Michael E. McIntyre
It is now 54 years since the late Sir James Lighthill published his famous pair of papers "On sound generated aerodynamically". The starting point was the fact that the equations of motion for unstratified, non-rotating flow can be rearranged into the form of a wave equation, (linear wave operator) Tij,ij where the right-hand side represents the nonlinear terms. The second spatial derivatives on the right reflect the absence of localized external forces, implying that the nonlinear terms represent internal eddy stresses. Most importantly, however, Lighthill argued that for vortical motion at small Mach number Tij can be regarded as known in principle. In other words, to good approximation one can neglect the back-reaction of spontaneous wave emission upon the vortical motion. It follows that one can use vorticity inversion to evolve the dynamics, evaluate Tij and thus, finally, to compute the wave emission from the wave equation a posteriori. It is therefore strongly arguable even if not rigorously provable (a) that spontaneous wave emission is inevitable, except in a few special cases of steady vortex motion, and (b) that at small Mach number the emission will be far weaker than predicted by any naive scale analysis. The remarkable weakness of the emission, and therefore of the back reaction on the vortex dynamics, results from the quadrupolar destructive interference implied by the second spatial derivatives.
The same applies almost word for word to the corresponding stratified rotating problems, including the Tij,ij form of the nonlinear terms (e.g., Ford et al, JAS 2000), except that the waves spontaneously emitted now include inertia-gravity waves as well as sound waves. Neglect of their back reaction corresponds to imposing a balance condition. Vorticity inversion is replaced by potential-vorticity (PV) inversion. The assumption of small Mach number is complemented by an assumption of small Froude or large Richardson number. That assumption is usually, but not necessarily, supplemented by a further assumption of small Rossby number. If the Rossby number is small, this weakens the spontaneous imbalance and emission still further (probably from algebraic to exponential smallness), strengthening Lighthill's argument.
None of this accounts for the mystery of weak spontaneous emission and uncannily accurate balance in some shallow-water cases, at least, in which Froude and Rossby numbers are not, in any sense, small. That mystery is partially but not wholly explained in terms of the short range of PV inversion operators in such cases. Some progress has been made in other directions, however, including an answer to the longstanding question of whether the most accurate balanced models necessarily exhibit velocity splitting (reflecting the fact that spontaneous imbalance and wave emission involve mass rearrangement). Recent studies with A. R. Mohebalhojeh of a new class of such models, the "hyperbalance equations", have indicated to my surprise that the answer is no, contrary to what Norton and I had plausibly conjectured in our 2000 JAS paper and, to my acute embarrassment, repeated in my 2003 Encyclopedia article on balance.