IFS Documentation front page

I Observations
II Assimilation
III Dynamics
IV Physics
V Ensemble
VI Technical
VII Waves

3.3 Nonlinear transfer

In Komen et al. (1994) the derivation of the source function , describing the nonlinear energy transfer, was given from first principles. For surface gravity waves the nonlinear energy transfer is caused by four resonantly interacting waves, obeying the usual resonance conditions for the angular frequency and the wave numbers. The evaluation of therefore requires an enormous amount of computation because a three dimensional integral needs to be evaluated. In the past several attempts have been made to try to obtain a more economical evaluation of the nonlinear transfer. The approach that was most succesful to date is the one by Hasselmann et al. (1985). The reason for this is that their parametrization is both fast and it respects the basic properties of the nonlinear transfer, such as conservation of momentum, energy and action, while it also produces the proper high-frequency spectrum.

To this end, Hasselmann et al. (1985) constructed a nonlinear interaction operator by considering only a small number of interaction configurations consisting of neighbouring and finite distance interactions. It was found that, in fact, the exact nonlinear transfer could be well simulated by just one mirror-image pair of intermediate range interactions configurations. In each configuration, two wavenumbers were taken as identical . The wavenumbers and are of different magnitude and lie at an angle to the wavenumber , as required by the resonance conditions. The second configuration is obtained from the first by reflecting the wavenumbers and with respect to the -axis. The scale and direction of the reference wavenumber are allowed to vary continuously in wavenumber space.

The simplified nonlinear operator is computed by applying the same symmetrical integration method as is used to integrate the exact transfer integral (see also Hasselmann and Hasselmann, 1985), except that the integration is taken over a two-dimensional continuum and two discrete interactions instead of five-dimensional interaction phase space. Just as in the exact case the interactions conserve energy, momentum and action. For the configurations

(3.19)

where , satisfactory agreement with the exact computations was achieved. From the resonance conditions the angles , of the wavenumbers and relative to are found to be , .

The discrete interaction approximation has its most simple form for the rate of change in time of the action density in wavenumber space. In agreement with the principle of detailed balance, we have

,

(3.20)

where , , are the rates of change in action at wavenumbers , , due to the discrete interactions within the infinitesimal interaction phase-space element and is a numerical constant. The net source function is obtained by summing equation (3.20) over all wavenumbers, directions and interaction configurations.

For a JONSWAP spectrum the approximate and exact transfer source functions have been compared in Komen et al. (1994). The nonlinear transfer rates agree reasonably well, except for the strong negative lobe of the discrete-interaction approximation. This feature is, however, less important for a satisfactory reproduction of wave growth than the correct determination of the positive lobe which controls the down shift of the spectral peak.

The usefulness of the discrete-interaction approximation follows from its correct reproduction of the growth curves for growing wind sea. This is shown in Fig. 3.2 where a comparison is given of fetch-limited growth curves for some important spectral parameters computed with the exact nonlinear transfer, or, alternatively, with the discrete-interaction approximation. Evidence of the stronger negative lobe of the discrete interaction approximation is seen through the somewhat smaller values of the Phillips constant . The broader spectral shape corresponds with the smaller values of peak enhancement for the parametrized case. On the other hand, the agreement of the more important scale parameters, the energy and the peak frequency is excellent (note that, as always, an asterix denotes nondimensionalisation of a variable through and the friction velocity ).

Figure 3.2 Comparison of fetch-growth curves for spectral parameters computed using the exact form and the discrete interaction approximation of . All variables are made dimensionless using and .

The above analysis is made for deep water. Numerical computations by Hasselmann and Hasselmann (1981) of the full Boltzmann integral for water of arbitrary depth have shown that there is an approximate relation between transfer rates in deep water and water of finite depth: for a given frequency-direction spectrum, the transfer for finite depth is identical to the transfer for infinite depth, except for a scaling factor :

,

(3.21)

where is the mean wavenumber. This scaling relation holds in the range , where the exact computations could be closely reproduced with the scaling factor

,

(3.22)

with . This approximation is used therefore in the WAM model.