ABSTRACT

Surface quasi-geostrophic (SQG) flows with a much larger horizontal scale than the Rossby radius of deformation are considered. A new version of the SQG model with two boundaries, which is reduced to a nonlinear system of partial differential equations, is proposed to describe the dynamics of such flows. This system describes the interaction between the barotropic and baroclinic components of the stream function and generalises the two-dimensional Euler equations for flows with a vertical velocity shear. The laws of conservation of both total and surface potential energies, which follow from this system, have been formulated. The solutions of a number of problems in the theory of baroclinic instability, which are in agreement with already known solutions, have been obtained within the framework of this system. It is shown that vertical shear flows are absolutely unstable, i.e. their instability is independent of the horizontal velocity profile structure. A generalised system of the SQG model equations, which additionally takes into account the β-effect and the Ekman bottom friction, has also been proposed. The transformation of jet flows due to the bottom friction and the influence of the β-effect on the stability of shear flows have been studied based on this system.

摘要

考虑了水平尺度比 Rossby 变形半径大得多的表面准地转 (SQG) 流。提出了一种新版本的具有两个边界的 SQG 模型，该模型简化为偏微分方程的非线性系统，用于描述此类流动的动力学。该系统描述了流函数的正压分量和斜压分量之间的相互作用，并概括了具有垂直速度剪切的流动的二维欧拉方程。从该系统得出的总势能守恒定律和表面势能守恒定律已经被制定出来。斜压不稳定性理论中的许多问题的解与已知解是一致的，都是在该系统的框架内得到的。结果表明，垂直剪切流是绝对不稳定的，即它们的不稳定性与水平速度剖面结构无关。SQG 模型方程的广义系统，它另外考虑了β效应和 Ekman 底部摩擦也已被提出。基于该系统研究了底部摩擦引起的射流的转变和β效应对剪切流稳定性的影响。