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

space via the control variable transform U, i.e.,
x ′ = Uv = UpUvUhv. (9.2)
The expansion U = UpUvUh represents the various stages of covariance modeling: horizontal
correlations Uh, vertical covariances Uv, and multivariate covariances Up.
The components of v are chosen so that their error cross-correlations are negligible, thus
permitting the matrix B to be block-diagonalized. The many varying applications (high/low
resolution, polar/tropical, etc.) of WRF-Var require a flexible choice of background error model.
This is achieved via a namelist option “cv options” as defined in Table 9.1.
cv options
Analysis
Increment
Change of
Variable
Vertical
Covariances
Horizontal
Correlations
Control
Variables
2 3 4 5
(original MM5) (NCEP) (Global) (Regional)
x ′ u ′ ,v ′ ,T ′ ,q ′ ,ps ′ (i, j, k)
Up ψ ′ ,χ ′ ,p ′ u,q ′ ψ ′ ,χ ′ u,T ′ u,r ′ ,p ′ su
Uv B = EΛE T RF B = EΛE T
Uh RF Spectral RF
v v(i, j, m) v(i, j, k) v(l, n, m) v(i, j, m)
Table 9.1: The definitions of the various stages of the control variable transform given by (9.2) for
the unified global/regional WRF-Var system. Indices (i, j, k) refer to grid-point space, index m
to vertical mode, and l, n to global spectral mode. The variables are: u, v: velocity components;
T : temperature; q: mixing ratio; ps: surface pressure; ψ: streamfunction; χ: velocity potential;
r: relative humidity. The subscript u indicates an unbalanced field. The acronym RF stands
for recursive filter.
Table 9.1 indicates that the only difference between global (cv options=4) and WRF regional
(cv options=5) versions of the WRF-Var control variable transform is in the horizontal error
correlations Uh. Note also, the only difference between the old MM5 background error model
(cv options=2) and WRF regional (cv options=5) is in the Up transform. The former imposes a
dynamical balance constraint via an unbalanced pressure control variable (Barker et al., 2004),
whereas in the new regional covariance model, balance is imposed via statistical regression (see
Section 9.3 for details). This choice of control variables is considered more appropriate for the
mass-based ARW solver.
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