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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 10 / 12





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the essentials can be found in McIntyre (2008), and a thorough and comprehensive development of the theory, including mathematical demonstrations that it has to take the generic form it does, can be found in Mohebalhojeh and McIntyre (2007a). Like their pre- decessors, going back to Hinkelmann’s forecast ini- tialization work of the 1950s, these highly accurate inversion operators depend on computing diagnostic estimates of time derivatives n/tn of the flow fields. So if a flow is exactly steady, then the time derivatives and their diagnostic estimates should all go to zero and the PV inversion should become exact, just as it does in trivial cases like steady circular vortices. But how does that square with the discovery of the new dipole solu- tions? Can one ‘‘invert’’ the PV field in such a way as to obtain the comoving gravity waves as well, as recently suggested for quite different reasons by Hakim (2008)?

The answer is probably yes and no. It should be no for diffusionless, ideal-fluid dynamics because it is unlikely that any inversion operator would make mathematical sense for a steady but singular flow field. Such a flow field would be expected from the nature of the wave capture process and the finite-amplitude validity, albeit dynamical instability, of plane inertia–gravity waves. Such a flow field would also violate the diffusionless ‘‘sign reversal property’’ discussed in Ford et al. (2000). The answer should be yes when the inversion operators are generalized to include nonideal flow with, for in- stance, artificial diffusivities or hyperdiffusivities such as those used in numerical models. In that case one expects to find ranges of small diffusivities and Rossby numbers R for which the departure from exact steadi- ness is exponentially small in R—the captured gravity waves having exponentially small amplitudes and the dipole’s decay being correspondingly slow—and for which a high-order PV inversion operator has an ex- ponentially small error and, in particular, delivers a close approximation to the complete flow field includ- ing the diffusively damped gravity waves approaching capture.

One further point seems worth making. The need to replace the slow manifold by the fuzzy ‘‘slow quasima- nifold,’’ whatever its thinness (section 4), seems to be a generic and deeply important insight, originally coming out of the cases studied by Lighthill, Errico, Warn, and others but now, it seems, having still wider relevance. For one thing, a generically fuzzy or chaotic structure is strongly indicated by dynamical systems theory, begin- ning with the standard example of the homoclinic neighborhood in perturbed simple pendulum problems, with the pendulum moving slowly near its unstable equilibrium in partial analogy to the vortical motion in the fluid problem. And in the fluid problem the chaotic



structure is no more than one naturally expects from the Lighthill picture together with the conspicuous un- steadiness, and the apparently chaotic evolution, of most cases of vortical motion. It is natural to expect at least this level of insight to apply to continuous stratification as well as to shallow water.

However, the new dipole examples—even more clearly than the quasi-steady frontal examples of Snyder et al. (1993)—have revealed an important variation on the generic theme, because they show that in strati- fied flows the spontaneous emission of gravity waves can be a steady process somewhat like mountain wave generation. If nothing else happens, the waves undergo capture and never escape from their ‘‘prison’’ within the dipole. But such idealized scenarios are over- whelmingly improbable in practice, important though they have turned out to be for broadening our theo- retical insight.

Imagine, then, a pair of dipoles in collision, or a di- pole interacting with almost any other vortex structure in its surroundings. It would not take much vortex-flow unsteadiness to allow some of the gravity waves to es- cape from their prison within each dipole and thereafter to become part of the ambient field of freely propa- gating gravity waves, much as in OSD95 and its suc- cessor studies. All these gravity waves can be regarded as contributing to the fuzziness of the slow quasimani- fold.

So now there comes yet another new insight: that the slow quasimanifold owes its thinness not only 1) to the Lighthill mechanism for large-scale gravity waves and 2) to jet flaccidity or fluviality in Gulf-stream-like cases with meander scales LD, but also 3) to the generic vulnerability of small-scale gravity waves to wave cap- ture. For such waves, escape from the prison of a pro- genitor vortex dipole is unlikely to result in prolonged freedom. Because of the tendency toward passive-tracer behavior, there will be a robust statistical bias toward subsequent wave capture. Such a bias is clear from random-straining models such as that of Haynes and Anglade (1997). It now seems that this, too, must be part of why atmospheric and oceanic flows often stay close to balance, and that it should count as a distinct third mechanism contributing thereto.

Acknowledgments. I thank Greg Hakim, Riwal

´ Plougonven, Chris Snyder, Jacques Vanneste, Alvaro

Viu´ dez, and Djoko Wirosoetisno for showing me pre- publication versions of their work and for many useful comments, and Tim Dunkerton and Pascale Lelong for their vision in organizing such a timely and stimulating First Workshop on Spontaneous Imbalance and for funding assistance. Djoko Wirosoetisno contributed

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