– equations / variables – vertical coordinate – terrain representation – grid staggering –...
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– Equations / variables – Vertical coordinate– Terrain representation– Grid staggering– Time integration scheme– Advection scheme– Boundary conditions– Map projections– Dynamics parameters
WRF Height-Coordinate Dynamical Solver
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z
W
x
U
t
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Ww
x
Uwg
z
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Wu
x
UufV
x
p
t
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,,, wWvVuUConservative variables:
Inviscid, 2-Dequations inCartesiancoordinates
Flux-Form Equations in Height Coordinates
where we have removed a hydrostatic base state g
z
p
pcR p Pressure termsdirectly related to
Flux-Form Equations in Height Coordinates
Acoustic mode separation
Recast Equations in terms of perturbation about time t
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z
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Flux-Form Equations in Height Coordinates
Moist Equations in Height-Coordinate Model
Moist Equations mddvv
dd RqR
RRp
d
1
z
W
x
UQ
z
W
x
U
t
z
Ww
x
Uw
qqgg
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p
gc
Rg
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Wu
x
Uu
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pfV
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t
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tm
tm
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tm
tmm
tt
lv
lvdd
t
m
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mt
v
mt
m
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ttt
m
dmt
m
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Height-Coordinate Model, Terrain Transform
Vertical coordinate transformation
bottop
bot
zz
zzzx
;,
zz
pp;WVU
;~~,,;~ vV
Redefine variables
and likewise for perturbation variables
Height-coordinate model, grid staggering
z
W
W
U U
x
lv qq ,,,
V
V
U U
x
y
lv qq ,,,
C-grid staggering
horizontal vertical
Height-Coordinate Model, Time Integration
3rd Order Runge-Kutta time integration
Rt
Rt
Rt
tt
t
tt
tt
1
1
2
3
advance
Amplification factor
241;;
41 tk
AAki nnt
Time-Split Leapfrog and Runge-Kutta Integration Schemes
Phase and amplitude errors for LF, RK3
Oscillation equationanalysis
ikt
Phase and amplitude errors for LF, RK3
Advection equationanalysis
xt U
5th and 6th order upwind-biased and centered schemes.Analysis for 4x wave.
Acoustic Integration in the Height Coordinate Model
2
1
2
1;
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Rz
W
x
U
t
Rz
W
x
U
t
Rgc
Rg
zR
t
W
Rx
Rt
U
tm
tmm
Wm
t
v
mt
m
d
Umt
m
d
Forward-backward scheme, first advance the horizontal momentum
Second, vertically-implicit integration of the acoustic and gravity wave terms
Advection in the Height Coordinate Model
2nd, 3rd, 4th, 5th and 6th order centered and upwind-biased schemesare available in the WRF model.
Example: 5th order scheme
UFUFxx
Uii
2
1
2
1
1
12132
32211
2
1
2
1
10560
1,1
60
1
15
2
60
37
iiiiii
iiiiiiii
Usign
UUF
where
For constant U, the 5th order flux divergence tendency becomes
TOHx
Cr
x
tU
x
Ut
x
Ut
iiiiiii
th
..6
6
60
6152015660
1321123
6
Advection in the Height Coordinate Model (cont.)
The odd-ordered flux divergence schemes are equivalent to the next higher ordered (even) flux-divergence scheme plus a dissipation term of the higher even order with a coefficient proportional to the Courant number.
Runge-Kutta loop (steps 1, 2, and 3) (i) advection, p-grad, buoyancy using (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 W, (ii) time-average U,V,W End acoustic loop Advance scalars using time-averaged U,V,WEnd Runge-Kutta loopOther physics (currently microphysics)
Begin time step
End time step
WRF Model Integration Procedure
,,t
,
Height Coordinate Model: Boundary Condition Options
1. Specified (Coarse grid, real-data applications).2. Open lateral boundaries (gravity-wave radiative).3. Symmetric lateral boundary condition (free-slip wall).4. Periodic lateral boundary conditions.5. Nested boundary conditions (not yet implemented).
Lateral boundary conditions
Top boundary conditions
1. Rigid lid.2. Gravity-wave radiative condition. 3. Absorbing upper layer (increased horizontal diffusion).4. Rayleigh damping upper layer (not yet implemented).
Bottom boundary conditions1. Free slip.2. Various B.L. implementations of surface drag, fluxes.
Height Coordinate Model: Coordinate Options
1. Cartesian geometry (idealized cases)2. Lambert Conformal3. Polar Stereographic4. Mercator
3rd order Runge-Kutta time step
Acoustic time step
Divergence damping coefficient: 0.1 recommended.
Vertically-implicit off-centering parameter: 0.1 recommended.
Advection scheme order: 5th order horizontal, 3rd order vertical recommended.
Height Coordinate Model: Dynamics Parameters
Courant number limited, 1D: 73.1
x
tUCr
Generally stable using a timestep approximately twice as large as used in a leapfrog model.
2D horizontal Courant number limited: 2
1
h
CC sr
stepsacousticofnumberRKsound t
