To be fussy (and as a physicist I am nothing if not fussy), one can either describe everything in fluid motion as waves simply because the medium is (somewhat) elastic and one can construct a wave equation to describe the propagation of pressure differences, or one can use the Navier-Stokes equations straight up and solve for bulk transport properties. We don't usually refer to the bulk transport as waves. When I stir my wort making beer and get it going in a nice cylindrical eddy in the cylindrical pot, decomposing this bulk transport in a wave description makes little sense, even though the motion is undoubtedly periodic, and it is difficult to see it as the outcome of a suitable transformation of the N-S equations into a real-valued second order PDE in space and effectively second order in time, which is what one usually "expects" for "waves". Second order in time leads to solutions that are either exponential (not waves) or harmonic (waves), with life getting more complicated to the extent that things are generally nonlinear in the N-S equations.
Similarly, I personally wouldn't describe the thermohaline circulation of the ocean as "a wave", or stable currents as "waves", or the flow of water downhill in a stream as "waves", or laminar flow in general as "waves" and am not sure that I'd even describe eddies and the onset of turbulence as waves, although there finally, in the vicinity of the conditions where laminar instabilities can grow and initiate turbulence, a wave description might start to be sensible as periodic propagating wave-like structures appear (even though they probably don't satisfy any sort of sensible wave equation)
Note that your example of shock waves is a good one, as they result when the overpressure in air waves exceeds one atmosphere, at which point (if not long before) the wave equation that was very nearly linear becomes very definitely nonlinear, as the wave underpressure is clipped at 0 atm (a vacuum) but the overpressure is unconstrained. The resulting nonlinear equations can support e.g. solitonic solutions, propagating hyperbolic secants plus a reverberations as the air subsides into normal waves again from nonlinearities in the dispersion. I'd still categorize these as "waves" as they represent a specific limiting behavior of the wave equation with nonlinearities.
So the real question is, are the waves discovered by the MIT volken describable by suitably approximated/linearized second order time equations with complex time solutions (granting that one will still have second order space equations describing the fluctuations away from equilibrium in the bulk medium) ? Or are they first order in time, describing bulk transport but without any elastic "wave" to the wave? Are they just currents in the ocean, or are they currents in the ocean with periods, with wavelengths, or even with solitonic properties e.g. shock fronts?
After all, we know already that the ocean supports waves with wavelengths constrained only by its physical and thermoisobaric geometry and boundaries. There is no "low frequency cutoff" per se in the wave equation that describes sound waves in the water that I know of. In much of the deep ocean, the speed of sound is around 1.5 km/sec, so a 10 Hz wave has a wavelength of 150 meters. A wave with wavelength 500 m has a frequency of 3 Hz. Of course waves with this sort of wavelength propagate in the free ocean in all 3 dimensions, so variations 500 meters "high" can and almost certainly do exist.
It is this last terminology that is very odd. In a transverse wave propagating on e.g. a one dimensional string, the wave amplitude can be described as being thus and such "high", where high is understood to be perpendicular to the direction of propagation. In surface waves in the water (a mix of longitudinal and transverse waves) the wave one can discuss the longitudinal and transverse wavelengths together or separately, but again given horizontal propagation on the gravity constrained surface, transverse is understood as "high", a wave is so and so many meters high.
Underwater waves are a different matter altogether. For one thing, there is no sharp interface like the water's top surface, constraining them. Sound waves are longitudinal, and one would not usually describe the amplitude as a "height" even if they were propagating or were standing waves constrained by the top and bottom boundary conditions in the vertical dimension or were a superposition of a horizontal travelling wave and a vertical stationary wave -- a sound wave propagating from right to left between two hard sheets might have modes that could be so described, but the vertical wavelengths are not really transverse wave amplitudes.
What one would like to do is interpret these words as "there exist transverse waves in the deep ocean with actual bulk transport of water up and down by distances of up to 250 meters (each way) around some undisplaced mean, with the displacement occurring in space and time. This is hard to imagine in a highly compressed dense fluid effectively constrained between two surfaces, because it has to satisfy a continuity equation -- one cannot move water up without more water flowing in to replace it on the bottom from somewhere, and without it flowing out of the way on top to somewhere.
Structures like this are not impossible to the N-S equation -- little is, really. One can imagine, for example, a set of convective rolls that propagate around in some closed path. In order for this to be "a wave", IMO the rolls cannot be stationary in their local medium, but actually have to propagate on top of any bulk transport of the medium in e.g. a current. Convective rolls in the Gulf Stream that are just going around in the moving frame but are moving relative to the stationary ground wouldn't count, as the bulk transport is driven by something (thermohaline density changes, atmospheric flow, coriolis forces, non-Markovian history, and boundary conditions) completely independent of the process that creates and amplifies the convective rolls.