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MAY 2009



Spontaneous Imbalance and Hybrid Vortex–Gravity Structures


Centre for Atmospheric Science at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

(Manuscript received 12 June 2007, in final form 29 September 2008)


After reviewing the background, this article discusses the recently discovered examples of hybrid propa- gating structures consisting of vortex dipoles and comoving gravity waves undergoing wave capture. It is shown how these examples fall outside the scope of the Lighthill theory of spontaneous imbalance and, concomitantly, outside the scope of shallow-water dynamics. Besides the fact that going from shallow-water to continuous stratification allows disparate vertical scales—small for inertia–gravity waves and large for vortical motion—the key points are 1) that by contrast with cases covered by the Lighthill theory, the wave source feels a substantial radiation reaction when Rossby numbers R * 1, so that the source cannot be prescribed in advance; 2) that examples of this sort may supply exceptions to the general rule that sponta- neous imbalance is exponentially small in R; and 3) that unsteady vortical motion in continuous stratification can stay close to balance thanks to three quite separate mechanisms. These are as follows: first, the near- suppression, by the Lighthill mechanism, of large-scale imbalance (inertia–gravity waves of large horizontal scale), where ‘‘large’’ means large relative to a Rossby deformation length LD characterizing the vortical motion; second, the flaccidity, and hence near-steadiness, of LD-wide jets that meander and form loops, Gulf- Stream-like, on streamwise scales LD; and third, the dissipation of small-scale imbalance by wave capture leading to wave breaking, which is generically probable in an environment of random shear and straining. Shallow-water models include the first two mechanisms but exclude the third.

1. Introduction

Lighthill’s celebrated paper of 19521 was the first to study spontaneous imbalance. It is relevant to some cases of spontaneous imbalance and not to others. Here I begin with a review of the Lighthill theory and its genesis, then go on to discuss some examples that vio- late its assumptions.

One such example is the hybrid vortex–gravity insta- bility discovered by Miles (1957), the first of many such hybrid instabilities known today. Another is the exam- ple discovered by O’Sullivan and Dunkerton (1995, hereafter OSD95), in which spontaneous imbalance in a

nonlinear baroclinic-wave life cycle of type 1 (LC1; e.g., Thorncroft et al. 1993 and references therein) produces internal inertia–gravity waves having small scales close to the grid scale of the numerical model. The small scales seem to put OSD95’s example at an opposite extreme to those in which the Lighthill theory is rele- vant. The Lighthill theory describes scenarios in which unsteady vortical motion spontaneously emits inertia– gravity waves having horizontal scales large in com- parison with the horizontal scales of the vortical motion. Thus, there was an understandable suspicion, at first, that gravity waves on scales close to the grid scale could, perhaps, be numerical artifacts.

1 The original publication, Lighthill (1952), is reproduced on pp. 1–24 of the Collected Papers, Vol. 3, M. Y. Hussaini, Ed., Oxford University Press (1997).

Corresponding author address: Department of Applied Mathe- matics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. E-mail: mem@damtp.cam.ac.uk

DOI: 10.1175/2008JAS2538.1

  • 2009 American Meteorological Society

However, with the growth of computer power it has become clear from very many subsequent studies that spontaneous imbalance of the kind found in OSD95 is a real fluid-dynamical phenomenon, not a numerical artifact. Conspicuously similar to OSD95 have been the examples recently discovered by Snyder et al. (2007, hereafter S07) and Viu´ dez (2006, 2007, 2008, hereafter respectively V06, V07, and V08), in which the vortical motions are, however, much simpler, consisting of

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