Advanced Research WRF (ARW) Technical Note - MMM - University ...

4.2 Filters for the Time-split RK3 scheme
Three filters are used in the ARW time-split RK3 scheme apart from those in the model physics:
three-dimensional divergence damping (an acoustic model filter); an external-mode filter that
damps vertically-integrated horizontal divergence; and off-centering of the vertically implicit
integration of the vertical momentum equation and geopoential equation. Each of these is
described in the following sections.
4.2.1 Divergence Damping
The damping of the full mass divergence is a filter for acoustic modes in the ARW solver. This
3D mass divergence damping is described in Skamarock and Klemp (1992). The filtering is
accomplished by forward weighting the pressure update in the acoustic step loop described in
Section 3.1.3, step (6). The linearized equation of state (3.5) is used to diagnose the pressure at
the new time τ after the U ′′ , V ′′ , µ ′′
d , and Θ′′ have been advanced. Divergence damping consists
of using a modified pressure in the computation of the pressure gradient terms in the horizontal
momentum equations in the acoustic steps (Equations (3.7) and (3.8)). Denoting the updated
value as p τ , the modified pressure p ∗τ used in (3.7) and (3.8) can be written as
p ∗τ = p τ + γd
p τ − p τ−∆τ , (4.8)
where γd is the damping coefficient. This modification is equivalent to inserting a horizontal
diffusion term into the equation for the 3D mass divergence, hence the name divergence damping.
A divergence damping coefficient γd = 0.1 is typically used in the ARW applications, independent
of time step or grid size.
4.2.2 External Mode Filter
The external modes in the solution are damped by filtering the vertically-integrated horizontal
divergence. This damping is accomplished by adding a term to the horizontal momentum
equations. The additional term added to (3.7) and (3.8) are
and
δτU ′′ = ... − γeδx(δτ−∆τµ ′′
d) (4.9)
δτV ′′ = ... − γeδy(δτ−∆τµ ′′
d). (4.10)
The quantity δτ−∆τµ is the vertically-integrated mass divergence (i.e., (3.20)) from the previous
acoustic step (that is computed using the time τ values of U and V ), and γe is the external
mode damping coefficient. An external mode damping coefficient γe = 0.01 is typically used in
the ARW applications, independent of time step or grid size.
4.2.3 Semi-Implicit Acoustic Step Off-centering
Forward-in-time weighting of the vertically-implicit acoustic-time-step terms damps instabilities
associated vertically-propagating sound waves. The forward weighting also damps instabilities
associated with sloping mode levels and horizontally propagating sound waves (see Durran and
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