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

transformation is isotropic. Anisotropic transformations, such as a latitude-longitude grid, can
be accommodated by defining separate map factors for the x and y transformations.
In the ARW’s computational space, ∆x and ∆y are constants. Orthogonal projections to
the sphere require that the physical distances between grid points in the projection vary with
position in the grid. To transform the governing equations, a map scale factor m is defined as
the ratio of the distance in computational space to the corresponding distance on the earth’s
surface:
(∆x, ∆y)
m =
. (2.22)
distance on the earth
The ARW solver includes the map-scale factors in the governing equations by redefining the
momentum variables as
U = µdu/m, V = µdv/m, W = µdw/m, Ω = µd ˙η/m.
Using these redefined momentum variables, the governing equations, including map factors and
rotational terms, can be written as
∂tU + m[∂x(Uu) + ∂y(V u)] + ∂η(Ωu) + µdα∂xp + (α/αd)∂ηp∂xφ = FU
∂tV + m[∂x(Uv) + ∂y(V v)] + ∂η(Ωv) + µdα∂yp + (α/αd)∂ηp∂yφ = FV
∂tW + m[∂x(Uw) + ∂y(V w)] + ∂η(Ωw) − m −1 g[(α/αd)∂ηp − µd] = FW
∂tΘ + m 2 [∂x(Uθ) + ∂y(V θ)] + m∂η(Ωθ) = FΘ
(2.23)
(2.24)
(2.25)
(2.26)
∂tµd + m 2 [Ux + Vy] + m∂η(Ω) = 0 (2.27)
∂tφ + µ −1
d [m2 (Uφx + V φy) + mΩφη − gW ] = 0 (2.28)
∂tQm + m 2 [∂x(Uqm) + ∂y(V qm)] + m∂η(Ωqm) = FQm, (2.29)
and, for completeness, the diagnostic relation for the dry inverse density
and the diagnostic equation for full pressure (vapor plus dry air)
∂ηφ = −αdµd, (2.30)
p = p0(Rdθm/p0αd) γ . (2.31)
The right-hand-side terms of the momentum equations (2.23) – (2.25) contain the Coriolis
and curvature terms along with mixing terms and physical forcings. The Coriolis and curvature
terms can be written as follows:
FUcor = + f + u ∂m
∂y
FVcor = − f + u ∂m
∂y
∂m
− v
∂x
− v ∂m
∂x
V − eW cos αr − uW
re
U + eW sin αr − vW
uU + vV
FWcor = +e(U cos αr − V sin αr) +
re
re
(2.32)
(2.33)
, (2.34)
where αr is the local rotation angle between the y-axis and the meridians, ψ is the latitude,
f = 2Ωe sin ψ, e = 2Ωe cos ψ, Ωe is the angular rotation rate of the earth, and re is the radius of
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