Large-time behavior of solutions to the Boussinesq equations with partial dissipation and influence of rotation
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2025
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| _version_ | 1866908320255705088 |
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| author | Jiang, Song Wang, Quan |
| author_facet | Jiang, Song Wang, Quan |
| contents | This paper investigates the stability and large-time behavior of solutions to the rotating Boussinesq system under the influence of a general gravitational potential $Ψ$, 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 $(ρ,u,v,w,p)=(ρ_s,0,v_s,0, p_s)$. These solutions are characterized by both geostrophic balance, given by $fv_s-\frac{\partial p_s}{\partial x}=ρ_s\frac{\partial Ψ}{\partial x}$ and hydrostatic balance, expressed as $-\frac{\partial p_s}{\partial z}=ρ_s\frac{\partial Ψ}{\partial z}$. Second, we establish that any steady-state solution satisfying the conditions $\nabla ρ_s=δ(x,z)\nabla Ψ$ with $v_s(x,z)=a_0x+a_1$ is linearly unstable when the conditions $δ(x,z)|_{(x_0,z_0)}>0$ and $(f+α_0)\leq 0$ are simultaneously satisfied. This instability under the condition $δ(x,z)|_{(x_0,z_0)}>0$ 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 $ρ=-γΨ+β$, where $γ$ and $β$ are positive constants, and the meridional velocity $v$ depends solely and linearly on $x$ variable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-time behavior of solutions to the Boussinesq equations with partial dissipation and influence of rotation Jiang, Song Wang, Quan Analysis of PDEs This paper investigates the stability and large-time behavior of solutions to the rotating Boussinesq system under the influence of a general gravitational potential $Ψ$, 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 $(ρ,u,v,w,p)=(ρ_s,0,v_s,0, p_s)$. These solutions are characterized by both geostrophic balance, given by $fv_s-\frac{\partial p_s}{\partial x}=ρ_s\frac{\partial Ψ}{\partial x}$ and hydrostatic balance, expressed as $-\frac{\partial p_s}{\partial z}=ρ_s\frac{\partial Ψ}{\partial z}$. Second, we establish that any steady-state solution satisfying the conditions $\nabla ρ_s=δ(x,z)\nabla Ψ$ with $v_s(x,z)=a_0x+a_1$ is linearly unstable when the conditions $δ(x,z)|_{(x_0,z_0)}>0$ and $(f+α_0)\leq 0$ are simultaneously satisfied. This instability under the condition $δ(x,z)|_{(x_0,z_0)}>0$ 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 $ρ=-γΨ+β$, where $γ$ and $β$ are positive constants, and the meridional velocity $v$ depends solely and linearly on $x$ variable. |
| title | Large-time behavior of solutions to the Boussinesq equations with partial dissipation and influence of rotation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.10827 |