370
7. Physically Insignificant Fast Waves
and F u , F w , Fb, and F p represent the forcing terms that are updated every l:i.t.
Damping coefficients of «, = 0.001(l:i.x)2/1:i.t" and a z = 0.001 (I:i.Z)2/1:i.t" removed all trace of noise and instability in the test problem shown in Fig. 7.5
without a supplemental Asselin filter.
The role played by divergence damping in stabilizing the small-time-step integration in the partial-splitting method can be appreciated by noting that if a single
damping coefficient a is used in all components of the momentum equation, the
divergence satisfies
-
a8
at
+ V
2P - aV
28 = G,
(7.87)
where G = -V· (v· Vv) + ab/az. Eliminating the pressure between (7.86) and
(7.87), one obtains
28
2 a8
at
a
2 - aV at -
2 2
2
aG
C s V 8 = fit - V Fp ,
The forcing on the right side of this equation will tend to produce divergence
-
mogeneous part of this equation, one obtains the classic equation for a damped
d 8
dt 2 + a«
2 d 8
di + CsK
2 28 = 0,
(7.88)
harmonie oscillator:
where 8(t) is the amplitude and K = ../k 2 + (.2 . The damping increases with wave
number and is particularly effective in eliminating the high-wave-number modes
at which the instability in the partial-splitting method occurs. Gravity waves, on
the other hand, are not significantly impacted by the divergence damper because
the velocity field in internal gravity waves is almost nondivergent. Skamarock
and Klemp (1992) have shown that divergence damping only slightly reduces the
amplitude of the gravity waves.
At this point it might appear that the partial-splitting approach is inferior to
the complete-splitting method considered previously, since filters are required to
stabilize the partially split approximation in situations where the completely split
sehe me performs quite nicely. Recall, however, that the completely split method
does not generate usable solutions to the compressible Boussinesq equations when
there is a vertical shear in the basic-state horizontal velocity impinging on the
gravity-wave generator. The same filtering strategies that stabilize the partially
split method in the no-shear problem remain effective in the presence of vertical
wind shear. This is demonstrated in Fig. 7.3d, which shows the pressure perturbations in the test case with vertical shear as computed by the partially split method
using a divergence damper with the values ofa x and a z given previously. Results
similar to those in Fig. 7.3d mayaIso be obtained using Asselin time filtering
with a = 0.1 in lieu of the divergence damper. The advantages of the partial
splitting method are not connected with its performance in the simplest test cases,
for which it can indeed be inferior to a completely split approximation, but in its
adaptability to more complex problems.
7. Physically Insignificant Fast Waves
and F u , F w , Fb, and F p represent the forcing terms that are updated every l:i.t.
Damping coefficients of «, = 0.001(l:i.x)2/1:i.t" and a z = 0.001 (I:i.Z)2/1:i.t" removed all trace of noise and instability in the test problem shown in Fig. 7.5
without a supplemental Asselin filter.
The role played by divergence damping in stabilizing the small-time-step integration in the partial-splitting method can be appreciated by noting that if a single
damping coefficient a is used in all components of the momentum equation, the
divergence satisfies
-
a8
at
+ V
2P - aV
28 = G,
(7.87)
where G = -V· (v· Vv) + ab/az. Eliminating the pressure between (7.86) and
(7.87), one obtains
28
2 a8
at
a
2 - aV at -
2 2
2
aG
C s V 8 = fit - V Fp ,
The forcing on the right side of this equation will tend to produce divergence
-
mogeneous part of this equation, one obtains the classic equation for a damped
d 8
dt 2 + a«
2 d 8
di + CsK
2 28 = 0,
(7.88)
harmonie oscillator:
where 8(t) is the amplitude and K = ../k 2 + (.2 . The damping increases with wave
number and is particularly effective in eliminating the high-wave-number modes
at which the instability in the partial-splitting method occurs. Gravity waves, on
the other hand, are not significantly impacted by the divergence damper because
the velocity field in internal gravity waves is almost nondivergent. Skamarock
and Klemp (1992) have shown that divergence damping only slightly reduces the
amplitude of the gravity waves.
At this point it might appear that the partial-splitting approach is inferior to
the complete-splitting method considered previously, since filters are required to
stabilize the partially split approximation in situations where the completely split
sehe me performs quite nicely. Recall, however, that the completely split method
does not generate usable solutions to the compressible Boussinesq equations when
there is a vertical shear in the basic-state horizontal velocity impinging on the
gravity-wave generator. The same filtering strategies that stabilize the partially
split method in the no-shear problem remain effective in the presence of vertical
wind shear. This is demonstrated in Fig. 7.3d, which shows the pressure perturbations in the test case with vertical shear as computed by the partially split method
using a divergence damper with the values ofa x and a z given previously. Results
similar to those in Fig. 7.3d mayaIso be obtained using Asselin time filtering
with a = 0.1 in lieu of the divergence damper. The advantages of the partial
splitting method are not connected with its performance in the simplest test cases,
for which it can indeed be inferior to a completely split approximation, but in its
adaptability to more complex problems.