5 - Oscillation and Decay Equations

Summary

Introduction

The governing equations of the evolution of the atmosphere are time-dependent equations. In this chapter, the problem of time differencing is taken up for detailed discussion without considering issues related to space differencing.

Properties of Time-differencing Schemes as Applied to the Oscillation Equation

Consider a general first-order differential equatio

The stability and other important properties of the various (two level and three level) time-differencing schemes that were introduced in Section 3.3 depend on the form of the function f(u,t) in Equation (5.1). In order to discuss these properties, one needs to prescribe the form of this function f(u,t). For applications in atmospheric models, it is of interest to consider the case where f = iwu, i.e., the ordinary differential equation (ODE), where f = f (u):

Equation (5.2) is known as the oscillation equation . It is possible for u to be complex, so that in general, the oscillation equation represents a system of two equations. The parameter w, the frequency of the system, is taken as real. The exact solution of Equation (5.2) is of the form

where u_o is the value of u at initial time t = 0. The amplitude u_o is an invariant of the system, i.e.,

The following are examples of more familiar equations that reduce to Equation (5.2). (i) One-dimensional linear advection equation:

(ii) Rotational motion. Pure inertial motion is described by the following equations

Multiplying Equation (5.6b) by i and adding it to Equation (5.6a), one gets

Defining U = u+iv, Equation (5.7) becomes

Equation (5.8) has the same form as Equation (5.2) but with w = -f .

The general solution of Equation (5.2) is u(t) = u_oe^iwt , or, for discrete values t = nΔt, the solution is u(nΔt) = u_oe^inwΔt.

If we consider the solution in the complex plane, its argument rotates by wΔt each time step and there is no change in amplitude. One can utilize the von Neumann method to analyze the properties of the various two time level and three time level difference schemes.