This shows that a wave which does not receive renewed energy input will dissipate energy by molecular friction, with the rate of dissipation

dW total dt

Of course, this is not the only mechanism by which waves lose energy. Energy is also lost by the creation of turbulence on a scale above the molecular level. This may involve interaction with the air, possibly enhanced by the breaking of wave crests, or oceanic interactions due to the Reynold stresses (2.59). Also, at the shore, surf formation and sand put into motion play a role in energy dissipation from the wave motion.

Once the wind has created a wave field, this may continue to exist for a while, even if the wind ceases. If only frictional dissipation of the type (2.81) is active, a wave of wavelength Xw = 10 m will take 70 h to be reduced to half the original amplitude, while the time is 100 times smaller for Xw = 1 m.

As mentioned above, the mechanisms by which a wave field is created by the wind field, and subsequently transfers its energy to other degrees of freedom, are not understood in detail. According to Pond (1971), about 80% of the momentum transfer from the wind may be going initially into wave formation, implying that only 20% goes directly into forming currents. Eventually, some of the energy in wave motion is transferred to the currents.

Early descriptions of wave formation by wind assumed that the energy transferred would be proportional to the wave velocity (most often taken as the phase velocity) and to the "steepness" of the wave, given by the ratio of height and wavelength, ka (see e.g. Neumann, 1949). The "steepness" would be interpreted as equivalent to the roughness length z0 appearing in the logarithmic velocity law (2.33), but, as discussed in connection with (2.42), measurements did not support such a relation very well.

A fruitful way of looking at the complex wave fields found in the real ocean has been to consider the field of amplitudes (2.68) associated with gravity waves as a random quantity with definite statistical properties. For a stationary situation, these statistical properties may be taken as fixed, whereas the creation of waves by wind and other interaction phenomena may be described in terms of slowly varying statistical properties, using perturbation theory. For a first approximation, the wave amplitude field may be assumed to be Gaussian, so that the probability of finding a given amplitude a is equal to (Pierson, 1955; see also Kinsman, 1965)

The variance, X 2, may be expressed as

/ pwX 2 = / pwg <a(x,t)2 > = J F(k) dk = / WtotaI, (2.82)

corresponding to the average potential energy or half the average total energy of the wave motion. The function F(k) is called the "energy spectrum" (and the generalisation from the case of propagation along the x-axis considered here to a three-dimensional wavenumber k can readily be made). The Gaussian distribution contains waves with arbitrarily large amplitude (but correspondingly small probability), whereas real waves will break if the amplitude exceeds a certain value. Still, the Gaussian distribution is useful for the discussion of many properties of ocean waves, including their formation and growth.

The interactions important for the structure of the wave field are found to include couplings between different spectral components of the wave field, as well as couplings to mean and turbulent flows of wind and currents.

As a basis for constructing the perturbation expansion of the interactions, a set of "normal mode" solutions may be chosen, comprising the elementary (lowest-order) solutions of the wave equations, i.e. (2.74), in the absence of the friction factor, as well as harmonic solutions for the external fields, e.g. V

- V* = V' for the fluctuating wind velocity or the analogous fluctuating part of the water current. The couplings to the atmospheric fluctuations are often described not in terms of the velocity fluctuations, V', but in terms of pressure fluctuations, P = P - P. The descriptions are equivalent due to the relation between thermodynamic pressure and velocity fluctuation (cf. section 2.3.1). The fluctuating fields, V' (or P ) for wind and V'w for currents, may be treated as random, e.g. assuming Gaussian distributions as used above for the lowest-order descriptions of the wave field.

Following Hasselmann (1967), the normal mode variables of the wave system itself are written an(k) = 2-1/2 (Pn(-k) -ian(k) qn(k)), a n(k) = 2-1/2 (pn(-k) + ian(k) qn{k)), (2.83)

in terms of the normal mode frequencies, an(k), where n enumerates the normal mode solutions, i.e. the solutions to the part of the classical Hamilton function, which is quadratic in the conjugate canonical variables qn(k) and (the momenta) pn(k). The coordinates qn(k) are the amplitudes of each normal mode [harmonic wave with a = exp(ikx)] in the lowest order solution. In this lowest order solution, the an(k) are each proportional to the corresponding factor exp(-ian(k)t).

The external fields may, if they are known, be decomposed into variables bn(k), each of which is proportional to exp(-ia>n(k)t). In this case, the equations of motion for the unknown an(k) (in the presence of couplings) are of the form d an ( k) = -ia n (k ) + Gext , (2.84)

where the Hamilton function H is a sum of terms with products of m normal mode variables an. (ki ) or 0t„. (ki ) , such that m = 2 is the uncoupled solution and m > 2 describes anharmonic wave-wave couplings. The external couplings Gext are represented by a similar summation, where, however, each term may be a product of m' variables an. (ki ) or a~n.(ki ) and (m-m') variables bn. (ki ) or bn. (ki ) . The external couplings in general do not fulfil the symmetry relations characteristic of the intrinsic wave-wave couplings, and they cannot be derived from a Hamilton function.

Of course, seen from a broader perspective, the wave motion will influence the external fields, so that couplings both ways must be considered. In this case, bn(k) will also be unknown variables to be determined by the solution, and a Hamilton function may be constructed for the entire system of waves, currents and winds, but with a much greater dimensionality.

If the couplings are considered to be small, the solution for the wave system may be considered not to disturb the external fields, and the solution may be obtained in terms of a perturbation series, an(k) = ^(k) + ^(k) + ••• , (2.81)

where each successive term may be evaluated by truncating the summations in the Hamilton function and Gext terms of (2.80) at a given order m.

The energy spectrum is analogous to (2.85) (but for a single normal mode type n),

Fn(k) = y2 <an(k) ~an( - k)>, involving the average of two an variables over a statistical ensemble. Hasselmann (1967) discusses the extent to which the energy spectra, defined this way, in turn determine the wave system.

The techniques described above are also being used extensively in the description of condensed matter and atomic nuclei.

In a number of simplified cases, perturbational solutions of the type (2.85) have been obtained, with the inclusion of anharmonic terms [also called "non-linear" terms, referring to the right-hand side of (2.81)] (Phillips, 1966; Hasselmann, 1962). Important studies of the wave formation by wind were made by Miles (1957) and by Phillips (1957). They formally correspond to the lowest-order treatment of two definite external fields in the above formalism.

In the model of Phillips (1957), the direct action of the fluctuating (turbulent) pressure field P in the atmosphere on the ocean surface is assumed to create waves. Such resonant waves will start out with high frequencies (wavelengths around 10-2 m), corresponding to the turbulent mixing length in the atmosphere, of the small eddies present near the ocean surface. The growth of the wave spectrum is linear, since d F(k)/d t is time independent and given by the (by assumption stationary and homogeneous) external P field.

After a period with linear growth, the mechanism suggested by Miles (1957) is then believed to come into play. It is a coupling between the mean wind field V* in the atmosphere and the wave field. Each wave component changes V* slightly, inducing a pressure perturbation which acts back on the wave component and makes it grow.

The energy transfer may be understood in terms of a "critical layer" of air, covering the region from the (wavy) ocean surface to the height where the mean wind velocity profile V(z)*, given by (2.33), for example, has reached the same speed as that of the surface waves, Uw (phase velocity). Within this "critical layer", energy is being removed from the mean wind flow by the Reynolds stresses [cf. (2.59)]. The rate of transfer may be derived from the last term in (2.55), after integration over z by parts, dW kln r_V *

One finds that the change in the wave spectrum, d F(k)/d t, implied by this mechanism is proportional to F(k) itself, so that the wave growth becomes exponential.

The Phillips—Miles theory firstly of linear and then of exponentially growing spectral components of waves, under the influence of a stationary wind field, would be expected to be a reasonable first-order description of wave initiation, whereas the later "equilibrium situation" will require a balancing mechanism to carry away the additional energy and momentum transferred from wind to waves. Such mechanisms could be the wave-current interaction, or the dissipation of wave energy into turbulent water motion (an eddy spectrum), which might then in a secondary process transfer the momentum to currents. However, when detailed experimental studies of wave growth became available, it was clear that the actual growth rate was much larger than predicted by the Phillips—Miles theory (Snyder and Cox, 1966; Barnett and Wilkerson, 1967; Barnett and Kenyon, 1975).

Hasselmann (1967) has shown that in a systematic perturbation expansion of the solutions to (2.84), other terms than those considered by Phillips and Miles contribute to the same order of approximation. They represent couplings between the wave field and the fluctuating part of the wind field, such that two normal mode components of the wave field, n and n', are involved together with one component, bn", of the V'-field. Through its disturbance of the mean wind field V*, the wave component n' interacts with the wind turbulence component n", causing a pressure fluctuation which in turn increases the amplitude of wave component n. The energy transfer into component n is positive, while the transfer from component n' may have either sign, causing Hasselmann to speculate that this mechanism may be responsible both for the enhanced initial growth rate and also in part for the dissipation of wave energy at a later stage, balancing continued energy input from the wind.

In this model, the loss of energy from a wave field would not only go into oceanic turbulence and currents, but also into atmospheric turbulence. A similar mechanism is involved in breaking of waves [formation of whitecaps and plungers when a wave has developed to the size of maximum height-to-wavelength ratio (2.76), and more energy is added}. Although the breaking process primarily transfers energy from the waves into the atmosphere, it has been suggested that the breaking of waves pushes the streamlines of the air flow upwards, whereby the pressure field, which in Phillips' theory can transfer new energy to the wave field, gets more strength (Banner and Melville, 1976; Cokelet, 1977). Thus, it may be that the dissipative breaking process provides a key to the explanation of the fast growth of wave spectra, contrary to the predictions of the Phillips—Miles theory.

The energy flux associated with wave motion

Now let us reconsider a single, spectral component of the wave field. The energy flux through a plane perpendicular to the direction of wave propagation, i.e. the power carried by the wave motion, is given in the linearised approximation by (Wehausen and Laitone, 1960)

For the harmonic wave (2.73) and (2.74), neglecting surface tension, this gives

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