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

The operator vertically interpolates variables on mass levels k to the w levels (k + 1).
It should
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be noted that the vertical grid is defined such that vertical interpolation from w levels to mass
levels reduces to a η
k = (ak+1/2 + ak−1/2)/2 (see Fig. 3.2).
The RHS terms in the discrete acoustic step equations for momentum (3.21), (3.22) and
(3.25) are discretized as
R t∗
U = − (µd x α x δxp ′ − µd x α ′xδx¯p) − (α/αd) x
(µd x δxφ ′η − δηp ′xηδxφ
η + µ ′ x
d δxφ η )
+ FUcor + advection + mixing + physics (3.29)
R t∗
V = − (µd y α y δyp ′ − µd y α ′yδy ¯p) − (α/αd) y
(µd y δyφ ′η − δηp ′yηδyφ
η + µ ′ y
d δyφ η )
+ FVcor + advection + mixing + physics
R
(3.30)
t∗
W = m −1 g(α/αd) η
[δηp ′ + ¯µdqm η ] − m −1 µ ′ dg
+ FWcor + advection + mixing + buoyancy + physics. (3.31)
3.2.2 Coriolis and Curvature Terms
The terms FUcor, FVcor, and FWcor in (3.29) – (3.31) represent Coriolis and curvature effects
in the equations. These terms in continuous form are given in (2.32) – (2.34). Their spatial
discretization is
Here the operators () xy
3.2.3 Advection
FUcor = + f x + uxδym − vyδxm x xy x xη
V − e W cos αr x xη
uW
−
FVcor = − f y + uxδym − vyδxm y xy y yη y vW
U + e W sin αr − yη
re
FWcor = +e(U xη cos αr − V yη xη xη yη yη
u U + v V
sin αr) +
.
= () xy
, and likewise for () xη
and () yη
.
The advection terms in the ARW solver are in the form of a flux divergence and are a subset of
the RHS terms in equations (3.13) – (3.18):
R t∗
Uadv = − m[∂x(Uu) + ∂y(V u)] + ∂η(Ωu) (3.32)
R t∗
Vadv = − m[∂x(Uv) + ∂y(V v)] + ∂η(Ωv) (3.33)
R t∗
µadv = − m2 [Ux + Vy] + mΩη (3.34)
R t∗
Θadv = − m2 [∂x(Uθ) + ∂y(V θ)] − m∂η(Ωθ) (3.35)
R t∗
Wadv = − m[∂x(Uw) + ∂y(V w)] + ∂η(Ωw) (3.36)
R t∗
φadv
re
= − µ−1
d [m2 (Uφx + V φy) + mΩφη]. (3.37)
For the mass conservation equation, the flux divergence is discretized using a 2nd-order centered
approximation:
R t∗
µadv = −m2 [δxU + δyV ] t∗
+ mδηΩ t∗
. (3.38)
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re