8.4 Wave-Absorbing Layers
431
The approximate local radiation condition (8.61) is implemented in essentially
the same manner as the nonlocal condition (8.55), except that instead of Fourier
transforming W, applying (8.55), and then inverse transforming to obtain the
boundary values for the pressure, (8.61) is solved to compute the grid-point values
of P directly from the grid-point values of w. A tridiagonal system for the gridpoint values of P along the top boundary is obtained when the second derivatives
in (8.61) are approximated by a standard three-point finite difference. Special conditions are required at those points adjacent to the lateral boundaries, where it can
be advantageous to use the less accurate approximation
This relation, which is derived from (8.55) using the approximation Ikl b) +
bzk z, does not require any assumption about the horizontal variation of W at the
lateral boundaries. Some assumption is nevertheless required about the variation
of P near the boundary, and satisfactory results have been obtained by setting
o P j ox to zero in the upper corners of the domain.
8.4 Wave-Absorbing Layers
One way to prevent outward-propagating disturbances from reftecting back into
the domain when they encounter the boundary is to place a wave-absorbing layer
at the edge of the domain. Wave-absorbing layers are conceptually simple and are
particularly attractive in applications for which appropriate radiation boundary
conditions have not been determined. Wave-absorbing layers also allow "largescale" time tendencies to be easily imposed at the lateral boundaries of the domain . These large-scale tendencies might be generated by a previous or concurrent
coarse-resolution simulation on a larger spatial domain. The chief disadvantage
of the absorbing-layer approach is that the absorber often needs to be rather thick
in order to be effective, and significant computational effort may be required to
compute the solution on the mesh points within a thick absorbing layer. Considerable engineering rnay also be required to ensure that a wave-absorbing layer
performs adequately in a given application.
Suppose that a radiation upper boundary condition for the two-dimensional linearized Boussinesq equations (8.37)-(8.40) is to be approximated using a waveabsorbing layer of thickness D. This absorbing layer can be created by defining a vertically varying viscosity a(z) and adding viscous terms of the form
a(z)oZujox Z, a(z)o2wjox 2, and a(z)o2bjox 2 to the right sides of (8.37), (8.38),
and (8.39), respectively. The viscosity is zero in the region z :s H within which
the solution is to be accurately approximated and increases gradually with height
throughout the layer H < z :s H + D . A simple rigid-lid condition, w = 0,
can be imposed at the top of the absorbing layer. The performance of this ab-
431
The approximate local radiation condition (8.61) is implemented in essentially
the same manner as the nonlocal condition (8.55), except that instead of Fourier
transforming W, applying (8.55), and then inverse transforming to obtain the
boundary values for the pressure, (8.61) is solved to compute the grid-point values
of P directly from the grid-point values of w. A tridiagonal system for the gridpoint values of P along the top boundary is obtained when the second derivatives
in (8.61) are approximated by a standard three-point finite difference. Special conditions are required at those points adjacent to the lateral boundaries, where it can
be advantageous to use the less accurate approximation
This relation, which is derived from (8.55) using the approximation Ikl b) +
bzk z, does not require any assumption about the horizontal variation of W at the
lateral boundaries. Some assumption is nevertheless required about the variation
of P near the boundary, and satisfactory results have been obtained by setting
o P j ox to zero in the upper corners of the domain.
8.4 Wave-Absorbing Layers
One way to prevent outward-propagating disturbances from reftecting back into
the domain when they encounter the boundary is to place a wave-absorbing layer
at the edge of the domain. Wave-absorbing layers are conceptually simple and are
particularly attractive in applications for which appropriate radiation boundary
conditions have not been determined. Wave-absorbing layers also allow "largescale" time tendencies to be easily imposed at the lateral boundaries of the domain . These large-scale tendencies might be generated by a previous or concurrent
coarse-resolution simulation on a larger spatial domain. The chief disadvantage
of the absorbing-layer approach is that the absorber often needs to be rather thick
in order to be effective, and significant computational effort may be required to
compute the solution on the mesh points within a thick absorbing layer. Considerable engineering rnay also be required to ensure that a wave-absorbing layer
performs adequately in a given application.
Suppose that a radiation upper boundary condition for the two-dimensional linearized Boussinesq equations (8.37)-(8.40) is to be approximated using a waveabsorbing layer of thickness D. This absorbing layer can be created by defining a vertically varying viscosity a(z) and adding viscous terms of the form
a(z)oZujox Z, a(z)o2wjox 2, and a(z)o2bjox 2 to the right sides of (8.37), (8.38),
and (8.39), respectively. The viscosity is zero in the region z :s H within which
the solution is to be accurately approximated and increases gradually with height
throughout the layer H < z :s H + D . A simple rigid-lid condition, w = 0,
can be imposed at the top of the absorbing layer. The performance of this ab-