JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 93, NO. C4, PAGES 3554-3562, APRIL 15, 1988

Modeling of Oceanic Fronts Using a Particle Method EDGAR G. PAVIA AND BENOIT CUSHMAN-ROISIN

Department of Oceanography, Florida State University, Tallahassee

A particle-in-cell method, which has been successfully applied to problems in plasma physics, is adapted to the problem of oceanic geostrophic fronts. The thrust of the present article is the devel- opment, exposition, and testing of the method by means of synthetic examples. The advantage of this approach is its direct applicability to a class of oceanic problems which, for the most part, have remained unsolved. The method is computationally straightforward and efficient. It is also robust (the overall solution does not depend on the details of the initial distribution of material points) and numerically accurate (invariants are well conserved).

1. INTRODUCTION

An important class of oceanic motion is the mesoscale ac- tivity with length scales of about 100 km and time scales of several days to weeks. At these spatial and temporal scales the rotation of the Earth is of paramount importance, and the Coriolis force dominates the balance of forces. This regime is called geostrophic turbulence and consists almost exclusively of eddies and fronts, mostly a combination of both. Eddy is hereby defined as recognizable, closed circulation, even if tem- porary. A front is understood to be a feature of vertical density-surface excursion comparable to a typical layer thick- ness. Because such displacements generate large pressure gradients, and pressure gradients force geostrophic motions, currents are present along the frontal line. If the frontal line is curved and closed onto itself, the front is also an eddy. Thus the studies of eddies and fronts on these scales are intertwined.

A theme in mesoscale oceanography concerns the properties and behavior of frontal eddies, such as Gulf Stream rings, including self-propulsion, tendency to become axisymmetric, interactions with one another, and response to external forc- ing (e.g., shear, strain, and cooling). Another topic of current interest is the shedding and recapture of eddies from and by a frontal jet such as the Gulf Stream. The assessment of the control exerted by the front during these events is of particular significance. Quasi-geostrophic models (the weak-front limit) have addressed these questions, and their results form the basis of our understanding of numerous mesoscale processes. However, studies using fully frontal models, and the determi- nation of the extent to which projections from quasi- geostrophic results can be applied to frontal eddies and frontal jets, remain to be completed.

One problem encountered in modeling oceanic geostrophic fronts is tracking the frontal line. Analytical methods [Stern, 1975; Nor, 1981; Cushman-Roisin, 1986a] make use of free streamlines (lines of discontinuities separating two regions of different properties). The applicability of these analytical solu- tions is limited to special, simplified cases. In contrast, numeri- cal methods offer the desired generality but do not lend them- selves easily to the treatment of discontinuities. Traditionally, numerical models of oceanic mesoscale fronts have been the result of compromises. On one hand, horizontal resolution has been sacrificed to accommodate a fine vertical representation [De Szoeke and Richman, 1984]. Limited horizontal resolution places emphasis on the vertical structure and diverts attention

Copyright 1988 by the American Geophysical Union.

Paper number 8C0041. 0148-0227/88/008C-0041 $05.00

from important horizontal processes such as frontal instabil- ities, eddy formation, and eddy merging. On the other hand, an alternate compromise is to restrict the applications to weak, quasi-geostrophic fronts in which vertical displacements are assumed to be small [e.g., Flierl, 1977; Pratt and Stern, 1986]. The extension of the results to strong fronts is limited, since strong fronts behave differently than weak fronts. An- other solution to the problem is the use of very high horizon- tal resolution [Adamec and Elsberry, 1985], but this is not a panacea because of the heavy computing cost. In the search for a method that makes reasonable computer demands, and compromises neither the horizontal resolution nor the vertical displacements, we turn to particle methods. These methods are naturally suited for the handling of free boundaries within the fluid [Salmon, 1983]. The present work develops and tests a particle method for the investigation of oceanic geostrophic fronts and eddies.

The frontal geostrophic model of Cushman-Roisin [1986a] is selected as a prototype to demonstrate the applicability of a particle method to ocean fronts. This model reduces the prob- lem of frontal evolution to a single equation for one layer thickness or, equivalently, a pressure field. It approximates the primitive equations when the cross-frontal length scale is suf- ficiently large (at least 3 times the radius of deformation). Consequently, some physical processes, namely, inertia-gravity oscillations, are neglected. The frontal geostrophic model is recapitulated by

- d h, (hVeh + «Vh. Vh) + Ot f? f-• yh (1) where h(x, y, t) is the thickness of the upper layer of the ocean (Figure 1), J(a, b)- axby- aybx the Jacobian operator, g' the reduced gravitational acceleration, and fo + fly the Coriolis parameter (see Cushman-Roisin [1986a] for additional infor- mation). The constants are set as follows: g'--0.01 m s -e, fo=7 x 10 -s s -a, and fi=0 or 2 x 10 -aa m -a s -a. These values are typical of Gulf Stream rings and other midqatitude eddies.

Equation (1) is equivalently written as the set

0'-• + •xx (hu) + •yy (hv) = 0 (2a)

g'e ( O•-xx ) g'• Oh (2b) u= --f--• J h, + f--• Y

g'2 (0h) g'• Oh (2C) 3554

PAVIA AND CUSHMAN-ROISIN: MODELING OCEANIC FRONTS USING A PARTICLE METHOD 3555

y

front

fluid at rest

Fig. 1. Schematic view of the oceanic frontal model.

where u and v play the role of horizontal velocity components and (2a) is a continuity equation. The latter form of the prob- lem makes it amenable to a particle method; this conversion task is undertaken in section 2. The method is tested and discussed in section 3. In section 4 the conclusions of this work are enumerated. Finally, the details of the numerical algorithm are confined to an appendix.

2. PARTICLE AND PARTICLE-IN-CELL METHODS

A particle method is a numerical technique of integration in which the fluid's continuous mass per unit area (or volume, circulation, or any conserved quantity), M(x, y, t), is approxi- mated by the distribution of a large but finite number of ma- terial points (or volume elements or point vortices). That is,

N

y, t)= M6(x- xO6(y- yO (3) k=l

where 6 is the Dirac delta function, xk(t ), yk(t) are the eastward and northward coordinates of the N computational particles, and M•, is their constant mass (or volume or circulation). The dynamics of the system prescribes the rules by which these particles move in time. The knowledge of their trajectories constitutes a satisfactory, approximate solution to the prob- lem.

As an extensive literature on the subject indicates, various techniques based on this type of discretization have been de- vised and tested over the years. Salient contributions related to the present purpose are those of Christiansen [1973], Harlow [1964], Harlow and Amsden [1971], Hockney et al. [1974], Meng and Thomson [1978], and Monaghan [1982]. This list is by no means exhaustive, since it is not our intent here to review the literature. A more comprehensive review is given by Leonard [1980]. Of particular interest is the work by Harlow [Harlow, 1964; Harlow and Amsden, 1971] outlining techniques that are appropriate for problems involving large distortions of a fluid.

It is important to note that genuine particle methods are not to be confused with marker-particle methods [e.g., Harlow and Welch, 1965], in which the particles are introduced only for visualization purposes and/or for implementation of

boundary conditions but are not used for the interior calcula- tions.

The oceanic problem posed in section 1 (equations (2)) can be transformed into a particle problem in the following manner. Since equation (2a) is a continuity equation for volume, a volume dV = h dS of an element of height h and cross section dS is conserved by the flow (u, v), given by (2b) and (2c), although neither h nor dS is conserved. Given a set of material points, each having an assigned volume, one can derive from their horizontal distribution an area which they each occupy. A division of the individual volume by the re- spective area provides the thickness h at the location of every material point. The horizontal distribution of h can then be used to calculate gradients from which the velocity compo- nents can be evaluated. The horizontal velocity can then be used to march the material points one time step forward. In their new positions the areas occupied by the material points and the height field will, in general, be different.

This marching method differs drastically from a point- vortex method, in which the velocity field is derived from an integration (of vorticity) rather than from a differentiation (of pressure). Additional progress toward the programing of a code requires specific choices and considerations of a more numerical sort, which leaves considerable freedom to the in- vestigator.

At this stage it is interesting to distinguish between pure Lagrangian particle methods and particle-in-cell methods. Pure Lagrangian particle methods are techniques in which the advecting velocities are evaluated directly at the particle posi- tion. This approach may be adequate when a representation of a free surface is desired [Harlow and Amsden, 1971] but usu- ally suffers from some major drawbacks. The method is not feasible for problems in which the evaluation of the trans- porting flow requires differentiation (rather than integration) of the transported quantity. Also, the number of computations required for this technique is typically proportional to the square of the number of material points [Leonard, 1980].

Particle-in-cell (PIC) methods have been developed and suc- cessfully applied in the context of plasma physics [Morse, 1970]. The objective of a PIC method is to retain a La- grangian treatment of the particle motion while the pressure field (layer thickness, here) is described on an Eulerian grid. In other words, the mass is transported from cell to cell in a Lagrangian fashion, while a fixed grid is used for the repre- sentation of the variables from which the derivatives are re-

quired. Such a procedure requires the following sequence of intermediate calculations at every time step: (1) the generation of grid values from the particle distribution (i.e., the volume of each particle is split into several parts, and each part is as- signed to neighboring grid points according to the particle's position), (2) the evaluation of the necessary derivatives (on the grid and by finite differences), (3) the calculation of the velocity components at the grid point locations, and finally (4) the interpolation of the velocities from the grid back to the Lagrangian points.

These intermediate steps transfer the information back and forth between the particle positions and the grid at every time step. They also permit automation of the calculations in a simple code and add stability to the numerical scheme by providing some smoothing (smearing of particle information). Another numerical advantage of PIC methods is that the number of computations is a multiple of the number of ma- terial points rather than a multiple of its square.

3556 PAVIA AND CUSHMAN-ROISIN' MODELING OCEANIC FRONTS USING A PARTICLE METHOD

-100 m

' 200 m

- 300 m

' 400 m

' 500 m

a)

-1 1 ms

b)

Fig. 2. (a) The depth profile of a circular eddy and (b) the associated velocity field as obtained from a set of particles placed along concentric annuli.

3. EXAMPLES

Several initial conditions have been selected in order to calibrate and test the technique outlined above. First, an opti- mal number of material points per grid cell is determined. This is done by studying a steady circular eddy in which, despite the rotation of the particles, the pressure field must remain unchanged. Next, the case of an elliptical eddy becoming axi- symmetric is selected to test the constancy of the invariants as the system undergoes significant changes. Finally, the beta effect is included to compare the numerical and analytical values of the westward drift rate.

3.1. Circular Front

For the first case a circular front (a model for a warm-core ring) with the height distribution

h(r)= H(1--•) (4) is imposed as initial condition. The depth at the center is H = 500 m, with the front located at r = R = 100 km and enclosing the volume •HR2/2. The particle distribution con- sists of a central particle surrounded by Q concentric annuli each composed of 3q (q = 1 to Q) particles equally spaced around the annulus. All particles are identical, and their total number is

Q

N= I + • 3q= I + •Q(Q + I) (5) q=!

The volume of each particle is dV = •HR2/2N. The Q radii of the annuli are chosen so that the volume of the Nq particles within and including the qth annulus is equal to the volume obtained from (4) integrated from the center outward to the qth annulus:

Rq = R{ 1 -- [1 -- (N,/N)] '/2}'/2 q = 1,..., g (6) where N• = 1 + 3q(q + 1)/2 and R o = 0.

The radial profile of the layer thickness obtained after a first conversion from particle positions to grid values is shown in Figure 2a. Note the slight departure from the parabolic profile of (4) in the vicinity of the front (h = 0). This is unavoidable because of the discrete nature of the algorithm, but rather than making the eddy less physical, it makes it more realistic. The corresponding velocity field is displayed in Figure 2b. The velocity increases linearly from the center outward, but in- stead of dropping abruptly from its maximum value to zero at the front as the analytical formulation would have it, it decays gradually over an interval of three grid spacings.

When/• = 0, the circular eddy front is an exact steady solu- tion (expression (4) is a solution of (1) with no time derivative), but the corresponding velocities are not zero. Therefore the computations should yield motions of particles with no change in the pressure field.

A 41 by 41 cell grid with Ax = Ay = 10 km (R = 10Ax) is used to test the sensitivity of the PIC method to the number of particles within each grid cell. According to the theory [Cushrnan-Roisin, 1986a] and for the values offo and g' given in section 1, the time scale of an unsteady feature of this depth and size is approximately 15 days. Therefore although the pressure field of this circular eddy should remain invariant, the model was run for 15-day simulations. On the basis of the results from these simulations we decided to use an average of 10-15 particles per cell (see Figure 3 for some comparative runs). A smaller number of particles leads to noticeable de- terioration of the solution due to jumps in gridded values each time a particle changes cell, while a greater number does not significantly improve the accuracy of the results. It was also found, by trial and error, that (I) = 0.15 (see equations (A3)) is the upper limit for the time step beyond which numerical instability appears. This means that particles do not move by more than 15% of a grid spacing in one time step and thus remain in the same cell for at least seven time steps.

PAVIA AND CUSHMAN-ROISIN' MODELING OCEANIC FRONTS USING A PARTICLE METHOD 3557

4 ppc

! i ! i ! ! ! ! ! ! ! ! ! ! ! I.

t=O

.

.

.

..

.

.

.

.

.

.

14 ppc

,

20 ppc

! ! i

Fig. 3.

t= 15 d t= 15 d

Depth contours of a circular eddy, with fl = 0 and after 15 days, using different number of particles per cell (ppc).

3.2. Elliptical Front (l• = O)

Closed elliptical eddy fronts were constructed for further testing in cases of substantial temporal variability. Initial con- ditions were obtained by squeezing the previous circular eddy in one direction and stretching it in the other.

According to the stability analysis of Cushman-Roisin [1986b], an elliptical eddy, in which the velocity increases linearly with distance from the center, is stable if its aspect ratio is less than 1.8. It is otherwise unstable to at least an azimuthal mode 3. On the other hand, numerical experiments with quasi-geostrophic dynamics [McCalpin, 1987] show that elliptical eddies, irrespective of their eccentricities, have a strong tendency to become axisymmetric.

Thus a very elliptical eddy (aspect ratio equal to 2.7) was used to test the PIC technique during a highly time-dependent evolution. We made this choice because the present model is frontal geostrophic, not quasi-geostrophic, and also because

the numerical discretization makes it impossible to simulate an eddy where the velocity increases linearly from the center outward with an abrupt discontinuity at the front (see Figure 2). At the same time we hoped to determine the fate of the eddy under the opposing tendencies of becoming unstable or axisymmetric.

With/• still set to zero the eddy was observed to develop a mode 3 asymmetry initially. This was, however, rapidly over- come by a tendency to become axisymmetric. The sequence of plots in Figures 4a and 4b shows, by means of particle posi- tions and pressure distribution, the progress of the eddy as it first distorts and then becomes axisymmetric and sheds lateral arms.

According to the analytical theory [Cushman-Roisin, 1986a] the moments of the mass field

I, = ffh" dS n = 0, 1, 2, -.. (7)

3558 PAVIA AND CUSHMAN-ROISIN: MODELING OCEANIC FRONTS USING A PARTICLE METHOD

a

t=0 t=3d

t=10d t=15 d

Fig. 4. Evolution of an initially elliptical eddy (/• = 0). (a) Location of the material points. (b) Depth contours.

should be invariant regardless of the initial conditions. Figure 5 shows that, with the exception of the first moment (Io), the moments are well conserved. Because of the particle repre- sentation of the problem the total volume, I x , is conserved exactly. The moment of inertia, I:, also related to the total energy [Cushman-Roisin, 1986a], is well conserved (within 11%), especially after a rapid adjustment that corresponds to a randomization of the particle positions from their initial annular distribution. The initial 6% jump is followed by a

slow decrease at a rate of 0.41% per day. The higher moment, I3, is better conserved (within less than 1%). Higher moments have not been calculated. Nevertheless, since the higher n, the more I, depends on the largest values of h, and since the center depth is constant, one can anticipate that as n increases, these integral quantities are even better conserved.

For n = 0 the moment of the mass field represents the hori- zontal area enclosed by the front. This area is easily calculated by counting the number of grid points where h is nonzero. As

PAVIA AND CUSHMAN-ROISIN: MODELING OCEANIC FRONTS USING A PARTICLE METHOD 3559

b

t : 0 t=3d

t= 10d

Fig. 4. (continued)

t= 15d

mentioned above, it should be constant in time. However, the PIC method tends to spread the front over three grid spacings (recall Figure 2), therefore enlarging the area within the con- tour h = 0 by an area approximately equal to the length of the front times three grid spacings. As the elliptical eddy becomes axisymmetric and develops arms, the front lengthens, and the area within it increases. Thus the 46% rise of I o in Figure 5 is merely a reflection of the increasing frontal length. To verify that this is the case, a modified first moment, Io', defined as the area enclosed by the contour h = 50 m, is also calculated.

Figure 5 shows that it is, indeed, better conserved (within 35%). To test the relationship between resolution and the deviation of the moments of the mass field, we performed a similar experiment with a 21 by 21 cell grid. In this latter case, I o increased by 84%. The other moments of the mass field also showed an almost twofold variation compared to the case with a 41 by 41 cell grid. As a rule, doubling the resolution (i.e., quadrupling the number of cells and particles) reduces by a factor of 2 the ratio of peripheral to interior points as well as the fluctuations in the moments of the mass field.

3560 PAVIA AND CUSHMAN-ROISIN'. MODELING OCEANIC FRONTS USING A PARTICLE METHOD

100% .

-lOO% o

I n =/fh n dx ds' h>O

tiae (das's)

Fig. 5. Variation of the first four moments of the mass field versus time during the evolution of an initially elliptical eddy' I o' is the surface enclosed by the contour h = 50 m.

3.3. Elliptical Front (fi = 2 x 10 -• m -• s -•) The same elliptical front eddy was then followed numeri-

cally with fi - 2 x 10- • • m- • s-•, as another test of the PIC method. Cushman-Roisin [1986a] predicts that, despite inter- nal changes, the eddy as a whole will translate westward at a

fig' ;;h 2 ds c - • ,•, 5.9 x 10- 3 m s- • (8)

2fø2 ;.Ihas

constant rate

The model was integrated with an average time step of 0.22 hour for a total of 15 days. During this period the eddy front drifted steadily westward at a speed c = 5.5 x 10 -3 m s- • This value is within 7% of the value predicted by the frontal geostrophic theory.

A comparison of the trajectories of the center of mass as obtained from the PIC method and the analytical theory is shown in Figure 6. Note that the numerical curve is almost a straight line, in good agreement with the theory which pre- dicts a constant drift despite changes within the eddy. The slower translation in the numerical model can be accounted

for by both numerical dissipation and truncation of the inte- grals of formula (8). However, it is not possible to separate these effects. As the center of mass drifted by approximately 7 km in the zonal direction, it moved only a few meters meridio-

nally, in good agreement with the analytical theory which predicts precise zonal translation.

4. CONCLUSIONS

A particle-in-cell method is developed to solve the frontal geostrophic equation [Cushman-Roisin, 1986a-[. The technique, which resembles some of the particle methods used in plasma physics, consists of tracking a finite number of material points within a rectangular grid. On the basis of several comparative runs with different numbers of particles, we decided to use an average of 10-15 material points per cell. This relatively small number of particles makes the technique inexpensive. The method is also accurate: results show that the moments of the mass field are generally well conserved.

Some reduction in accuracy is caused by the spread of the front (where the layer thickness vanishes) over an interval of three grid spacings. This can be remedied easily by increasing the spatial resolution. However, a formula for the required resolution can be developed only after a large number of tests have been performed.

In one example a highly elliptical eddy initially developed an azimuthal mode 3 distortion and then became axisyme- metric and shed lateral arms. This mixed behavior reflects

both the instability predicted by the analytical theory [Cushman-Roisin, 1986b] and the tendency toward becoming axisymmetric predicted by numerical studies of quasi- geostrophic eddies [McCalpin, 198TI.

On a beta plane the theory [Cushman-Roisin, 1986a] pre- dicts a constant drift to the west. The numerical calculations by the PIC method verify both the westward translation (a nonzero meridional displacement is negligible) and its con- stant rate (despite great adjustments occurring within the eddy). In this example the value of the westward drift obtained by the PIC method is 7% less than that predicted analytically. This moderate deceleration reflects the dissipation inherent to the numerical model, which the analytical theory does not include. It also reflects the truncation error in the estimation

of the integrals forming the analytical formula. We should recall, however, that the smearing and dissi-

pation effects of the PIC method are greatly outweighed by the benefits of using an underlying grid. For example, the interpolation (from the particles to the grid and back to the material points) provides good control on the amount of smoothing being applied. This two-step process avoids the

Fig. 6.

Z00 Km ....

195 Km•

Numerical

5 15

time (days)

Westward drift of the center of mass of an initially elliptical eddy (fi = 2 x 10- x x m-x s-x). The upper curve was obtained numerically by the PIC method, while the lower one is the analytical prediction for the same eddy.

PAVIA AND CUSHMAN-RoISIN.' MODELING OCEANIC FRONTS USING A PARTICLE METHOD 3561

computational difficulties arising in pure Lagrangian methods. Also, the number of operations increases only linearly with the number of material points, since the velocities are calculated at the grid. Another advantage over pure Lagrangian methods is that the PIC method can be extended to several layers. Furthermore, the smearing and dissipation can always be re- duced by decreasing the spacing between grid points.

Finally, we conclude that the particle-in-cell method is well suited for the study of oceanic geostrophic fronts. First, the problem of tracking the frontal line is solved by using a finite number of particles, where the front is just the envelope of the moving material points. Second, by combining Lagrangian and Eulerian features the front line can undergo great distor- tions without numerical difficulties. The major drawback of the model is its inability to handle solid boundaries. However, the wide variety of problems that it can solve, such as eddy merging and jet instability, makes this a valuable tool that may further advance the numerical investigation of frontal dynamical processes.

APPENDIX: THE NUMERICAL ALGORITHM

This appendix describes the numerical algorithm of the PIC method developed for oceanic geostrophic fronts and eddies. Some of the features are common to methods devised for

other applications. The first step of the algorithm is to assign volumes from the

particles to the grid points. According to (3) we have at some time and for each particle the volume and the position: M k and (xk, Y0, for k -- 1 to N. The coordinates are transformed into x•- i Ax + cSx•, y• =j Ay + cSyk, where Ax (Ay) is the distance between grid points in the x (y) direction, (i, j) is the grid point nearest to (x•, Yk), and cSx• (SYk) is a positive or negative offset fraction with absolute value less than or equal to Ax/2 (Ay/2). Then (x•, Y0 is interpreted as the location of the center of a fluid element spread over a square cross section Ax by Ay. This element will most often lie within four contigu- ous grid cells (see Figure 7) and is thus split into four parts. Each of the four corresponding fractions of the mass is equally shared among the four grid points surrounding the cell in which it lies. This scheme shares Mk among the nine grid points closest to (x•, Y0, unless x• or y• lies (or both lie) exactly at the center of a grid cell, in which case Mk is shared only among six or four points. Such spreading of information mini- mizes the statistical fluctuations that normally result from scattered particles and is typical of previous PIC techniques [Hockney et al., 1974]. The idea is simply to make the passage of a particle from one grid cell to the next a smooth transition. The unavoidable smearing of information is compensated for by a fine grid and a large number of material points.

Mathematically, the mass-sharing weight factors are given by

wherem=i- 1, i, ori+ 1;p=j- 1, j, orj+ 1;and•isthe not-Kronecker delta (•,•- 0 for cr--fl; •,•- 1 otherwise). Toward the end of the procedure, once the velocities have been computed at the grid points, the same weights will be applied to determine the velocities of the material points:

• Ax •

i-1

I

Ay

-Yk

i+1

J+l

j-1

Fig. 7. A fluid element spread over a square cross section (dashed outline). The material point (solid circle) is at the center of the ele- ment. Note that in the particular case depicted, 6x is negative, while ,Sy is positive.

i•l j+ 1 lgk(Xk' Yk) = Z Wrnplgrnp rn:i-1 p=j-1

i+ 1 j+ 1

l')k(Xk'Yk)--" • Z WinpUmp m=i-1 p=j-1

(A2)

where limp and Vm. are the eastward and northward velocity components at the (m, p) grid point. Note that (A1) yields only positive weights, unlike other nine-point Lagrangian interpo- lation schemes (see, for example, Tripoli and Krishnamufti [1975, equations (3.1)-(3.2)]). This is an important feature that prevents negative volumes.

Once the contributions of all the individual volumes have been added at every grid point, an elementary division by the cell area, Ax Ay, provides the grid values for the layer thick- ness h. This field is smoothed with a Hanning filter. Grid points where h = 0, i.e., those outside the cloud(s) of particles, are excluded from the smoothing. This is done so that they do not contaminate the nonzero values of h within the flow

region, and therefore the smoothing weights are correspond- ingly modified in the vicinity of these points. This smoothing is required to obtain stable estimates of the derivatives of the pressure field; its effect is dissipation of energy in the short waves. The dissipation rate for a particular example is quanti- fied in section 3.2.

In order to evaluate the right-hand sides of (2b) and (2c) the horizontal pressure gradients h• and hy are first computed using centered finite difference approximation. The Jacobians are then calculated using the Arakawa formulation which con- serves enstrophy and wave number [Haltiner and Williams, 1980]. The boundary conditions are periodic in both direc- tions. Thus the computational grid's geometry is a torus. Periodic boundary conditions are well suited for problems which involve one or several isolated fluid patches and/or an infinitely long front.

To increase the efficiency of the algorithm, a variable time

3562 PAVIA AND CUSHMAN-ROISIN.' MODELING OCEANIC FRONTS USING A PARTICLE METHOD

step is calculated:

At n -- (I) Az/V n V n = max {[(uijn) 2 -•- (1)ijn)2]l/2} (A3)

where ß is a positive constant less than unity, Az is the smaller of Ax and Ay, and V n is the maximum flow speed at the nth time step. In order to obtain a second-order accuracy in time, a predictor-corrector technique is used:

Xk'= Xk n -{- Atnut• n yk'= yt• n + Atnvt• n (A4)

Xk n+l = Xk n -{- At n (glk n -{- U•')/2

y•n+X = y•n + Atn (1)k n _{_ vk')/2 (A5)

where Uk n and 1)k n are computed with the present positions (xt• n, yt• n) and u•' and v•' are computed using the first estimate (xk', y•') of the new positions.

In summary, the procedure is as follows: the volume ele- ments are first distributed onto the grid with the weights (A1) and converted into layer thickness values. The velocities are then calculated at each grid point with a discretized version of (2b) and (2c), and a new time step At n (A3) is computed. Next the interpolation back to the particle positions is done with (A2) to get Idk n and v• n. First estimates of the new positions are obtained from (A4). The cycle is then repeated, now using these to obtain u•' and v•'. The new positions of the material points are finally evaluated with (A5).

Acknowledgments. This research was supported in part by the Office of Naval Research contract N00014-82-C-0404 and by Florida State University through time granted on its Cyber 205 Supercompu- ter. The authors are also indebted to the anonymous reviewers for valuable suggestions and to D. Kopriva, M. Lopez, and A. Pares- Sierra for helpful comments.

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(Received November 23, 1987; accepted December 30, 1987.)