Nonlinear inviscid damping and shear-buoyancy instability in the 2D Boussinesq equations

Nonlinear inviscid damping and shear-buoyancy instability in the 2D Boussinesq equations

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Roberta Bianchini, IAC

Zoom link: https://princeton.zoom.us/j/4745473988


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.