The linear normal modes of the atmosphere fall into two categories, the
low frequency Rossby waves and the high frequency gravity waves. The
elastic pendulum is a simple mechanical system having low frequency and
high frequency oscillations. Its motion is governed by four coupled
nonlinear ordinary differential equations. We study the dynamics of
this system, drawing analogies between its behaviour and that of the
atmosphere. The linear normal mode structure of the system is
analysed, the procedure of initialization is described and the
existence and character of the slow manifold is discussed.
Prologue
The concepts of initialization, filtering and the slow manifold can be clearly illustrated by considering the dynamics of a simple mechanical system governed by a set of ordinary differential equations. The elastic pendulum or swinging spring depicted in Fig. 1 comprises a heavy bob suspended by a light elastic rod which may stretch but not bend. The bob is free to move in a vertical plane. The oscillations of this system are of two types, distinguished by their physical restoring mechanisms. For an appropriate choice of parameters, the elastic oscillations have much higher frequency than the rotation or libration of the bob. We consider the elastic oscillations to be analogues of the high frequency gravity waves in the atmosphere. Similarly, the low frequency rotational motions are considered to correspond to the rotational or Rossby-Haurwitz waves. We will refer to the elastic and rotational motions as ``fast'' and ``slow'' respectively.
The linear analysis of this mechanical system is straightforward. When nonlinear effects are included there is coupling between the two types of motion, and analytical methods are incapable of providing the solution. To obtain insight into the characteristics of the motion in this case we must turn to some powerful and general results from dynamical systems theory. The pendulum equations contain a small parameter, e, the ratio of the frequencies of the slow and fast oscillations. The problem may be formulated in terms of a perturbed Hamiltonian, the size of the perturbation depending on e. Then the Kolmogorov-Arnold-Moser or KAM theorem implies certain restraints on the nature of the solution. The validity of the conclusions drawn will be supported by numerical simulations.
Lorenz (1986) constructed a highly simplified model, comprising five ordinary differential equations, based on a truncated spectral expansion of the shallow water equations. He identified the variables corresponding to the high frequency oscillations as representing the gravity wave activity and defined the slow manifold to be an invariant sub-manifold of the five-dimensional phase space in which high frequency oscillations are permanently absent. In Bokhove and Shepherd (1996: BS96) Lorenz's model is further reduced, to a system of four ordinary differential equations. A similar reduction is made by Camassa (1995). These equations are structurally similar to the equations for a nonlinear pendulum coupled to a linear harmonic oscillator. The system is amenable to the application of Hamiltonian perturbation theory. For small values of the perturbation or coupling parameter, one may identify an invariant manifold on which the high frequency activity is unequivocally zero. This manifold is nonlinearly stable: a small gravity wave disturbance about it will remain permanently bounded. However, the manifold is not defined continuously throughout phase space, but is fractal in structure. Numerical experiments in BS96 showed that, as the perturbation parameter increases, the extent of the manifold decreases until, ultimately, it disappears entirely.
It turns out that the simple mechanical system considered in this report is governed by mathematical equations having a structure very similar to Lorenz's model. Both can be described in terms of a system with two modes of behaviour, a linear harmonic oscillator and a nonlinear pendulum. The precise details of the coupling between the oscillator and pendulum differ in the two cases; but the conclusions of the KAM theorem do not depend upon these details. Therefore, much of the discussion in BS96-in particular, their conclusions about the existence of a slowest invariant manifold-can be applied directly to the elastic pendulum considered herein.
Outline of Contents
In Section 2 the Hamiltonian equations for the elastic pendulum are set down. The linear solutons are examined and the procedures for linear and nonlinear initialization are discussed. The concept of the slow manifold is introduced and illustrated by some numerical integrations. In Section 3 the ideas underlying the KAM theorem are presented and its main conclusions are summarised. Some general consequences of the theorem-in celestial mechanics, particle physics and statistical mechanics-are briefly described. The application of KAM to the elastic pendulum occupies Section 4. The implications for the existence of a slow manifold for this system are discussed. We conclude that, for small values of the frequency ratio e, it is possible, for almost all values of the slow variables, to define appropriate values of the fast variables in such a way that the solution has no high frequency oscillations. The exceptional cases give the slow manifold a fractal structure; but, for small e, they form a set of negligible measure. These conclusions are supported by numerical experiments which are described in Section 5. Poincaré sections showing regions of regular motion and regions of chaos are plotted. For small e the solutions are predominantly regular and the core solution representing purely slow motion can be clearly seen. As e grows the solutions become more complex, until a stage is reached where the distinction between fast and slow time scales no longer makes sense. In Section 6 we digress to consider the case of resonance when the fast and slow time scales are in the ratio two-to-one, and show how energy is transformed back and forth periodically between the swinging and springing modes. In the final part of the paper, Section 7, we discuss miscellaneous aspects of the problem and touch upon some unresolved issues.