# The wave model WAVEWATCH III

WAVEWATCH III is a numerical model that integrates the wave action equation. It is based on a spectral representation of the sea state.... What?

At selected locations of the ocean  (these locations form the "spatial grid" of the model), the wave field is decomposed into a "spectrum": we define the energy of many wave "components". Each component is a regular wave train, like corrugated iron. The sum of many wave trains going in different directions θ and with a different frequency f gives the full sea state.  On the right figure, there is a black diagram showing and example of a spectrum in frequency only. For each frequency, we further break up the wave energy in directions, as shown in the right diagram in which the color gives the amount of energy of each component (to be precise, the spectral density).

The energy of the waves can be added up: the total energy in this example is the area in black, which is E=0.6 m2 , times the water density (about 1000 kg/m3), times the gravity g (nearly 10 m/s2)... which gives Et=6 kiloJoules per unit of ocean surface. With those waves, a square column of water with 1 m width contains on average 6 kJ of wave energy.

This energy changes in time and space as the waves are generated by the wind, propagate, and dissipate.

In order to track the evolution of this energy (for example for weather forecasting), the model WAVEWATCH III computes the evolution of the wave spectrum from one ocean location to another, by solving the Wave Action Equation (WAE),

local evolution in time +[ propagation  + turning of the waves ] = generation  + exchange of energy - dissipation

This equation is written for EACH spectral component (typically 24 direction x 32 frequencies), and all these equations are coupled by the terms:

• "turning of the waves" (energy from one direction is shifted to energy from another direction)
• "exchange of energy" : the different wave trains permanently exchange energy between them and with currents
• dissipation: the dissipation of one wave component is a complex function of the energy of all components
• generation: the amount of energy given by the wind to one wave train can also depend on the other wave trains

Models such as WAVEWATCH III differ in their method from the old "ray tracing models" that conserved energy due to the last 3 interactions. When the energy of a wave component is nearly conserved ray tracing is a more simple method for tracking the energy ... but this is not the case in general.

It is customary to summarize the sea state by a few numbers. The most common is the significant wave height Hs=4√E  .  The maps shown above with wave heights are computed with WAVEWATCH III: at each point of the ocean the full spectrum E(f,θ) was computed and from that spectrum we computed Hs.

The main reason for using such a rich spectral decomposition, introduced in the 1950s by Gelci and his colleagues, is that the wavesdisperse: each component travels at a different speed and in a different direction.  In very shallow water the waves tend to travel at the same speed and other methods can be much more efficient.

The cost of a wave model can be much larger than that of an ocean circulation model with the same horizontal resolution because the wave model usually carries many more variables (the spectral densities) compared to the velocity, temperature and salinity at different levels.

The accuracy of wave models critically depends on :

- the accuracy of the forcing fields (winds, currents, sea ice, bathymetry, bottom roughness ...)

- the accuracy of the parameterizations that represent the generation, exchange and dissipation  processes

- the quality of the numerical methods used to solve the WAE.

For recent reviews see Rascle and Ardhuin (Ocean Modelling 2013) and Roland and Ardhuin (Ocean Dynamics 2014)