# The wave model

TitleThe wave model
Publication TypeEducation material
Date Published2002
Secondary TitleMeteorological Training Course Lecture Series
AuthorJanssen, PAEM
PublisherECMWF
Keywordslecture notes, NWP
Abstract

In this set of lectures I would like to give a brief overview of the state-of-the-art of ocean wave modelling, ranging from a derivation of the evolution equation of the wave spectrum from the Navier-Stokes equations to the practice of wave forecasting. From validation studies using satellite data from Geosat and ERS-1 it appears that present-day wave models are reliable. In addition, it is known from experience that wave results depend in a sensitive manner on the quality of the driving surface winds. For these reasons ocean wave forecasting has certain beneÞts for atmospheric modelling. Two examples of these benefits are given. The program of these lectures is, therefore, as follows:

1. Derivation of the energy balance equation
1.1 Preliminaries-Dynamical equations, dispersion relation in deep and shallow-water, group velocity, energy density, Hamiltonian and Lagrangian for potential flow. The averaged Lagrangian.
1.2 Energy balance equation (adiabatic part) - The need for a statistical description of ocean waves: the wave spectrum. - From the averaged Lagrangian we show that the adiabatic rate of change of the wave spectrum is determined by advection (with the group velocity) and refraction.
1.3 Energy balance equation (physics) - From now on we only consider deep water. It is shown that, in addition to adiabatic effects, the rate of change of the wave spectrum is determined by: (a) energy transfer from wind; (b) non-linear wavewave interactions; (c) dissipation by white capping.

2. The WAM model - The WAM model is the first model that solves the energy balance equation, including non-linear wavewave interactions.
2.1 Energy balance for wind sea - Distinction between wind sea and swell. Empirical growth curves: fetch and duration limitation. Energy balance for wind sea according to the WAM model. Evolution of wave spectrum. Comparison with observations from JONSWAP.
2.2 Wave forecasting - Quality of wind field (SWADE). Validation of wind and wave analysis using ERS-1 altimeter data and buoy data. - Quality of wave forecast, forecast skill depends on whether sea state is dominated by wind sea or swell.

3. Benefits for atmospheric modelling
3.1 Use as a diagnostic tool - The apparent over-activity of the atmospheric model during the forecast is reflected in too high forecasted wave height. This is shown by comparing monthly mean wave forecasts with the verifying analysis.
3.2 Coupled wind-wave modelling - The energy transfer from atmosphere to ocean is sea state dependent. To be consistent one has to couple wind and waves to take the sea state dependent slowing down of the wind into account. Such a coupled wind-wave model gives an improved climate in the Northern Hemisphere. Also, the wind field in the tropics is affected, to a considerable extent, (e.g. in monsoon area and warm pool east of Indonesia).

###### Contents

1. Derivation of the energy balance equation
2. The WAM model
3. Benefits for atmospheric modelling
4. References

URLhttps://www.ecmwf.int/node/16951