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

In the current version of the ARW, the advection of vector quantities (momentum) and scalars
is performed using the RK3 time integration as outlined in Fig. 3.1. The spatial discretization
used in this approach is outlined in the next section. For many applications it is desirable to
use positive definite or monotonic advection schemes for scalar transport. In the next major
release of the ARW we will be including a forward-in-time scheme for scalar transport that has
positive definite and monotonic options. We describe that scheme in the section following the
description of the RK3 advection.
RK3 Advection
2 nd through 6 th order accurate spatial discretizations of the flux divergence are available in the
ARW for momentum, scalars and geopotential using the RK3 time-integration scheme (scalar
advection option 1, step 7 in the time-split integration sequence in Fig. 3.1). The discrete
operators can be illustrated by considering the flux divergence equation for a scalar q in its
discrete form:
R t∗
qadv = −m2 [δx(Uq xadv ) + δy(V q yadv )] − mδη(Ωq ηadv ). (3.39)
As in the pressure gradient discretization, the discrete operator is defined as
δx(Uq xadv −1
) = ∆x (Uq xadv
)i+1/2 − (Uq xadv
)i−1/2 . (3.40)
The different order advection schemes correspond to different definitions for the operator q xadv.
The even order operators (2 nd , 4 th , and 6 th ) are
2 nd order: (q xadv )i−1/2 = 1
2 (qi + qi−1)
4 th order: (q xadv )i−1/2 = 7
12 (qi + qi−1) − 1
12 (qi+1 + qi−2)
6 th order: (q xadv )i−1/2 = 37
60 (qi + qi−1) − 2
15 (qi+1 + qi−2) + 1
60 (qi+2 + qi−3),
and the odd order operators (3 rd and 5 th ) are
3 rd order: (q xadv )i−1/2 = (q xadv ) 4 th
i−1/2
+ sign(U) 1
12
(qi+1 − qi−2) − 3(qi − qi−1)
5 th order: (q xadv
)i−1/2 = (q xadv 6
) th
i−1/2
− sign(U) 1
(qi+2 − qi−3) − 5(qi+1 − qi−2) + 10(qi − qi−1)
60
.
The even-order advection operators are spatially centered and thus contain no implicit diffusion
outside of the diffusion inherent in the RK3 time integration. The odd-order schemes are
upwind-biased, and the spatial discretization is inherently diffusive. As is evident in their formulation,
the odd-order schemes are comprised of the next higher (even) order centered scheme
plus an upwind term that, for a constant transport mass flux, is a diffusion term of that next
higher (even) order with a hyper-viscosity proportional to the Courant number (Cr). Further
details concerning RK3 advection can be found in Wicker and Skamarock (2002)
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