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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 2 / 12





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propagating vortex dipoles. In further contrast with the Lighthill theory, the gravity wave emission is revealed as a steady, mountain-wave-like process in the sense that the waves have zero phase speed in the comoving reference frame. Before that, Snyder et al. (1993) had studied gravity wave emission from a collapsed surface front held steady by numerical diffusivities—already showing the possibility of an essentially steady wave emission process—and Plougonven and Snyder (2005) had shown by careful analysis that the reason for the small scales in examples like OSD95 is not the wave emission process at all, but rather the subsequent re- fraction leading to ‘‘wave capture.’’ This turns out to be true of the dipole examples as well.

Wave capture explains the tendency for the gravity wave scales to approach the grid scale in the OSD95 and dipole cases. Wave capture is the counterpart of critical- layer absorption that results not from vertical shear alone but from the straining of wave crests by large- scale horizontal deformation fields, modified by vertical shear (Jones 1969; Badulin and Shrira 1993). A compre- hensive review may be found in B ¨uhler and McIntyre (2005). The horizontal straining shrinks the wavelength exponentially fast, rather than algebraically. The wave packet behavior becomes passive-tracer-like as the group velocity goes to zero. Mathematically, the flow fields tend toward a singular limit. More physically relevant is that the linearized wave theory predicts its own break- down in a manner suggesting the onset of wave breaking in reality.

The new dipole examples have turned out to be of key importance in that they have allowed the wave source region and mechanism to be identified unequivocally, as summarized in section 6 below. The horizontal and vertical scales of the source region are broadly compa- rable to the scales of the dipole itself and are much larger than the scales of the most conspicuous gravity waves, which are those undergoing capture at some re- move from the source. However, for order-unity Rossby and Richardson numbers there is no scale separation within the source region. The wave source is therefore strongly influenced by the radiation reaction exerted on the source region by the waves. In the dipole examples this radiation reaction has a recognizable fingerprint, in that it destroys the fore–aft symmetry of what would otherwise be a balanced omega or vertical-velocity field. The fore–aft asymmetry becomes conspicuous— completely reshaping the omega field—as soon as Rossby and Richardson numbers attain order-unity values. In terms of the mountain wave analogy, the radiation re- action drastically reshapes the mountain. One cannot specify the mountain shape in advance. By contrast, the main point of the Lighthill theory is that, when the



theory applies, the radiation reaction is so weak that one can, in principle, prescribe the wave source in advance.

There is a final twist in the tail of this tale. There is a sense in which consistent high-order potential vorticity (PV) inversion operators (hyperbalance inversion op- erators) appear capable of delivering not only the vor- tical motion but also the comoving gravity waves, as explained in the concluding remarks below. This makes a peculiarly unexpected connection with the general- ized, Bayesian PV inversion operators proposed, for quite different reasons, by Hakim (2008).

This article is dedicated to my former student Rupert Ford, whose meteoric career was tragically cut short on 30 March 2001 (McIntyre 2001, 2008), and to my former colleague Sir James Lighthill who died on 17 July 1998, in typically magnificent style, on one of his ‘‘adventure swims’’ around the island of Sark. Both were extraor- dinary thinkers who made far-reaching contributions to our subject, and both were persons of exemplary warmth, generosity, and scientific integrity.

2. Lighthill’s theory of acoustic imbalance

While still a terrifyingly bright young man—fresh from wartime aerodynamics after his journey through pure mathematics with schoolmate Freeman Dyson—James Lighthill (1952) singlehandedly put his finger on a key aspect of the phenomenon of spontaneous imbalance.

Addressing the problem of noise emitted by jet air- craft, Lighthill studied the simplest thought experiment in which the phenomenon arises. For unstratified, non- rotating, compressible flow in an unbounded domain with no gravity or other external force, he asked how a freely evolving vortical flow occupying some finite region might emit sound waves, even when the Mach number M 5 U/cs 1. Here, U is a typical flow speed and cs is the sound speed. By combining simple mathematics with a careful and powerful heuristic argument based on physical insight, Lighthill showed that practically any unsteady vortical flow will spontaneously emit sound for any value of M, however small. He also showed that through destructive interference the sound emission is surprisingly weak when M 1, far weaker than one would estimate from naive order-of-magnitude analyses. The ideas are well known and are reviewed, for instance, in Ford et al. (2000, section 2). From an atmosphere– ocean dynamics perspective, the emission of sound may be viewed as the simplest possible example of sponta- neous imbalance. In Lighthill’s original problem, the balance is elastostatic. Sound waves represent the only possible kind of imbalance.

If the spontaneous emission of sound can be neglected

  • that is, if one can neglect the imbalance—then one can

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