Nonlinear wave interactions in Rotating Shallow Water Equations on the Sphere: Theory and multi-wave applications

One of the challenges in weather forecasting is the understanding of the nonlinear interactions between the fast and slow dynamics in the atmosphere. This is related to both numerical problems, such as the choice of a stable time step, and modelling and understanding the dynamics of atmospheric phenomena, such as the Madden-Julian Oscillation. Using a Rotating Shallow Water model on the sphere, in which both fast (inertia-gravity) and slow (Rossby-Haurwitz) waves occur, the nonlinear interactions in reduced models containing three, four and five waves were analysed using Hough harmonics spectral decomposition. Considering a Galerkin expansion as a solution of the nonlinear system, equations for the dynamics of each mode were derived, along with necessary conditions in the zonal and meridional structure of the modes for three interacting waves. In this talk, we will show results of three, four and five wave system interaction, discussing the energy transfers between Rossby-Haurwitz and gravity waves. We will particularly illustrate how we can observe relevant slow oscillations emerging from fast wave dynamics in realistic parameter ranges.