A new low dispersion version of DieCAST ocean model is shown to converge fully and much more rapidly than more conventional approaches in a low amplitude transient wind flat bottom lake with stratification patterned after the Great Lakes during summer. Simulations of the test problem using 5.000, 2.500, 1.250 and 0.625 kilometers indicate good convergence.

Schwab, et al (1995) show that the Princeton Ocean Model and the more conventional "a" grid DieCAST Ocean Model give similar results in a prototype and realistic Great Lake simulations. The present results suggest that the new reduced dispersion "a" grid version may be substantially better.

The wind ramps up for a day, then shuts off after two days. This is a good numerical test for coastal ocean models because: it is simple (100 m deep, 100 km diameter circular flat bottom lake); it includes dominant geophysical dynamic effects (buoyancy and rotation); and there are good theories for the linear (low amplitude wind forcing) problem. It includes Kelvin edge waves that are challenging to resolve and maintain against numerical dissipation and dispersion.

Although this is a good test for coastal ocean models, the dominant eddies, fronts and boundary currents of deep ocean circulations are not addressed. These also influence coastal regions leading to shelfbreak fronts and offshore spurts. These deep water dynamics have different physics than the transient Poincare and Kelvin wave dynamics of the present test problem. In addition to the low dispersion and dissipation numerics of the present problem, deep water applications require accurate baroclinic pressure gradients with large amplitude topography and time varying stratification.

The present study uses a linear equation-of-state and about 50 percent larger Rossby radius than the cases studied by Schwab, et al (1995) using 5.00, 2.50 and 1.25 km resolution. Thus, the present study allows a larger time step, but somewhat less resolution is required for grid convergence.

Dietrich (1996) compares the new reduced dispersion "a" grid DieCAST model with earlier more conventional versions in three other test problems: passive scalar advection; non-hydrostatic sloshing of internal waves; and real Gulf of Mexico circulation. Earlier resolution sensitivity tests with the DieCAST model include: tests with baroclinic instability (Dietrich, et al, 1990); comparisons of second and fourth order Coriolis treatments used by Arakawa "c" grid versions of the DieCAST model with the conventional Arakawa scheme (Dietrich, 1993); and 20 km and 1/12 deg resolution Gulf of Mexico studies that have been extensively validated by satellite and conventional observations (Dietrich and Lin, 1994; Dietrich and Ko, 1994; Dietrich, et al, 1996).

The present application focuses on the resolution sensitivity of fast (non-geostrophic) internal modes including Poincare and Kelvin edge waves in a basin patterned after the Great Lakes.

Table 1 below suggests that the new reduced dispersion "a" grid model may be substantially better than two more conventional approaches for the fast Poincare and Kelvin edge wave modes that occur in lake responses to transient wind events. It gives close to the correct linear Kelvin wave phase speed (about 8 days for propagation around the lake) even with 2.5 km resolution. The 5 km Rossby radius of deformation is a dominant scale. It also has very small dissipation. Indeed, the warm front of the marginally resolved Kelvin edge wave is clearly seen for eight(!) revolutions around the basin with 2.5 km resolution (80X80 grid). The two more conventional schemes have much more diffuse fronts.

The actual patterns are rapidly changing with time scales of order 1-8 days dominating. Yet the patterns of the two highest resolution cases are highly correlated in the three higher resolution cases with the reduced dispersion Scheme 3B, and including very detailed oscillations and fronts. A postscript file of results using the reduced dispersion numerics with 1.25 and 0.625 km resolution shows this high correlation, and is available on request.

In the higher resolution low dispersion cases, the Kelvin wave warm front tightens greatly after day 20 which is easily seen in this color animation. Such tightening is consistent with the fact that Kelvin wave warm fronts are much more commonly observed than cold fronts (Schwab, et al, 1995). As this happens, along-shore scale of the trailing warm water decreases rapidly until its along-shore scale roughly matches its off-shore scale, which is the natural Rossby radius of deformation. It thus becomes a half-cylinder warm core trapped against the boundary moving at the natural Kelvin wave speed.

This tight 5 km diameter half-cylinder pattern propagates with little dissipation in the 0.625 km resolution case with the reduced dispersion numerics. With 1.25 km resolution it is very similar, occurring at identical location (all aspects of the total pattern are highly correlated), but dissipates more rapidly than the 0.625 km resolution result. That is the reason that the large temperature difference is maintained at day 30 in the 0.625 km resolution result (Table 1). All other aspects of the two highest resolution simulations are very similar through day 30.

The schemes indicated in Table 1 below are as follows.

• 1B: an Arakawa "c" grid scheme used in the early DieCAST model studies except the Coriolis terms are implicit trapezoidal and solved separately from the backward implicit vertical diffusion

• 3A: the original "a" grid scheme validated by a wide variety of Gulf of Mexico observations except Coriolis terms are explicit

• 3B: a modified version of 3A with reduced numerical dispersion

Table 1 includes vertical resolution sensitivity, but this is not of much relevance in comparing the new schemes, which differ only by horizontal interpolations approaches.

All versions have a free surface bt submodel that is inactivated for this comparison.

All of these schemes do well with the deep water physics as demonstrated in many applications including the Gulf of Mexico.

Scheme 3A: EPS=1.00 (conventional original DieCAST "a" grid scheme)

Scheme 3B: EPS=0.03 (reduced dispersion "a" grid scheme blended with 3 percent of the more dispersive scheme 3A)

REFERENCES

Schwab, D.J., D. Beletsky, W.P. O'Connor, and D.E. Dietrich (1995). Numerical Simulation of Internal Klevin Waves and Coastal Upwelling Events. Proceedings ASCE Fourth International Esturine and Coastal Modeling Conference, San Diego, CA, 26-28 Oct 1995.

Dietrich, D.E. (1996). Reynolds Number and Resolution Sensitivity Studies with Gulf Of Mexico Eddies. Journal of Geophysical Research. (Under Review.)

Dietrich, D.E., P.J. Roache, and M.G. Marietta (1990). Convergence Studies with Sandia Ocean Modeling System, Int. J. Numer. Methods Fluids, 11, pp. 127-150.

Dietrich, D.E. (1993). On Modeling Flows with Low Rossby Number. Canadian Journal Atmosphere-Ocean, 31, 57-71.

Dietrich, D.E. and C.A. Lin (1994). Numerical Studies of Eddy Shedding in the Gulf of Mexico. Journal of Geophysical Research, 99, C4, 15 Apr. 1994, 7599-7615.

Dietrich, D.E. and D-S. Ko (1994). A Semi-Collocated Ocean Model Based on the SOMS Approach. International J. Num. Methods in Fluids, 19, 1103-1113.

Dietrich, D.E., C.A. Lin, A. Mestas-Nunez, and D-S. Ko (1996). A High Resolution Numerical Study of Gulf of Mexico Fronts and Eddies. Journal of Geophysical Research (Under Revision).

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