Balance regimes for the stability of a jet in an f-plane shallow water system

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime (Ro < 5), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr² / Ro, where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime (Ro > 5), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr². Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.