A Proof for the Equivalence of Two Upper Bounds for the Growth of Disturbances from Barotropic Instability

Abstract

A previous study proposed two methods for calculating the upper bound of the growth of disturbances from barotropic instability of a zonal flow in a two-dimensional incompressible fluid on a rotating sphere. The study conjectured that these two upper bounds are equivalent. One method was based on the conservation of the domain-averaged pseudomomentum density, and the other solved a minimization problem under the constraints of the conservations of all Casimir invariants and the total absolute angular momentum. In this study, this conjecture is verified, i.e., a proof is presented for their equivalence by developing an annealing-like procedure to reach the absolute vorticity profile that corresponds to the upper bound. The procedure also provides a more efficient method to calculate the upper bound.