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IFS Documentation
front page
Table of contents
Chapter 1. Introduction
Chapter 2. The kinematic part of the
energy balance equation
Chapter 3. Parametrization of source terms and the energy balance
in a growing wind sea
Chapter 4. An optimal interpolation scheme for assimilating altimeter data
into the WAM model
Chapter 5 Numerical scheme
Chapter 6 The WAM-model software
Chapter 7 Wind-wave interaction at ECMWF
REFERENCES
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We now turn to the important case of spherical coordinates. When one transforms
from one set of coordinates to another there is no guarantee that the flow
remains divergence-free. However, noting that equation (2.2) holds for any rectangular
coordinate system, the generalization of the standard Cartesian geometry
transport equation to spherical geometry (see also
Groves and Melcer, 1961 and WAMDI , 1988) is straightforward. To that
end let us consider the spectral action density  with respect to angular frequency  and direction  (measured clockwise relative to true north) as a function of
latitude  and longitude  . The reason for the choice of frequency as the independent
variable (instead of, for example, the wavenumber  ) is that for a fixed topography and current the frequency
 is conserved when following a wave group, therefore
the transport equation simplifies. In general, the conservation equation
for  thus reads
and since  the term involving the derivative with respect to  drops out in the case of time-independent current
and bottom. The action density  is related to the normal spectral density  with respect to a local Cartesian frame  through  =  , or
where  is the radius of the earth. Substitution of (2.10) into (2.9) yields
where, with  the magnitude of the group velocity,
represent the rates of change of the position and propagation
direction of a wave packet. Equation (2.11) is the basic transport
equation which we will use in the numerical wave prediction model. The remainder
of this section is devoted to a discussion of some of the properties of
(2.11). We first discuss some
peculiarities of (2.11) for the infinite depth
case in the absence of currents and next we discuss the special cases of
shoaling and refraction due to bottom topography and currents.
From (2.12a)-(2.12d) we infer that in spherical
coordinates the flow is not divergence-free. Considering the case of no
depth refraction and no explicit time dependence, the divergence of the
flow becomes
which is nonzero because the wave direction, measured with
respect to true north, changes while the wave group propagates over the
globe along a great circle. As a consequence wave groups propagate along
a great circle. This type of refraction is therefore entirely apparent and
only related to the choice of coordinate system.
Let us now discuss finite depth effects in the absence of currents by considering
some simple topographies. We first of all discuss shoaling of waves for
the case of wave propagation parallel to the direction of the depth gradient.
In this case, depth refraction does not contribute to the rate of change
of wave direction  because, with equation (2.3b),  . In addition, we take the wave direction  to be zero so that the longitude is constant ( ) and  . For time-independent topography (hence  ) the transport equation becomes
where
and the group speed only depends on latitude  . Restricting our attention to steady waves we immediately
find conservation of the action-density flux in the latitude direction,
or,
If, in addition, it is assumed that the variation of depth with latitude
occurs on a much shorter scale than the variation of  , the latter term may be taken constant for present purposes.
It is then found that the action density is inversely proportional to the
group speed  ,
and if the depth is decreasing for increasing latitude,
conservation of flux requires an increase of the action density as the group
speed decreases for decreasing depth. This phenomenon, which occurs in coastal
areas, is called shoaling. Its most dramatic consequences may be seen when
tidal waves, generated by earthquakes, approach the coast resulting in tsunamis.
It should be emphasized though, that in the final stages of a tsunami the
kinetic description of waves, as presented here, breaks down because of
strong nonlinearity.
The second example of finite depth effects that we discuss is refraction.
We again assume no current and a time-independent topography. In the steady
state the action balance equation becomes
where
In principle, equation (2.18) can be solved by means
of the method of characteristics. We will not give the details of this,
but we would like to point out the role of the  term for the simple case of waves propagating along the
shore. Consider, therefore, waves propagating in a northerly direction (hence
 ) parallel to the coast. Suppose that the depth only depends
on longitude such that it decreases towards the shore. The rate of change
of wave direction is then positive as
since  . Therefore, waves which are propagating initially parallel
to the coast will turn towards the coast. This illustrates that, in general,
wave rays will bend towards shallower water resulting in, for example, focussing
phenomena and caustics. In this way a sea mountain plays a similar role
for gravity waves as a lens for light waves.
Finally, we discuss some current effects on wave evolution. First of all,
a horizontal shear may result in wave refraction; the rate of change of
wave direction follows from (2.18) by taking the current
into account,
where  and  are the components of the water current in latitudinal and
longitudinal directions. Considering the same example as in the case of
depth refraction, we note that the rate of change of the direction of waves
propagating initially along the shore is given by
which is positive for an along-shore current which decreases
towards the coast. In that event the waves will turn towards the shore.
The most dramatic effects may be found, when the waves propagate against
the current. For sufficiently large current, wave propagation is prohibited
and wave reflection occurs. This may be seen as follows. Consider waves
propagating to the right against a slowly varying current  . At  the current vanishes, decreasing monotonically to some negative
value for  . Let us generate at  a wave with a certain frequency value  . Following the waves, we know from (2.8), that for time-independent
circumstances the angular frequency of the waves is constant, hence for
increasing strength of the current the wavenumber increases as well. Now,
whether the surface wave will arrive at  or not depends on the magnitude of the dimensionless
frequency  (where  is the maximum strength of the current); for  propagation up to  is possible, whereas in the opposite case propagation is prohibited.
Considering deep water waves only, the dispersion relation reads
and the group velocity  vanishes for  so that the value of  at the extremum is  . At the location where the current has maximum strength the
critical angular frequency  is the smallest. Let us denote this minimum value of  by  ( ). If now the oscillation frequency  is in the entire domain of consideration, then the group speed
is always finite and propagation is possible (this of course corresponds
to the condition  ), but in the opposite case propagation is prohibited beyond
a certain point in the domain. What actually happens at that critical point
is still under debate. Because of the vanishing group velocity, a large
increase of energy at that location may be expected suggesting that wave
breaking plays a role. On the other hand, it may be argued that near such
a critical point the usual geometrical optics approximation breaks down
and that tunnelling and wave reflection occurs (Shyu and Phillips, 1990).
A kinetic description of waves which is based on geometrical optics then
breaks down as well. This problem is not solved in the WAM model. In order
to avoid problems with singularities and nonuniqueness (note that for finite
 one frequency  corresponds to two wavenumbers) we merely transform to the
intrinsic frequency  (instead of frequency  ) because a unique relation between  and wavenumber  exists.
To conclude this section we note that a global third generation wave model
solves the action balance equation in spherical coordinates. By combining
previous results of this section, the action balance equation reads
where
and
and  is the dispersion relation given in (2.4).
Before discussing possible numerical schemes to approximate the left hand
side of equation (2.24) we shall first of all
discuss the parametrization of the source term  , where  is given by
representing the physics of wind input, wave-wave interactions
and dissipation due to whitecapping and bottom friction.
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