(0)9. 0 The numerical models that are used to study the global circulations of the world's oceans are usually based on the primitive equations, a truncation of the Navier-Stokes equations in which both the Boussinesq and the hydrostatic approximations are enforced. In the hydrostatic approximation, the vertical acceleration is ignored in comparison with other terms of the vertical momentum equation. The justification for ignoring this term is based on scale analysis of the equations. However, the fact that a term in the governing (partial-differential) equations is small does not necessarily imply that its effect on the solution is small, especially if the integration time is long enough. In addition, the nonhydrostatic effects may lead to transitions between long-term states which are themselves predominantly hydrostatic.
The main purpose of this paper has been to estimate the size of the differences introduced by the hydrostatic assumption in the solutions of a representative ocean system, and the time scales on which these errors become significant. We estimate these directly by comparison of numerical simulations over realistic climate time scales--approximately one hundred years. Furthermore, we do not compare the detailed solutions, which have inherent uncertainty due to the chaotic nature of the flow, but rather the domain-averaged frequency spectra of kinetic and potential energies which are more descriptive of long term climate.
The hydrostatic approximation is employed in numerical simulations for two principal reasons. First, the numerical algorithm is simplified and made more efficient--a three-dimensional elliptic equation for pressure is replaced by a two-dimensional elliptic equation. Second, simulation of the nonhydrostatic effects requires a large increase in horizontal resolution. In view of the latter, the nonhydrostatic ocean models of, e.g., Marshall et al. [20] could only be used to estimate the importance of nonhydrostatic effects in simulations with horizontal grid scales of the order of a few kilometers or less. It is not feasible to run such fine resolution of the global ocean for hundreds of years, now or in the foreseeable future. It is therefore understandable that there have not been previous quantifications of the nonhydrostatic effects in large scale flows. (The nonhydrostatic ocean model considered by Mahadevan et al. [21], on the lines of Browning et al.[22], addresses issues of well-posedness of the governing equations at open boundaries, and problems associated with diagnosing the vertical velocity. As indicated in their paper, the nonhydrostatic terms in that model are unphysical to the extent they are scaled by arbitrary parameters.)
To circumvent these limitations of computer speed and memory, our strategy in this paper has been to utilize shallow water models in which the nonhydrostatic effects appear as rigorously derived dispersive terms in the horizontal momentum equations. These new dispersive terms do not introduce any additional variables into the hydrostatic shallow water equations, and so may be considered as a model for the effects of the unresolved or underrepresented vertical component of velocity. Thus the dispersive shallow water model allows us to consider (physical) nonhydrostatic effects in realistic large-scale flows at very modest (about 30%) increases in computation over the hydrostatic case.
We have chosen the one and one-half layer wind-driven ocean basin as a problem representative of long term ocean dynamics. With appropriate choice of the wind profile, the flow develops into two gyres. Furthermore, the nonlinear interaction of these gyres leads to the formation of eddies, whose complicated dynamics contribute in large part to the variability of the system.
We have compared the frequency spectra of the basin-averaged potential and kinetic energies generated by models with and without the dispersive terms that represent nonhydrostatic effects. We have used maximum entropy methods and singular spectrum analysis, which are superior tools for analyzing short and noisy time series compared with the usual Fourier analysis. (Because of the predominance of low-frequency variability in the wind-driven double-gyre system, we note that a hundred year simulation is still relatively short : McCalpin [23] and McCalpin and Haidvogel [24].) We find considerable differences in the low-frequency variability between the hydrostatic and nonhydrostatic solutions, implying that nonhydrostatic effects are significant on time scales of tens of years. We have quantified these differences by further comparisons with hydrostatic runs in which the strengths of other subgrid parameterizations, the Rayleigh friction and Laplacian viscosity, are varied. We find that the size of the differences in the energy spectra at low frequencies that result from including nonhydrostatic effects are comparable to the changes resulting from large variations in the strength of these other parametrizations.