(0)14. 0
Fig. 1. Comparison of the instantaneous flow patterns. Top row, Hydrostatic Shallow Water (SW1); Middle row, Green-Naghdi model (GN1); Bottom row, Green-Naghdi model with horizontal rotation (GNH1). Contours of the depth of the surface layer are drawn at equal intervals of 31.25m. The three pictures in each row are at intervals of 4 months and are shown for year 52 after all the models have been initialized with the same state. The average depth of the basin is about 500m. Note the instability of the jet originating at the separation point of the WBCs in all the cases and the large disruption of the mean position of the jet due to the interaction of the westward propagating eddies with the WBCs.
Fig. 2. Comparison of the mean flow pattern. Contour plot of the 110 year mean of the depth field. Top: Hydrostatic Shallow Water (SW1). Bottom: Green-Naghdi model with horizontal rotation (GNH1). The averaging was done at one hour intervals over the 110 years. Dotted contours indicate cyclonic gyres with depressed sea surface. Continuous contours indicate anticyclonic gyres with elevated sea surface. The close similarity (with maximum pointwise differences between 3% and 6%) of the time-mean contours for these two cases and cases SW2, SW3, and GN1 (not shown) imply the similarity among the base flows.
Fig. 3. Comparison of the overall spectra for cases SW1, SW2, SW3, GN1, and GNH1. The responses at the lowest two decades of frequency are shown on a linear scale in the inset. The domain-averaged quantities in each case are sampled once every four hours over 110 years and the method of Maximum Entropy is used with the same number of poles for each of the five cases. Power spectral density of (a) Domain-averaged horizontal kinetic energy (300 poles). Lowering Rayleigh friction shifts the entire spectrum upwards (cf. SW1 and SW3). Introducing nonhydrostatic terms (GN1 and GNH1) or changing eddy viscosity changes the low frequency part of the spectrum while causing only minor changes at higher frequencies. (b) Potential energy (300 poles). As with the kinetic energy, introducing nonhydrostatic terms causes changes in the low frequency variability (cf. SW1, GN1, and GNH1), the sizes of which are comparable to the sizes of changes obtained by large variations of parameters controlling subgrid scale processes (cf. SW1, SW2, and SW3) (c) Same as (a) except that 400 poles were used in the Maximum Entropy estimate. (d) Horizontal kinetic energy (300 poles). Cases SW4 and GNH2 compared to case SW1. For cases SW4 and GNH2, the Rayleigh friction and the eddy-viscosity was set to zero. The increased variability (note the log scale) at all frequencies (compared to SW1 in dotted line) arises from the more inviscid nature of the flow. Differences between SW4 and GNH2 are significantly greater than between SW1 and GNH1 (see Fig. 3a) showing that the nonhydrostatic terms are more important in more inviscid and lesser damped flows.
Fig. 4. The histogram of kinetic energy distribution for cases SW1 (Solid curve peaking at an x-value of about 8), GNH1 (Dot-dashed curve peaking at an x-value of about 8), SW4 (Solid curve peaking at an x-value of about 20), and GNH2 (Dot-dashed curve peaking at an x-value of about 20). The small peaks in the distributions for cases SW4 and GNH2 are due to the initial conditions.
Fig. 5. (a) The eigenvalue spectra (y-axis on the left) of the EOF basis (mode number on the x-axis) and the cumulative variance it contains (y-axis on the right). The domain-averaged potential energy sampled 10 times a year over 110 years for case SW1 is analyzed. Asterisks and open triangles are for the maximum lag of 8 years. Plus signs and open diamonds are for the maximum lay of 30 years. 10 modes in the 8-year case and 30 modes in the 30-year case roughly contain 90% of the variance. (b) The spectra of the reconstructed time series. Solid line: 30 mode reconstruction of the 30 year maximum lag. Dashed line: 10 mode reconstruction of the 8 year maximum lag. The close correspondence between the spectra of the two independent reconstructions gives credibility to the spectral estimate. The peak at 0.8 cpy is apparently due to the interaction of energetic eddies with the WBCs.
Fig. 6. Low frequency spectra of the 30 mode reconstructions of the domain-averaged potential energy for cases SW1, SW2, SW3, GN1, and GNH1. Eliminating variability due to higher modes (accounting for the last 10% of the variability) exposes remnants of dominant periodicities. The variations in the locations of the peaks in the spectrum due to including the nonhydrostatic terms are again as large as those obtained by large changes of subgrid scale parameters in the hydrostatic shallow water case.