Inviscid Geophyiscal Models

Main narrative

Via physics-based approximations and asymptotic analyses, the discipline of GFD has developed and adopted a hierarchy of models specifically for large-scale geophysical flows. These models span from the simple strictly two-dimensional planar flows all the way up to the non-hydrostatic Boussinesq model for fully three-dimensional flows. The goals of performing rigorous mathematical analysis on these models are two folds. The first one is to establish the well-posedness of the models, or, more precisely, the conditions under which these models are well-posed. Only well-posed models can be trusted as decent approximations to the reality and can be worthy of being simulated. The other goal, which is equally important, is to explore and understand the long-term dynamics of these models. Knowledge obtained through theoretical analyses can serve as a guide for numerical simulations, and make the latter much easier. Along this line, we have established the global well-posedness for the barotropic quasi-geostrophic (QG) equation under a free surface (Chen 2017 submitted). Compared with the two-dimensional Euler equation for strictly planar flows, the QG model considered in this work adds some mild variations in the third dimension by incorporating the top surface fluctuation into the potential vorticity. A slightly more sophisticated model, the multi-layer QG model, has also been considered (Chen 2018, submitted). Here, the interactions between the vertical layers require more elaborate estimates on the streamfunctions.

Chronicle of developments

1. In Chen (2017) we established the global well-posedness of the inviscid barotropic QG equation on a bounded domain.

2. Chen (2018) dealt with the global well-posedness of the multi-layer QG equation. Here, due to the layer interactions, more elaborate estimates were required.