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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 4 / 12





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because, for one thing, the far field into which inertia– gravity waves are radiated is now at rest in the rotating frame rather than in an inertial frame. Furthermore, there are now two small parameters. The vortical mo- tion is characterized by a Rossby number R as well as by a Froude number F, and there are a variety of asymp- totic limits corresponding to R or F, or both, being small in one sense or another. The full Lighthill picture has been shown by detailed asymptotic analysis to apply in the limit where F ! 0 faster than R, that is, with F R (Ford et al. 2000), in which case the dimensionless spontaneous imbalance is still O(F 4) as F ! 0. How- ever, in complementary cases where R ! 0 with F * R, the spontaneous imbalance tends to become exponen- tially small in R (Ford 1994; Ford et al. 2002; Vanneste

´ and Yavneh 2004; Wirosoetisno 2004; Olafsdo´ ttir et al.

2008; Temam and Wirosoetisno 2007; R. Temam and

  • D.

    Wirosoetisno 2008, manuscript submitted to SIAM

  • J.

    Math. Anal., hereafter TW).3

Ford et al. (2002) argued that this exponential smallness—associated with vortical-flow unsteadiness, with smooth time dependence implying an exponen- tially decaying spectral tail—makes Lighthill’s ideas still more powerful. Not only is the back-reaction still weaker, but on top of that, in some cases at least, the destructive–interference aspect remains significant. The inertia–gravity wave frequencies emitted tend to be so close to the inertial frequency f that horizontal wave- lengths l LD, where LD is the Rossby deformation length. Vortical flows often have horizontal scales ; LD not only in the shallow-water system but also in strati- fied cases in which there is a dominant vertical scale.4

´ The recent work of Olafsdo´ ttir et al. (2008) confirms the

l LD behavior in a clear-cut stratified case in which a weak ellipsoidal vortex of horizontal dimension LD is sheared horizontally by a background flow. In that case, l scales as R21LD as R ! 0. Most importantly, the weakness of the back-reaction continues to mean that PV inversion can be accurate—indeed, yet more accu-

3 In case the reader wishes to consult this manuscript, it is available at the entry numbered 0808.2878 in the well-known on- line database at arXiv.

rate than before—with exponentially rather than alge- braically small error.

4. The slow quasimanifold

Lighthill’s ideas, and their extension to rotating sys- tems, remain important also in their complementary aspect. The wave equation with an unsteady wave- source term shows that spontaneous imbalance, weak though it may be, must be generically nonzero if we discount the unlikely event that destructive interference is perfect. Robustly, therefore, albeit heuristically, Lighthill’s arguments are enough to show that there can be no such thing as an invariant slow manifold of the primitive equations, in the strict sense of Leith and Lorenz. This aspect was emphasized and carefully dis- cussed by Ford et al. (2000, 2002). The idea of a slow manifold—a sharply defined hypersurface within phase space, with zero thickness—must be replaced by the idea of a fuzzy ‘‘slow quasimanifold,’’ a chaotic layer or stochastic layer of finite thickness, of the generic sort familiar from studies of low-order dynamical systems, such as the perturbed simple pendulum.

As did Rupert Ford, I prefer to avoid self-contradic- tory terms such as ‘‘fuzzy manifold,’’ ‘‘hairy bald head,’’ ‘‘asymmetric symmetric baroclinic instability,’’ ‘‘ageo- strophic geostrophic adjustment,’’ and so on, despite the ubiquity of such terms in human language (e.g., McIntyre 1997)—hence the term ‘‘slow quasimanifold’’ advocated in Ford et al. (2000). Accuracy of balance and inversion means only that the layer is thin, not that it is actually a manifold.

Historically, understanding could well have been impeded, as often happens, by the persistence of self- contradictory language together with the Humpty Dumpty credo that words are unimportant. The ‘‘fuzzy slow manifold’’—the ‘‘hairy bald head’’ of atmosphere– ocean dynamics—was recognized independently, long ago, by some researchers at least (Errico 1982; Warn 1997).5 Back then it seems that Errico and Warn had no knowledge of Lighthill’s original work. However, Erri- co’s sections 9 and 10 take us very close to Lighthill’s ideas, including the insight that the inertia–gravity or fast modes of the standard normal-mode description

4 This is especially clear for continuously stratified cases in which the buoyancy frequency N is not too strongly variable. Theory and laboratory experiment with N ; constant indicate that not only the velocity and buoyancy fields but also the vortex cores themselves, the PV anomalies, tend to have horizontal-to-vertical aspect ratios of the order of Prandtl’s ratio N/f. Other aspect ratios are usually unstable (e.g., Miyazaki and Fukumoto 1992; Dritschel and de la Torre Jua´rez 1996; Billant and Chomaz 2000). In other words, the horizontal scale ;LD when LD is defined as the vertical scale times N/f.

respond directly to high frequencies existing within the quasi-geostrophic solution, i.e., to the ‘‘tail’’ of the geo- strophic power spectrum (Errico 1982, p. 585b).

5 The correct date for Warn’s pioneering contribution is 1983. However, bureaucracy prevents me from acknowledging this in the list of references. Warn’s contribution was rejected for publication in 1983, being perhaps too far ahead of its time.

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