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Theories of Wave Growth

Phase (a): The Onset of Waves

We consider firstly the initial generation of waves over a flat water surface, independently of the simultaneous generation of a surface drift current. The key theoretical result is that the initial wavelength which can be excited on the air–water interface is a wave of wavelength 17mm, which is the capillary gravity wave of minimum phase speed 230mms-1, controlled by gravity and surface tension. The classical Kelvin–Helmholtz analysis completed in 1871, which relies on random natural disturbances present on the water surface, shows that this wave can only be excited by a velocity shear across the sea surface exceeding 6.5ms-1.

Observations, however, show that waves are generated at much lower wind speeds, of order 1–2ms-1. In order to resolve this dilemma, another mechanism was proposed by Phillips in 1957. It takes into account the turbulent structure of wind flow. Turbulent pressure pulsations in the air create infinitesimal hollows and ridges in the water surface, which, once the pressure pulsation is removed, may start propagating as free waves (similarly to the waves from a thrown stone). If the phase speed of such free waves is the same as the advection speed of the pressure pulsations by the wind, a resonant coupling can occur which will then lead these waves to grow beyond the infinitesimal stage. The first wave to be generated as the wind speed increases is likely to be the wave of minimum phase speed, propagating at an angle to the wind direction. Laboratory observations indicate that at slightly higher wind speeds, wave growth results from a shear flow instability mechanism. These two processes acting in the open ocean give rise to cat’s paws, which are groups of capillary-gravity wavelets (ripples) generated by wind gusts.

These results are applicable for clean water surfaces. In the presence of surfactants (surface-active agents), which lower the surface tension, ripple growth is inhibited, and at a sufficiently high surfactant concentration it may be totally suppressed. Phytoplankton are a major source of surfactants that produce surface films, and hence slicks, which are regions of relatively smooth sea surface.

Phase (b): Mature Growth

Once the finite-height waves exist, other and much more efficient processes take over the air–sea interaction.

Jeffreys in 1924 and 1925 pioneered the analytical research of the wind input to the existing waves by employing effects of the wave-induced pressure pulsations in the air. Potential theory predicts such pressure fluctuations to be in antiphase with the waves, which results in zero average momentum/ energy flux. Jeffreys hypothesised a wind-sheltering effect due to presence of the waves which causes a shift of the induced pressure maximum toward the windward wave face and brings about positive flux from the wind to the waves.

The original theory of Jeffreys was based on an assumed phenomenon of the air-flow separation over wave crests. Experiments conducted between 1930 and 1950 with wind blown over solid waves found such an effect to be small and the theory fell into a long disrepute. Jeffreys’ sheltering ideas are now coming back, with both experimental and theoretical evidence lending support to his qualitative conclusions.

The period from 1957 until the beginning of the new century was dominated by the Miles theory (MT) of wave generation. This linear and quasi-laminar theory, originally suggested by Miles, was later modified by Janssen to allow for feedback changes of the airflow due to growing wind-wave seas. MT regards the air turbulence to be important only in forming the mean boundary-layer wind profile. In such a profile, a critical height exists where the wind speed equals the phase speed of the waves (Figure Mean streamlines). Wave-induced air motion at this height leads to waterslope-coherent air-pressure perturbations at the water surface and hence to energy transfer to the waves.

MT however fails to comprehensively describe known features of the air–sea interaction. For example, for adverse winds the critical height does not exist and therefore no wind-wave energy transfer is expected, but attenuation of waves by such winds is observed. Therefore, a number of nonlinear and fully turbulent alternatives have been developed over the past 40 years.

Figure Mean streamlines in the turbulent flow over waves according to the MT, in a frame of reference moving with the wave. The critical layer occurs at the height (Z) where the wave speed (C) equals the wind speed (U(Z)). Reproduced from Phillips OM (1966) The Dynamics of the Upper Ocean, figure 4.3. Cambridge, UK: Cambridge University Press, with permission from Cambridge University Press.
Figure  Mean streamlines in the turbulent flow over waves according to the MT, in a frame of reference moving with the wave. The critical layer occurs at the height (Z) where the wave speed (C) equals the wind speed (U(Z)). Reproduced from Phillips OM (1966) The Dynamics of the Upper Ocean, figure 4.3. Cambridge, UK: Cambridge University Press, with permission from Cambridge University Press.

One of the most consistent fully turbulent approaches is the two-layer theory first suggested by Townsend, and advanced by Belcher and Hunt (TBH). TBH revives the sheltering idea in a new form: by considering perturbations of the turbulent shear stresses, which are asymmetric along the wave profile. While still in need of experimental verification, particularly for realistic non-monochromatic threedimensional wave fields, this theory has been extensively and successfully utilized in phase-resolvent numerical simulations of the air–sea interaction by Makin and Kudryavtsev. TBH and similar theories attract serious attention because the nature of the air–sea interface is often nonlinear and always fully turbulent.

Air–sea interaction is also superimposed by a variety of physical phenomena, which alter the wave growth. Wave breaking appears to cause air-flow separation, which brings the ideas of Jeffreys back in their original form; and gustiness and nonstationarity of the wind, the presence of swell and wave groups, nonlinearity of wave shapes, modulation of surface roughness by the longer waves have all been found to cause either a reduction or an enhancement of the wind-wave input.

These processes of active wave generation give rise to the windsea in which a simple measure of the sea state, relevant to wave growth, is the wave age (c=u) where c is the wave speed of the dominant waves, and u is the friction velocity in the air (the square root of the wind stress divided by the air density). The age of the windsea increases with fetch (the distance from the coast over which the wind is blowing), and the windsea becomes ‘fully developed’, that is, the energy flux from the wind and the dissipation flux are in balance, at a wave age of about 35. Empirical relations for the properties of the fully developed sea in terms of the wind speed (U) at 10m (approximately the height of the bridge on large ships) given by Toba are: Hs=0.30U2/g and Ts=8.6U/g in which Ts (=2Πc/g) is the significant wave period and g is the acceleration of gravity. As the fetch increases, Hs and Ts both increase toward their fully developed values, and the wave spectrum spreads to lower frequencies. Older seas of wave age greater than 35 can also exist after the wind has moderated.

The observations of the velocity structure in the atmospheric boundary layer by Hristov, Miller, and Friehe have shown directly the existence of the MT critical layer mechanism for fast-moving waves of wave age about 30. It is not yet known whether it operates for younger wave age, where a quasilaminar theory may not be appropriate.

Phase (c): Very High Wind-Speed Wave Environments

The processes discussed in the previous two subsections are all grounded in two-layer fluid dynamics in which there exists a sharp interface between the two fluids. In recent times, it has been realized that this model is inadequate, especially at very high wind speeds. The link between the two phases is the breaking wave. In moderate winds (less than about 25ms-1) the sea state is characterized by whitecapping due to the production of foam in a roller on the wave crests, and also foam streaks on the sea surface, whereas at very high wind speeds (greater than about 30ms-1, Beaufort force 12) the air is filled with foam.

This transition arises from the structure of the breaking waves. In moderate winds, the roller remains attached to the parent wave and dissipates by the formation of foam streaks down its forward face, the trailing face of the wave remaining almost foam free. In this situation the airflow separates over the troughs and reattaches at the crests of the wave, producing Jeffreys-like phase shifts between the pressure and the underlying wave surface which enhance the energy flux to the wave. At very high wind speeds, on the other hand, the foam detaches from the wave crests, and is jetted forward into the air where it disperses vertically and horizontally
before returning to the water surface. This process implies a return of momentum to the atmosphere, and hence the sea surface drag coefficient (which is an overall measure of the efficiency of momentum transfer from the atmosphere to the ocean both to waves and turbulence), which has been rising in phase (b), becomes ‘capped’ and possibly even reduces in phase (c). The all-pervasive presence of spray in extreme winds has prompted the anecdotal statement that ‘‘in hurricane conditions the air is too thick to breathe and too thin to swim in.’’

In summary, at very high wind speeds, the airflow effectively streams over the wave elements, which are reduced to acting as sources of spray. The spray then stabilizes the wind profile, and caps the sea surface drag coefficient, and interestingly, this feedback most likely allows the hurricanes to exist in the first place. This analysis has been greatly stimulated by the dropwindsonde observations of Powell, Vickery, and Reinhold in which wind profiles in hurricanes were measured for the first time, and also subsequently by experiments in high-wind-speed wind-wave tanks.

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