We show that physically interesting steady states of the Rotating Shallow Water equations are characterized by a minimax principle. The objective functional is A-thetaH where A is the quadratic enstrophy, H is the energy and theta is a positive constant. The inner maximization is subject to a pointwise constraint on the potential vorticity (PV) while the outer minimization is over all vorticity fields. In physical terms, the inner maximization represents geostrophic adjustment, while the outer minimization represents relaxation to a steady state through PV mixing. The key idea behind the principle is the separation of time scales between the fast inertial-gravity waves and the slow vortical modes, which implies that during geostrophic adjustment, the vorticity field remains frozen, while during vortical mixing the energy remains constant. The inner maximization problem is solved analytically by an asymptotic expansion in Rossby number epsilon, thus obtaining a first order correction to the quasigeostrophic(QG) fields. The outer minimization problem is then solved numerically for the 1-D case using the corrected fields. The resulting steady flows are therefore analogues of quasigeostrophic steady states at finite epsilon.; The first order Rossby number effect is examined for zonal shear flows in parameter regimes relevant to the oceans and to the atmosphere of Jupiter, and include the beta effect and bottom topography. Some of the striking results at finite Rossby number include the cyclone-anticyclone asymmetry, when anticyclones are found to be much more prevalent than cyclones. Also as an example, for two different bands on Jupiter, certain jet structures are found be more robust than others when first order Rossby number corrections are included.
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