(0)3. 0 We consider a fairly large domain, 2800 kms in the north-south direction, and 3600 kms in the east-west direction, so that the dynamics of the unstable midlatitude jet is not influenced by the eastern boundary. The initial undisturbed depth of the upper layer is 500 meters and the reduced gravity parameter, g' is 0.031 m2/s, corresponding to the (previously assumed) density difference of 0.3% between the upper and lower layers. The corresponding Rossby deformation radius is about 50 kms at the center of the domain. Thus, the domain of computation is roughly 70 Rossby radii in the East-West and 55 Rossby radii in the North-South directions. The domain is spanned by 256 points in each direction. The wind stress is chosen as the profile ,where A corresponds to a wind stress amplitude of 0.08 N/m2 and D is the domain size in the north-south direction. The simulations are carried out on a beta plane, i.e., , with f0 = and .If not stated otherwise, the coefficient of kinematic eddy-viscosity is fixed at 10 m2/s, and the coefficient of interfacial Rayleigh friction is set at .All three systems are initialized (at day 0) with a state obtained from an initial ten-year spin-up run with the same values of and as above.
In Fig. 1, we show color-coded snapshots of three quantites obtained from the system GNH (or system 3) at two different times. The three different quantities are i) the layer-depth, ii) the magnitude of the GN nonhydrostatic accelaration terms (i.e.the nonhydrostatic accelaration terms not related to the horizontal component of earth's rotation) and iii) the magnitude of the nonhydrostatic accelaration terms due to the horizontal component of earth's rotation. A logarithmic color scale has been chosen to better illustrate the regions of activity in all the three fields. It is clear that the dynamically active regions--the regions of changing colors in the depth field--are also the regions of most nonhydrostatic acitivity at both the instants of time shown. Further, as may have been expected, the nonhydrostatic accelaration terms due to the horizontal component of earth's rotation is active over much larger regions of the flow as compared to the GN terms.
The layer-depth fields in Fig. 1 also show the complex patterns that arise due of the instability of the jet and the ensuing eddy-shedding and merger activity. Eddies may be shed at different east-west locations; some of these eddies interact strongly with the jet, and some may even be recaptured into the jet itself. Or, some of the westward propagating eddies may interact with the western boundary currents (WBCs) and result in a vacillation of the separation point of the WBCs and cause large latitudinal excursions of the midlatitude jet. All these phenomena enrich the variability of the system, but preclude a simple phenomenological explanation of the main features of the flow. Equally importantly they demonstrate the difficulty of comparing the different systems: the unstable jet dynamics causes a large degree of variability in each of the systems.
In Fig. 2, we illustrate in quantitative terms the sizes of the nonhydrostatic accelaration terms pictured in Fig. 1. The quantity plotted Fig. 2a is either the x or y component of the ratio of the (veectorial) GN nonhydrostatic accelaration at a point to the hydrostatic accelaration at the same point, the point being the location of the maximum of the former field. The x-axis is time in years. Thus, a small value of this quantity at time t would mean that the flow field at time t is hydrostatically balanced to a large extent. The quantity we have chosen to measure the extent of nonhydrostasy is better than say the ratio of the scalar acclerations since we are measuring the ratio of terms as they appear in the equations without unnecessary averaging. The quantity plotted Fig. 2b is the corresponding analogue for the quasi-hydrostatic terms. The breaking of the hydrostatic balance is evident in both the plots.