conspicuous gravity waves are the nearly plane waves inside the separatrix. In the wave-capture limit, the wavelength tends to zero and the wave crests tilt toward the isotachs, as verified in finer detail in S07’s Fig. 10.
The bottom left panel, after V07’s Fig. 2, shows in grayscale the vertical velocity field in the bottom half of a different dipole flow. Here there are no solid boundaries. Instead of a PV delta function, there are continuously distributed PV anomalies occupying a substantial volume within the three-dimensional dipole structure. The vertical section is slightly skewed from the streamwise vertical midplane but shows a view that would be qualitatively similar to the streamwise vertical midplane view. The complete picture in the vertical midplane is therefore qualitatively like the bottom left panel together with its sign-reversed reflection in a horizontal mirror at z 5 0, in other words, qualitatively like the two left-hand panels viewed together. Both the small-scale gravity waves and the central pattern of vertical velocities, marked by the white arrows, are similar in the two cases despite the very different PV distributions.
In the horizontal sections on the right (see caption), the shading again shows the vertical velocity field. In both cases the most conspicuous gravity waves are be- ing advected cyclonically and anticyclonically around each half of the dipole, precisely as expected from a wave-capture scenario. The cyclonic–anticyclonic asym- metry is related to the fact that Rossby numbers R are not small. Indeed, R values are close to unity in a natural quantitative sense to be made precise in the next section.
Now the most conspicuous gravity waves are nearly plane waves in a locally valid approximation. They therefore have well-defined group velocities and their propagation can be well described by ray theory. In the vertical midplane the group velocities relative to the comoving frame, including the contribution from ad- vection by the background flow, are directed across the wave crests and away from the central region. Apart from the diminishing wavelengths this is qualitatively the same as in classical mountain-wave problems with uniform flow and uniform buoyancy frequency N, sat- isfying a radiation condition at infinity.
indicated by the spacing of the white arrows down to the smallest scales approaching the separatrix.
As indicated schematically by the white arrows, the velocity field in the source region above z 5 0 shows a simple pattern of excess descent behind the center of the dipole and excess ascent ahead, and vice versa below z 5 0, where ‘‘excess’’ means additional to the larger-scale quadrupolar pattern of vertical motion characteristic of the quasigeostrophic limit. This point is most clearly brought out in S07’s Fig. 11 (in the top row of color plots, not reproduced here). So on the streamwise vertical midplane of the dipole where the quasigeostrophic ver- tical motion vanishes, the streamlines and stratification surfaces bend toward the horizontal midplane or surface z 5 0, at the center of which the rightward flow speed |v| in the comoving frame is close to maximal, as seen from the lowermost (7 m s21) contour at the top left in Fig. 1 above. The bending of the stratification surfaces toward z 5 0, common to S07 and V07, is clearly a robust feature and is just what one would expect dynamically from the Bernoulli relation, together with hydrostatic balance and the crowding of streamlines around the central point on z 5 0.
Now the Bernoulli and other inertial effects are not enough in themselves to cause substantial imbalance and wave emission. This point is clear from the known shallow-water examples, including those mentioned at the end of section 4. Bernoulli effects are strongly pre- sent in shallow-water motion at order-unity Froude and Rossby numbers F, R, for instance in the jet exit regions in MN00’s Fig. 2a. However, in those examples they manifest themselves hardly at all as imbalance but rather, almost entirely, as nonlinear modifications to the balance condition and PV inversion operator. The same thing is seen in classic two-layer studies such as that of Van Tuyl and Young (1982), in which the comoving ‘‘gravity–inertia signal’’ is essentially the same nonlin- ear modification to balance, with negligible gravity wave emission. Bernoulli and other inertial effects are fully represented by nonlinear terms such as = (v =v) within all the most accurate balance and inversion op- erators. So a strong Bernoulli effect does not, of itself, necessarily imply substantial imbalance and a strong gravity wave source.
We can therefore identify the source region unam- biguously. It is the central region marked by the white arrows. We can also unambiguously deduce that the back- reflection required for resonant-cavity behavior is absent. Resonance has no role. In the standard ray-theoretical manner the propagation is accurately one-way, from the central source region toward the region of wave capture near the forward separatrix. The scale varies smoothly and continuously all the way from the half wavelength
What most plainly distinguishes the present problem from the shallow-water and two-layer examples is the different wave dispersion relation for continuous strat- ification, which allows not only wave capture but also the freedom, in the wave field, to fit the central flow scales indicated by the half-wavelength spacing of the white arrows and the concomitant vertical scale with aspect ratio f/N. This permits wave emission with no spatial destructive interference as well as permitting the