5 0.44 and 1.1. The value RLagr
5 1.1 applies
to V07’s case reproduced in the lower half of Fig. 1
above.7 In S07 the case with the largest RLagr
value (not re-
produced here) has a value 1.5 times greater than for the case in Fig. 1 above. In the former case, showing very strong fore–aft asymmetry in the vertical velocity field
(S07 Fig. 11, top right), we have RLagr over 1.1.
5 0.75 3 1.5, just
We remark finally that some dipole examples of this sort may violate the usual rule that the strength of spon- taneous imbalance diminishes exponentially as R ! 0. The reasons are as follows.
As R ! 0, particle travel times across the central re- gion increase relative to p/f. Then quasi-steady, moun- tain-wave-like gravity wave emission has to rely on the diminishing spatial scales in the velocity and buoy- ancy fields that then correspond to intrinsic wave fre- quencies * f. So the amplitude of wave emission will depend on a spatial projection integral whose integrand consists of a rapidly oscillating factor, representing the diminishing wave scales, multiplied by a slowly varying factor coming from the velocity and buoyancy fields of the vortex dipole. Thus, as R diminishes from order-unity values toward zero, destructive interference re-enters the problem, wave emission weakens, and to that extent a Lighthill-type treatment becomes appropriate again de- spite the motion being quasi-steady, and despite the scale
7 For that case, RLagr is estimated at the nominal depth z 5 20.21 in V07’s dimensionless units, corresponding to the upper heavy line in the bottom left panel of Fig. 1 above. The half wavelength was measured from V07’s Fig. 1a (not reproduced here), whose dimensionless length and width are both 20p units. The half wavelength is about 12 units. V07’s Fig. 11a shows hori- zontal velocity vectors and isopleths at the same nominal depth, z 5 20.21, in a plot whose dimensionless length and width are 45.5 units. From this information it can be determined that the flow through the central half wavelength is accurately along the streamwise midplane and lies mostly within the |v| 5 2.5 isopleth, the innermost isopleth surrounding a central maximum |v| value of 3.0 units. To go into the comoving frame, one must subtract ;0.2 units, so that on the streamwise midplane the values 2.5 and 3 become roughly 2.3 and 2.8 units. Taking the half-wavelength particle travel time as th 5 12/2.6 units, and noting from V07 (p. 362) that p/f 5 5 units exactly, we get RLagr 5 5/(12/2.6) 5 1.1. The
case with RLagr
5 0.44 is most thoroughly described in V08, where
Fig. 16 presents particle travel-time isochrones. The travel time through a quarter-wavelength in the central source region, be- tween the center of the region and the first vertical velocity ex- tremum, can be read off from the figure as about 1.6 isochrone intervals. This can be seen to be 1.6O2/4 in units of the inertia period, after correcting a typographical error in the caption, where O2/2 should be O2/4. So the time through a central half wavelength i s 6 . 4 O 2 / 4 h a l f i n e r t i a p e r i o d s , m a k i n g R L a g r 5 ( 6 . 4 O 2 / 4 ) 2 1 5 0 . 4 4 . ´ I am grateful to Dr A. Viu´ dez for supplying the numerical infor-
mation reproduced in this footnote.
disparity being in the opposite sense, with gravity wave scales small relative to vortex scales.
The projection integral will diminish with R in a manner depending on the smoothness of the dipole’s velocity field. With a completely smooth velocity field, perhaps smoothed by artificial viscosity, the integral, over the whole spatial domain, if infinite or periodic, may be expected to diminish exponentially. Indeed, such heuristics are consistent with a rigorous upper bound O [exp(2const. R21/4)] on spontaneous imbalance, recently established by TW on the assumption of diffusive flow (for buoyancy as well as momentum), together with compatible smoothness conditions (Gevrey regularity) and triply periodic boundary conditions as in V06, V07, and V08.
However, with nondiffusive, ideal-fluid flow and a velocity field that is less smooth (such as would be ex- pected, for instance, with isentropic distributions of PV that have jump discontinuities), the projection integral might instead diminish algebraically with R. Here, the R dependence would be that of a set of ideal-fluid cases computed over a finite time interval before the onset of wave breaking via wave capture.
This speculation is based on the idea that the smoothness of the actual velocity and mass fields should be related to the smoothness of the fields produced by a PV inversion operator. Such operators are well known to be nonlocal smoothing operators but are, of course, incapable of producing velocity and mass fields that are infinitely smooth at a PV discontinuity. The projection integral expresses the nonlocal aspects of the wave emission dynamics and so might be expected to depend on the dipole’s global fields and not just on, for instance, the shapes of particle trajectories, which in a steady flow must follow PV contours and may well be smoother than the velocity field itself. Taking these ideas further would be a severe mathematical challenge.
8. Concluding remarks
All spontaneous imbalance scenarios must involve a radiation reaction of one kind or another. Lighthill’s achievement was to find a set of cases in which the ef- fective source strengths are insensitive to the radiation reactions they provoke, with all the conceptual and computational simplifications that follow. By contrast, the non-shallow-water dipole examples involve source strengths that become sensitive to radiation reactions as Rossby numbers R approach unity.
There is yet another intriguing twist. A recent theo- retical discovery, the hyperbalance equations, has given us the first fully consistent PV inversion operators of arbitrarily high formal accuracy. A quick review of