Rotational and gravitational influences on the degree of balance in the shallow-water equations

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Abstract
Numerical solutions of the Shallow-Water Equations with random, balanced initial conditions are analyzed for their persistence of balance. The standard for assessing balance is provided by the pressure and velocity potential fields determined diagnostically from the instantaneous streamfunction through the momentum divergence (i.e., gradient-wind balance) and omega relations of the Balance Equations. The amplitude, wavenumber and frequency content of both the balanced and unbalanced components are examined for a wide range of Rossby R, and Froude F, numbers. These properties are found to be generally consistent with the estimates from a formal scaling analysis for small R and F, and the scaling estimates are apt even when these parameters are not particularly small. The unbalanced component is dominated by higher order advective motion for R ≪ 1 and by nearly linear inertia-gravity waves, whose frequencies are larger than those of the balanced component, for R ≫ l. These waves are forced by nonlinear interaction with advective terms formally of O(F 2) which are neglected in the Balance Equations. These results support the hypotheses that the Balance Equations are an accurate model for the advective component of the flow (also called the slow manifold), broadly throughout the regimes of significant environmental rotation and/or stable stratification, and that the dynamical coupling is weak with the small-amplitude, fast wave component, especially so for R ≪ l.