Simulating Upwelling in a Large Lake Using Slippery Sacksjensen/SSmodel.pdf · carried out to test...

66 VOLUME 132M O N T H L Y W E A T H E R R E V I E W

q 2004 American Meteorological Society

Simulating Upwelling in a Large Lake Using Slippery Sacks*

PATRICK T. HAERTEL

Department of Atmospheric Sciences, University of North Dakota, Grand Forks, North Dakota

DAVID A. RANDALL

Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

TOMMY G. JENSEN

International Pacific Research Center, University of Hawaii at Manoa, Honolulu, Hawaii

(Manuscript received 15 January 2003, in final form 26 June 2003)

ABSTRACT

A Lagrangian numerical model is used to simulate upwelling in an idealized large lake. This simulation iscarried out to test the model’s potential for simulating lake and ocean circulations.

The model is based on the slippery sack (SS) method that was recently developed by the authors. It representsthe lake as a pile of conforming sacks. The motions of the sacks are determined using Newtonian dynamics.The model uses gravity wave retardation to allow for long time steps and has pseudo-Eulerian vertical mixing.

The lake is exposed to northerly winds for 29 h. Upwelling develops in the eastern edge of the basin, andafter the winds shut off, upwelling fronts propagate around the lake. This case was previously simulated usinga height- and sigma-coordinate ocean model. The SS model produces circulations that are similar to thoseproduced by the other models, but the SS simulation exhibits less mixing.

1. Introduction

In this study we modify our slippery sack (SS) free-surface fluid model (Haertel and Randall 2002, hereafterHR02) and use it to simulate upwelling in an idealizedlarge lake. This simulation is carried out to test themodel’s potential for simulating lake and ocean circu-lations. In this section we review the SS method, discussthe potential merits of an SS ocean model, and explainwhy we select lake upwelling as a test problem.

a. The SS method

The SS method is a Lagrangian numerical methodthat has similarities with both smoothed particle hydro-dynamics (Monaghan 1992) and oceanic applications ofthe particle-in-cell method (e.g., Pavia and Cushman-Roisin 1988). Under the SS method a fluid is represented

* School of Ocean and Earth Science and Technology ContributionNumber 6237 and International Pacific Research Center ContributionNumber 225.

Corresponding author address: Dr. Patrick T. Haertel, Dept. ofAtmospheric Sciences, University of North Dakota, Box 9006, GrandForks, ND 58202.E-mail: [email protected]

as a pile of conforming sacks (e.g., Fig. 1). The sacksare assigned a stacking order that specifies their relativevertical positions and is usually defined so that densityis a nonincreasing function of height. Horizontal posi-tions xi and velocities vi of sacks are determined usingNewtonian dynamics:

dxi 5 v , (1)idt

Fdv pi i1 f k 3 v 5 1 D , (2)i vidt Mi

where i is the sack index, t is time, f is the Coriolisparameter, k is the unit vector in the vertical, F is thepi

horizontal force on sack i resulting from pressure, Dvi

is the diffusive tendency of velocity, which representsmomentum transports by subsack-scale processes, andMi is the mass of sack i. Equations (1) and (2) are easilystepped forward in time (e.g., by Adams–Bashforth timedifferencing); the challenge of solving them is diag-nosing F and D .p vi i

Each slippery sack is assumed to have a horizontalmass distribution mi(x9) that is constant with respect totime in the sack’s frame of reference (x9 denotes hori-zontal position relative to the sack center). Each sack isalso assumed to have a spatially uniform in situ densityri. It follows that a sack’s vertical thickness Hi is

JANUARY 2004 67H A E R T E L E T A L .

FIG. 1. A pile of slippery sacks in a basin.

FIG. 2. The normalized velocity difference from a high-resolutionfinite-difference solution as a function of sack size for SS simulationsof internal and external gravity waves in a two-layer system (fromHR02). The dotted line has a slope of 22.

m (x 2 x )i iH (x) 5 , (3)i ri

and the horizontal force on a sack resulting from hy-drostatic pressure is

k

F 5 g=H (r 2 r )HOp E i j i ji [j5i11

k

1 r b 1 H dA, (4)Oi j1 2]j51

where the integral is evaluated over the horizontal pro-jection of sack i, g is gravity, k is the total number ofsacks, sacks with lower indices lie below sacks withhigher indices, b is the height of the bottom topography,and A is the horizontal area measure [this equation is avariant of Eq. (7) in HR02 that was adapted to includebottom topography]. Equation (4) may be approximatedwith a Riemann sum, which leads to the conservationof energy in the limit as the time step approaches zeroand requires O(k) operations to evaluate for k sacks(HR02).

Equations (1)–(4) are not derived from equations ofmotion for a continuous fluid; rather, they are derivedby making a few assumptions about sack properties andapplying Newton’s second law to each sack (HR02).However, it turns out that solutions to (1)–(4) approx-imate solutions to the equations of motion for an in-compressible, hydrostatic fluid having a free surface.HR02 use (1)–(4) to simulate a nonlinear deformation,internal and external gravity waves, and Rossby waves,and they compare the SS solutions to analytic and high-resolution finite-difference solutions for continuous flu-ids. For each test case, the SS solutions rapidly convergeto the control solution as sack sizes are reduced. Forexample, Fig. 2 shows the dependence of the normalizedvelocity difference from the control for SS simulationsof gravity waves within a two-layer system. The velocitydifference is approximately proportional to the squareof sack width; that is, the SS method is found to besecond-order accurate.

In this study we incorporate diffusion, time-depen-

dent sack densities, and tracers other than density intothe SS method. In addition to adding the term D tovi

(2), we include the following generic tracer diffusionequation:

dsi 5 D , (5)sidt

where s denotes an arbitrary tracer and Ds denotes thediffusive tendency of s (see section 3 for a discussionof calculating diffusive tendencies). When (5) is usedto diffuse active tracers such as temperature and salinity,sack densities can change, and they are diagnosed withan equation of state. For example, for the upwellingsimulation, which is for a freshwater lake, temperatureis diffused and density is diagnosed using the UnitedNations Educational, Scientific, and Cultural Organi-zation (UNESCO) equation of state for pure water (e.g.,Gill 1982, p. 599). While we have never tested an equa-tion of state having a pressure dependence, in theory,incorporating such a dependence would be straightfor-ward and, in that case, (1)–(5) would approximate theprimitive equations. Note that (1)–(5) do not invoke theBoussinesq approximation; a sack’s horizontal accel-eration depends on its density.

b. The potential merits of an SS ocean model

The primary advantage of the SS model over existingocean models is that it has negligible advection errors.To illustrate this point we present an SS simulation ofa simple tracer advection problem and compare it to afinite-difference solution. Consider a two-dimensionalpool of water 1 m deep and 20 m across in a periodicdomain. Suppose that there is a tracer within the waterhaving a Gaussian distribution in x with a radius of 2m. Suppose also that the water has a uniform velocityof 1 m s21 in the positive x direction and that there isno tracer diffusion. This problem has an analytic so-lution; the tracer distribution maintains its shape as it


FIG. 3. Slippery sack and finite-difference simulations of traceradvection. (a) Sack outlines at 0 s. (b)–(d) Tracer distributions at 0,5, and 20 s, respectively, for the SS (solid) and finite-difference(dotted) solutions.

moves with the water. Since the domain is periodic, after20 s the tracer distribution is identical to the initial con-dition.

For the SS simulation of this problem we representthe water as a pile of 40 sacks (Fig. 3a), each having amaximum depth of 1 m, a radius of 0.5 m, an initialvelocity of 1 m s21, and a tracer concentration equal tothe value of the Gaussian distribution at the sack’s center(Fig. 3b). We solve (1)–(5) using forward time differ-encing with a time step of 0.001 s (the time differencingintroduces no errors in this case, but a small time stepis used so that identical time differencing may be usedfor the finite-difference solution to produce negligibletime-differencing errors). The SS simulation behavesjust like the analytic solution (Figs. 3b–d); the tracerdistribution maintains its shape, moves in the positivex direction at 1 m s21, and at 20 s (Fig. 3d) it is thesame as the initial distribution (Fig. 3b). Althoughround-off errors produce tiny deviations in sack posi-tions at 20 s from their initial positions, the histogramof the tracer is exactly conserved. In contrast, a finite-difference solution having 40 grid points, second-ordercentered spatial differencing, and the same time differ-encing degrades steadily with time (Figs. 3c,d). It ex-

hibits unphysical negative tracer values and a reducedtracer maximum, and the tracer is advected too slowly.

In addition to having negligible advection errors, theSS method also has the following appealing properties:1) it does not require horizontal diffusion or filteringfor numerical stability, 2) it represents lower and lateralboundaries in a simple and natural way, and 3) its com-putational efficiency is competitive with Eulerian mod-els. These properties would appear to make the SS meth-od well suited for lake and ocean simulations, whichare computationally intensive and are known to be ad-versely affected by advection errors, excessive mixing,and boundary-related errors (e.g., Roberts et al. 1996;Pacanowski and Gnanadesikan 1998; Griffies et al.2000; Beletsky and Schwab 2001). Of course, there arepotential disadvantages of an SS ocean model as well,and these are discussed in section 5.

c. Lake upwelling: A good test problem

In this paper we use our SS model to simulate up-welling in an idealized large lake. The case we selectwas previously simulated by Beletsky et al. (1997) usingboth a z- and s-coordinate ocean model. The lake, whichsits in a parabolic basin, is exposed to a surface windstress forcing that lasts 29 h. Upwelling develops inresponse to the forcing, and, after the forcing shuts off,upwelling fronts propagate around the lake. This caseprovides a good test of the SS method’s potential forsimulating lake and ocean circulations for the followingreasons: 1) it is a simple example of the response of abody of water to a surface wind stress forcing, 2) thelake has variable bottom topography, 3) the responseinvolves internal gravity wave dynamics and boundarycurrents, and 4) unlike an ocean simulation, the up-welling simulation does not require open boundary con-ditions.

In order to adapt the SS model for the upwellingsimulation we make it more computationally efficientand we implement vertical diffusion. These changes aredescribed in sections 2 and 3, respectively. The up-welling simulation is presented in section 4, section 5is a discussion, and section 6 is a summary.

2. Computing efficiently

The following ‘‘back-of-the-envelope’’ calculationsuggests that the SS model should require about 1–2times as much computer processing per sack per timestep as a typical Eulerian model requires per grid pointper time step. The SS model spends most of its timecalculating the pressure force on sacks. This requiresevaluating the sack thickness gradient 36 times per sack[we approximate (4) using a Riemann sum having sixpoints per sack width in each horizontal dimension].The analog in an Eulerian model, evaluating the pressuregradient at a grid point, requires only a single calculationfor each grid point. The latter makes up a few percent


of the total computations required for a grid point (whichinclude calculating advective and diffusive fluxes formomentum and tracers in all directions). Thus, the SSmodel should require about the same amount of time toevaluate the pressure force on a sack as a typical Eu-lerian model requires to complete all calculations for agrid point. Excluding the pressure-force calculation, theSS model requires fewer computations per sack than anEulerian model requires per grid point; for the SS modeladvection is trivial and horizontal diffusion is not re-quired for many applications.

The SS model described in HR02 was not even closeto being as fast as the above calculation predicts, butsince that paper was written we have made four changesto the model that have made it about 600 times fasterfor most lake and ocean applications, and its speed isnow approaching that predicted. These changes are de-scribed below.

a. Split time differencing

The time derivative in (2) may be partitioned intothree components: the Coriolis acceleration, the pressureacceleration, and the diffusive tendency. Formerly, theSS model used second-order Adams–Bashforth time dif-ferencing for all components. The current model usesthird-order Adams–Bashforth time differencing for theCoriolis and pressure accelerations and forward timedifferencing for the diffusive tendency. For the up-welling simulation, the Coriolis and pressure time stepis held fixed at 200 s, and the viscous time step is thesame during weak winds, but it is reduced (i.e., sub-cycled) to as low as 50 s during the strongest windswhen the coefficient of vertical diffusion is the highest.This allows the model to calculate the pressure force,the most computationally costly part of a time step, asfew times as possible.

b. Polynomial mass-distribution function

The sack mass-distribution function used by HR02 isa cosine-squared bell function, and it was selected be-cause it has several appealing properties: 1) it is con-tinuous, 2) it has a continuous horizontal gradient thatvanishes at the sack’s edges, and 3) it is easy to constructpiles that are perfectly level using sacks with this massdistribution. However, this function is computationallyexpensive because computers approximate trigonomet-ric functions using power series that require many float-ing point operations to evaluate. The following massdistribution has a similar shape and the same appealingproperties, but it requires fewer floating point operationsto evaluate:

Mi 2m(x9, y9) 5 [1 1 (2|x9| 2 3)(x9) ]r rx y

23 [1 1 (2|y9| 2 3)(y9) ], (6)

where rx and ry are the sack radii in the x and y direc-tions, respectively; x9 5 (x 2 xi)/rx, y9 5 (y 2 yi)/ry;and m is nonzero only for | x9 | , 1 and | y9 | , 1.Evaluating this function and its horizontal gradient at apoint requires evaluating two third-order polynomialsand two second-order polynomials, a task that modernprocessors accomplish very quickly (e.g., a 2-gHz Pen-tium processor can do this about 40 million times persecond). Tests have revealed that using this mass dis-tribution instead of the cosine-squared bell function bellchanges solutions little, as long the Riemann sum thatapproximates (4) has a sufficient horizontal resolution(at least six points per sack width in each horizontaldimension).

c. Gravity wave retardation

Under the SS method a pile of sacks has a free surfaceand supports the rapid oscillations of external gravitywaves. In order for these waves to be stable one mustuse a pressure-acceleration time step on the order of thetime it takes a wave to cross a sack radius. Therefore,by reducing the phase speed of external gravity waves,one can increase the maximum stable time step. In thissection we show that gravity waves can be slowed underthe SS method by gravity wave retardation (GWR),which Jensen (1996, 2001, 2003) used to lengthen timesteps by factors of 4–16 in several ocean simulations.One advantage GWR has over mode splitting, an alter-native used in some ocean models (e.g., Bleck and Smith1990; Ezer and Mellor 1997), is that GWR does notrequire separate solutions for internal and externalmodes.

To implement GWR one simply reduces the portionof the pressure force associated with the external modeby a constant factor. In (4) the pressure force is decom-posed into internal and external components, associatedwith the first and second terms in brackets, respectively.Multiplying the external component by a positive con-stant g yields

k

F 5 g=H (r 2 r )HOp E i j i ji [j5i11

k

1 gr b 1 H dA. (7)Oi j1 2]j51

Setting g , 1 slows the phase speed of the externalmode by the multiple while leaving the phaseÏgspeeds of internal modes unchanged.

To illustrate the effects of GWR we repeat the gravitywave simulation presented in HR02, using both g 5 1(no GWR) and g 5 1/2. For this test a two-dimensionalfluid comprising two 1-m deep layers with densities r1

5 1100 kg m23 and r2 5 1000 kg m23 is initializedwith a lower-layer momentum perturbation that has aGaussian radius of 1 m and an amplitude of 1 cm s21.Gravity is set to 1 m s22, and there is no viscosity or


FIG. 4. An internal gravity wave simulation that illustrates theeffects of GWR. Velocity of the lower (solid) and upper (dashed)layers at 5 s for (a) g 5 1 and (b) g 5 1/2.

rotation. We represent the fluid using 40 sacks for eachlayer, each having a radius of 1/2 m, and we approximate(3) using a Riemann sum with a resolution of 1/6 m.

For g 5 1, the solution (Fig. 4a) is very similar tothe one presented in HR02 (see their Figs. 5d,f ). Thisis what we expect, since the only differences betweenthe two simulations are the magnitude of the momentumperturbation, the shape of the mass-distribution func-tion, and the resolution of the Riemann sum used toapproximate the pressure equation. The momentum per-turbation projects onto both internal and external gravitywaves. By 5 s, they have propagated away from thecenter of the domain where the momentum perturbationwas originally located (Fig. 4a). The velocity pertur-bations in the two layers are in (out of ) phase for theexternal (internal) waves, as predicted by linear theory(e.g., Gill 1982, p. 119). When we repeat the simulationsetting g 5 1/2 the external waves are slowed by themultiple , but the internal waves propagate at theÏ1/2same speed (Fig. 4b).

One side effect of GWR is that surface height per-turbations associated with balanced external-mode cir-culations are increased by the factor g 21. For most lakeand ocean applications surface height perturbations aresmall (tens of centimeters in amplitude), and this am-plification is not a problem as long as g is not too small.For comparison with observations one need only rescalethe model surface height perturbations by multiplyingby g (Jensen 1996). Errors in velocities caused by GWRhave been found to be small in the open ocean for bothbaroclinic (Jensen 2001) and barotropic (Jensen 2003)modes for g . 0.01. Tobis (1996) investigated the im-pact of GWR on baroclinic instability and found thatsolutions were fairly accurate provided that the ratio ofthe modified external to internal radius of deformationremained large, which was the case for g . 1/64. For

the upwelling simulation we use g 5 0.02, which allowsincreasing the time step by a factor of about 7.

Gravity wave retardation has not yet been thoroughlytested for coastal waters. The increased surface eleva-tion response would most likely influence the magnitudeof strong currents in shallow areas (perhaps forcing oneto use an even larger value of g than those discussedabove). However, recall that implicit methods also resultin large errors when used with a large Courant number(Fig. 6.1 of Mesinger and Arakawa 1976, p. 56). Intheory, GWR should not have a large effect on the prop-agation of shelf waves or topographic waves. For ex-ample, the phase speed of the topographic wave is in-dependent of gravity, while the longshore velocity de-pends on the product of gravity and the surface heightgradient (Csanady 1982, p. 131), and this product isunaltered by GWR. The effects of GWR on storm surgeshave not yet been investigated.

d. Coding/compiling changes

The SS model is coded in C11. A sack’s position,velocity, temperature, salinity, mass, and density arerepresented with the user-defined type sack. A linkedlist of pointers to sacks is maintained, and this list issorted by sack density to maintain neutral or stable strat-ification. Since HR02 was written, we have made thefollowing changes to model’s code and the way it iscompiled: First, we no longer represent sack positionand velocity with user-defined types. The mathematicaloperations on the built-in types are evaluated much morequickly. Second, the linked list of sack pointers is nolonger sorted every time step (i.e., a weakly unstablestratification is allowed for short periods of time). Forthe upwelling simulation we found that only sortingonce per hour changed the simulation little. Third, ratherthan evaluating (7) for one horizontal position at a time,we now evaluate it for one sack at a time; that is, thevalue of each of the sums at each horizontal position ismaintained in an array. This means that most of theretrievals of sack variables are from the memory cache.Fourth, we now compile the model using the Intel com-piler, which produces optimal code for the Pentium 4processor on which the model is run.

3. Vertical diffusion

One way to parameterize vertical mixing of momen-tum and tracers by subgrid-scale processes in a lake orocean model is to calculate height-dependent coeffi-cients of vertical viscosity and tracer diffusivity fromvertical profiles of horizontal velocity and temperature/salinity or density (e.g., Pacanowski and Philander1981; Large et al. 1994; Beletsky et al. 1997). In thissection we present an implementation of vertical dif-fusion for the SS method that was developed to facilitatethe use of such parameterizations. The representation ofdiffusion presented here is used to calculate diffusive


FIG. 5. The diffusion–advection simulation. The tracer concentra-tion for the finite-difference solution is contoured with an interval ofone tracer unit. Centers of sacks in the SS solution are marked withboxes having sizes proportional to the tracer values. The times shownare (a) 0, (b) 5, and (c) 15 s.

tendencies for both velocity [D in (2)] and tracersvi

[D in (5)] in the SS model.si

a. Pseudo-Eulerian diffusion

Including vertical diffusion in an Eulerian fluid modelis straightforward. For example, a diffusive tendency]q/]t can be set equal to the difference between diffusivefluxes into and out of a grid box divided by the amountof mass in the box:

]q 1i 5 (Q 2 Q ), (8)i21/2 i11/2]t Mi

where i is the vertical index of a grid point, the halfindexes denote values at locations halfway between gridpoints (i.e., at the top and bottom of the grid box), Mi

denotes the total mass in the grid box, and Q denotesthe vertical diffusive flux, which may be approximatedas follows:

q 2 qi11 iQ 5 2k r dA, (9)i11/2 i11/2 i11/21 2Dz

where k is the coefficient of diffusion and Dz denotesthe difference in the height of grid points i and i 1 1.

To implement vertical diffusion under the SS methodwe simply divide the model domain into vertical col-umns, associate each sack with the column that its centerlies in, and apply the above finite-difference approxi-mation to each column of sacks. We set the columnwidth equal to the sack radius in each dimension, Dzequal to half of the sum of the maximum vertical thick-nesses of sacks i and i 1 1 (i.e., the vertical distancethat would separate the sack centers if they were per-fectly aligned), and ri11/2 5 (ri 1 ri11)/2. Differentdiffusion coefficients are used for momentum and trac-ers (temperature and salinity), and a sack’s density isrecalculated after each Coriolis/pressure time step (de-fined in section 2) using an equation of state.

b. Diffusion–advection simulation

To illustrate the above implementation of diffusion, weapply it to a simple problem that involves both diffusionand advection. Consider a two-dimensional fluid 10 mdeep and 20 m across on a periodic domain. Suppose thatthe x velocity of the fluid is 1/2 m s21 at the top, 21/2m s21 at the bottom, and varies linearly in between. Nowsuppose a tracer is released in the center of the fluid,initially having a Gaussian distribution with a radius of 2m in each dimension and a maximum value of 10 units.

We compare an SS and a finite-difference solution tothis problem. The SS solution uses 200 sacks, each hav-ing a radius and a depth of 1 m. A sack’s velocity isprescribed to be the value of the linear velocity profileat the vertical midpoint of the sack. The finite-differencesolution uses 200 points with Dx 5 Dz 5 1 m. Bothsolutions use (7)–(8) to represent diffusion, forward

time differencing with a very small time step (0.001 s),and k 5 1 m2 s21. The finite-difference solution usescentered differencing to calculate both horizontal andvertical gradients of the tracer.

For both solutions the tracer coverage is sheared hor-izontally as it spreads vertically (Fig. 5). Owing to ad-vection errors the finite-difference solution containssmall regions having negative tracer values (Fig. 5b).In the SS solution the tracer is advected slightly morerapidly (Fig. 5c); this difference is also attributable toadvection errors in the finite-difference simulation (seealso Figs. 3c,d). Overall, however, the two solutions arequite similar, suggesting that the SS pseudo-Eulerianvertical diffusion behaves similarly to the standard Eu-lerian finite-difference representation of vertical diffu-sion. We do not, however, suggest that the former is asaccurate as the latter. The SS pseudo-Eulerian diffusionhas errors proportional to the first order of the columnspacing and the sack height, whereas errors for the Eu-lerian finite-difference representation of diffusion de-pend only on the vertical grid spacing Dz and are pro-portional to Dz and (Dz)2 for nonuniform and uniform


FIG. 6. East–west vertical cross sections of temperature (8C)through the center of the lake for the initialization simulation: (a) 0and (b) 30 days.

grids, respectively. To the extent that the diffusion isused to parameterize mixing processes that are not wellunderstood, using a first-order accurate diffusionscheme for the SS model is not a problem; that is, theuncertainty in the diffusion parameters is greater thanthe numerical error.

4. The upwelling simulation

In this section we present an SS simulation of up-welling in an idealized large lake. This case was pre-viously simulated using the Princeton Ocean Model(POM; Blumberg and Mellor 1987) and the Dietrich/Center for Air Sea Technology (DieCAST; Dietrich andKo 1994) model by Beletsky et al. (1997). We comparethe SS simulation to these previous simulations.

a. The setting

The lake is 100 m deep, has a diameter of 100 km,and sits in a parabolic basin. The initial temperaturedistribution is a function of depth only, 208C above 5m, 58C below 15 m, with a constant temperature gradientbetween 5 and 15 m. The lake is initially motionlessand is exposed to northerly winds for 29 h. The am-plitude of the wind stress starts at zero, ramps up to 0.3N m22 over 18 h, maintains this value for 6 h, and rampsto zero over the next 5 h.

b. Initialization

The SS model is initialized by the following procedure.The lake’s domain, a box 100 km across in each hori-zontal dimension and 100 m deep, is divided into rect-angular columns having widths dx 5 dy 5 2.5 km. Eachcolumn is divided into a stack of 28 boxes with thefollowing vertical dimensions: 6 at 10 m, 2 at 5 m, 4 at2.5 m, 4 at 1.5 m, 6 at 4/3 m, and 6 at 1 m (listed inascending order). Each box whose center lies above thelake’s bottom is converted to a sack having the samevolume as the box and the same horizontal position andtemperature as the center of the box. The sack is assignedthe horizontal mass distribution defined by (6) with rx 5ry 5 2.5 km. When the sacks are stacked in the parabolicbasin, the lake’s surface and isotherms are quite wavy(Fig. 6a). There are two causes for the waves: 1) thecollection of boxes is a bumpy approximation of theparabolic lake and 2) converting boxes to sacks redis-tributes mass in the horizontal. To remove the waves, a30-day preliminary simulation is run with no wind forc-ing or Coriolis force and with Newtonion damping ofvelocity with a time scale of 1 day. The sacks settle andthe isotherms flatten (Fig. 6b), and the resulting distri-bution of sacks is used for the initial condition in theupwelling simulation. Note that this initial condition in-cludes small residual waves. To assess the stability ofthese waves we ran the model for 15 days without dif-fusion, wind stress forcing, or damping (but with GWR),

starting with this initial condition, and found that theamplitude of the waves remained very small, suggestingthat the waves are either stable or perhaps very weaklyunstable.

The effective horizontal resolution of the SS simu-lation is half that of the POM and DieCAST simulations.The SS and DieCAST simulations have similar verticalresolutions, slightly more than twice that of the POMsimulation (Beletsky et al. 1997).

c. Numerical details

In order to calculate F we approximate (7) using api

Riemann sum with a resolution of 5/6 km, and we set


FIG. 7. East–west vertical cross sections of temperature (8C)through the center of the lake at 29 h: the (a) SS, (b) POM, and (c)DieCAST simulations. Surface height data were not available for thePOM simulation, so a contour was inserted along z 5 0. The POMand DieCAST figures are adapted from Beletsky et al. (1997).

g 5 0.02. Coefficients of vertical viscosity and tracerdiffusion are calculated using the Richardson numberformulation presented in Beletsky et al. (1997) with a5 25 3 1023 m3 kg21 s and K0 5 1025 m2 s21. Theseare the same values used in the DieCAST simulation(W. P. O’Connor 2002, personal communication). Qua-dratic bottom friction is included with a coefficient of0.002. The model is run without horizontal diffusion.

d. Results

The wind stress forcing produces an Ekman layer thattransports surface water toward the west. By 29 h thesurface layer has deepened in the western edge of the

basin, and upwelling has developed in the eastern edge(Fig. 7a). A little numerical noise is apparent in thetemperature contours (Fig. 7a), but considering that theSS model has no horizontal diffusion and no filteringto remove such noise, its magnitude is surprisinglysmall. The temperature distribution in the SS simulationis similar to the temperature distributions produced bythe DieCAST model and the POM (Figs. 7b,c). In theSS solution, however, the water is a little cooler in theeastern edge of the basin; the 68C isotherm intersectsthe surface a short distance offshore (Fig. 7a), whereasin the POM simulation this isotherm intersects the bot-tom (Fig. 7b) and in the DieCAST simulation it is closerto the shore (Fig. 7c). This difference between the SS,


FIG. 8. Horizontal cross sections of temperature (8C) at 10 m: (a) the SS simulation at 29 h, (b) the POM simulation at 29 h, (c) theDieCAST simulation at 29 h, (d) the SS simulation at 120 h, (e) the POM simulation at 120 h, and (f ) the DieCAST simulation at 120 h.A running-mean filter with the width of a sack radius in each dimension has been applied to the SS fields. The POM and DieCAST figuresare adapted from Beletsky et al. (1997).

POM, and DieCAST simulations is also apparent onhorizontal cross sections of temperature at the depth of10 m; the 68C isotherm encloses a larger area in the SSsimulation (Fig. 8a) than it does in the other simulations(Figs. 8b,c). The SS solution probably differs in thisway from the POM and DieCAST solutions because ithas no horizontal diffusion and no numerical verticaldiffusion, whereas the POM and DieCAST simulationseach have both parameterized and numerical horizontaldiffusion in addition to numerical vertical diffusion.

The similarity between the SS, POM, and DieCASTsolutions persists over time. Figures 8d–f show the tem-perature distribution at 10 m at 120 h for each model.

In each of the three simulations the upwelling frontshave propagated counterclockwise around the lake, andthe gross structure of the temperature distribution is sim-ilar. However, once again the SS solution preserves moreof the water masses with temperature extremes; boththe 68 and 188C isotherms enclose larger areas in theSS solution (Fig. 8d) than they do in the POM andDieCAST solutions (Figs. 8e,f).

The comparison of the SS, POM, and DieCAST sim-ulations suggests that the SS model produces circula-tions similar to those produced by the other models butwith less mixing. This result is encouraging since anoverly diffuse thermocline has been recognized as a


FIG. 8. (Continued )

significant problem in numerical lake models (Beletskyand Schwab 2001). We remain optimistic about the SSmodel’s usefulness for simulating lake and ocean cir-culations, especially for cases in which excessive mixingis detrimental.

5. Discussion

In this section we compare the SS model to othermodels in light of the results presented in this paper.We also mention how we plan to modify the SS modelin the near future.

a. Advantages

The SS model has advantages over other models withregard to how it handles physical processes and bound-aries, and its computational efficiency is competitivewith other models.

1) ADVECTION

The primary advantage of the SS model is that it hasnegligible advection errors. In particular, there is nodegradation of temperature, salinity, or momentum dis-tributions. While this characteristic of the model cer-tainly distinguishes it from height- and sigma-coordi-nate ocean models, it also distinguishes it from isopyc-nal models (Bleck 1998), which have horizontal advec-tion errors.

2) VERTICAL MIXING

The SS model currently uses pseudo-Eulerian verticalmixing. This means that any vertical mixing schemeused in a height-coordinate model may be adapted to

the SS model. One advantage this gives the SS modelover isopycnal models is the ability to resolve the mixedlayer.

3) BOTTOM TOPOGRAPHY

The SS model represents continuous bottom topog-raphy in a simple way. This distinguishes it from height-coordinate models that either use step topography orsomewhat complicated approximations of continuoustopography (e.g., Pacanowski and Gnanadeskikan1998).

4) LATERAL BOUNDARIES

Most ocean models have fixed lateral boundaries.This is unrealistic in the sense that if the water is suf-ficiently perturbed by a large-scale forcing (e.g., as intides) it can slosh and significantly change its horizontalboundaries. Models that do allow wetting and dryingover tidal flats require rather complicated and costlynumerical schemes (e.g., Kowalik and Murty 1993). Incontrast, the SS model can naturally represent such wet-ting and drying simply by moving sacks up and downa sloping sea bed.

5) COMPUTATIONAL EFFICIENCY

The SS model requires slightly more computationsper sack than Eulerian models require per grid point,but because it has no numerical diffusion and does notrequire parameterized horizontal diffusion for numericalstability, it can be run at a lower horizontal resolutionand complete simulations in a comparable amount oftime. For example, the POM upwelling simulation,which had a duration of 15 days (Beletsky et al. 1997)


would take on the order of 10 min to run on a 2-GHzPentium processor (D. J. Schwab 2002, personal com-munication). When the SS simulation is run to 15 daysit requires 17 min to complete on such a processor. TheSS simulation has about half as many sacks as the POMsimulation has grid points. However, even at a lowerresolution the SS model does a better job of preservingtemperature extremes (Figs. 8d,e). Moreover, we are stillin the process of optimizing the SS model and expectit to become faster.

b. Disadvantages

The results presented here also suggest that SS modelhas several disadvantages relative to other ocean mod-els. While none of these disadvantages is particularlysevere, we emphasize that the SS model has only un-dergone limited testing at this point, so that the mostsignificant challenges for the model are probably yet tobe discovered. For example, we have not yet tested theSS model for an analog to pressure-gradient errors (Ha-ney 1991), which could result from the approximationof the integral in (7) with a Riemann sum.

1) SIDE EFFECTS OF GWR

As we note in section 2, GWR both slows externalgravity waves and magnifies surface height perturba-tions. For many applications these changes are incon-sequential. However, for some applications, such ascoastal ocean modeling, such changes may cause prob-lems, and it might be necessary to use GWR only in alimited way, resulting in reduced computational effi-ciency.

2) NUMERICAL NOISE

Numerical noise develops during the upwelling sim-ulation and appears as high-frequency waves in iso-therms (Fig. 7a). The amplitude of this noise increasesalong with the surface wind stress until 18 h, remainsnearly constant from 18 to 29 h, and gradually dimin-ishes after 29 h. The noise in the SS simulation is com-parable to that in the POM simulation (Fig. 7b) andgreater than that in the DieCAST simulation (Fig. 7c).However, recall that the DieCAST simulation also ex-hibits the most mixing. Sensitivity tests suggest that theamplitude of the noise in the SS simulation is approx-imately proportional to the vertical thickness of a sack.While the effects of the noise on the SS upwelling sim-ulation are largely cosmetic, such noise might be moreof a problem for simulations with prolonged and spa-tially variable wind stress forcings.

3) INITIALIZATION

Representing an arbitrary fluid shape with a pile ofsacks constrained to have a mass distribution such as

(6) is, in general, a challenging problem. For the up-welling simulation the initial condition includes levelisopycnal surfaces, which are easily produced by allow-ing gravity to act on the fluid in the presence of dis-sipation without rotation. However, initial conditionsinvolving horizontal gradients in height or isopycnalsurfaces may prove to be more difficult to reproduceaccurately.

c. Planned improvements for the SS model

In the near future we plan to add the following fea-tures to the SS model.

1) DIVIDING AND MERGING SACKS

In the SS upwelling simulation we use sacks withsmall vertical thicknesses for the near-surface layer sothat this layer is highly resolved. These vertically thinsacks move westward and pile up along the westernshore, leaving the near-surface layer along the easternshore poorly resolved. In the future we will address thiskind of problem by allowing sacks to divide. A maxi-mum vertical thickness will be specified as a functionof height, and when a sack’s vertical thickness exceedsthe maximum it will be divided into two perfectly over-lapping sacks, each having half the vertical thickness.Since dividing sacks will increase the total number ofsacks, for long-term simulations we also plan to mergeoverlapping vertically thin sacks in deep water.

2) PARALLEL PROCESSING

The SS equations of motion are local; solving themfor a given sack only requires information about othersacks within a sack width in each dimension. Therefore,the SS method is well suited to parallel programming,and we have already begun working on a parallel versionof it. Each processor will keep track of sacks within ahorizontal subdomain assigned to it. Processors willpass information to each other about sacks within a sackwidth of subdomain boundaries.

3) SPHERICAL GEOMETRY

The SS method appears to be easily adapted to spher-ical geometry. Longitude and latitude may be used ashorizontal coordinates for sack positions, and the mass-distribution function may be defined in terms of latitudeand longitude deviations from the sack center, wherethe sack radius in each dimension is a fixed number ofdegrees. Doing so results in a distribution function thathas the same form as (6) with the addition of the multiple1/cos(f), where f is is the sack’s latitude. This imple-mentation of spherical geometry is suitable for domainsthat do not contain poles; we are also exploring a ge-ometry for global SS simulations.


6. Summary

In this study we modify our SS model and use it tosimulate upwelling in an idealized large lake. Four mod-ifications make the model more computationally effi-cient: split time differencing, polynomial mass distri-butions for sacks, gravity wave retardation, and coding/compiling optimization. A fifth modification, includingpseudo-Eulerian vertical diffusion, facilitates the use ofparameterizations for vertical mixing by subsack-scaleprocesses. The upwelling simulation is similar to twosimulations previously carried out with height- and sig-ma-coordinate ocean models, but it exhibits less mixing.

The successful completion of the upwelling simula-tion demonstrates the SS model’s potential for simulat-ing lake and ocean circulations. The SS model has sev-eral appealing features, including negligible advectionerrors, no need for horizontal diffusion or filtering fornumerical stability, simple and natural bottom topog-raphy, and lateral boundaries, and it is computationallycompetitive with other ocean models. The model hasseveral disadvantages as well related to side effects ofgravity wave retardation: the presence of numericalnoise and the initialization process. We continue to mod-ify the SS model and plan to apply it in ocean simu-lations in the near future.

Acknowledgments. This research was supported byDepartment of Energy Cooperative Agreement DE-FC02-ER63163 and NOAA Grant GC01-351. We thankDima Beletsky for providing the data for the POM andDieCAST simulations, and we thank two anonymousreviewers and George Kiladis for their comments.

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