Zoom link: https://princeton.zoom.us/j/4745473988(link is external)

In this talk, we consider the 2D inviscid Boussinesq equations near the stably stratified Couette flow, for an initial Gevrey perturbation of size ε, and we study their long-time properties. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an O(t^{1/2}) inviscid damping, while the vorticity and density gradient grow as O(t^{1/2}). The result holds at least until the natural, nonlinear timescale t≈ε^{-2}. The proof relies on two main ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem.
 

This is a recent joint work with J. Bedrossian, M. Coti Zelati and M. Dolce.