Numerical Study of Linear Stability

This chapter deals with the method of normal modes widely used for the linear stability study of atmospheric and hydrodynamic flows on a sphere. It is the most effective and constructive technique allowing to find exponentially growing perturbations at the initial stage of the instability when the perturbation is still small and their behavior is well described by the linearized equation. However, the linear approximation ceases to function when the unstable perturbation gets big enough, and therefore the stability by linear approximation does not guarantee the nonlinear stability in the sense of Liapunov (to small but finite perturbations). Nevertheless, the instability by linear approximation means the Liapunov instability. In the framework of this method, the original nonlinear equation is linearized with respect to very small (infinitesimal) perturbations of the basic flow, and the perturbations (normal modes) are sought in the special form as the product of a time function and a function of spatial variables. The time function determines the exponential growth or decay of the mode, while the spatial function is the mode amplitude. The search for unstable perturbations reduces to the solution of the eigenvalue problem for the linearized operator. The numerical solution of the eigenvalue problem raises certain questions of spectral approximation. Indeed, how can we estimate the accuracy with which normal modes on a sphere are calculated? How does the structure and growth rate of unstable modes depend on the approximation of the main flow and perturbations, and also on the degree of the Laplace operator representing the term of turbulent viscosity? With the aim to demonstrate the importance of such questions, it will suffice to mention that unavoidable numerical errors do not allow to analyze the algebraic growth of disturbances, because the Jordan blocks of the stability matrix can be destroyed under any infinitesimal perturbation. Therefore, the stability matrix is always simple in structure, i.e. its eigenvectors are linearly independent, and hence only the exponential growth of infinitesimal perturbations can be analyzed. Thus, the numerical analysis cannot give complete information about the dynamics of disturbances, especially if the instability is weak. The method of normal modes is described in Sect. 8.1. In the case of a viscous fluid, the spectrum of the linearized operator is analyzed in Sect. 8.2. In Sect. 8.3, one estimate is derived in terms of the graph norm of operator. This estimate will be used later in Sect. 8.5. A discrete eigenvalue problem is obtained by truncating the Fourier series of spherical harmonics, representing the basic flow and disturbances, to finite spherical sums of degrees K and N, respectively. The spectral approximation theory for closed operators is briefly described in Sect. 8.4 and is used in Sect. 8.5 to estimate the rate of convergence of the eigenvalues and eigenfunctions of discrete problem to the corresponding eigenvalues and eigenfunctions of the original differential problem. It is shown that the convergence takes place if the truncation numbers K and N tend to infinity keeping the ratio N∕K fixed. Besides, the convergence rate increases with the smoothness of basic flow and with the power s of Laplace operator in the turbulent viscosity term of the vorticity equation. At the same time, the dependence of the convergence rate on the turbulent viscosity coefficient is weak and expressed only through the constant T of estimate (8.3.1). It should be noted that unlike viscous flows, the numerical analysis of linear stability of ideal flows is generally a more difficult problem. Indeed, in the case of a viscous fluid, the linearized operator has a compact resolvent, and, hence isolated eigenvalues of finite multiplicity (see below Theorem 8.2.1 ). Besides, the only accumulation point of eigenvalues may be at infinity. Unlike this, in the case of an ideal fluid, the spectrum of the linearized operator may have a continuous part and finite accumulation points. This problem is discussed in Sect. 8.6, and the stability matrix in the basis of spherical harmonics is introduced in Sect. 8.7. Since the block diagonal structure simplifies the solution of the eigenvalue problem, we consider in Sect. 8.8 special types of stationary flows whose stability matrices have a block diagonal structure. Finally, the structure of the stability matrix of a Legendre-polynomial flow is studied in Sect. 8.9.