7.2 The Semi-implicit Method
355
whenever
(IUIK + N)l:i.t < I,
which is the same condition obtained for the stability of the gravity waves using
leapfrog differencing. Thus, as suggested previously, the semi-implicit scheme
allows the compressible equations goveming low-Mach-number ftow to be integrated with a much larger time step than that allowed by fully explicit schemes.
This increase in efficiency comes at a price; whenever the time step is much larger
than that allowed by the CFL condition for sound waves, the sound waves are artificially decelerated by a factor of cos(wl:i.t). This error is directly analogous to
that considered in Section 7.2.1, in which spurious decelerations were produced
by fully implicit schemes using very large time steps. Nevertheless, in many practical applications the errors in the sound waves are of no consequence, and the
quality of the solution is entirely determined by the accuracy with which the
slower-moving waves are approximated.
How does semi-implicit differencing inftuence the accuracy of the gravity-wave
modes? The only inftuence is exerted through the reduction in the speed of sound
in the third term in the denominator of (7.53) . This term is generally small and
has little effect on the gravity waves unless wl:i.t is far from zero and the waves
are sufficiently long that Ikl + III N fcs . It is actually rather hard to satisfy both
of these requirements simultaneously. First, since wl:i.t 1 for the stability of the
mode in question, Cs can never drop below 0.54c s . Second, the maximum value
of l:i.t is limited by the frequency of the most rapidly moving wave Wrn . In most
applications the frequencies of the long waves are much lower than w m , so for all
the long waves, wl:i.t « Wrnl:i.t I , and thus C s cs . As a example where the
deviation of Cs from Cs is maximized, consider a basic state with N = 0.02 s-I,
C s = 318 ms:", and U = 0, together with the mode (k, l) = (N fcs , O.IN fcs)
and time steps in the range 0
l:i.t
1f N . The approximate solution obtained
using leapfrog time-differencing exhibits an accelerative phase error that reaches
11% when l:i.t = 1f N . This accelerative phase-speed error is reduced by the semiimplicit method to a -5.7% decelerative error when N = l:i.t. The wave in this
example is a rather pathological mode with horizontal and vertical wave1engths
of 100 and 1000 km, respective1y. The difference between the leapfrog and semiimplicit gravity-wave solutions is much smaller in most realistic examp1es.
The semi-implicit differencing scheme (7.48)-(7.51) provides a way to circumvent the CFL stability criterion for sound-wave propagation without losing
accuracy in simulation of the gravity-wave modes . In global -scale atmospheric
models the gravity waves may actually be of minor physical significance, and
the features of primary interest may evolve on an even slower time scale.f If the
fastest-moving gravity-wave modes do not need to be accurately represented, it
is possible to generalize the preceding semi-implicit scheme to allow even larger
5In particular, the most important features may consist of slow-moving Rossby waves, which appear as additional solutions to the Euler equations when latitudinal variations in the Coriolis force are
included in the horizontal-momentum equation s.
355
whenever
(IUIK + N)l:i.t < I,
which is the same condition obtained for the stability of the gravity waves using
leapfrog differencing. Thus, as suggested previously, the semi-implicit scheme
allows the compressible equations goveming low-Mach-number ftow to be integrated with a much larger time step than that allowed by fully explicit schemes.
This increase in efficiency comes at a price; whenever the time step is much larger
than that allowed by the CFL condition for sound waves, the sound waves are artificially decelerated by a factor of cos(wl:i.t). This error is directly analogous to
that considered in Section 7.2.1, in which spurious decelerations were produced
by fully implicit schemes using very large time steps. Nevertheless, in many practical applications the errors in the sound waves are of no consequence, and the
quality of the solution is entirely determined by the accuracy with which the
slower-moving waves are approximated.
How does semi-implicit differencing inftuence the accuracy of the gravity-wave
modes? The only inftuence is exerted through the reduction in the speed of sound
in the third term in the denominator of (7.53) . This term is generally small and
has little effect on the gravity waves unless wl:i.t is far from zero and the waves
are sufficiently long that Ikl + III N fcs . It is actually rather hard to satisfy both
of these requirements simultaneously. First, since wl:i.t 1 for the stability of the
mode in question, Cs can never drop below 0.54c s . Second, the maximum value
of l:i.t is limited by the frequency of the most rapidly moving wave Wrn . In most
applications the frequencies of the long waves are much lower than w m , so for all
the long waves, wl:i.t « Wrnl:i.t I , and thus C s cs . As a example where the
deviation of Cs from Cs is maximized, consider a basic state with N = 0.02 s-I,
C s = 318 ms:", and U = 0, together with the mode (k, l) = (N fcs , O.IN fcs)
and time steps in the range 0
l:i.t
1f N . The approximate solution obtained
using leapfrog time-differencing exhibits an accelerative phase error that reaches
11% when l:i.t = 1f N . This accelerative phase-speed error is reduced by the semiimplicit method to a -5.7% decelerative error when N = l:i.t. The wave in this
example is a rather pathological mode with horizontal and vertical wave1engths
of 100 and 1000 km, respective1y. The difference between the leapfrog and semiimplicit gravity-wave solutions is much smaller in most realistic examp1es.
The semi-implicit differencing scheme (7.48)-(7.51) provides a way to circumvent the CFL stability criterion for sound-wave propagation without losing
accuracy in simulation of the gravity-wave modes . In global -scale atmospheric
models the gravity waves may actually be of minor physical significance, and
the features of primary interest may evolve on an even slower time scale.f If the
fastest-moving gravity-wave modes do not need to be accurately represented, it
is possible to generalize the preceding semi-implicit scheme to allow even larger
5In particular, the most important features may consist of slow-moving Rossby waves, which appear as additional solutions to the Euler equations when latitudinal variations in the Coriolis force are
included in the horizontal-momentum equation s.