Implicit Normal Mode Initialization

Abstract

It is shown that nonlinear normal mode initialization (NMI) can be implemented without knowing the normal modes of a model. The implicit form of nonlinear NMI is particularly useful for models whose normal modes cannot readily be found; for example, if the underlying linear equations are nonseparable. An implicit nonlinear NMI scheme is formulated for the shallow-water equations on a polar stereographic projection. The linear equations which define the implicit normal modes include most of the beta terms as well as variable Coriolis parameter and map scale factor. Even in this nonseparable case, the equivalence between implicit and conventional nonlinear NMI is shown to be exact. The scheme is implemented in a regional model on a quasi-hemispheric domain, which uses a finite-element discretization on a nonuniform grid. The well-posed lateral boundary conditions of this model lead to consistent boundary conditions for the initialization. Results are presented not only for the implicit form of Machenhauer's nonlinear NMI technique, but also for the implicit form of Tribbia's corresponding second-order scheme which results in an even better initial balance.