameter decreases, theerror eventually levels off and no longer continues todecrease. The point where the error starts to level offdecreases as γ decreases. This gives an overall rela-tionship between the three parameters that need to bechosen using the shallow water Brinkman penalizationmethod.
It is important to note that these results are for afixed ratio of φ and ηpen (α = ηpen/φ = 10−2). It wasfound in Liu and Vasilyev (2007) that φ > ηpen resultedin the best convergence because the φ parameter ismore forgiving and results in smaller associated errors
10−3
10−2
10−1
100
10−4
10−3
10−2
10−1
100
Porosity, φ
L2 E
rror
(L 2)
=4.46e−3
=2.23e−2
=4.46e−2
O(φγγγ
)
Fig. 7 Plot of L2 error versus porosity parameter, φ for α = 1 ×10−2 and for three different γ . The purple arrows are pointing tothe optimal φ for each of the given γ parameters tested
1210 Ocean Dynamics (2012) 62:1201–1215
10−4
10−3
10−2
10−1
100
10−7
10−6
10−5
10−4
L2 Error
Porosity, φ
L2 E
rror
(L 2)
O(φ0.5)
Fig. 8 Plot of L2 error versus porosity parameter, φ for α = 1and for γ = 4.46 × 10−4
than with the ηpen parameter. For comparison, severalcases were tested with α = 1 and for γ = 4.46 × 10−4.These results are shown in Fig. 8. This shows that theerror convergence is not as strong when the φ and ηpen
are equal.
4.3 Practical implementation
To set the Brinkman penalization parameters for theshallow water formulation, the procedure is as follows.First, set the sharpness of the Brinkman mask, i.e., thenumerical thickness of the smoothed coastal bound-
ary. Ensure that the maximum allowable resolution issufficient to resolve it. For the adaptive wavelet colloca-tion method, five to ten points is enough to resolve δpen.Assuming the sea surface height initial conditions areknown, and therefore, the length scale of the problemis known, γ = δpen/
√δη can be calculated. Using Fig. 7
as a guideline, the optimal φ can be estimated. To getηpen, maintaining the ratio, α = ηpen/φ = 10−2, is bestfor minimizing error (not to mention, this ratio hasbeen tested extensively). However, ηpen < φ will giveexcellent Brinkman results. Although there are variousways to set up these parameters, this procedure servesas a general guideline to pick parameters in a way thatwill minimize errors.
4.4 Cost of Brinkman penalization
There is added cost associated with the implementationof Brinkman penalization. Depending on the δpen used,the computational cost to resolve the Brinkman bound-ary may be higher than a conventional boundary wall.This is unavoidable but does contribute to improvingthe accuracy at the boundaries, so in many cases it iswell worth it. The other aspect of Brinkman penaliza-tion is the additional domain space. Adding a Brinkmanzone means the computational domain needs to belarger, which also increases the computational cost.However, this can easily be minimized by making thezone as small as possible. It is not cost-effective to useBrinkman penalization to define straight boundaries. Itis a technique to accurately represent complex, variable
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
0.5
Sea Surface Height
x
y
-0.5-0.5
0.5
Sea Surface Height
x
y
-0.5
0.5
0.75-0.75 -0.5 0.5
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
Velocity
0
0.005
0.01
0.015
Velocity
x
y
-0.5
0.5
0.75-0.75 -0.5 0.5
0.5x
y
-0.5-0.5
0.5
0
0.005
0.01
0.015
Adaptive Grid
x
y
-0.5
0.5
0.75-0.75 -0.5 0.5
0.5
Adaptive Grid
x
y
-0.5-0.5
0.5
Brin
kman
Pen
aliz
atio
n
Trad
ition
al B
ound
arie
s
Fig. 9 2-D plots of sea surface height, velocity, and the adaptive grid for a wind-driven double gyre circulation using Brinkmanpenalization
Ocean Dynamics (2012) 62:1201–1215 1211
10−1
100
10−1
100
Porosity Parameter, φ
Max
imum
Err
or
O(φ)
Fig. 10 2-D convergence study showing max error versus poros-ity parameter for 2-D wind-driven single gyre in a square basin.The dotted line shows O(φ) convergence, for comparison
geometry boundaries, in which case the added compu-tational cost should be expected.
4.5 Two dimensional wind-driven double gyrewith Brinkman penalization
Various two dimensional studies have verified the newformulation of Brinkman penalization. The first case isa 2-D wind-driven double gyre test case in a rectangulardomain. This test case was solved using the adaptivewavelet collocation method with the Brinkman penal-ization method used for flat-bottom rectangular do-main. The results using Brinkman were compared tothe same test case solved using the adaptive waveletcollocation method but applying conventional no-slip
boundary conditions on the side walls. Although no-slipboundary conditions are not always used in Munk gyreproblems, it has been done extensively (Fox-Kemperand Pedlosky 2004; Fox-Kemper 2004). It is well knownthat at higher Reynolds number, subtle variations inviscous treatments near the boundaries can stronglyaffect the basin-wide flow (Fox-Kemper and Pedlosky2004; Fox-Kemper 2004). Direct comparison to thequasigeostrophic simulations in those papers with theBrinkman method is underway. Lastly, analytic solu-tions similar to this test case are given in Pedlosky(1996); however, those solutions are linear and quasi-geostrophic, not the nonlinear shallow water solutionpresented here. Therefore, comparison to numericalboundary conditions is simpler. These two results showqualitative agreement, as can be seen in Fig. 9.
A complete two dimensional convergence study isnot necessary, since a thorough 1-D convergence studyhas been completed. However, to verify that 2-D con-vergence is similar, a small range of parameters aretested using a 2-D wind-driven single gyre test case.Figure 10 shows that the convergence is slightly lessthan O(φ).
Studies of variable bathymetry cannot be comparedto conventional boundary conditions but can verify therobustness of the method. There is a short transienttime for this steady state solution to adjust to theBrinkman penalization. This is the case for a rectangu-lar domain, as well. It takes the solution slightly longerto adjust for a nonrectangular domain but still on a timescale much shorter than the time scale of the problem.With better initial conditions, this can be even furtheravoided. For boundary currents, it is the steady statesolution that is of interest, so it is even less of an issue.Figure 11 shows the solution and grid for a wind-drivengyre in a nonrectangular domain. The grid shows that
Velocity
x
y
Adaptive Grid
x
y
-0.75 0.75
0.5
-0.5-0.75 0.75
0.5
-0.5
Fig. 11 2-D plots of velocity and the adaptive grid for a wind-driven double gyre circulation using Brinkman penalization for variablebathymetry
1212 Ocean Dynamics (2012) 62:1201–1215
0 1
0z,
dep
th
x
0 1
0
x
0 1
0
x
0 1
0
x
Fig. 12 Demonstration of no-slip boundary conditions with andwithout Brinkman penalization. The test case shown here isa constant horizontal wind force on the top surface (decaying
exponentially downward) with no-slip boundary conditions onthe sidewalls, which creates an x–z circulation pattern convenientfor testing bottom boundary conditions
the method is not only adapting the boundary regionbut is also adapting to the dominating circulating gyrestructure.
5 Volume penalization for modeling bathymetry
The ocean bathymetry is a highly varying, intricate, andcomplex surface. Using current techniques for repre-sentation of this bottom boundary results in a surfacethat is either too crude (stair-step representation) ortoo expensive (body-fitted meshes). The incompress-ible formulation of Brinkman penalization (Arquis andCaltagirone 1984) is implemented by adding the term,−(χ/ηpen)u, to the momentum equations. When ap-plied to the hydrostatic primitive equations, the follow-ing equations result:
∂η
∂t+
(uh
∣∣∣∣z=zmax
)· ∇hη =
(w
∣∣∣∣z=zmax
), (32)
∂uh
∂t+u · ∇uh+ 1
Rof k̂ × uh =−∇h P+ 1
Re∇2uh− χ
ηpenu,
(33)
∂w
∂z= −∇huh,
(w
∣∣∣∣z=zbottom
= 0
), (34)
∂ P∂z
= − 1
Fr2,
(P
∣∣∣∣z=zmax
= η
Fr2
). (35)
The added Brinkman term forces the velocity to bezero in the Brinkman zone. As a first attempt to utilizevolume penalization to represent ocean boundaries,Brinkman penalization for no-slip boundary conditions
is used as a benchmark case. Results are shown inFig. 12 compared against no-slip wall conditions di-rectly applied to the boundaries.
In addition, the convergence of the no-slip Brinkmanpenalization is verified. Figure 13 shows the strong errorconvergence with decreasing penalization parameter.
Comparison of the solution profiles demonstratesaccurate representation of no-slip conditions using thismethod. However, for the large scale ocean modelingof interest, no-slip boundary conditions are not neces-sary or realistic. To avoid resolving the boundary layerassociated with no-slip conditions, it is convenient toextend this methodology to slip conditions, ∂u/∂z =κu. This boundary condition is somewhat more generalthan a strict slip boundary condition of ∂u/∂z = 0.However, the latter would require us to impose anadditional boundary layer drag parameterization such
10−5 10−4 10−3 10−2 10−110−10
10−9
10−8
10−7
10−6
10−5Error Convergence
ηpen
Err
or
Fig. 13 A plot of error convergence for decreasing penalizationparameter for no-slip conditions
Ocean Dynamics (2012) 62:1201–1215 1213
10−1
0Velocity−no Penalization
z, d
epth
10
−1
0Velocity−Penalization
x
x
01−1
0Grid−no Penalization
x
0 1
−1
0Grid−Penalization
x
1
Fig. 14 Demonstration of slip boundary conditions with and without Brinkman penalization for κ = 1
as Ekman layer drag (see, e.g., Pedlosky 1987, chap. 4.5)or quadratic drag (e.g., Arbic and Scott 2008); the formused here is a simple step in that direction. The ideabehind this method is very similar to the no-slip case.The term added has a time scale much smaller than thetime scale of the problem, so the boundary conditionsare applied on this small time scale, ηpen.
Two different volume penalization methods are de-veloped and tested. The full set of penalized governingequations are
∂η
∂t+
(uh
∣∣∣∣z=zmax
)· ∇hη =
(w
∣∣∣∣z=zmax
), (36)
(a)
∂uh
∂t+ u · ∇uh + 1
Rof k̂ × uh
= −∇h P + 1
Re∇2uh + χ
ηpen
(−κ
∂uh
∂z+ ∂2uh
∂z2
),
(37)
(b)
∂uh
∂t+ u · ∇uh + 1
Rof k̂ × uh
= −∇h P + 1
Re∇2uh + χ
ηpen
(−κ2uh + ∂2uh
∂z2
),
(38)
∂w
∂z= −∇huh,
(w
∣∣∣∣z=zbottom(xh)
)= 0, (39)
∂ P∂z
= − 1
Fr2,
(P
∣∣∣∣z=zmax
= η
Fr2
), (40)
where (a) and (b) are two different volume penalizationoptions.
For the flat-bottom test problem, both methods con-verge equally well. The test used is a wind-forced cir-culation in the zonal and depth direction. The windforces only the top surface, but there is a smooth tran-sition to zero through the top half of the domain toavoid numerical instability. This case demonstrates theadvantage of slip conditions, while also showing what
10−1
0Velocity−no Penalization
z, d
epth
10
−1
0Velocity−Penalization
x
x
0−1
0Grid−no Penalization
x
0 1
−1
0Grid−Penalization
x−8 −6 −4 −2 0 2
x 10−3
1
Fig. 15 Demonstration of slip boundary conditions with and without Brinkman penalization for κ = 10
1214 Ocean Dynamics (2012) 62:1201–1215
slip conditions look like at the boundary for a simplecirculation problem. The velocity profile is extremelysensitive to the value of κ . Figures 14 and 15 show plotsof velocity and the adaptive grid for both κ = 1 andκ = 10. Additionally, as a result of the steeper slopein a higher κ value, there is a finer resolution near thepenalization zone.
This new volume penalization formulation enforcesslip boundary conditions. The advantage of such aformulation is the computational savings associatedwith not needing to resolve the boundary layer at theboundaries, which is necessary with the conventionalBrinkman penalization no-slip formulation. This samevolume penalization has also been applied to the non-hydrostatic equations and initial testing has indicatedthat it performs similarly.
6 Concluding remarks
Various formulations of Brinkman penalization hasbeen developed and tested, all solved using the adap-tive wavelet collocation method. The shallow waterformulation of Brinkman penalization has been thor-oughly discussed such that applying it to new anddifferent ocean problems using any numerical methodcan be done. The key to successfully using this penaliza-tion is to carefully and intentionally choose all numeri-cal parameters, as discussed in this paper. Even thoughBrinkman penalization can be used in combinationwith any numerical methodology, the use of adaptivemesh refinement techniques such as the wavelet collo-cation method considerably improves its versatility androbustness, mainly due to the completely automateddefinition and adequate resolution of complex geom-etry. Lastly, one particular advantage of Brinkman pe-nalization is the ability to represent not only complexbut also evolving bathymetry.
The preliminary work on volume penalization forapplying slip conditions at the bottom bathymetryboundary for use in the nonhydrostatic and hydro-static primitive equations was also presented. Furtherwork includes generalizing this to a normal derivativecondition rather than a vertical derivative condition, aswell as testing in three dimensions.
Acknowledgements This work was supported by DOE-CCPP(DE-FG02-07ER64468). BFK was supported by NSF FRG0855010. Also, thanks to Scott Reckinger.
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