7.3 Fractional-Step Methods
369
-10
x (km )
x (km)
15
FIGURE 7.5. (a) contours of U + u at intervals ofO.l ms- I and ljI at intervals ofO.1 s-I
at t = 8000 s. (b) as in (a) except that P is contoured at intervals of 0.5 m 2s-2 •
ing partially split approximation include divergence damping and forward biasing
the trapezoidal integral of the vertical derivative terms (7.83) and (7.85). Forward
biasing the trapezoidal integration is accomplished without additional computational effort by replacing those terms of the form ( 2
(7.86)
where
where 0 :::: E :::: I. A value of E = 0.2 provides an effective filter that does not
noticeably modify the gravity waves (Durran and Klemp 1983).
Since trapezoidal time-differencing is used only to approximate the vertical
derivatives, forward biasing those derivatives will not damp horizontally propagating sound waves. Skamarock and Klemp (1992)
recommended including a"divergence damper"
in the momentum equations such that the system of equations that is integrated on the small time step becomes
e« ap
es
at
- + - - a
ax x ax
-=F",
u,
aw ap
es
- + - - b - a - = F
at az
zaz
w,
ab
2
at
-+N W=Fb,
ap 2
-
at +c s 8 = F p ,
au aw
ax az