Numerical methods of estimating bounds on the non-linear saturation of barotropic instability

Two numerical methods are presented for calculating rigorous upper bounds on the finite-amplitude growth of barotropic instabilities to zonal jets on a rotating sphere. One of the methods is based on Shepherd (1988)'s analytic method, which uses the conservation law of domain-averaged pseudomomentum density. A variational minimization problem is solved numerically with a quasi-Newton method after discretization. The other is the authors' original method to solve a minimization problem under the constraints of the conservation laws of all Casimir invariants and total absolute angular momentum. The convex simplex method, which has been used in operations research, is applied to solve a quadratic programming problem. The two methods are applied to estimate the upper bounds for several profiles of the initial unstable jet and the bounds are compared with the results of non-linear time integrations from the unstable jet with a high-resolution model (Ishioka and Yoden, 1994). The two bounds are found to be almost completely identical to each other. Evidence from high-resolution numerical experiments is that the bounds overestimate the actual wave-enstrophy achieved in numerical experiments by a factor of 1.2 to 2.3.