Large-time behavior of solutions to the Boussinesq equations with partial dissipation and influence of rotation

Abstract

This paper investigates the stability and large-time behavior of solutions to the rotating Boussinesq system under the influence of a general gravitational potential $\hat{I}¨$, which is widely used to model the dynamics of stratified geophysical fluids on the $f-$plane. Our main results are threefold: First, by imposing physically realistic boundary conditions and viscosity constraints, we prove that the solutions of the system smust necessarily take the following steady-state form $(\ddot{I},u,v,w,p)=(\ddot{I}{}_s,0,v_s,0,p_s)$. These solutions are characterized by both geostrophic balance, given by $f{v}_s-\frac{\partial p_s}{\partial x}=\ddot{I}{}_s\frac{\partial \hat{I}¨}{\partial x}$ and hydrostatic balance, expressed as $-\frac{\partial p_s}{\partial z}=\ddot{I}{}_s\frac{\partial \hat{I}¨}{\partial z}$. Second, we establish that any steady-state solution satisfying the conditions $\nabla \ddot{I}{}_s=\hat{I}´(x,z)\nabla \hat{I}¨$ with $v_s(x,z)=a_{0}x+a_1$ is linearly unstable when the conditions $\hat{I}´(x,z){|}_{(x_0,z_0)}>0$ and $(f+\hat{I}{\pm }_0)\le 0$ are simultaneously satisfied. This instability under the condition $\hat{I}´(x,z){|}_{(x_0,z_0)}>0$ corresponds to the well-known Rayleigh-Taylor instability. Third, although the inherent Rayleigh-Taylor instability could potentially amplify the velocity around unstable steady-state solutions (heavier density over lighter one), we rigorously demonstrate that for any sufficiently smooth initial data, the solutions of the system asymptotically converge to a neighborhood of a steady-state solution in which both the zonal and vertical velocity components vanish. Finally, under a moderate additional assumption, we demonstrate that the system converges to a specific steady-state solution. In this state, the density profile is given by $\ddot{I}=-{\hat{I}}^3\hat{I}¨+{\hat{I}}^2$, where ${\hat{I}}^3$ and ${\hat{I}}^2$ are positive constants, and the meridional velocity $v$ depends solely and linearly on $x$ variable.