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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
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5.3 Advective terms and refraction
The advective and refraction terms in the energy balance equation have been written in flux form. We shall only consider, as an example, the one-dimensional advection equation
 ,
|
(5.10) |
with flux
, since the generalization to four dimensions 
, 
, 
and 
is obvious. Two alternative propagation schemes were tested, namely
a first order upwinding scheme and a second order leap frog scheme (for
an account of the numerical schemes of the advection form of the energy
balance equation see WAMDI , 1988). The first-order scheme is characterized
by a higher numerical diffusion, with an effective diffusion coefficient
, where 
denotes grid spacing and 
is the time step. For numerical stability the time step must
satisfy the inequality 
, so that 
. The advection term of the second-order
scheme has a smaller, inherent, numerical diffusion, but suffers from the
drawback that it generates unphysical negative energies in regions of sharp
gradients. This can be alleviated by including explicit diffusion terms.
In practice, the explicit diffusion required to remove the negative side
lobes in the second order scheme, is of the same order as the implicit numerical
diffusion of the first order scheme, so that the effective diffusion is
generally comparable for both schemes. As shown in WAMDI (1988) both schemes have similar propagation and diffusion properties. An advantage of the second order scheme is that the lateral diffusion is less dependent on the propagation direction than in the first order scheme, which shows significant differences in the diffusion characteristics for waves travelling due south-north or west-east compared with directions in between. The first order scheme has the additional problem that there is excessive shadowing behind islands when waves are propagating along the coordinate axes. However, these undesirable features in the first order upwinding scheme may be alleviated by rotating the spectra by half its angular resolution, in such a way that no spectral direction coincides with the principle axes of the spatial grid. In general, the differences between the model results using first or second order propagations methods were found to be small, but there is a preference for the first order scheme because of its efficiency and simplicity.
Historically, the main motivation for considering the second order scheme in addition to the first order scheme was not to reduce diffusion, but to be able to control it. In contrast to most other numerical advection problems, an optimal propagation scheme for a spectral wave model is not designed to minimize the numerical diffusion, but rather to match it to the finite dispersion associated with the finite frequency-direction spectral resolution of the model (SWAMP , 1985, appendix B). In this context, it should be pointed out that an ideal propagation scheme would give poor results for sufficiently large propagation times, since it would not account for the dispersion associated with the finite resolution in frequency and direction (the so-called garden sprinkler effect). Now, the dispersion due to the different propagation velocities of the different wave components within a finite frequency-direction bin increases linearly with respect to propagation time or distance, whereas most propagation schemes yield a spreading of the wave groups which increases with the square root of the propagation time or distance. However, Booij and Holthuijsen (1987) have shown that linear spreading rates may be achieved by introducing a variable diffusion coefficient proportional to the age of the wave packets. This idea has been tested in the context of a third generation wave model by Chi Wai Li (1992) and Tolman (2000) who uses an averaged age of the wave packets per ocean basin.
To summarize our discussion, we have chosen the first order upwinding scheme because it is the simplest scheme to implement (requiring less computer time and memory) and because in practice it gives reasonable results. Applied to the simple advection scheme in flux form (5.10) we obtained the following discretization, where for the definition of grid points we refer to Fig. 5.1 .
The rate of change of the spectrum
in the 
th grid point is given by
 ,
|
(5.11) |
where
is the grid spacing and 
the propagation time step, and
 ,
|
(5.12) |
where
is the mean group velocity and the flux at 
is obtained from (5.12) by replacing 
with 
. The absolute values of the mean speeds arise
because of the upwinding scheme. For example, for flow going from the left
to the right the speeds are positive and, as a consequence, the evaluation
of the gradient of the flux involves the spectra at grid points 
and 
. We furthermore remark that one could consider using a semi-Lagrangian scheme for advection. This scheme is gaining popularity in meteorology because it does not suffer from the numerical instabilities which arise in conventional discretization schemes when the time step is so large that the Courant-Friedrichs-Levy (CFL) criterion is violated. The wave model community has, so far, not worried too much about this problem because advection is a relatively inexpensive part of the computations. In addition, in most applications, the propagation time is larger or equal to the source time step, which is usually 20 min. According to the CFL criterion, short propagation time steps (less than, say, 10 min) are only required for very high resolution (
). But in these circumstances the advection will
induce changes in the physics on a short time scale, so that it is advisable
to decrease the source time step accordingly. Therefore, in the WAM model,
the source time step is always less than or equal to the propagation time
step.
We finally comment on the so-called pole problem in the case of of the use of spherical coordinates. When moving towards the poles, the distance in the latitudinal direction decreases. Clearly, close to the poles violation of the \ CFL criterion occurs. In the ECMWF version of the WAM model this problem is solved by choosing an irregular spherical grid in such a way that the distance in the latitude direction is more or less fixed to its value at the equator. An example of such a grid from the present operational ECMWF WAM model is shown in Fig. 5.2 . The advection scheme is still formulated in terms of spherical coordinates but the gradient in the longitudinal fluxes is evaluated by linear interpolation of the fluxes from the closest neighbours. The additional advantages of the use of an irregular spherical grid is a reduction in the total number of grid points by 30%, giving a substantial reduction in the cpu consumption.
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23.04.2002 |
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