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Spontaneous Imbalance and Hybrid Vortex–Gravity Structures - page 3 / 12





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MAY 2009


describe the dynamics in the classical way as a purely vortical flow. What does that mean? The crucial feature is that one can invert the vorticity field at each instant to obtain the velocity, pressure, and density fields.

In an unbounded domain, one can do this via the Biot– Savart integral [e.g., Batchelor 1967, Eq. (2.4.11)] or re- finements thereof. One can equally well invert what might be called the acoustic PV. This is the vorticity di- vided by the mass density. It has the simplest evolution equation and visualizability [e.g., Batchelor 1967, Eq. (5.3.6)]. Invertibility means that there is a purely diag- nostic functional relation—nonlocal, of course—between the vorticity or PV field and everything else. Knowledge of the vorticity or PV field at one instant implies knowledge of everything else at that instant. One can therefore use the standard language of aerodynamics and speak of the velocity field ‘‘induced’’ at each instant by a given vorticity field (e.g., Lighthill 1963). Thinking thus in terms of vorticity or PV inversion makes explicit the most basic peculiarity of vortex dynamics—the point missed when only local balances of terms are considered—the apparent action at a distance whereby a vortex contrib- utes instantaneously to the motion of other vortices.

Of course such action or influence cannot really travel faster than sound. But Lighthill’s arguments showed, in effect, that using balance and invertibility to describe the vortex dynamics, as if one did have instantaneous action at a distance, can be far more accurate than naive order-of-magnitude analysis would predict. And, as Lighthill was careful to point out, that is the very reason why his ideas make sense. The prediction of destructive interference and therefore weak sound emission de- pends on being able to suppose that the vortical flow can be regarded as known, in principle, independently of the sound emission:

All the evidence of experiment, and of the theory to be developed below, is that the sound produced is so weak relative to the motions producing it that no significant back-reaction can be expected . . . (Lighthill 1952)

This means that one can in principle compute the vor- tical flow evolution to high accuracy—and the weaker the spontaneous emission, the higher the accuracy—then afterward compute the emission with the nonlinear terms treated as known source terms in a wave equation. The destructive interference arises from the long wave- length l of the sound emitted, relative to the scales of the vortical motion, and the special form of the source terms for all cases of vortex motion. Each term takes the form of a second spatial derivative [Lighthill 1952, Eq. (4c); Ford et al. 2000, Eq. (2)]. Thus, the whole picture is self-consistent—the more so, the weaker the emission (Ford et al. 2002).



3. Shallow-water rotating flow

The ideas of balance, imbalance, and PV inversion carry over at once to shallow water dynamics with only trivial changes of wording. Sound waves are replaced by gravity waves. The Mach number M 5 U/cs 1 is replaced by the Froude number, in its standard sense F 5 U/c 1, where c is the gravity wave speed. The acoustic PV is replaced by the Rossby PV, absolute vorticity divided by layer depth (Rossby 1936).2

It is well known that the shallow-water equations are identical to the two-dimensional equations for a perfect gas with g 5 2. Here, g is the ratio of specific heats, whose numerical value is an unimportant detail. So Lighthill’s ideas carry over without modification as long as there are no Coriolis terms. For two-dimensional flow, the weakening of spontaneous imbalance and spontaneous gravity wave emission as F diminishes can be expressed by saying that if c is held constant while U is diminished, then the gravity wave power radiated goes as U7. The corresponding dimensionless measure of imbalance in two dimensions is F 4 (Ford et al. 2000).

If Coriolis terms are introduced, then Lighthill’s ideas still carry over qualitatively (Ford et al. 2000, 2002). Pure gravity waves are replaced by inertia–gravity waves. The problem is no longer identical to Lighthill’s

2 A reviewer has insisted that the PV concept dates from well before 1936. That is arguably the case provided that one uses the Lagrangian description of fluid dynamics. Then the material in- variance of PV for ideal fluid flow is a straightforward corollary of nineteenth-century vortex dynamical theory. Most simply, the Rossby–Ertel PV is proportional to the absolute Kelvin circulation around an infinitesimally small closed material contour C lying on a stratification surface, isentropic or isopycnal—or on the free surface of a shallow-water system—divided by the mass of the infinitesimal material fluid element whose perimeter is C. For continuous stratification this element is bounded above and below by neighboring stratification surfaces [for more detail, see McIn- tyre (2003)]. Thus, PV invariance follows almost trivially from mass conservation and Kelvin’s circulation theorem. By using the full machinery of the Lagrangian description, one may also relate the PV to a quantity from nineteenth-century theory called ‘‘Beltrami’s material vorticity’’ (Viu´ dez 2001). This is a way of describing the three-dimensional vorticity field mapped into La- grangian label space. The mapping is defined as a kinematically possible fluid flow with the vortex lines and stratification surfaces frozen into the fluid. Of course the usefulness of the PV concept as first published by Rossby, both for shallow water (Rossby 1936) and for multilayer and continuous stratification (Rossby 1940), lies in being able to avoid the Lagrangian description. This is useful because, for one thing, PV inversion is quintessentially an Eulerian or field-theoretic procedure. We may note that 1936 is also the year in which Lighthill began his high-school mathematical journey with Dyson at the age of 12, and 1940 the year in which they were both prevented, by their youth, from going on to Cambridge de- spite having won scholarships.

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