ARW Dynamics and Numerics

42 Pages·2012·2.57 MB·English

by  Kelly Keene

The Advanced Research WRF
(ARW) Dynamics Solver
Bill Skamarock
[email protected]
Jimy Dudhia
[email protected]
WRF Tutorial July 2012
WRF ARW Tech Note
A Description of the Advanced Research WRF Version 3
http://www.mmm.ucar.edu/wrf/users/pub-doc.html
WRF Tutorial July 2012
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 July 2012
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 July 2012
2D Flux-Form Moist Equations in ARW
Moist Equations:
Diagnostic relations:
WRF Tutorial July 2012
Time Integration in ARW
3rd Order Runge-Kutta time integration
advance
Amplification factor
WRF Tutorial July 2012
Phase and amplitude errors for LF, RK3
Oscillation
equation
analysis
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t
ik
WRF Tutorial July 2012
Time-Split Runge-Kutta Integration Scheme
U = L (U) + L (U)
t fast slow
3rd order Runge-Kutta, 3 steps
L (Ut) U* •  RK3 is 3rd order accurate for
s
linear eqns, 2nd order
t t+dt/3 t+dt accurate for nonlinear eqns.
•  Stable for centered and
L (U*) U**
s upwind advection schemes.
•  Stable for Courant number
t t+dt/2 t+dt
Udt/dx < 1.73
•  Three L (U) evaluations per
L (U**) Ut+dt slow
s
timestep.
t t+dt
WRF Tutorial July 2012
WRF ARW Model Integration Procedure
Begin time step
Runge-Kutta loop (steps 1, 2, and 3)
(i) advection, p-grad, buoyancy using (t, *, **)
(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 m, Q, then w, f

(ii) time-average U,V, W
End acoustic loop
Advance scalars using time-averaged U,V, W
End Runge-Kutta loop
Adjustment physics (currently microphysics)
End time step
WRF Tutorial July 2012
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 ’’.
WRF Tutorial July 2012