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

and lead to the acoustic time-step equations:
δτU ′′ + µ t∗
α t∗
∂xp ′′τ t
+ (µ ∗
∂x¯p)α ′′τ
+ (α/αd)[µ t∗
∂xφ ′′τ
+ (∂xφ t∗
)(∂ηp ′′ − µ ′′ ) τ ] = RU t∗
δτV ′′ + µ t∗
α t∗
∂yp ′′τ t
+ (µ ∗
∂y ¯p)α ′′τ
+ (α/αd)[µ t∗
∂yφ ′′τ
+ (∂yφ t∗
)(∂ηp ′′ − µ ′′ ) τ ] = RV t∗
δτΘ ′′ + m 2 [∂x(U ′′ θ t∗
δτW ′′ − m −1 g
δτµ ′′
d + m 2 [∂xU ′′ + ∂yV ′′ ] τ+∆τ + m∂ηΩ ′′τ+∆τ = Rµ t∗
) + ∂y(V ′′ θ t∗
(α/αd) t∗∂η(C∂ηφ ′′
c2 s
) + ∂η
αt∗ Θ ′′
Θt∗
δτφ ′′ + 1
µ t∗
)] τ+∆τ + m∂η(Ω ′′τ+∆τ θ t∗
d
− µ ′′
d
τ
) = RΘ t∗
= RW t∗
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
[mΩ ′′τ+∆τ φη − gW ′′τ ] = Rφ t∗
. (3.12)
The RHS terms in (3.7) – (3.12) are fixed for the acoustic steps that comprise the time integration
of each RK3 sub-step (i.e., (3.1) – (3.3)), and are given by
R t∗
U = − m[∂x(Uu) + ∂y(V u)] + ∂η(Ωu) − (µdα∂xp ′ − µdα ′ ∂x¯p)
− (α/αd)(µd∂xφ ′ − ∂ηp ′ ∂xφ + µ ′ d∂xφ) + FU
R t∗
V = − m[∂x(Uv) + ∂y(V v)] + ∂η(Ωv) − (µdα∂yp ′ − µdα ′ ∂y ¯p)
− (α/αd)(µd∂yφ ′ − ∂ηp ′ ∂yφ + µ ′ d∂yφ) + FV
(3.13)
(3.14)
R t∗
µd = − m2 [∂xU + ∂yV ] + m∂ηΩ (3.15)
R t∗
Θ = − m 2 [∂x(Uθ) + ∂y(V θ)] − m∂η(Ωθ) + FΘ
R t∗
W = − m[∂x(Uw) + ∂y(V w)] + ∂η(Ωw)
+ m −1 g(α/αd)[∂ηp ′ + ¯µd(qv + qc + qr)] − m −1 µ ′ dg + FW
(3.16)
(3.17)
R t∗
φ = − µ −1
d [m2 (Uφx + V φy) + mΩφη − gW ], (3.18)
where all variables in (3.13) – (3.18) are evaluated at time t ∗ (i.e., using Φ t , Φ ∗ , or Φ ∗∗ for the
appropriate RK3 sub-step in (3.1) – (3.3)). Equations (3.7) – (3.12) utilize the discrete acoustic
time-step operator
δτa = aτ+∆τ − aτ ,
∆τ
where ∆τ is the acoustic time step, and an acoustic time-step averaging operator
a τ =
1 + β
2 aτ+∆τ 1 − β
+
2 aτ , (3.19)
where β is a user-specified parameter (see Section 4.2.3).
The integration over the acoustic time steps proceeds as follows. Beginning with the small
time-step variables at time τ, (3.7) and (3.8) are stepped forward to obtain U ′′τ+∆τ and V ′′τ+∆τ .
Both µ ′′τ+∆τ and Ω ′′τ+∆τ are then calculated from (3.9). This is accomplished by first integrating
(3.9) vertically from the surface to the material surface at the top of the domain, which removes
the ∂ηΩ ′′ term such that
δτµd = m 2
0
[∂xU
1
′′ + ∂yV ′′ ] τ+∆τ dη. (3.20)
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