The Advanced Research WRF (ARW) Dynamics Solver

15 Pages·2014·0.89 MB·English

by  Jimy Dudhia

The Advanced Research WRF
(ARW) Dynamics Solver
WRF Tutorial January 2014
ARW Dynamical Solver
–  Terrain representation
–  Vertical coordinate
–  Equations / variables
–  Time integration scheme
–  Grid staggering
–  Advection scheme
–  Time step parameters
–  Filters
–  Boundary conditions
–  Nesting
–  Map projections
WRF Tutorial January 2014
Vertical Coordinate and Prognostic Variables
π
Hydrostatic pressure
Column mass µ = π − π
s t
(per unit area)
( )
π −π
t
Vertical coordinate η=
µ
(cid:0)
Layer mass µΔη= Δπ = gρΔz
(per unit area)
(cid:0)
Conserved state (prognostic) variables:
(cid:0)
µ, U = µu, V = µv, W = µw, Θ = µθ
φ = gz
Non-conserved state variable:
WRF Tutorial January 2014
2D Flux-Form Moist Equations in ARW
Moist Equations:
Diagnostic relations:
WRF Tutorial January 2014
Time Integration in ARW
3rd Order Runge-Kutta time integration
advance
Amplification factor
WRF Tutorial January 2014
WRF ARW Model Integration Procedure
Begin time step
Runge-Kutta loop (steps 1, 2, and 3)
(i) advection, p-grad, buoyancy using (φt, φ*, φ**)
(ii) physics if step 1, save for steps 2 and 3
(iii) mixing, other non-RK dynamics, save…
(iv) assemble dynamics tendencies
Acoustic step loop
(i) advance U,V, then µ, Θ, then w, φ

(ii) time-average U,V, Ω

End acoustic loop
Advance scalars using time-averaged U,V, Ω

End Runge-Kutta loop
Adjustment physics (currently microphysics)

End time step
WRF Tutorial January 2014
Flux-Form Perturbation Equations
Introduce the
perturbation variables:
Note –
likewise
Reduces horizontal pressure-gradient errors.
For small time steps, recast variables as perturbations from time t
Allows vertical pressure gradient to be expressed in terms of φ’’.
WRF Tutorial January 2014
Small Time Step Integration of Acoustic/Gravity Wave Terms
(Without expanding variables into perturbation form)
•  Forward-backward differencing on U, Θ, and µ equations
•  Vertically implicit differencing on W and φ equations
WRF Tutorial January 2014
ARW model, grid staggering
C-grid staggering
y z
V W, φ
µ,θ,q ,q µ,θ,q ,q
U U U U
v l v l
(cid:0) V (cid:0) W, φ
x x
horizontal vertical
WRF Tutorial January 2014
Advection in the ARW Model
2nd, 3rd, 4th, 5th and 6th order centered and upwind-biased schemes
are available in the ARW model.
Example: 5th order scheme
⎛ ⎞
( )
∂ Uψ 1
( ) ( )
= ⎜ F Uψ −F Uψ ⎟
1 1
∂x Δx
⎝ i+ i− ⎠
2 2
where
⎧ 37 2 1 ⎫
( ) ( ) ( ) ( )
F Uψ = U ⎨ ψ +ψ − ψ +ψ + ψ +ψ ⎬
1 (cid:0) 1 i i−1 i+1 i−2 i+2 i−3
⎩ 60 15 60 ⎭
i− i−
2 2
1
{ }
( ) ( ) ( ) ( )
−sign 1,U ψ −ψ −5 ψ −ψ +10 ψ −ψ
i+2 i−3 i+1 i−2 i i−1
60
WRF Tutorial January 2014
(cid:0)