It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

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In a previous study (Longuet-Higgins & Cleaver 1994) we calculated the stability of the flow near the crest of a steep, irrotational wave, the ‘almost-highest’ wave, considered as an isolated wave crest. In the present paper we consider the modification of this inner flow when it is matched to the flow in the rest of the wave, and obtain the normal-mode perturbations of the modified inner flow. It is found that there is just one exponentially growing mode. Its rate of growth β is a decreasing function of the matching parameter ε and hence a decreasing function of the wave steepness ak. When compared numerically to the rates of growth of the lowest superharmonic instability in a deep-water wave as calculated by Tanaka (1983) it is found that the present theory provides a satisfactory asymptote to the previously calculated values of the growth rate. This suggests that the instability of the lowest superharmonic is essentially due to the flow near the crest of the wave.

A simple way to model stratiíication of the ocean or atmosphere is in terms of two superposed homogeneous layers of different density. Ef€ects of cooling of the upper layer, such as that which occurs during bottom-water formation in the ocean, can be simulated by mass transfer from the upper layer to the lower layer. A model is constructed to see What effect such a mass transfer has on the flow when the mass transfer is confined to a limited region. The main effects are (i) doming of the interface, which maintains pressure gradients in balance with the velocity fieid, (ii) cyclonic rotation in the upper layer due to conservation of angular mornentum of particles king drawn toward the sink, yet anticyclonic vorticity for those particles outside the mass transfer region due to shrinking of vortex lines drawn up over the dome. (iii) generally anticyclonic rotation in the lower layer due to particles tending to maintain their angular momentum while being pushed ,outwar+, but some cyclonic rotation near the centre of mass transfer, due to momentum transfer from the upper layer. Similar effects to these are seen in the Greenland Sea where bottom water formation OCCUIS. Results of the same sort are also found in a laboratory model of the process.

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