
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 nearshore 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
lefthand 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 righthand 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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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
$z=\zeta (x,y,t)$.
The fact that a wave
field is regular is expressed by
$\zeta (x,y,t+T)=\zeta (x,y,t)$
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 socalled 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:
$\zeta (x,y,t)=a\; cos(\omega t\; \; k\; x)$
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:
$\sigma 2=\; g\; k\; tanh(k\; d)$,
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
$\zeta (x,y,t)=a\ast cos(\omega t\; \; k$_{x}  k_{y} y)
where $k$_{x} = k∗cos θ
and $k$_{y} = k∗sin θ
are the components of the wave
number in x and ydirection resp., and $\theta $
is the wave direction.
In this course we adhere to the Cartesian definition of direction, i.e.
measured from the positive xaxis 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 nonuniform and/or timedependent depth the above
expressions for regular waves cannot be used any more. We now use:
$\zeta (x,y,t)\; =\; a(x,y,t)\ast cos(\psi (x,y,t))$,
where $\psi $ is the phase function.
The gradient of this phase function is equal to k, i.e.
$(\psi $_{,x})^{2}+(ψ_{,y})^{2}=k^{2}
Furthermore:
$\omega \; =\; \psi $_{,t}
$\theta \; =\; arctg(\psi $_{,x}/ψ_{,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} = U_{x}+c_{g}cos(θ)
$dy/dt\; =\; c$_{y} = U_{y}+c_{g}sin(θ)
$d\omega /dt\; =\; k$_{x} ∗ ∂U_{x}/∂t +
k_{y} ∗ ∂U_{y}/∂t 
∂σ/∂d ∗ ∂d/∂t
$d\theta /dt\; =\; k$_{x}/k ∗
∂U_{x}/∂n 
k_{y}/k ∗ ∂U_{y}/∂n 
1/k ∗ ∂σ/∂d
∗ ∂d/∂n
In this parameter representation the information travels along the rays with
the socalled group velocity, this is the velocity with which the wave energy
propagates.
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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 randomphase
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
$E(\omega ,\theta )$
is a measure of the contribution of
one wave component
$(\omega ,\theta )$
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
$(\omega $_{i},θ_{j});
the amplitude of the wave associated
with bin (i,j) is
$a$_{i,j}=
[8 E(ω_{i},θ_{j})ΔωΔθ]^{1/2}
The wave field is described by:
$\zeta (x,y,t)=\Sigma \{a$_{i,j}∗
cos(ψ_{i,j} + ω_{i} t  k x
cos(θ_{j})  k y sin(θ_{j})}
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 2d spectrum, i.e.
$E(\sigma ,\theta )=\sigma \; N(\sigma ,\theta )$
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 socalled 1d spectrum,
which is a function of frequency only. This spectrum is obtained by integrating E over direction, i.e.
$E(\sigma )\; =$_{0}∫^{2π} E(σ,θ) dθ.
The 1d spectrum is shown in the lower panel. This 1d 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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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
$(\sigma ,\theta )$space instead of
$(\omega ,\theta )$space.
Action density N is related to energy density E by:
$N(\sigma ,\theta )=E(\sigma ,\theta )/\sigma $.
Since we now consider a larger
scale N and E are variable in space and time, i.e. we consider
$N(x,y,t,\sigma ,\theta )$.
Action propagates in $(x,y,t,\sigma ,\theta )$ space
with the propagation velocities which appear in the ray equations.
Thus the action balance equation reads:
$\partial N/\partial t\; +\; \partial (c$_{x}N)/∂x
+ ∂(c_{y}N)/∂y
+ ∂(c_{θ}N)/∂θ
+ ∂(c_{σ}N)/∂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
c_{x} and c_{y}.
The term with the derivative with respect to θ is the refraction term;
it causes the change of propagation direction. c_{θ} 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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The SWAN
model distinguishes two types of nonlinear interaction terms, the 3wave
interactions or triads and the 4wave 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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