Third Generation Ocean Wave Model
The forecasting of ocean waves dates back to the attempts made during the second world war by Sverderup and Munk to predict waves in the English channel, for the invasion of Normandy in 1945. During this study they introduced the statistical expression, significant wave height, which was intended to express the wave height a trained observer would report for a given sea state. In the fifties, the wave energy spectrum was introduced by Pierson in 1957. This was based on the assumption that the sea surface may be represented as a Fourier series of superimposed waves with different wave lengths and with statistically random phases. The energy spectrum would then represent the mean wave energy at each of these Fourier modes. From the wave energy spectrum, several statistical parameters describing the sea state may be extracted. Examples are significant wave height, mean wave direction, mean wave length period etc. When the wave energy spectrum was introduced, an equation describing the evolution of the spectrum in time could be developed. This equation which express the conservation laws for wave energy in the ocean, and forms the basis for all numerical wave prediction models.
Before the wave model WAM will be described, a short summary of the basic theory behind wave modeling will be given.
The waves considered in numerical wave prediction, is assumed to be “short” in the sense that they are not affected by the rotation of the earth. Also the effects of density stratification of the ocean will be assumed to have no impact of the ocean waves. We will start with describing the basic laws of wave motion in fluids, and then give an interpretation of the individual terms in the wave energy equation.
2.1 Wave propagation
When considering the propagation of waves on the surface of a fluid, it is important to recognize the difference between the velocity of one individual wave, and the group velocity, which expresses the velocity of the wave energy for a particular wave length. The surface elevation caused by one Fourier component may be expressed as (for simplicity we assume only one dimension of propagation)
h = sin (k x - w t )
where k = 2p / l is the wave number and l is the wave length. w = 2p / T is the angular frequency and T is the wave period. The speed of one individual wave will in this case be
while the group velocity is defined as the rate of change of angular frequency to the wave number
For deep water waves, the dispersion relation yields
where g is the acceleration of gravity. In this case the relation between the wave speed and the group velocity becomes
c = c
In deep water the wave energy will propagates with half the wave speed of the individual waves
However, for waves that are long compared with the ocean dept., the wave propagation speed will be independent of the wave length. In this case the relation between group and wave velocity will be
c = c =
i.e. the wave energy will propagate with the same speed as the individual wave modes.
Definition: Swell is waves generated outside the area where they are observed.
Generally, wave energy enters the ocean as short waves and ripples. Through the process of non-linear interaction between the wave modes in the spectrum, the wave energy will be transferred to longer waves, finally ending up as long swells at the shore. The frictional dissipation of energy in long waves (the order of 100 m and more) is extremely small, and wave energy generated in one part of the ocean may easily travel across the world oceans, ending up as swells at the shore on a different continent. When considering such waves one must take into account how waves behave on a globe. Generally, deep water wave propagation on a sphere will be along a great circle path. If the wave length is comparable with the ocean depth, waves propagating over a sloping bottom will be bent toward shallow areas.
Long waves, compared to the ocean depth, will always be bent toward shallow areas when they propagate over a sloping bottom.
The modeling of swells propagating over large distances is difficult in numerical wave prediction models.
2.2 Conservation of wave energy
2.2.1 The wave energy spectrum
The main outcome of any wave prediction model is the energy spectrum. As mentioned, it is based upon the idea that the ocean surface could be represented by a series of harmonic functions with different angles and frequencies. As an example for one dimensional motion
Here, the mean wave energy in one single frequency interval df will in this case be
The spectral energy pr. Frequency interval is then defined as
and must be interpreted as the mean energy in one place (x,y) of a Fourier component with one particular frequency, propagating in one particular direction.
When outloning the equation for the development of the wave spectrum, the basic assumption is that in absence of frictional dissipation, wind, bottom variation and nonlinear interaction, the wave spectrum will change according to
which simply is an advection equation describing the change of the energy spectrum because of advection of wave energy with the group velocity. The effects of wind, friction and non-linear interaction are now introduced as source/sink terms in this equation
A brief description of these sink and source terms will be given below
2.2.2 Wind input
The wind input source term represents the work done by the wind on the ocean surface to produce waves. The wind generation of waves takes place in the high frequency part of the spectrum, i.e. it produces the relatively short waves (the order of a few meters and less, which can be observed when wind is blowing on the surface. The basic theory behind this term was developed by Miles in 1975. And it assumes a linear relationship between wave energy and the rate of change of energy,
which gives an exponential growth of wave energy with time
According to Miles the growth rate is dependent on the curvature of the wind profile at the point where the wind velocity equals the wave speed. For positive values of the growth rate the wind will give at net input of wave energy to the ocean. In the wave model WAM this growth rate is always either positive or zero. In other words, the wind input term will only It is important to note that in the real world, the growth rate may also have negative values. This means that the flow of energy is from the waves to the wind, i.e. that waves may generate wind. An example of this is very long waves or wind blowing in the opposite direction of the wind.
2.2.3 Dissipation of energy
Wave energy may be lost from the ocean in two different ways; wave breaking and frictional dissipation caused by velocity differences. White capping and breaking of waves takes energy from the waves and transfer some of it into current, the rest is dissipated, which means that mechanical energy is lost and water is heated up. The physical processes that takes place during wave breaking and white capping is extremely difficult to model. And in wave models these processes are parametrized by using data from several measurements.
On the other hand, the dissipation of energy that takes place within the fluid because of velocity differences may by modeled in a way similar to that of wind input, as a linear relationship between the wave energy and the rate of change of energy.
2.2.4 Non-linear interaction
If wind input and frictional dissipation was the only processes that was acting to change the energy spectrum, ocean waves would consist of only short surface waves. Apparently, the ocean also consist of long swells which could not be generated by the wind directly. Such long waves are the result of energy cascades that takes energy from the short wind waves and feeds the longer waves with energy. When the wave amplitude becomes large, three waves with different wave lengths may interact through mechanical resonance and create a fourth wave length. Only a limited combination of waves makes this possible. The conditions that has to be fulfilled to make resonant interaction possible are
A classification of wave models into first, second and third generation wave models is also used, which takes into account the method of handling the nonlinear source term
Third generation wave models are similar in structure, representing the state-of-the-art knowledge of the physics of the wave evolution. For the WAM models the wind input term, ,was adopted from Snyder et.el.(1981) with a scaling instead of . The dissipation source function corresponds to the form proposed by Komen et. (1984), in which the dissipation has been tuned to reproduce the observed fetch-limited wave growth and to eventually generate the fully developed Pierson-Moskowitz spectrum. The non-linear wave interactions are calculated using the Discrete-Interaction Approximation of Hasselmann et al (1985). The model can be used both as a deep water and a shallow water model.