三维分层平面Couette流的稳定性：一种新的升力机制与位势涡度
Stability of 3D Stratified Plane Couette Flow: A New Lift-up Mechanism and Potential Vorticity
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中文总结 AI 辅助
本文研究垂直分层背景下三维平面Couette流的稳定性，发现非平行几何产生新的升力机制，通过位势涡度重构建立无粘阻尼，并证明强分层可抑制瞬态增长，实现非线性稳定性。
中文摘要 AI 辅助
自1931年Taylor和Goldstein的开创性工作为分层剪切流的研究奠定基础以来，非平行构型受到的关注相对较少。本文研究垂直分层背景$\eta^*=\alpha z$下，三维Boussinesq系统在平面Couette流$V^*=(y,0,0)$附近的稳定性。剪切方向与分层方向相互垂直。这种几何结构产生了平行构型中不存在的两种结构效应：非零模系统中的额外非局部耦合，该耦合使得系统在没有重构的情况下无法具有有效的对称化结构；以及零模的非平行/波-剪切耦合的升力机制。由升力效应引起的$L^p$瞬态增长可以通过足够强的分层（当$p>2$时）或在适当的初始数据假设下（当$p\geq 2$时）被抑制。为了研究非零模系统，我们引入了一种基于线性化位势涡度（PV）的重构方法，位势涡度是地球物理流体动力学中的一个基本量，以恢复更好的能量结构。我们建立了$u_{1,\neq}$和$u_{2,\neq}$的无粘阻尼。我们进一步将此方法应用于倾斜Couette流的线性稳定性研究。基于通过PV的新公式，我们还研究了$\mathbb{T}\times\mathbb{R}\times\mathbb{T}$上Sobolev扰动下平面Couette流的非线性稳定性。我们考虑具有有限加权Sobolev范数的初始数据，其中权重选择用于在线性层面抑制升力效应，使得瞬态增长仅通过非线性相互作用产生，并且非线性升力可以通过强分层被抑制。
英文摘要
Since the pioneering work of Taylor and Goldstein in 1931 laid the foundation for the study of stratified shear flows, non-parallel configurations have received comparatively little attention. In this article, we study the stability of the three-dimensional Boussinesq system near the plane Couette flow $V^*=(y,0,0)$ under the vertically stratified background $η^*=αz$. The shear and stratification directions are perpendicular. This geometry produces two structural effects that are absent in the parallel configuration: an additional nonlocal coupling in the nonzero-mode system, which prevents the system from having a valid symmetrization structure without reformulation, and a non-parallel/wave-shear coupled lift-up mechanism for the zero modes. The $L^p$ transient growth due to the lift-up effect can be suppressed either by sufficiently strong stratification (when $p>2$), or under suitable assumptions on the initial data (when $p\geq 2$). To study the nonzero-mode system, we introduce a reformulation based on the linearized potential vorticity (PV), a fundamental quantity in geophysical fluid dynamics, to recover a better energy structure. We establish the inviscid damping of $u_{1,\neq}$ and $u_{2,\neq}$. We further apply this method to study the linear stability of the tilted Couette flow. Based on the new formulation through PV, we also investigate the nonlinear stability of the plane Couette flow for Sobolev perturbations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We consider initial data with finite weighted Sobolev norm, where the weight is chosen to suppress the lift-up effect at the linear level, so that transient growth arises only through nonlinear interactions and the nonlinear lift-up can be suppressed by strong stratification.
发表机构
- University of Bath（巴斯大学）
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