regular waves
spectrum
action balance equation
propagation terms
source terms
nonlinear interaction terms

## Appearance of waves in nature

The SWAN model is developed to simulate waves in the near-shore zone. This zone extends from the coast to several tens of kilometers into the sea. The present course serves as an introduction to (potential) users of SWAN. It consists of an introduction to wave dynamics, to the use of numerical simulation models in general, and to the use of SWAN.
Waves at sea are in most cases generated by the wind. There are other causes for the generation of waves, such as ships or earthquakes, but we do not consider these in this course.
The wave field on the sea is irregular in the sense that it is approximated by a summation of regular waves of different frequencies. All these regular wave fields propagate at different speeds so that the appearance of the sea surface is constantly changing.

Irregular waves are described by the energy or action density spectrum. Essentially the action density is the contribution of waves in a certain direction and with a certain frequency to the total wave action. The action density is a function of space and time (on a scale large compared with wave length and period) and of spectral coordinates (wave frequency and direction).
Spectral wave models are based on the action balance equation, since wave action is a conserved quantity in absence of wave generation (by the wind) and dissipation. The left-hand side of this equation contains propagation terms, propagation both in geographical space and in spectral space. In this context refraction is considered as propagation in spectral space. The right-hand side of the equation contains source terms, i.e. terms which model the generation and dissipation of wave energy. In contrast with the propagation terms most of the source terms are empirical in nature and contain empirical "constants". SWAN has default values for almost all of these constants; these values are mostly based on literature, and have been obtained by studying laboratory experiments or field observations.
Due to the empirical nature of parts of the model a verification is needed for every new application of the model. The chapter on usage of numerical models also describes how to calibrate and validate a simulation model.

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## Regular Waves

We call a wave field a regular wave field if it repeats itself in time. In general we describe the vertical displacement of the sea surface as a function of horizontal coordinates x and y, and time t, i.e. the surface is located at $z=\zeta \left(x,y,t\right)$. The fact that a wave field is regular is expressed by $\zeta \left(x,y,t+T\right)=\zeta \left(x,y,t\right)$ for all values of x, y and t.
This T is called the period of the waves. In fact, if there is such a T there are many more since 2T, 3T etc. are also periods. So, if we talk about the period of the waves we mean the smallest positive T. The frequency of the waves is f=1/T, its unit is Hz=1/s; the so-called angular frequency is $\omega =2\pi /T$, its unit is rad/s.
The propagation speed of the waves depends on the period, at least in deeper water (depth large compared with the wave length). The waves with the longer period propagate faster than the ones with a smaller period.
The classical example of a regular wave on constant depth (and current velocity) is the sinusoidal wave: $\zeta \left(x,y,t\right)=a cos\left(\omega t - k x\right)$ where a is the amplitude, $\omega$ is the angular frequency (as measured at a fixed location in space), and k is the wave number ($k=2\pi /\lambda$ with $\lambda$ the wave length). k depends on the frequency and the depth by: $\sigma 2= g k tanh\left(k d\right)$, where d is the depth and $\sigma =\omega -k.U$ is the (angular) frequency measured by an observer moving with the current velocity U.
A sinusoidal wave in two space dimensions is described by
$\zeta \left(x,y,t\right)=a\ast cos\left(\omega t - k$x - ky y) where $k$x = k∗cos θ and $k$y = k∗sin θ are the components of the wave number in x- and y-direction resp., and $\theta$ is the wave direction.
In this course we adhere to the Cartesian definition of direction, i.e. measured from the positive x-axis in counterclockwise direction. This is also the definition which is used in SWAN internally. SWAN has possibilities for other definitions of direction; then during input and output values for direction are transformed.
The phase speed of the waves (the speed with which the shape of the waves propagates) is $\lambda /T= \omega /k$.
In a situation with non-uniform and/or time-dependent depth the above expressions for regular waves cannot be used any more. We now use:
$\zeta \left(x,y,t\right) = a\left(x,y,t\right)\ast cos\left(\psi \left(x,y,t\right)\right)$, where $\psi$ is the phase function.
The gradient of this phase function is equal to k, i.e. $\left(\psi$,x)2+(ψ,y)2=k2
Furthermore:
$\omega = \psi$,t
$\theta = arctg\left(\psi$,x,y)
Here the subscript ",x" denotes partial differentiation with respect to x, etc.

It can be shown (see detailed explanation) that the phase function and some related quantities can be determined along wave rays in (x,y,t)-space. The ray equations read:
$dx/dt = c$x = Ux+cgcos(θ)
$dy/dt = c$y = Uy+cgsin(θ)
$d\omega /dt = k$x ∗ ∂Ux/∂t + ky ∗ ∂Uy/∂t - ∂σ/∂d ∗ ∂d/∂t
$d\theta /dt = -k$x/k ∗ ∂Ux/∂n - ky/k ∗ ∂Uy/∂n - 1/k ∗ ∂σ/∂d ∗ ∂d/∂n
In this parameter representation the information travels along the rays with the so-called group velocity, this is the velocity with which the wave energy propagates.

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## The Wave Spectrum

An irregular wave field can be represented by the summation of a large number of regular sinusoidal wave components. As long as nonlinear effects are small all these components travel at different speeds so that their phases can be assumed to be uncorrelated. Spectral models are therefore often called random-phase models.
A spectrum is valid for a region a few wave lengths in size, assuming that the wave field is almost uniform over this region. Consequently depth and current velocity are also assumed to be almost uniform. It is noted here that the application of spectral models in the zone very close to the coast is subject to debate because here the depth varies significantly over one wave length.
In the spectral representation the energy density E is a function of frequency $\omega$ and direction $\theta$. The energy density $E\left(\omega ,\theta \right)$ is a measure of the contribution of one wave component $\left(\omega ,\theta \right)$ to the total wave energy.
We illustrate the significance of the energy spectrum by a method to reconstruct the wave field from a given spectrum. As a first step the spectral plane is covered by a grid. The central point of the bin (i,j) is $\left(\omega$ij); the amplitude of the wave associated with bin (i,j) is
$a$i,j= [8 E(ωij)ΔωΔθ]1/2
The wave field is described by:
$\zeta \left(x,y,t\right)=\Sigma \left\{a$i,j∗ cos(ψi,j + ωi t - k x cos(θj) - k y sin(θj)}
where $\psi$i,j is a random number chosen uniformly between 0 and $2\pi$. This shows how the concept of random phase enters the procedure.

An example of a spectrum computed by the SWAN program is shown in the figure. The top panel shows the 2-d spectrum, i.e. $E\left(\sigma ,\theta \right)=\sigma N\left(\sigma ,\theta \right)$ where E is the energy density (of which isolines are shown), and N is the action density (which is the quantity computed by SWAN originally). The wind direction is also shown in the figure, and usually the average wave direction is roughly the same as the wind direction. The wind vector is only the local wind, and in large regions wind directions will vary, which is one of the causes of deviation between wind and wave direction. Moreover this location is in an estuary (the Haringvliet in the Netherlands) so that part of the spectrum may be cut off.
For comparison with measurements this spectrum is often reduced to the so-called 1-d spectrum, which is a function of frequency only. This spectrum is obtained by integrating E over direction, i.e. $E\left(\sigma \right) =$0 E(σ,θ) dθ. The 1-d spectrum is shown in the lower panel. This 1-d spectrum shows two peaks, a feature that often occurs in shallow areas.
Overall parameters of the local wave field are in the lower right of the figure. These are significant wave height, peak frequency, average frequency (both Hz), average wave direction and directional spread (both in degrees).

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## Action Balance Equation

The above paragraph shows how the wave field can be reconstructed locally from the spectrum. On a larger scale the spectra can be computed using balance principles. In a general case with current the wave energy is not conserved. Wave action is.
In order to avoid ambiguity in presence of currents we now consider the density in $\left(\sigma ,\theta \right)$-space instead of $\left(\omega ,\theta \right)$-space.
Action density N is related to energy density E by: $N\left(\sigma ,\theta \right)=E\left(\sigma ,\theta \right)/\sigma$. Since we now consider a larger scale N and E are variable in space and time, i.e. we consider $N\left(x,y,t,\sigma ,\theta \right)$.
Action propagates in $\left(x,y,t,\sigma ,\theta \right)$ space with the propagation velocities which appear in the ray equations.
Thus the action balance equation reads:
$\partial N/\partial t + \partial \left(c$xN)/∂x + ∂(cyN)/∂y + ∂(cθN)/∂θ + ∂(cσN)/∂σ = S(x,y,t,σ,θ)

If this equation is discretized, the derivatives with respect to x and y can be interpreted as transport from one cell to the next (see the figure).

The right hand side of the action balance equation is called the source term; it consists of various contributions. The next section describes the contributions in the SWAN model.
The left hand size of the action balance equation shows the propagation terms. The terms with derivatives with respect to x and y take care of the propagation in space; the (action) propagation velocities are cx and cy. The term with the derivative with respect to θ is the refraction term; it causes the change of propagation direction. cθ depends on the bottom slope and on the spatial derivatives of the current velocity. The term with the derivative with respect to σ causes a change of frequency; this term is zero if the depth is stationary and if the current is zero.

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## Source Terms in the Action Balance Equation

The source terms in the action balance equation can be divided into the terms which result in exchange of energy between spectral bins, the interaction terms, and the generation and dissipation terms proper.
There is one generation term, the wind input source term. Its value depends on the local wind velocity, and on the spectrum itself. It consists of a linear growth term which is dominant in the first phase of wave growth, and an exponential growth term, where the source term is proportional with the action density itself.
There are several dissipation terms. The term which is active already in the area where the waves are generated is the whitecapping. This source term depends mainly on the steepness of the waves.
As the waves propagate into shallow water the bottom friction dissipation is becoming important. This term depends mainly on the orbital motion of the waves near the bottom, and on the bottom roughness.
Very close to the shore surf breaking becomes dominant. This source terms depends on the ratio between significant wave height and depth.
Click for details on the source terms.

Summary of the source terms:

wind source term
main parameter: wind velocity (linear growth)
main parameter: U/c (exponential growth)
whitecapping dissipation
main parameter: wave steepness
bottom friction
main parameter: orbital velocity near the bottom
surf breaking
main parameter: wave height over depth ratio

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## Nonlinear Interaction Terms

The SWAN model distinguishes two types of nonlinear interaction terms, the 3-wave interactions or triads and the 4-wave interactions or quadruplets. Both have the effect to exchange energy between spectral components which are in (near-) resonance. With the triads this means that wave number and frequency of one component is equal to the sum of the wave numbers and to the sum of the frequencies of the other two components. With the quadruplets it means that the sum of wave numbers and the sum of the frequencies of one pair of components is equal to the same for the other pair. The default formulation of the quadruplet source term is the DIA by Hasselmann (1962, 1963a, 1963b). Other options are available; more accurate formulations often require very long computation times but they are very useful to verify.
The triad source term (if it is active) is stronger than the quadruplet source term but it is active only in shallow water. In SWAN it results in transfer of energy to higher frequencies. The formulation of the triad source term is taken from Eldeberky and Battjes (1995) and Eldeberky (1996).
The quadruplet source term results in transfer of energy to lower frequencies.

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