Thesis title: Well-posedness and long-time dynamics for a deterministic and stochastic quasi-geostrophic ocean-atmosphere model with heat exchange
In this work we study from a mathematical perspective a climate model which describes the interaction between atmosphere and ocean at mid latitudes and at synoptic scale. The model is composed of a two-layer quasi-geostrophic atmosphere, coupled both thermally and mechanically to a quasi-geostrophic shallow-water ocean layer with a deep and quiescent layer below. The equations of the model are nonlinear because of the presence of transport and polynomial terms.
First we describe the physics of the model, the geometry of the ocean-atmosphere domain and the boundary conditions. Then we discuss well-posedness showing existence and uniqueness of the Weak Solution. While existence is proved using standard techniques, uniqueness requires a non-trivial estimate in which the asymmetry in the regularity between the streamfunctions and the ocean temperature plays a determinant role. We also show that the Weak Solutions are Lipschitz continuous with respect to all the radiation parameters. Requiring extra assumptions related to the boundary conditions for the temperature, we show existence and uniqueness of more regular solutions than the Weak ones, and in a periodic domain we can find smooth solutions and show positivity of the temperature functions.
Afterwards we show the existence of a finite dimensional global attractor, the injectivity of the dynamics on it, and the existence of a finite set of determining modes that can be chosen to not depend on the ocean temperature. This result follows from a balance between the heat conduction coefficient and the short-wave radiation function, which holds for physically realistic values of the parameters.
In the last part of the thesis we present a stochastic version of the model for a random force given by an additive noise, white in time and colored in space.
We show existence and uniqueness of the Weak and Strong Solutions for the stochastic model, using a standard approach. Finally, we prove the existence of an invariant measure and we introduce some techniques that can be used to show its uniqueness, presenting a partial result in this direction.