Iterative Methods for Solving the Nonlinear Balance Equation with Optimal Truncation

Abstract

Two types of existing iterative methods for solving the nonlinear balance equation (NBE) are revisited. In the first type, the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution. In the second type, the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively. The two methods are rederived by expanding the solution asymptotically upon a small Rossby number, and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution. For each rederived method, two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration. Upon testing with analytically formulated wavering jet flows on the synoptic, sub-synoptic and meso-α scales, the iterative procedure designed for the first method with the Poisson solver, named M1a, is found to be the most accurate and efficient. For the synoptic wavering jet flow in which the NBE is entirely elliptic, M1a is extremely accurate. For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic, M1a is sufficiently accurate. For the meso-α wavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed, M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow, but not for the anti-cyclonically curved part.