Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations |
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Authors: | Jacob Bedrossian Roberta Bianchini Michele Coti Zelati Michele Dolce |
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Affiliation: | 1. Department of Mathematics, University of California, Los Angeles, California, USA;2. IAC, Consiglio Nazionale delle Ricerche, Rome, Italy;3. Department of Mathematics, Imperial College London, London, UK;4. Institute of Mathematics, EPFL, Lausanne, Switzerland |
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Abstract: | We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. 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 inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several 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 developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth. |
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