Peter A. E. M. Janssen

On phase velocity and growth rate of wind-induced gravity-capillary waves

The generation and growth of gravity-capillary waves by wind are considered using linear instability theory to describe the process. The growth rate of the initial wavelets, the first waves to be generated by wind, is found to be proportional to the cube of the friction velocity in air. The effect of changes in the shape of the profiles of wind and wind-induced current is also considered; the growth rate is found to be very sensitive to the shape of the wind profile while the influence of changes in the current profile is much smaller. The values of the current and the current shear at the interface are much more important for determining the phase velocity correctly than the shape of either the wind or the current profile.

Quasilinear approximation for the spectrum of wind-generated water waves

According to Miles’ theory of wind-wave generation, water waves grow if the curvature of the wind profile at the critical height is negative. As a result, the wind profile changes in time owing to the transfer of energy to the waves. In the quasilinear approximation (where the interaction of the waves with one another is neglected) equations for the coupled air–water system are obtained by means of a multiple-time-scale analysis. In this way the validity of Miles’ calculations is extended, thereby allowing a study of the large-time behaviour. While the water waves grow owing to the energy transfer from the air flow, the waves in turn modify the flow in such a way that for large times the curvature of the velocity profile vanishes. The amplitude of the waves is then limited because the energy transfer is quenched. In the high-frequency range the asymptotic wave spectrum is given by a ‘–4’ law in the frequency domain rather than the ‘classical’ ‘–5’ law.

Modulational instability and the Fermi‐Pasta‐Ulam recurrence

The long‐time behavior of the modulational instability of the nonlinear Schrodinger equation is investigated. Linear stability analysis shows that a finite amplitude uniform wave train is unstable to infinitesimal modulational perturbations with sufficiently long wavelengths while it is stable for perturbations with short wavelengths. Near the threshold for instability, the long‐time behavior of the unstable modulation is obtained by means of the multiple time scale technique. As a result, the Fermi–Pasta–Ulam recurrence is rediscovered, in agreement with recent experiments and with a numerical solution of the problem at hand.

Nonlinear evolution of the transverse instability of plane‐envelope solitons

The nonlinear evolution of the transverse instability of plane envelope soliton solutions of the nonlinear Schrodinger equation is investigated. For the case where the spatial derivatives in the two‐dimensional nonlinear Schrodinger equation are elliptic a critical transverse wavenumber is found above which the soliton is stable. Application of the multiple time scale method near this critical transverse wavenumber gives a dynamical equation for the nonlinear evolution of the transverse instability. Nonlinearity is found to enhance the growth of the linearly unstable mode. The results are discussed in connection with soliton collapse.

Friction velocity scaling in wind wave generation

This note is devoted to the problem of the appropriate scaling of parameters relevant for sea waves, such as wave height, peak frequency, duration, and fetch. In the past, the growth of sea waves has often been analysed in terms of the wind velocity at a fixed height, despite the fact that many authors have stressed the importance of scaling with the friction velocity. This problem would be immaterial if the ratio between the friction velocity and the wind speed at a fixed height were a constant. There is, however, ample evidence that this ratio increases with wind speed (Smith and Banke, 1975; Smith, 1980), in agreement with dimensional considerations by Charnock (1955) on the friction height. As a result, the scaling problem is an important one. In this note we conjecture that the correct procedure is to scale wave parameters with friction velocity, and we discuss experimental evidence for the correctness of this conjecture. Comparing two independent datasets (‘JONSWAP’ and ‘KNMI’), we find some evidence supporting our ideas. Further confirmation remains desirable, however, and suggestions are made as to how this might be obtained.

Effect of atmospheric stability on the growth of surface gravity waves

In this paper we study the effect of atmospheric stability on the growth of surface gravity waves. To that end we numerically solved the Taylor-Goldstein equation for wind profiles which deviate from a logarithmic form because stratification affects the turbulent momentum transport. Using Charnocks relation for the roughness height z0 of the wind profile, it is argued that the growth rate of the wave depends on the dimensionless phase velocity c/u* (where u* is the friction velocity) and a measure of the effect of atmospheric stability, namely the dimensionless Obukhov length gL/u*2, whereas it only depends weakly on gzt/u*2 (where zt is the roughness height of the temperature profile). Remarkably for a given value of u*/c, the growth rate is larger for a stable stratification (L > 0) than for an unstable one (L < 0). We explain why this is the case. If, on the other hand, one considers the growth rate as a function of c/U10 (where U10 is the windspeed at 10 m), the situation reverses for c/U10 < 1. For practical application in wave prediction models, we propose a new parameterization of the growth rate of the waves which is an improvement of the Snyder et al. (1981) proposal because the effect of stability is taken into account.

Consequences of the Effect of Surface Gravity Waves on the Mean Air Flow

The effect of wind-generated gravity waves on the air flow is discussed using quasi-linear theory of wind-wave generation. In this theory both the effects of the waves and the effect of air turbulence on the mean wind profile is taken into account, but effects of turbulence on the wave-induced air motion are disregarded. In addition, effects of wave breaking on the air-sea momentum transfer are not considered. Nevertheless, this relatively simple model of the momentum transport from air to water is shown to produce realistic results.

The Determination of the Surface Stress in an Atmospheric Model

Abstract By forcing a third-generation wave-prediction model with surface stresses from the European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric model, it was discovered that lower wave heights were generated than by forcing with the ECMWF surface winds. The apparent inconsistency between surface stresses and surface winds in the atmospheric model turns out to be time-step dependent. A similar conclusion may be inferred from results of the WAMDI group. Apparently, a number of atmospheric models have inaccuracies in the boundary-layer scheme near the surface. In this paper it is argued that the reason for the inaccuracies is related to the numerical integration scheme that is used in these models. It is shown that a numerical scheme that treats physics and dynamics separately has an equilibrium that is time-step dependent. An alternative scheme—namely, simultaneous, implicit treatment of both physics and dynamics—removes this deficiency. Possible consequences for atmospheric-, wave-, and ...

Shallow Water Intercomparison of Wave Models — Part III

This paper describes the intercomparison and verification of results from three operational shallow water wave prediction models in their hindcasts of a severe storm period. The three models, BMO (Meteorological Office), GONO (KNMI) and HYPAS (Max-Planck-Institut) have already been compared theoretically and via idealised experiments with constant winds in Parts I and II of the SWIM project (SWIM Group, 1985). The work reported here attempts to unravel the complicated processes involved in the prediction of shallow water waves in a complex synoptic situation by running all three models in parallel with common wind fields and similar computation grids and bottom topography. The case chosen, two North Sea storms in November 1981, is fully described in SWIM (1984). Results presented here show that all models produce broadly similar results and acceptable shallow water energy levels at three verification sites, despite the differences in formulation and results shown in Parts I and II of this project.

Shallow Water Intercomparison of Wave Models — Part I Three Different Concepts to Model Surface Waves in Finite Water Depth

The SWIM-project is an extension into water of finite depth of the SWAMP-study (1982, 1984), which investigated the deep water performance of different wave prediction models.