New nonlinear normal modes flow decomposition based on instantaneous phase speeds in the normal modes framework

Normal modes of hydrostatic primitive equations on the sphere decompose the linearized flow into the slow propagating Rossby, fast propagating inertio-gravity (IG), and equatorial Kelvin and MRG waves, characterized by intermediate speeds. However, this simple characterization is no longer valid in the nonlinear system, where wave-wave, wave-mean flow interactions, and adiabatic forcing can significantly alter the wave speeds.

A number of methods aimed at determining the slowly evolving component of the nonlinear flow (so called slow manifold) were proposed under the umbrella terms ”nonlinear normal mode decomposition” (NNMD) or "nonlinear normal mode initialization" (NNMI). In the classical NNMD, Rossby waves are considered slow a priory, while the slow IG part consists of the the unbalanced component of the flow slaved to the Rossby modes. More precisely, in classical NNMD, slow IG waves are those that would be stationary in a nonlinear flow consisting of the slow modes only. While this approach is very successful in suppressing high-speed gravity waves, it also suppresses slowly propagating linearly unbalanced flow and large scale tropical circulation.

We propose a new method of flow decomposition into the slow and fast components based on computing instantaneous phase speeds of normal modes in the nonlinear system. The method does not make assumptions on the composition of the slow manifold. Instead, the decomposition is obtained from a constraint optimization problem that minimizes the norm of the fast component, while requiring that the slow manifold does not contain modes propagating faster than the selected cutoff speed. We apply the method to reanalysis data, demonstrate its efficiency, and analyze the composition of the slow manifold as function of the cutoff speed. In particular, when the cutoff is chosen to be approximately equal to the fastest linear Rossby wave speed, most of the tropical circulation and significant part of unbalanced modes are retained in the slow manifold.