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.

摘 要

本文回顾了求解非线性平衡方程 (NBE) 的两种迭代方法, 一种是针对相对小的初始猜值情况, 把NBE转化成线性方程再迭代求解线性方程; 另一种把NBE改写成二次方程式, 认为解是平方根的正值分量, 再代回Poisson方程迭代求解. 本研究以罗斯贝数 (Ro) 为小参数, 通过渐进展开非线性平衡方程 (NBE) 的解, 增加最优截断条件, 重新推导了以往的两种迭代求解方法, 并在求解边条件下Poisson方程的每一步迭代过程中, 采用原有的超张弛算法和积分形式的高精度求解算法. 通过四组不同Ro分别代表的天气尺度、次天气尺度和中-尺度天气系统的数值试验, 检验了不同迭代方法的求解精度和效率. 结果表明, 采用积分形式的高精度求解算法的第一种类型迭代法, 求解误差显著降低, 求解效率大幅提高. 即使在气旋式弯曲的中-尺度急流中, Ro增大到了0.4, 而且NBE变成可能无解的双曲型方程, 在急流气旋式弯曲部分依然能求得高精度的解, 但在反气旋式弯曲部分的解仍有待改进.