Abstract

Based on the equation for potential vorticity in the quasi-geostrophic approximation, an analysis of stable and unstable perturbations of ocean currents of a finite transverse scale with a vertical linear velocity profile (Couette-type flows) is presented. The model takes into account the influence of vertical diffusion of buoyancy, friction, and the beta effect (the change in the Coriolis parameter with latitude). The analysis is based on the small perturbation method. The problem depends on several physical parameters and reduces to solving a spectral non-self-adjoint problem for a fourth-order equation with a small parameter at the highest derivative. Asymptotic expansions of eigenfunctions and eigenvalues are constructed for small values of the wavenumber \(k\). Using the continuation in the parameter \(k\), trajectories of the eigenvalues are calculated, which made it possible to compare the influence of the beta effect on unstable perturbations of the first and higher order modes. It is shown that the flow instability depends in a complex way on the physical parameters of the flow.

中文翻译：

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Beta效应对洋流不稳定扰动谱特征的影响

摘要

基于准地转近似中的潜在涡度方程，提出了具有垂直线速度分布（Couette型流）的有限横向尺度洋流的稳定和不稳定扰动分析。该模型考虑了浮力，摩擦力和贝塔效应（科里奥利参数随纬度的变化）的垂直扩散的影响。该分析基于小扰动方法。该问题取决于几个物理参数，并简化为求解具有最高导数的小参数的四阶方程的谱非自伴问题。对于波数\（k \）的小值，构造本征函数和特征值的渐近展开。在参数中使用延续\（k \），计算出特征值的轨迹，这使得可以比较贝塔效应对一阶和高阶模态的不稳定摄动的影响。结果表明，流动的不稳定性以复杂的方式取决于流动的物理参数。