8.3 Two-Dimensional Stratified Flow
423
In order to satisfy (8.47) exactly, a reflected wave of amplitude r must also be
present, in which case the total solution may be expressed as
1fr(x, Z, t) = ei(kx+mz-wt) + rei(kx+mz-iiJt) .
The frequencies and vertical wave numbers of the incident and reflected waves
must be identical, or the preceding expression will not satisfy (8.47) for arbitrary
values of Z and t . Since m= m and w= w, the dispersion relation implies that
k = -k. Using these results to substitute
1fr(x, Z, t) = ei(kx+mz-wt) + rei(-kx+mz-wt)
into (8.47) and defining C= co] k, one obtains
Irl=TI IC-Cjl
p
- .
j=1 c+Cj
If the waves are hydrostatic, C = N Im, and the reflection coefficient may be
alternatively expressed as
IrI = TI Im - mI,
(8.48)
j = 1 m +mj
where m j is the vertical wave number of a hydrostatic wave moving parallel to the
x-axis at speed Cj. The reflections generated by the Higdon boundary condition
are plotted as a function of the vertical wave number of the incident wave in
Fig. 8.6. The waves are assumed hydrostatic, so that the reflection coefficient
is determined by (8.48) . Perfeet transmission is achieved whenever the vertical
wave number of the incident wave matches one of the mi- When the perfectly
transmitted waves in the two-operator scheme are chosen such that m2 = 3m I,
the range of vertical wave numbers that are transmitted with minimal reflection
is much larger than that obtained using the standard one-way wave equation. The
range of wave numbers that undergo minimal spurious reflection can be further
increased by using the three-operator scheme, which was configured such that
m2 = 3m I and m3 = 9m I in the example plotted in Fig. 8.6.
The Higdon boundary condition can be implemented in the numerical approximations to the momentum and buoyancy equations (8.37)-(8.39) using the finitedifference formula (8.36). The numerical formulation of the boundary condition
for pressure is less obvious, because (8.36) is implicit whenever p :::: 2. This implicitness need not reduce the efficiency of the numerical integration when the
solution in the interior can be updated prior to the evaluation of the boundary
condition, as will be the case if (8.37)-(8.39) are approximated using explicit
time-differencing. The pressure, however, is evaluated by solving a Poisson equation, and the derivation of a stable pressure boundary condition for the Poisson
equation from (8.36) may be nontrivial. An alternative approach for formulating
the pressure boundary condition is investigated in Problems 8 and 9.
j
423
In order to satisfy (8.47) exactly, a reflected wave of amplitude r must also be
present, in which case the total solution may be expressed as
1fr(x, Z, t) = ei(kx+mz-wt) + rei(kx+mz-iiJt) .
The frequencies and vertical wave numbers of the incident and reflected waves
must be identical, or the preceding expression will not satisfy (8.47) for arbitrary
values of Z and t . Since m= m and w= w, the dispersion relation implies that
k = -k. Using these results to substitute
1fr(x, Z, t) = ei(kx+mz-wt) + rei(-kx+mz-wt)
into (8.47) and defining C= co] k, one obtains
Irl=TI IC-Cjl
p
- .
j=1 c+Cj
If the waves are hydrostatic, C = N Im, and the reflection coefficient may be
alternatively expressed as
IrI = TI Im - mI,
(8.48)
j = 1 m +mj
where m j is the vertical wave number of a hydrostatic wave moving parallel to the
x-axis at speed Cj. The reflections generated by the Higdon boundary condition
are plotted as a function of the vertical wave number of the incident wave in
Fig. 8.6. The waves are assumed hydrostatic, so that the reflection coefficient
is determined by (8.48) . Perfeet transmission is achieved whenever the vertical
wave number of the incident wave matches one of the mi- When the perfectly
transmitted waves in the two-operator scheme are chosen such that m2 = 3m I,
the range of vertical wave numbers that are transmitted with minimal reflection
is much larger than that obtained using the standard one-way wave equation. The
range of wave numbers that undergo minimal spurious reflection can be further
increased by using the three-operator scheme, which was configured such that
m2 = 3m I and m3 = 9m I in the example plotted in Fig. 8.6.
The Higdon boundary condition can be implemented in the numerical approximations to the momentum and buoyancy equations (8.37)-(8.39) using the finitedifference formula (8.36). The numerical formulation of the boundary condition
for pressure is less obvious, because (8.36) is implicit whenever p :::: 2. This implicitness need not reduce the efficiency of the numerical integration when the
solution in the interior can be updated prior to the evaluation of the boundary
condition, as will be the case if (8.37)-(8.39) are approximated using explicit
time-differencing. The pressure, however, is evaluated by solving a Poisson equation, and the derivation of a stable pressure boundary condition for the Poisson
equation from (8.36) may be nontrivial. An alternative approach for formulating
the pressure boundary condition is investigated in Problems 8 and 9.
j