Fig. 2 shows a contour plot of the 100-year mean of the thickness of the surface layer, for systems 1 and 3. It is clear from these pictures and that for system 2 (not shown), that the mean state is roughly the same for all three systems, with differences of only 3% to 6%. In the next section, we concentrate on differences in the variabilities of the three systems by analyzing the time series of their domain-averaged kinetic and potential energies.
As is evident from the governing equations, there are two dissipative processes in the problem parametrized by , the Laplacian eddy-viscosity, and , the Rayleigh friction. In runs SW1, GN1, and GNH1, these two parameters are the same and are set at and .To place into perspective the differences we find among these three runs, we consider two other runs of the shallow water system: SW2 in which the Laplacian viscosity is changed to 50 m2s-1, and SW3 in which the Rayleigh friction is changed to .
Figs. 3a and 3b show the spectral estimates for the kinetic and potential energies for the five cases. The power spectral density is estimated using the Maximum Entropy (ME) method with 300 poles. The data points are spaced 4 hours apart and the time series spans 110 years. The trends in these power spectra persist when using a larger (see Fig. 3c) or smaller (not shown) number of poles. Errors in the estimation of the spectra are mostly systematic, i.e., are related to the model assumed for the processes rather than statistical, and hence are not relevant for the discussions that follow. That is to say, if there were substantial statistical uncertainity associated with the estimation of each spectrum, it would be unlikely that these spectra would all fall on one another so closely at the intermediate frequencies.
While such a simple description of the frequency spectra does not do justice to the complicated dynamics of the jet, one may say in general that there are energy-bearing frequency scales and dissipation scales, separated by an intermediate range of frequency scales. The overall similarity of frequency spectra across the different runs confirms the similarity of the base state of the flow. On comparing the spectra for SW1 and SW3, one sees that the lower Rayleigh friction in SW3 leads to higher variability in kinetic energy. This occurs systematically at all frequencies, since the Rayleigh friction itself is scale independent. In contrast, the differences introduced either by the Green-Naghdi terms or by a different value of eddy viscosity do not result in such systematic changes to the spectra. (These terms constitute singular perturbations of the inviscid shallow water equations.) The full frequency range is shown for completeness. The broad peak near the 24 hour period is due to certain modes that are excited by the initial conditions and persist for long times. Interestingly, at these frequencies (1/day), the shallow water runs all seem to exhibit one common response, while the two Green-Naghdi runs exhibit a different (smaller) common response to the modes excited by the initialization.
Comparison of the potential energy spectra for cases SW1, SW2, GN1, and GNH1 in Figure 3b shows that the most significant differences in these spectra occur--perhaps surprisingly--at the low-frequencies. The low-frequency end of the spectrum is shown on a log-linear plot in the inset, to give a better idea of the magnitude of the differences between the spectra there. In these plots, the two Green-Naghdi runs show significantly higher variability in potential energy relative to SW1. (For reference, also compare cases SW1, SW2, and SW3 in the same plots.) We emphasize the large magnitudes of the differences in potential energy spectra which occur when the nonhydrostatic terms are included. The magnitudes of these differences in spectra at the low-frequency end indicate that the long term effects of the nonhydrostatic terms are not negligible. In fact, their inclusion results in differences in potential energy spectra at low frequencies whose sizes are comparable to those obtained by making large variations of parameters such as Rayleigh friction and eddy viscosity, representing subgrid scale losses. The kinetic energy spectra of Fig 3a appear to lend further credence to this observation.
Thus, while these spectra may not imply that including nonhydrostatic terms is important in climate simulations, they do annul the naive argument that nonhydrostatic terms should produce negligible long term contributions, simply because scale analysis indicates they may have small coefficients. The cumulative differences at low frequency due to nonhydrostatic terms in the double-gyre spectra shown in Figures 3a and 3b demonstrate that the long term contributions of these terms are by no means negligible.
In Fig. 3d, we show a comparison of the spectral estimates of the domain-averaged potential energy for systems 1 and 3, when both the Laplacian viscosity and the Rayleigh friction are set to zero. (For reference, case SW1 is shown in dotted line.) In these runs, energy is constantly fed into the system through windstress and is removed by applying Rayleigh friction at the domain boundaries. Both the systems (SW4 and GNH2) are considerably more energetic than SW1. This is further clarified in Fig. 4, where we have plotted the histogram of the kinetic energy time series of cases SW1, GNH1, SW4, and GNH2 used in the spectral analysis of Fig. 3. We note that the small peaks at the low values of the kinetic energy in the plots for cases SW4 and GNH2 are due to the initial conditions. (Recall that the same initial condition for each of the runs was obtained from an initial ten year spin-up run with values of and corresponding to case SW1.) The spectra of the two cases SW4 and GNH2 are based on 60 year runs, and the differences due to the nonhydrostatic terms, i.e., the differences between SW4 and GNH2 at these timescales are much larger than between SW1 and GNH1. Extrapolation of the spectrum to the 100 year periods using the ME method shows a similar trend. This comparison of spectra demonstrates that nonhydrostatic corrections are significantly more important in the more inviscid regions of the flow.