On the global stability and large time behavior of solutions of the Boussinesq equations

TL;DR

This paper analyzes the stability and long-term behavior of solutions to the 2D viscous Boussinesq equations, revealing conditions for equilibrium, instability, and convergence to hydrostatic states in stratified flows.

Contribution

It characterizes the steady states, establishes conditions for their instability, and demonstrates convergence to hydrostatic equilibrium despite Rayleigh--Taylor instability.

Findings

Hydrostatic equilibria are only of a specific form satisfying certain gradient relations.

Hydrostatic equilibria are linearly unstable if a certain condition on density and potential holds.

Solutions tend to converge to hydrostatic states even when unstable.

Abstract

We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, $(\mathbf{u},\rho,p) = (0, \rho_s, p_s)$, where the pressure gradient satisfies $\nabla p_s = -\rho_s \nabla f$. Moreover, the relation between $\rho_s$ and $f$ is constrained by $(\partial_y \rho_s, -\partial_x \rho_s) \cdot (\partial_x f, \partial_y f) = 0$, which allows us to write $\nabla \rho_s = h(x,y) \nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0, \rho_s, p_s)$ is linearly unstable if $h(x_0, y_0) > 0$ at some point $(x, y) = (x_0, y_0)$. This instability coincides with the classical Rayleigh--Taylor instability. Third, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh--Taylor instability makes perturbations around the unstable equilibrium grow exponentially in time, the system ultimately converges to a state of hydrostatic equilibrium. The analysis is carried out for perturbations about an arbitrary hydrostatic equilibrium, covering both stable and unstable configurations. Finally, we derive a necessary and sufficient condition on the initial density perturbation under which the density converges to a profile of the form $-\gamma f + \beta$ with constants $\gamma, \beta > 0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability.