424
8. Nonreftecting Boundary Conditions
1.0
-0.2
0.8
[r ]
0 .6
0.4
0 L-L ---'--- _ _L -
- ' -
.J-.I
o m /3
m
3m
6m
Incidcnt Wavc numbcr
FIGURE 8.6. Reftection coefficients generated by the Higdon lateral boundary condition
(8.47) as a function of the incident vertical wave number fOT p equal to one (dotted curve),
two (dashed) , and three (solid). The waves are assumed to be hydrostatic.
8.3.2 Upper Boundary Conditions
Open upper or lower boundary conditions for the incompressible Boussinesq
equations cannot be formulated in the same manner as the lateral boundary conditions because the vertical trace speed of an internal gravity wave is frequently
directed opposite to its vertical group velocity. The difference in the direction of
the vertical trace speed and group velocity is particularly evident when the waves
are hydrostatic and U = O. In such circumstances the dispersion relation simpli -
fies to w = ±Nk/m; the vertical trace speed is
w
Nk
m
-=±z' m
(8.49)
and the vertical group velocity is
8w
Nk
-8 ==f z '
(8.50)
m
m
which is equal in magnitude to the trace speed but opposite in sign .
Let us attempt to follow the same procedure used in the preceding section to
obtain an approximate open boundary condition at the upper boundary. As a first
step, the correct dispersion relation is approximated as io = cm, where c is a
constant equal to the estimated vertical trace speed of the dominant mode in the
physical system. The relat ion t» = cm is the exact dispersion relation for a one-