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The Advanced Research Weather Research and Forecasting Model WRF-ARW Stephanie Evan
```The Advanced Research Weather
Research and Forecasting Model
WRF-ARW
Stephanie Evan
1
WRF is used both for research (the Advanced Research
WRF) and operational forecasting (the Nonhydrostatic
Mesoscale Model WRF).


They are both Eulerian Mass dynamical cores with terrain-
following vertical coordinates.

ARW is developed by NCAR/MMM (Mesoscale and
Microscale Meteorology Division).

NMM is developed by NCEP/EMC (Environmental Modeling
Center).
2
Terrain representation-Vertical Coordinate
Terrain following, hydrostatic, pressure vertical coordinate
p h − p ht
η=
where μ = p hs − p ht
μ
p h , p hs , p ht represent the hydrostatic component of
the pressure and values of pressure along the surface and top boundaries.
μ(x, y) represents the mass per unit area within the column in the
model domain at (x, y).
3
Grid staggering
Conservation of mass :
µ
τ+1
−µ
⇒
∆t
u

F =  v *µ
ω

τ
∂ µ ∂ ( µ u ) ∂ (µ v ) ∂ (µ ω )
+
+
+
= 0
∂t
∂x
∂y
∂η
= (Fx − dx / 2 − Fx + dx / 2 ) + (Fy − dy / 2 − Fy + dy / 2 ) + (Fη − dη / 2 − Fη − dη / 2 )
The same mass fluxes are used for neighboring grid cells : mass is conserved
locally and globally.
4
ARW Equations/Variables

 V = µ d v = (µ d u , µ d v , µ d w ) = ( U , V , W )

The flux form variables are :  Θ = µ d θ

•
 Ω = µ d η
Where µ d represents the mass of the dry air in the column.
Moist Euler equations :
∂ U ∂ ( Uu ) ∂ ( Vu ) ∂ (Ω u )
∂p α ∂p ∂ϕ
x − momentum eq :
+
+
+
+ µ dα
+
= FU
∂t
∂x
∂y
∂η
∂x α d ∂η ∂x
y − momentum eq :
∂ V ∂ ( Uv ) ∂ ( Vv ) ∂ (Ω v )
∂p α ∂p ∂ϕ
+
+
+
+ µ dα
+
= FV
∂t
∂x
∂y
∂η
∂y α d ∂η ∂y
z − momentum eq :
∂ W ∂ ( Uw ) ∂ ( Vw ) ∂ (Ω w )
α ∂p
+
+
+
+ g (µ d −
) = FW
∂t
∂x
∂y
∂η
α d ∂η
∂ Θ ∂ ( Uθ ) ∂ ( Vθ ) ∂ (Ω w )
+
+
+
= FΘ
∂t
∂x
∂y
∂η
∂µ
∂U ∂V ∂Ω
Conservation of mass : d +
+
+
= 0
∂t
∂x ∂y ∂η
∂ Q m ∂ ( Uq m ) ∂ ( Vq m ) ∂ (Ω q m )
Conservation of water :
+
+
+
= FQm
∂t
∂x
∂y
∂η
Q m = µ d q m with q m mixing ratios for water vapor, cloud, rain, ice...etc.
Conservation of heat :
α d is the inverse density of the dry air and α is the inverse density of the full parcel .
∂ϕ
Hydrostatic equation :
= − µ dα d
∂η
γ
R θ 
Equation of state : p = p 0  d m  where γ = c p /c v = 1.4 and p 0 = 1000hPa
 p 0α d 
with θ m = θ [1 + ( R v / R d )q v ]
5
3rd order Runge Kutta time integration
∂ϕ
= R(ϕ )
∂t
∆t
ϕ* = ϕτ +
R(ϕ τ )
3
∆t
**
τ
ϕ = ϕ +
R(ϕ * )
2
ϕ τ + 1 = ϕ τ + ∆ tR(ϕ ** )
Begin time step
Begin RK3 Loop: Steps 1, 2, and 3
1. Advection, Pressure gradient force, buoyancy (φt, φ*, φ**)
2. Physics, if step 1, save for steps 2 and 3
3. mixing, non-RK dynamics, save...
4. assemble dynamics tendencies
Acoustic step loop
1. Advance U, V, then µ, Θ, then w, φ
2. Time average U, V, Ω
End acoustic step loop
End RK3 Loop
Adjustment parameterizations
End Time Step
6
Advection in WRF
∂ϕ
∂ (Uϕ )
∂ϕ
= −
= −U
∂t
∂x
∂x
3rd order Runge - Kutta time integration scheme :
∂ϕ
given by :
∂t
U∆ t τ
ϕ *n = ϕ nτ −
( ϕ n + 1/2 − ϕ nτ − 1/2 )
3∆ x
U∆ t *
**
τ
ϕn = ϕn−
( ϕ n + 1/2 − ϕ *n − 1/2 )
2∆ x
U∆ t **
ϕ nτ + 1 = ϕ nτ −
( ϕ n + 1/2 − ϕ *n*− 1/2 )
Δx
And
ϕ iτ+ 1/2 − ϕ iτ− 1/2
∂ϕ
=
∂x
Δx
with
ϕ iτ+ 1/2 = 0.5( ϕ iτ+ 1 + ϕ iτ ) and ϕ iτ− 1/2 = 0.5( ϕ iτ− 1 + ϕ iτ )
∂ϕ
ϕ iτ+ 1 − ϕ iτ− 1
=
∂x
2∆ x
7
ϕ nτ + 1
2
C
C
= ϕ nτ − (ϕ nτ + 1 − ϕ nτ − 1 ) +
(ϕ nτ + 2 − 2ϕ nτ + ϕ nτ - 2 )
2
8
C3 τ
−
(ϕ n + 3 − 3ϕ nτ + 1 + 3ϕ nτ -1 − ϕ nτ - 3 )
48
Assume ϕ
τ
n
∧
= ϕ exp(i(kn∆ x + ω τ ∆ t ))
2
C
C
e iω ∆ t = 1 − (e ik∆ x − e -ik∆ x ) +
(e 2ik∆ x − 2 + e - 2ik∆ x )
2
8
C 3 3ik∆ x
−
(e
− 3e ik∆ x + 3e -ik∆ x − e - 3ik∆ x )
48
2
3
3
C
C
C
e iω ∆ t = 1 +
[cos( 2k∆ x ) − 1] + i[(
− C) sin(k∆ x ) −
sin( 3k∆ x )]
4
8
24
With ω = ω r + iω i and λ = ± e -iω i ∆ t Eq (1) can be written as :
C
C

 λ cos(ω r ∆ t ) = [1 + 2 sin(k∆ x )][1 − 2 sin(k∆ x )]

3
C
2
 λ sin( ω ∆ t ) = sin(k∆ x )[
sin
(k∆ x ) − C]
r

6
8
Phase and Amplitude Errors
λ
2
3
C
C
C
= [1 +
sin(k∆ x)]2 [1 −
sin(k∆ x )]2 + sin 2 (k∆ x )[
sin 2 (k∆ x ) − C]2
6
2
2


C3
2


sin(k∆ x)[
sin (k∆ x ) − C]
cϕ
1
6

=
tan − 1 
C
C


U
kU∆ t
[
1
+
sin(
k
∆
x
)][
1
−
sin(
k
∆
x
)]


2
2


2Δx
λ
4Δx
10Δx
20Δx
2Δx
Cφ/U 4Δx
10Δx
20Δx
0.001
1.000
1.000
1.000
1.000
0.000
0.637
0.935
0.984
0.010
1.000
1.000
1.000
1.000
0.000
0.637
0.935
0.984
0.200
1.000
1.000
1.000
1.000
0.000
0.637
0.935
0.984
0.300
1.000
1.000
1.000
1.000
0.000
0.637
0.936
0.984
0.400
1.000
0.999
1.000
1.000
0.000
0.637
0.936
0.984
0.500
1.000
0.998
1.000
1.000
0.000
0.638
0.936
0.984
C
0.600
1.000
0.995
0.999
1.000
0.000
0.639
0.936
0.984
0.700
1.000
0.992
0.999
1.000
0.000
0.641
0.936
0.984
0.800
1.000
0.986
0.998
1.000
0.000
0.645
0.937
0.984
0.900
1.000
0.980
0.997
1.000
0.000
0.650
0.938
0.984
1.000
1.000
0.972
0.996
1.000
0.000
0.656
0.939
0.984
1.200
1.000
0.954
0.991
0.999
0.000
0.675
0.943
0.984
Amplitude is conserved but 2Δx waves don't propagate
RK3 allows larger Courant numbers thus larger time steps.
1.500
1.000
0.946
0.981
0.998
0.000
0.610
0.953
0.985
9
Comparison with the leapfrog scheme
Spatial order
Time scheme
3rd
4th
5th
6th
Leapfrog Unstable
0.72 Unstable
0.62
RK2
0.88 Unstable
0.3 Unstable
RK3
1.61
1.26
1.42
1.08
Maximum stable Courant Numbers for 1-D linear advection.
From Wicker and Skamarock (2002).
CRK3≈ 2 Cleapfrog => ΔtRK3≈ 2 Δtleapfrog
For MM5 Δt(s)=3*Δx (km)
For WRF Δt(s)=6*Δx (km)
However it is recommended to use a time step that is ~25% less than
that given by the previous expression.
10
φi+1/2 can be approximated by even and odd ordered approximations.
2nd order : ϕ i − 1 / 2 = 0.5(ϕ i + ϕ i − 1 )
7
(ϕ i + ϕ i − 1 ) −
12
37
=
(ϕ i + ϕ i − 1 ) −
60
1
(ϕ i + 1 + ϕ i − 2 )
12
2
1
6th order : ϕ i − 1 / 2
(ϕ i + 1 + ϕ i − 2 ) +
(ϕ i + 2 + ϕ i − 3 )
15
60
1
3rd order : ϕ i − 1 / 2 = ϕ 4i −th1 / 2 + sign( U ) [(ϕ i + 1 − ϕ i − 2 ) − 3(ϕ i − ϕ i − 1 ) ]
12
1
5th order : ϕ i − 1 / 2 = ϕ 6i −th1 / 2 − sign( U ) [( ϕ i + 2 − ϕ i − 3 ) − 5(ϕ i + 1 − ϕ i − 2 )
60
+ 10( ϕ i + 2 + ϕ i − 3 )]
4th order : ϕ i − 1 / 2 =
Even-Order schemes = centered scheme, no implicit diffusion.

Odd-Order schemes = centered scheme + diffusive term.

The coefficient in the diffusive term is proportional to the speed of the
advecting wind.

In light wind conditions the diffusion is weak, thus 2dx and 4dx
features are not removed properly : explicit diffusion is needed.

11
Sensitivity of the WRF Model to Advection and diffusion schemes
(Kusaka et al., 2005)
24h Simulation of Heavy Rainfall using 3rd to 6th order horizontal advection
schemes and 2nd to 6th order numerical diffusion.
2 nested domains : outer domain is 3600x3360km (dx=12km) and inner domain is
1320x1200km (dx=4km).
Observations
Simulation
12
3S = 3rd
4S = 4th
5S = 4th
6S = 6th
4C = 4th
6C = 6th
4F = 4th
6F = 6th
order Adv scheme + 4th order diffusion (implicit)
order Adv scheme
order Adv scheme + 6th order diffusion (implicit)
order Adv scheme
order Adv scheme + 2nd order diffusion (explicit)
order Adv scheme + 2nd order diffusion (explicit)
order Adv scheme + 4th order diffusion (explicit)
order Adv scheme + 6th order diffusion (explicit)
4th and 6th order:
- Noisy
- Buildup of energy at
short wavelengths.
13
Cases 4C and 6C: noise is damped but the diffusion negates the
advantage of the high-order scheme.


Case 4F: noise is damped but smoothing is stronger than 3S and 5S

Case 6F: noise is damped + more detailed structure, similar to 5S.
5th order advection scheme is the one used by default in WRF and
the one recommended by this study.
14
Explicit Numerical Diffusion in the WRF Model (1)
(Knievel et al., 2007)
Diffusion is implicit when using an odd-ordered advection schemes but can be
insufficient to remove 2dx and 4dx features (in case of light wind conditions). Thus an
explicit 6th-order numerical diffusion scheme can be used to remove short-wave
numerical noise.
∂ϕ
Xue (2000) :
∂t
= S + α ∇ 2η ϕ α = 2-6 p − 1 ∆ t − 1β
- α : coefficient for diffusion.
- p = number of directions = 2
- β (from 0 to 1) is a parameter which specifies the amount of diffusion
applied in one time step (e.g β=0.20 would reduce the amplitude of 2dx
features by 20% in one time step).
Diffusion acts on potential temperature, winds, moisture variables, passive
scalars and on subgrid turbulence kinetic energy.
The diffusive term is calculated along the η-surfaces (problem if complex
topography)
NB The 6th order numerical diffusion is not used by default.
15
2nd order Diffusion in WRF (2)
∂ϕ
= K h ∇ 2 ϕ + K V ∇ 2η ϕ
∂t


If PBL parameterization is used, vertical diffusion is done by the PBL scheme only.
Otherwise diffusion in 3 dimensions (horizontal+vertical).
Horizontal diffusion can be made along η-surfaces (which is not recommended if
the topography is complex) or in physical space (x,y,z). In the first case gradients
are simply taken along coordinate surfaces whereas in the second case gradients
use full metric terms to more accurately compute horizontal gradients in sloped

coordinates.
4 different ways to specify Kh the horizontal eddy viscosity:
1- Kh = constant
2- Kh,v given by the Turbulent Kinetic Energy Kh,v=Ck l e1/2
3- Kh and Kv given by the 3-D Smagorinsky first order
closure (given by the deformation and stability).
4- Kh given by horizontal Smagorinsky first order closure
(Kh given by the deformation based on horizontal wind).
16
The effective resolution of WRF
(Skamarock, 2004)
Kinetic energy spectra possess a
wavenumber dependence of k-3 for
large scales.


3-D turbulence theory predicts a k-5/3
behavior for small scales.

Departure from observed spectra
can be used to define WRF's effective
resolution.
17
Modes with shorter wavelengths are not well-resolved in the model.
The effective resolution of the model can be defined as the wavelength
where a model's spectrum begins to decay relative to the observed
spectrum or relative to a spectrum from higher-resolution simulation.
18
Effective resolution determined from
forecast-derived spectra for the BAMEXconfigured WRF model at 22, 10 and 4km
horizontal grid spacing.
Effective resolution=7Δx
19
Comparison of WRF forecasts with observations
Supercell thunderstorms observed at 23 UTC 30 May 2003 above Wisconsin and
Illinois
20
WRF Boundary conditions
Bottom BC:
➔

SST=constant or can be updated.

Provided by a Land-Surface model
Lateral BC:

Periodic

Open

Symmetric

Specified (nudging), used for real data-cases. When nesting is used, on the
coarse grid (outer domain) the specified BC applies to U, V, θ, φ, μ and water
vapor. Vertical velocity has a zero gradient boundary condition (Wn=Wn-1).
Top BC:

Rayleigh Damping Layer (for idealized case): U, V, θ, w are gradually relaxed to a
predetermined reference state.

Vertical-Velocity damping: apply a Rayleigh damping term in the vertical
momentum equation.

Gravity wave absorbing layer: Horizontal and vertical eddy viscosities are
increased in the absorbing layer.
21
WRF Parameterizations
(from ARW version 2 User's Guide)

Microphysics (7 schemes)

Kessler scheme, used for idealized cloud modeling studies. Warm-rain
only no ice.

Lin et al. scheme: 5-class microphysics including graupel, includes icesedimentation and time-split fall terms

WSM 3-class scheme: 3-class microphysics with ice, ice processes below
0 deg C

WSM-5 class scheme: 5-class microphysics with ice, supercooled water
and snow melt

Ferrier scheme: Supercooled liquid water and ice melt, variable density
for precipitation ice (snow/graupel/sleet)

WSM 6-class scheme: 6-class microphysics with graupel

Thompson et al. graupel scheme: 6-class microphysics with graupel, ice
number concentration also predicted
Longwave Radiation (3 schemes)

Rapid Radiative Transfer Model scheme: uses look-up tables, accounts for
multiple bands, trace gases and microphysics species.

GFDL scheme: Eta operational radiation scheme. Multi-Band scheme with
CO2, O3 and microphysics effects

CAM scheme: Allows aerosols and trace gases.

22
Shortwave Radiation (3 schemes):
Dudhia scheme: Simple downward calculation, clear-sky scattering, water vapor
absorption, cloud albedo and absorption

Goddart shortwave: Spectral method, interacts with clouds, ozone effects.

CAM3 shortwave: spectral method, interacts with clouds, ozone effects, can interact
with aerosols and trace gases.


Surface Layer (2 schemes):

MM5 similarity: Based on Monin-Obukhov

Eta similarity: Used in Eta model. Based on Monin-Obukhov

Land Surface (3 schemes):

5-layer thermal diffusion: Soil temperature only scheme, using five layers

Noah Land Surface Model: Unified NCEP/NCAR/AFWA scheme with soil temperature and
moisture in four layers, fractional snow cover and frozen soil physics

RUC Land Surface Model: RUC operational scheme with soil temperature and moisture
in six layers, multi-layer snow and frozen soil physics

Planetary Boundary layer (2 schemes):

Yonsei University scheme: Non-local-K scheme with explicit entrainment layer and
parabolic K profile in unstable mixed layer

Mellor-Yamada-Janjic scheme: Eta operational scheme. One-dimensional prognostic
turbulent kinetic energy scheme with local vertical mixing

23

Cumulus scheme (3 schemes):

Kain-Fritsch scheme: Deep and shallow convection sub-grid scheme using a mass flux
approach with downdrafts and CAPE removal time scale

Betts-Miller-Janjic scheme. Operational Eta scheme. Column moist adjustment scheme
relaxing towards a well-mixed profile

Grell-Devenyi ensemble: Multiple-closure, explicit updrafts/downdrafts.
Total= 7 x 3 x 3 x 2 x 3 x 2 x 3 = 2268 possible configurations
24
Conclusions

Model Equations: Compressible, nonhydrostatic Euler equations.

Dimensionality: 3D

Grid: Arakawa C-grid

Vertical resolution: Variable, stretched.

Model domain: Globally relocatable, multiple-level nests

Solution technique: 3rd order Runge-Kutta scheme, Advection terms are in the
form of a flux divergence. 2nd to 6th order centered and upwind biased
schemes: 5th order recommended.

Lateral boundary condition: Specified, open, symmetric, periodic, nested.

Top boundary condition: Rigid or absorbing upper layer (increased horizontal
diffusion).

Surface boundary: Friction, land-use categories.
WRF-ARW ≠ WRF-NMM (version of WRF used for forecasting)
A lot of options to fix in “namelist.input”
25
References
Hiroyuki Kusaka, Andrew Crook, Jason C. Knievel and Jimy Dudhia:
Sensitivity of the WRF Model to Advection and Diffusion Schemes for
Simulation of Heavy Rainfall along the Baiu Front, SOLA, Vol. 1, pp.177-180,
2005.


Knievel, J. C., G. H. Bryan, and J. P. Hacker: Explicit numerical diffusion in
the WRF Model. Mon. Wea. Rev., 135, 3808-3824, 2007.

Skamarock, W. C., and J. B. Klemp: A time-split nonhydrostatic atmospheric
model for research and NWP applications. J. Comp. Phys., special issue on
environmental modeling, 2007.

Skamarock, W. C.: Evaluating Mesoscale NWP Models Using Kinetic Energy
Spectra. Mon. Wea., Rev., 132, 3019-3032, 2004.

Wicker, L. J., and W. C. Skamarock: Time splitting methods for elastic
models using forward time schemes. Mon. Wea. Rev., 130, 2088-2097, 2002.
26
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