8.3 Two-DimensionalStratified Flow
425
way wave equation of the form
(
8
_ 8 )
-+e- 8t
8z
"" =0,
(8.51)
In order to avoid reflections from the upper boundary, the value of cused in this
one-way wave equation must be the trace speed for a mode whose group velocity is directed upward . According to (8.49) and (8.50) , any mode with a positive
group velocity will have a negative trace speed, so all the acceptable values for c
are negative. Yet only one-sided numerical approximations to the spatial derivative can be evaluated at the upper boundary, and these one-sided differences yield
unstable finite-difference approximations to (8.51) whenever c < 0, because the
numerical domain of dependence does not include the domain of dependence of
the true solution.
Boundary Conditions That Are Nonloeal in Space, but Loeal in Time
The problems with the one-way wave equation (8.51) can be avoided by following the strategy of Klemp and Durran (1983) and Bougeault (1983), in which a
radiation upper boundary condition is imposed through a diagnostic relationship
between the pressure and the vertical velocity at the top boundary of the domain .
This relationship is derived by substituting a wave of the form (8.41) into the
horizontal momentum equation (8.37) to yield
- i (w - Uk)u + ikP = O.
Substituting the same wave into the continuity equation (8.40), one obtains
(8.52)
iku + i mii: = O.
Eliminating ubetween the two preceding equations gives
(8.53)
After making the hydrostatic approximation, the dispersion relation (8.42) can be
written
w-Uk
N
- - - = ± -.
k
m
The vertical group velocity for hydrostatic waves, which is given by (8.50), will
depend on both the sign of k and the choice of the positive or negative root. The
choice of sign required to limit the dispersion relation to those waves with upward
group velocity is
w-Uk
k
N
= - sgn(k)m '
Substituting the preceding into (8.53) yields
(8.54)
N
p=-w.
A
Ikl
(8.55)
425
way wave equation of the form
(
8
_ 8 )
-+e- 8t
8z
"" =0,
(8.51)
In order to avoid reflections from the upper boundary, the value of cused in this
one-way wave equation must be the trace speed for a mode whose group velocity is directed upward . According to (8.49) and (8.50) , any mode with a positive
group velocity will have a negative trace speed, so all the acceptable values for c
are negative. Yet only one-sided numerical approximations to the spatial derivative can be evaluated at the upper boundary, and these one-sided differences yield
unstable finite-difference approximations to (8.51) whenever c < 0, because the
numerical domain of dependence does not include the domain of dependence of
the true solution.
Boundary Conditions That Are Nonloeal in Space, but Loeal in Time
The problems with the one-way wave equation (8.51) can be avoided by following the strategy of Klemp and Durran (1983) and Bougeault (1983), in which a
radiation upper boundary condition is imposed through a diagnostic relationship
between the pressure and the vertical velocity at the top boundary of the domain .
This relationship is derived by substituting a wave of the form (8.41) into the
horizontal momentum equation (8.37) to yield
- i (w - Uk)u + ikP = O.
Substituting the same wave into the continuity equation (8.40), one obtains
(8.52)
iku + i mii: = O.
Eliminating ubetween the two preceding equations gives
(8.53)
After making the hydrostatic approximation, the dispersion relation (8.42) can be
written
w-Uk
N
- - - = ± -.
k
m
The vertical group velocity for hydrostatic waves, which is given by (8.50), will
depend on both the sign of k and the choice of the positive or negative root. The
choice of sign required to limit the dispersion relation to those waves with upward
group velocity is
w-Uk
k
N
= - sgn(k)m '
Substituting the preceding into (8.53) yields
(8.54)
N
p=-w.
A
Ikl
(8.55)