422
8. NonreflectingBoundary Conditions
space-time grid to determine both Cand a1/l/at. Moreover, (8.45) has no apriori
validity in the continuously stratified problem, since there is generally no correct
value of Cto diagnose via (8.46).
Durran et al. (1993) investigated the effectiveness with which (8.46) diagnosed
the phase velocity of numerically simulated shallow-water waves, for which Chas
the well-defined value of JgH. Except during the passage of a trough or crest,
a simple finite-difference approximation to (8.46) proved capable of diagnosing
a reasonable approximation to JiH in a control simulation on a very large periodic domain in which all waves propagating past the point of the calculation were
really traveling in the positive x direction. Attempts to perform the same diagnostic calculation for cin conjunction with the imposition of a radiation boundary
condition in a second simulation were, however, a complete failure. Small errors
in the initial diagnosis of cgenerated weak reflected waves. These reflected waves
generated increasing errors in the calculation of cbecause (8.45) does not apply at
locations where both rightward- and leftward-propagating waves are present. The
additional error in cincreased the amplitude of the spurious reflected waves and
induced a positive feedback that rapidly destroyed the reliability of the ccalculation. The majority of the values computed from (8.46) were outside the stability
limits for the upstream method and had to be reset to either zero or I1t/ I1x. Durran et al. (1993) also considered tests in which the physical system supported
several different modes moving at different trace speeds and concluded that it is
best to use a fixed C.
When the mean horizontal wind is not zero, cis replaced by the Doppler-shifted
trace speeds V +cand V - cat the upstream and downstream boundaries, respectively. One should also ensure that these Doppler-shifted trace speeds are actually
directed out of the domain by choosing Icl > lVI at the inflow boundary. In
some applications the dominant upstream-propagating mode may have a different intrinsic trace speed from the dominant downstream-propagating mode , and
in such circumstances it can be useful to specify the Doppler-shifted trace speeds
as V + CI and V - C2 without requiring CI = C2 .
Higdon (1994) has suggested that an improved approximation to the radiation
boundary condition for dispersive waves can be obtained by replacing the basic
one-way wave equation (8.45) with the product of aseries of one-way operators
ofthe form
[ fr
j = 1 Bt
ax
1/1 = 0,
(8.47)
where the set of cj is chosen to span the range of x trace speeds associated with
the waves appearing at x = L. The preceding is a generalization of Higdon's radiation boundary condition for shallow-water waves (8.35). Although this scheme
does not seem to have been used as a boundary condition for limited-area models
of stably stratified flow, Higdon has successfully used to it simulate dispersive
shallow-water waves on an f -plane.
The spurious reflection generated by the Higdon boundary condition can be
determined by considering a unit-amplitude wave striking the boundary at x = L.
8. NonreflectingBoundary Conditions
space-time grid to determine both Cand a1/l/at. Moreover, (8.45) has no apriori
validity in the continuously stratified problem, since there is generally no correct
value of Cto diagnose via (8.46).
Durran et al. (1993) investigated the effectiveness with which (8.46) diagnosed
the phase velocity of numerically simulated shallow-water waves, for which Chas
the well-defined value of JgH. Except during the passage of a trough or crest,
a simple finite-difference approximation to (8.46) proved capable of diagnosing
a reasonable approximation to JiH in a control simulation on a very large periodic domain in which all waves propagating past the point of the calculation were
really traveling in the positive x direction. Attempts to perform the same diagnostic calculation for cin conjunction with the imposition of a radiation boundary
condition in a second simulation were, however, a complete failure. Small errors
in the initial diagnosis of cgenerated weak reflected waves. These reflected waves
generated increasing errors in the calculation of cbecause (8.45) does not apply at
locations where both rightward- and leftward-propagating waves are present. The
additional error in cincreased the amplitude of the spurious reflected waves and
induced a positive feedback that rapidly destroyed the reliability of the ccalculation. The majority of the values computed from (8.46) were outside the stability
limits for the upstream method and had to be reset to either zero or I1t/ I1x. Durran et al. (1993) also considered tests in which the physical system supported
several different modes moving at different trace speeds and concluded that it is
best to use a fixed C.
When the mean horizontal wind is not zero, cis replaced by the Doppler-shifted
trace speeds V +cand V - cat the upstream and downstream boundaries, respectively. One should also ensure that these Doppler-shifted trace speeds are actually
directed out of the domain by choosing Icl > lVI at the inflow boundary. In
some applications the dominant upstream-propagating mode may have a different intrinsic trace speed from the dominant downstream-propagating mode , and
in such circumstances it can be useful to specify the Doppler-shifted trace speeds
as V + CI and V - C2 without requiring CI = C2 .
Higdon (1994) has suggested that an improved approximation to the radiation
boundary condition for dispersive waves can be obtained by replacing the basic
one-way wave equation (8.45) with the product of aseries of one-way operators
ofthe form
[ fr
j = 1 Bt
ax
1/1 = 0,
(8.47)
where the set of cj is chosen to span the range of x trace speeds associated with
the waves appearing at x = L. The preceding is a generalization of Higdon's radiation boundary condition for shallow-water waves (8.35). Although this scheme
does not seem to have been used as a boundary condition for limited-area models
of stably stratified flow, Higdon has successfully used to it simulate dispersive
shallow-water waves on an f -plane.
The spurious reflection generated by the Higdon boundary condition can be
determined by considering a unit-amplitude wave striking the boundary at x = L.