On Imbalance Generated by Vortical Flows in a Two-Layer Spherical Boussinesq Primitive Equation Model

Abstract

Abstract The spontaneous adjustment emission of inertia–gravity waves is investigated by examining the amount of imbalance generated during the evolution of unstable jets in an isentropic two-layer primitive equation model on the sphere. To determine the balance and thus the imbalance, potential vorticity (PV) inversion by means of the first-, second-, and third-order plain-δδ, plain-δγ, and plain-γγ balance relations, as well as the Bolin–Charney balance relations, is applied. Five sets of experiments are carried out by (i) changing the upper-layer potential temperature with a fixed initial jet speed and (ii) changing the initial jet speed with a fixed upper-layer potential temperature. For each set, the relation between the optimal measure of imbalance with the corresponding measure of balanced vortical flow is sought at the peak of instability. The minimal imbalance obtained by 1 of the above 10 PV inversion procedures is considered as the optimal. The focus herein is on the magnitude of imbalance and its scaling with various measures of balanced flow. It is shown that for the magnitude of imbalance it is always possible to find the proper measure with an exponentially small scaling. For the imbalance-to-balance ratio, however, a power law is obtained when either the jet or the stratification is weak. The latter difference in scaling behavior of the imbalance and imbalance-to-balance ratio may be an artifact of the inevitable numerical errors whose effects are felt more strongly when the signature of vortical flow is weak.