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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 8 / 12





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waves to escape as though they satisfied a radiation condition, as already remarked.

Along with the order-unity R values this state of things implies, in turn, that there is a substantial back- reaction upon the central source region in the form of a radiation reaction, like the power drawn by an effi- cient radio antenna. In summary, 1) the order-unity importance of Bernoulli and other inertial effects and 2) the approximate spatial and temporal scale matching between the source and the emitted waves (temporal as well as spatial when R ; 1) together imply that there is a radiation reaction on the central source region having a substantial, leading-order ef- fect on the dynamics.

By itself, the Bernoulli effect in the streamwise midplane would produce a pattern of motion and stratification-surface distortion having fore–aft symme- try. This is the symmetry suggested by looking only at the white arrows and thin contours in Fig. 1. Indeed, such fore–aft symmetry is an exact property of balanced, purely vortical motion induced by PV anomalies with the same symmetry. It is related to the general time sym- metry or ‘‘sign reversal property’’ discussed in Ford et al. (2000). So the fore–aft asymmetry in the vertical-motion fields actually found in the central region—conspicuous in the top right panel especially—must be a consequence of the radiation reaction. That conclusion is further supported by the top row of S07’s Fig. 11, showing the vertical velocity field in five cases with R values 0.125, 0.5, 0.75, 1.0, and 1.5 times the value for Fig. 1 above. The first case shows almost perfect fore–aft symmetry and the others increasing fore–aft asymmetry, very strong in the last case.

This says, then, that when R * 1 the wave emission is not an affair of master and slave. It is an affair of bootstrapping. That is, it depends on an intimate, two- way interplay between the inertial effects in the source region and the radiation reaction on that region, in- troducing a local arrow of time. The source emits the radiation, but the radiation reaction reshapes the source. It is as if one had a mountain-wave problem in which the mountain were elastic and substantially changed its shape in response to the surface-pressure field, in turn creating a large change in the vertical velocity field.

So we have here something that is about as far from a Lighthill scenario as it is possible to imagine. With the source strength so intimately dependent on the radia- tion reaction, the source strength cannot be considered to be known in advance. This insight is, of course, consistent with the standard remark that when R * 1—see next section—one should not expect to be able to distinguish balance from imbalance (i.e., to distin-



guish vortical motion from gravity wave motion) even in principle. The distinction becomes meaningless and the slow quasimanifold can no longer be regarded as thin in any sense. No mathematical device, no manip- ulation of the equations, however ingenious, can ever hope to produce a unique and clear-cut separation between balance and imbalance within such a source region. Arguably, a tendency to forget this fact has impeded understanding in past decades.

Further support for the picture just sketched comes from the work of V07 and V08. It was found there that even the iterative ramping procedure of Viu´ dez and Dritschel (2004) called ‘‘optimal PV balance’’— perhaps the closest to an objective balancing proce- dure one is ever likely to get—failed to disentangle the flow within the central source region into balanced and imbalanced parts. (See the discussion on p. 364 of V07, noting incidentally that ‘‘upper’’ means slightly below the horizontal midplane z 5 0.) In section 8 below we remark that a corresponding ambiguity shows up in the equations defining high-order balance and PV inver- sion operators.

7. The dependence on R

Let us look more closely at the role of the Rossby number R and the most natural way to define R in this problem. To get temporal as well as spatial scale matching in the central source region—an efficient radio antenna, so to speak—the particle travel time through a half wavelength of the pattern (say, th) must satisfy th & p / f. That is, the travel time needs to be about half an inertial period or less. So it is natural to define R for this purpose in the standard Lagrangian sense (e.g., Hoskins 1975; Koch and Dorian 1988; Z ¨ulicke and Peters 2006, 2008 and references therein), as

R 5 RLagr


p thf


and to anticipate that RLagr * 1 should characterize efficient radiation and a substantial radiation reaction on the central region not only qualitatively but also quantitatively to moderate accuracy.

From the spacing of the white arrows in S07’s case at top left in Fig. 1 we see that the horizontal half wavelength there is close to 250 km. The particle ve- locity through the lowest part of the central region, be- tween the bottom two contours, is around 6 m s21. If we t a k e t h 5 2 5 0 3 1 0 3 / 6 5 4 . 2 3 1 0 4 s t h e n , w i t h f 5 1 . 0 3 21 1024 (S07, p. 4419a), we have RLagr 5 p/(th f) 5 s

p/4.2 5 0.75. The two cases described in V07 and V08

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