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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 5 / 12





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MAY 2009



Errico’s section 9 tests the idea that one can first de- scribe the vortical motion by itself and then afterward treat it as a known source for inertia–gravity waves, essentially as in Lighthill. Warn’s paper was the first to point out, explicitly, that the evidence from Errico’s work argued against the existence of a slow manifold.

Just how thin the slow quasimanifold can be in shallow- water flows, how nearly manifold-like, with order-unity values of F and R, well outside the parameter regimes to which Lighthill’s ideas are manifestly applicable, remains astonishing and still poses a challenge to our understanding. High-order shallow-water balanced models based on advecting the exact Rossby PV were first shown by Norton (1988) to maintain an uncanny accuracy over several eddy turnaround times, as judged by their ability to track primitive equation evolution even for order-unity values of F and R; see also McIntyre and Norton (1990, 2000, hereafter MN00), and Mohebalhojeh and McIntyre (2007b). This behav- ior was a total surprise when it was first discovered. However, the uncanny accuracy—and, by implication, the weakness of spontaneous imbalance in these par- ticular flows despite order-unity F and R—can with hindsight be related to the flaccidity and slow evolution of jets meandering Gulf-Stream-like, or river-like, on scales much larger than the Rossby deformation length LD (e.g., Nycander et al. 1993 and references therein; McIntyre 2008 and references therein). The relative thinness ;LD and near-steadiness of such jets helps to keep their Lagrangian time scales considerably lon- ger than inertia-wave time scales f21, even for order- unity values of the Rossby number R based on jet width LD. This is so far outside the scope of Lighthill theory that the jet flaccidity, or fluviality, characteristic of the regime R ; F and apparently persisting out to R ; F ; 1, is probably best considered as a sepa- rate, second mechanism for keeping flows close to balance.

very many others).6 Here the typical scenario is an ex- ponentially growing instability consisting of a counter- propagating, phase-locked pair of waves, one being a Rossby and the other a gravity wave. However, in some cases there are also neutrally stable, steadily propagat- ing modes at certain wavelengths. Because of the steadiness or near-steadiness, it is again clear that we have examples of spontaneous imbalance that fall out- side any Lighthill-type scenario.

Lighthill himself cautioned that there would be ex- ceptions to his theory. Here is how he continues the passage quoted in section 2 above:

. . . no significant back-reaction can be expected unless there is . . . a resonator present to amplify the sound

or, in the present context, the gravity wave. That is, the self-consistency of Lighthill’s picture depends on ex- cluding resonance phenomena. The hybrid instabilities just mentioned can, of course, be recognized as reso- nance phenomena. The Miles (1957) instability is a clear- cut example. The gravity wave is a surface gravity wave on water, and the Rossby wave propagates on a vertical vorticity gradient in the air above. Once they are phase- locked together, each wave resonantly excites the other. So if we say that the Rossby wave excites the gravity wave, then we must admit that, equally, the gravity wave exerts a significant back-reaction on the Rossby wave.

6. Vortex dipoles in continuous stratification

The dipole examples of S07 and of V06, V07, and V08—and by implication the OSD95 example and its successors—belong in yet another category. Continuous stable stratification is an essential factor. The sponta- neous imbalance in these examples is quite different from that in classical Lighthill-type scenarios and quite different, also, from that in the hybrid instabilities and their neutral modes.

5. Some classic hybrid vortex–gravity structures

It is crucial to Lighthill’s picture that the vortical motion be unsteady. If only for that reason, the propa- gating dipole vortex–gravity structures described in S07 and in V06, V07, and V08 must fall entirely out- side any such picture. The propagation of those struc- tures is very close to being steady. Before discussing them we may remark that other such hybrid structures have long been known in which, however, the vortical part itself takes the form of a wave—a Rossby wave in the natural, generalized sense of the term, propagating on a gradient of vorticity or PV (e.g., Miles 1957 and

Rather than an equal partnership between the vortical and gravity wave parts, each of which resonantly excites the other, we have what looks at first sight like a master– slave relation—‘‘slave’’ in the general sense and not the slow-manifold sense—and, to that extent, something more like a Lighthill scenario. There is no resonance

6 The idea of explaining shear instabilities in terms of phase- locked counterpropagating waves has a long history, going back at least as far as Taylor (1931) and Lighthill (1963, p. 93). Taylor’s classic paper is reproduced on pp. 219–239 of The Scientific Papers of Sir Geoffrey Ingram Taylor, vol. II, ed. G. K. Batchelor, Cam- bridge University Press (1960). The idea has been greatly devel- oped in recent years; see, for example, Ford (1994), Methven et al. (2005), and references therein.

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