Inertia–Gravity Wave Generation by Balanced Motion: Revisiting the Lorenz–Krishnamurthy Model

Abstract

The spontaneous generation of inertia?gravity waves by balanced motion at low Rossby number is examined using Lorenz's five-component model. The mostly numerical analysis by Lorenz and Krishnamurthy of a particular (homoclinic) balanced solution is complemented here by an asymptotic analysis. An exponential?asymptotic technique provides an estimate for the amplitude of the fast inertia?gravity oscillations that are generated spontaneously, through what is shown to be a Stokes phenomenon. This estimate is given by 2π???2 exp[?π/(2?)], where ? ? 1 is proportional to the Rossby number and the prefactor ? is determined from recurrence relations. The nonlinear dependence of ? on the O(1) rotational Froude number indicates that the feedback of the inertia?gravity waves on the balanced motion directly affects their amplitude. Numerical experiments confirm the analytic results. Optimally truncated slaving relations are used to separate the exponentially small inertia?gravity oscillations from the (much larger) slow contribution to the dependent variables. This makes it possible to examine the switching on of the oscillations in detail; it is shown to be described by an error function of t/?1/2 as predicted theoretically. The results derived for the homoclinic solution of Lorenz and Krishnamurthy are extended to more general, periodic, solutions.