Seminar Analysis and Mathematical Physics: Michele Dolce (EPFL)

Long-time behaviour in the 2D Euler-Boussinesq equations near a stably stratified Couette flow

Fluids in the ocean are often inhomogeneous, incompressible and, in relevant physical regimes, can be described by the 2D Euler-Boussinesq system. Equilibrium states are then commonly observed to be stably stratified, namely the density increases with depth. We are interested in considering the case when also a background shear flow is present. In the talk, I will describe quantitative results for small perturbations around a stably stratified Couette flow. The density variation and velocity undergo an O(1/(t^{1/2})) inviscid damping while the vorticity and density gradient grow as O(t^{1/2}) in L^2. This is precisely quantified at the linear level. For the nonlinear problem, the result holds on the optimal time-scale on which a perturbative regime can be considered. Namely, given an initial perturbation of size O(eps), it is expected that the linear regime is observed up to a time-scale O(eps^{-1}). However, we are able to control the dynamics all the way up to O(eps^{-2}), where the perturbation become of size O(1) due to the linear instability.

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