Convergence Properties of Machenhauer's Initialization Scheme

Abstract

The convergence properties of Machenhauer?s nonlinar normal-mode initialization scheme are explored. Only adiabatic initialization is considered. Several models are used, including an f-plane model. a numerical weather prediction model, and simple linear models with analytic solutions. The last are used to estimate a radius of convergence for Machenhauer's scheme. It is fist demonstrated with the f-plane model that Machenhaur's scheme may be approximated by one that is linear in gravity-mode coefficients. The components which diverge when the scheme is applied are shown to be linear combinations of gravity modes which interact due to advection. Those combinations which diverge fall into two categories those whom phase speed more than doubles as a result of advection, and those whose direction of propagation changes due to advection. These results agree with those of the simpler model of Ballish. Consideration of Ballish's model suggests that the inclusion of under-relaxation in Machenhauer's scheme, as suggested by Kitade, improves the convergence of eastward waves, but not that of westward waves. Experiments with both the f-plane and numerical weather prediction model also yield this result. Therefore, improvement in convergence by using under-relaxation will depend strongly on which modes are initialized.