大气运动不稳定的变分原理

VARIATIONAL PRINCIPLE OF INSTABILITY OF ATMOSPHERIC MOTIONS

摘要:

本文利用非线性基本方程组和变分原理研究旋转大气运动的不稳定性问题。文中所用方法是普遍适用的,对各种可能的基流模型都可获得其不稳定性判据。例如:适合于正压或斜压、分层或连续模型以及准地转模式或原始方程组;基流可以是带状或非带状的(即平行流或非平行流)、定常的或非定常的。虽然基流具有多样性,但它们在由相应的不变泛函所决定的空间内都是驻点(临界点)。如果在相应的泛函中将角动量守恒包括进去,那么基流可以是非定常流。无论线性或非线性二阶变分都给出不稳定性判据。但尤其值得一提的是,本文第一次得到关于非定常基流、地形扰动流和非地转流的不稳定性判据。同时本文也指出了线性理论和我们发展的变分原理所得到的不稳定性判据之间的差别,该差别说明用非线性基本方程组的重要性。在最后一节,将该理论推广到流体不具有有限能量,因而变分原理不能直接应用的情况,如β-平面,然而广义的Liapounoff函数仍然可以在变分考虑下得到。

Abstract:

Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this palter is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroelinic,layer or continuous, quasi-geostrophic or primitive equations, the basic flow can be zonal or nonzonal, steady or unsteady. Although the basic flows possess a great deal of variety,they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional. The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper. It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations. In the appendix the theory is extended to cases such as in a 13-plane where the fluid does not possess finite total energy,hence the variational principle can not be directly applied. However, a generalized Liapounoff norm can still be obtained on the basis of variational consideration.