The application of a particular transformation to the triadic integral results in an expression having the following advantages: (a) it satisfies global energy conservation in an evident fashion, independently of the accuracy of the energy-spectrum calculations; (b) it allows an economy of computational time; (c) it shows certain symmetries in the behaviour of non-local interactions; (d) it provides, for the non-local interactions contribution to the transfer function, an expression which is simpler and more compact than those existing, in spite of being more nearly complete; (e) finally the calculated energy-transfer distribution is in good agreement with the experimental findings through a very large range of Rλ.

Aspects of the nonlinear dynamics of waves shoaling between 9 and 1 m water depths are elucidated via the bispectrum. Bispectral-signal levels are generally high, indicating significant nonlinear coupling. In 9 m depth, the biphases of interactions involving frequencies at, and higher than, the peak of the energy spectra are suggestive of Stokes-like nonlinearities (Hasselman, Munk & MacDonald 1963). Further shoaling gradually modifies these biphases to values consistent with a wave profile that is pitched shoreward, relative to a vertical axis. Bicoherence and biphase observations with a double-peaked (swell and wind-wave) power spectrum provide evidence for excitation of modes at intermediate frequencies via difference interactions, as well as the sum interactions responsible for harmonic growth. Shoreward-propagating low-frequency (surf-beat) energy is shown to have statistically significant coupling to higher-frequency modes within the power-spectral peak. In 18 m depth, the biphase of these interactions is close to 180°, a value consistent with the classical concept of bound long waves. In shallower water, however, substantial biphase evolution occurs, and there is no longer a unique phase relationship between surf beat and the envelope of high-frequency waves. The contributions to sea-surface-elevation skewness and asymmetry (with respect to a vertical axis) from interactions among various wave triads are given by the real and imaginary parts of the bispectrum, respectively. In very shallow water, coupling between surf beat and higher-frequency waves results in a skewness with sign opposite to, and about 40% of the magnitude of, the skewness resulting from interactions between the power-spectral-peak frequency and higher frequencies.

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.

The vorticity distribution in a Rossby-wave nonlinear critical layer, given by the Stewartson–Warn–Warn solution, may be strongly modified by the action of shear instability. In a companion paper (Killworth & McIntyre 1985) it was shown, using linear theory, that unstable modes indeed existed. Here, using numerical methods, the nonlinear evolution of unstable disturbances is followed up to the time at which their growth ceases. By such a time there has been considerable redistribution of vorticity in the critical layer.

Vorticity fronts can form in a shear flow as the result of fast patches of fluid catching up with slower ones. This process and its consequences are studied in an inviscid two-dimensional model consisting of piecewise uniform-vorticity layers. Calculations using the method of contour dynamics for ‘intrusive’ initial states indicate that the leading edge of the front evolves into a robust structure whose propagation speed can be accounted for by a simple shock-joining theory. Behind the leading edge several different effects can occur depending upon the relative amplitude of the intrusion. These effects include lee-wave generation with possible wave breaking and folding of the front. A critical value of the frontal slope, above which wave breaking occurs, is suggested.