Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

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 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 . 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, i.e. tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Traveling waves near Couette flow for the 2D Euler equation

“…For the analogous problem in the viscous setting or even in the compressible case, we refer to [5,3,4,19,53,18,1] for a comprehensive but not exhaustive list of references. Similar analysis have been also carried out recently for the Boussinesq system, see for example [9,10,25,55,66,67].…”

Global axisymmetric Euler flows with rotation

“…(2) We may view Theorem 1.1 as a global stability result (in the class of axisymmetric perturbations satisfying (1.3)) for uniformly rotating solutions U rot = r e θ in cylindrical coordinates (r, θ, z) to the incompressible 3d Euler equations (1.1). From this perspective, our result connects with the study of stability of infinite energy solutions to the 2d Euler equations, such as shear flows [5,38,39,52] or stratified configurations [4], even though the stability mechanism ("phase mixing") in these settings is different. However, to the best of our knowledge there are no such results for the Euler equations in 3d.…”

Small scale formation for the 2D Boussinesq equation

“…Thus the infinite-in-time growth estimate (3.23) directly implies that any such steady state (with zero density) is nonlinearly unstable, in the sense that for any 0 < k 0 1, an arbitrarily small perturbation ρ 0 = k 0 cos(x 1 ), ω 0 = ω s leads to lim t→∞ ω(t) L 1 = ∞. See [3,5,14,17,41,45,48] for more results on stability/instability of steady states of the inviscid or viscous Boussinesq equations.…”

“…In the absence of thermal diffusion, the first global-intime regularity results were obtained by Hou-Li [29] in the space (u, ρ) ∈ H m (R 2 ) × H m−1 (R 2 ) for m ≥ 3, and Chae [6] in the space H m (R 2 ) × H m (R 2 ) for m ≥ 3. When Ω ⊂ R 2 is a bounded domain, Lai-Pan-Zhao [36] proved global well-posedness of solutions in H 3 (Ω)×H 3 (Ω) with noslip boundary condition, and showed that the kinetic energy is uniformly bounded in time. The function space was improved by Hu-Kukavica-Ziane [30] to (u, ρ) ∈ H m (Ω)×H m−1 (Ω) for m ≥ 2, where Ω is either a bounded domain or R 2 , T 2 .…”