Nonlinear Partial Differential Equations

Date: November 16, 2021

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Antoine Remond-Tiedrez, University of Wisconsin - Madison

Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundary

Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations. These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by latent heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings). In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.