The introduction of random perturbations by noise in partial differential equations has proven extremely useful to understand more about long-time behaviour in complex systems like atmosphere and ocean dynamics or global temperature. Considering additional transport by noise in fluid models has been shown to induce convergence to stationary solutions with enhanced dissipation, under specific conditions. On the other hand the presence of simple additive forcing by noise helps to find a stationary distribution (invariant measure) for the system and understand how this distribution changes with respect to changes in model parameters (response theory). I will discuss these approaches with a two-layer quasi-geostrophic model as example.

In analyzing complex systems modeled by stochastic partial differential equations (SPDEs), such as certain turbulent fluid flows, an important question concerns their long-time behavior. In particular, one is typically interested in determining how long it takes for the system to settle into statistical equilibrium, and in investigating efficient numerical schemes for approximating such long-time statistics. In this talk, I will present two general results in this direction, and illustrate them with applications to the 2D stochastic Navier-Stokes equations. Specifically, our results provide a general set of conditions that guarantee: (i) Wasserstein contraction for a given Markov semigroup and, consequently, exponential mixing rates; and (ii) uniform-in-time weak convergence for a parametrized family of Markov semigroups towards a limiting dynamic. Most importantly, our approach does not require gradient bounds for the underlying Markov semigroup as in previous works, and thus provides a flexible formulation for a broad range of applications. This is based on joint work with Nathan Glatt-Holtz (Tulane U).

The 2D Boussinesq equations describe the evolution of a heat-conducting fluid. In this talk, I consider the long-time behavior of these equations near a combination of a shear flow and thermal stratification in the setting without dissipation. Unlike the viscous case, here the coupling by buoyancy here gives rise to linear instability and associated norm inflation on critical time-scales. Moreover, nonlinear resonances, called echoes, turn out to be highly frequency-dependent.

Consider the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios, a well-studied model for GFD and foundation to the dynamic core of operational weather forecast. The quasi-geostrophic approximation of this system, introduced by Charney, captures the slow dynamics governing potential vorticity (PV), up to an error at the order of the Rossby/Froude numbers. Fast inertia-gravity waves are filtered out based on the convention that they are "too fast", hence having negligible effect on the slow PV part. GFD literature however has seen counterexamples. In this mathematical study using rigorous PDE analysis, we introduce a novel treatment of near resonance (NR) that no longer fully neglects fast waves, and instead capture the important portion of the nonlinear wave interactions selected using NR criteria. We prove global existence for the proposed NR approximation with a key technique being a sharp counting of the relevant number of nonlinear interactions -- such counting is tied to the nonlinear estimates on the advection terms. An additional regularity advantage arises from a careful examination of some slow/fast mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.

We have a similar NR approximation applied to the rotating Navier-Stokes equations (without stratification) but with crucial differences which seem to make this an easier case.

Joint work with Zisis N. Sakellaris