A proof for the equivalence of two upper bounds for the growth of disturbances from barotropic instability

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 domainaveraged 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. © 2013,meteorological society of japan.