Do Rossby-wave critical layers absorb, reflect, or over-reflect?

The Stewartson–Warn–Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat's eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale δq, i.e. the initial absolute vorticity gradient dq0/dy times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate du0/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat's-eye pattern, probably leading to chaotic motion there, with very large ‘eddy-viscosity’ values. Those values correspond to critical-layer Reynolds numbers δ−1 [Lt ] 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen. The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection) might be prolonged indefinitely is ruled out, in many cases of interest (even if the ‘eddy viscosity’ is large), by the existence of a rigorous, general upper bound on the magnitude of the time-integrated absorptivity α(t). The bound is uniformly valid for all time t. The absorptivity α(t) is defined as the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, no matter how large t may become, |α(t)| cannot greatly exceed the rate of absorption predicted by linear theory multiplied by the timescale on which linear theory breaks down, say the time for the cat's-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that |α(t)| cannot greatly exceed the initial absolute vorticity gradient dq0/dy times the cube of the widthscale of the critical layer. In typical cases, therefore, a brief answer to the question posed in the title is that the critical layer absorbs at first, at a rate ∞ dq0/dy, whereas after linear theory breaks down the critical layer becomes a perfect reflector in the long-time average. If absolute vorticity gradients vanish throughout the critical layer then the bound is zero, implying perfect reflection for all t. The general conditions for the bound to apply are that the wave amplitude and critical-layer width are uniformly bounded for all t, the motion is two-dimensional, and vorticity is neither created nor destroyed within the critical layer, nor transported into or out of it by diffusion, by advection, or by other means. Vorticity may, however, be diffused or turbulently transported within the critical layer, provided that the region within which the transport acts is of bounded width and the range of values of vorticity within that region remains bounded. There are no other restrictions on wave amplitude, none on wavelength, and no assumptions about flow details within the critical layer nor about the initial vorticity profile qo(y), apart from an assumption that q0(y) has singularities no worse than a finite number of jump discontinuities. The proof, in its most general form, makes use of anew finite-amplitude conservation theorem for disturbances to parallel shear flows, generalizing the classical results of Taylor, Eliassen & Palm, and others.