1. WRF-ARW Basics Fundamentals of Numerical Weather Prediction
Real vs. artificial atmosphere
Map projections
Horizontal grid staggering
Vertical coordinate systems
Definitions & Acronyms
Flavors of WRF
ARW core
NMM core
Other Numerical Weather Prediction Models
MM5
ARPS
Global Icosahedral
WRF Model Governing Equations
Vertical coordinate and grid discretization
Time integration
Microphysics
Current Defects of WRF

2. Real vs. Artificial Atmosphere
True analytical solutions are unknown!
Numerical models are discrete approximations of a continuous fluid.

3. Map Projections x = r cos  y = r 
Example of a regional high resolution grid (projection of a spherical surface onto a 2D plane) nested within a global (lat,lon) grid with spherical coordinates
x = r cos  y = r 
Differences in map projections require caution when dealing with flow of information across grid boundaries.
WRF offers polar stereographic, Lambert conformal, Mercator and rotated Lat-Lon map projections.

4. Arakawa “A” Grid Unstaggered grid - all variables defined everywhere.
Poor performance, first grid geometry employed in NWP models.
Noisy - large errors, short waves propagate energy in wrong direction, additional smoothing required.
Poorest at geostrophic adjustment - wave energy trapped, heights remain too high.
Can use a 2x larger time step than staggered grids.

5. Arakawa “B” Grid Staggered, velocity at corners.
Preferred at coarse resolution.
Superior for poorly resolved inertia-gravity waves.
Good for geostrophy, Rossby waves: collocation of velocity points.
Bad for gravity waves: computational checkerboard mode.
Used by MM5 model.

6. Arakawa “C” Grid
Staggered, mass at center, normal velocity, fluxes at grid cell faces, vorticity at corners.
Preferred at fine resolution.
Superior for gravity waves.
Good for well resolved inertia-gravity waves.
Simulates Kelvin waves (shoulder on boundary) well.
Bad for poorly resolved waves: Rossby waves (computational checkerboard mode) and inertia-gravity waves due to averaging the Coriolis force.
Used by WRF-ARW, ARPS, CMAQ models.

7. Arakawa “D” Grid
Staggered, mass at center, tangential velocity along grid faces.
Poorest performance, worst dispersion properties, rarely used.
Noisy - large errors, short waves propagate energy in wrong direction.

8. Arakawa “E” Grid Semi-staggered grid.
Equivalent to superposition of 2 C-grids, then rotated 45 degrees.
Center set to translated (lat,lon) = (0,0) to prevent distortion near edges, poles.
Developed for Eta step-mountain coordinate to enhance blocking, overcome PGF errors caused by sigma coordinates.
Controls the cascade of energy toward smaller scales.
Used by WRF-NMM and Eta models.

12. Definitions & Acronyms
WRF: Weather Research & Forecasting numerical weather prediction model
ARW: Advanced Research WRF [nee Eulerian Model (EM)] core
NMM: Nonhydrostatic Mesoscale Model core
WPS: WRF Preprocessing System (3 components) - prepares real atmospheric data for input to WRF
WRF-VAR: Variational 3D/4D data assimilation system (not used for this class)
IDV: Integrated Data Viewer - Java application for interactive visualization of WRF model output

13. Flavors of WRF (ARW) ARW solver (research - NCAR, Boulder, Colorado)
Fully compressible, nonhydrostatic equations with hydrostatic option
Arakawa-C horizontal grid staggering
Mass-based terrain following vertical coordinate
Vertical grid spacing can vary with height
Top is a constant pressure surface
Scalar-conserving flux form for prognostic model variables
2nd to 6th-order advection options in horizontal &vertical
One-way, two-way and movable nest options
Runge-Kutta 2nd & 3rd-order time integration options
Time-splitting
Large time step for advection
Small time step for acoustic and internal gravity waves
Small step horizontally explicit, vertically implicit
Divergence damping for suppressing sound waves
Full physics options for land surface, PBL, radiation, microphysics and cumulus parameterization
WRF-chem under development:

14. Flavors of WRF (NMM)
NMM solver (operational - NCEP, Camp Springs, Maryland)
Fully compressible, nonhydrostatic equations with reduced hydrostatic option
Arakawa-E horizontal grid staggering, rotated latitude-longitude
Hybrid sigma-pressure vertical coordinate
Conservative, positive definite, flux-corrected scheme used for horizontal and vertical advection of TKE and water species
2nd-order spatial that conserves a number of 1st-order and quadratic quantities, including energy and enstrophy
One-way, two-way and movable nesting options
Time-integration schemes: forward-backward for horizontally propagating fast waves, implicit for vertically propagating sound waves, Adams-Bashforth for horizontal advection and Coriolis force, and Crank-Nicholson for vertical advection
Divergence damping & E subgrid coupling for suppressing sound waves
Full physics options for land surface, PBL, radiation, microphysics (only Ferrier scheme) and cumulus parameterization
Note: Many ARW core options are not yet implemented! Nesting still under development
NMM core will be used for HWRF (hurricane version of WRF), operational in summer of 2007

15. Other NWP Models (MM5) MM5 (research - PSU/NCAR, Boulder, Colorado)
Progenitor of WRF-ARW, mature NWP model with extensive configuration options
Support terminated, no future enhancements by NCAR’s MMM division
Nonhydrostatic and hydrostatic frameworks
Arakawa-B horizontal grid staggering
Terrain following sigma vertical coordinate
Unsophisticated advective transport schemes cause dispersion, dissipation, poor mass conservation, lack of shape preservation
Outdated Leapfrog time integration scheme
One-way and two-way (including movable) nesting options
4-dimensional data assimilation via nudging (Newtonian relaxation), 3D-VAR, and adjoint model
Full physics options for land surface, PBL, radiation, microphysics and cumulus parameterization

16. Other NWP Models (ARPS)
ARPS (research - CAPS/OU, Norman, Oklahoma)
Advanced Regional Prediction System
Sophisticated NWP model with capabilities similar to WRF
Primarily used for tornado simulations at ultra-high (25 meter) resolutions and assimilation of experimental radar data at mesoscale
Elegant, source code, easy to read/understand/modify, ideal for research projects, very helpful scientists at CAPS
Arakawa-C horizontal grid staggering
Currently lacks full mass conservation and Runge-Kutta time integration scheme
ARPS Data Assimilation System (ADAS) under active development/enhancement (MPI version soon), faster & more flexible than WRF-SI, employed in LEAD NSF cyber-infrastructure project
wrf2arps and arps2wrf data set conversion programs available

17. Global Icosahedral Model

18. WRF Model Governing Equations (Eulerian Flux Form)
Momentum:
∂U/∂t + (∇ · Vu) − ∂(pφη)/∂x + ∂(pφx)/∂η = FU
∂V/∂t + (∇ · Vv) − ∂(pφη)/∂y + ∂(pφy)/∂η = FV
∂W/∂t + (∇ · Vw) − g(∂p/∂η − μ) = FW
Potential Temperature:
Diagnostic Hydrostatic (inverse density a):
∂Θ/∂t + (∇ · Vθ) = FΘ
∂φ/∂η = -μ
Continuity:
where:
μ = column mass
V = μv = (U,V,W)
Ω = μ d(η)/dt
Θ = μθ
∂μ/∂t + (∇ · V) = 0
Geopotential Height:
∂φ/∂t + μ−1[(V · ∇φ) − gW] = 0

19. WRF Vertical Coordinate h

20. Vertical Grid Discretization

21. Runge-Kutta Time Integration
Φ∗ = Φt + t/3 R(Φt )
Φ∗∗ = Φt + t/2 R(Φ∗)
Φt+t = Φt + t R(Φ∗∗)
“2.5” Order Scheme
Linear: rd order
Non-linear: 2nd order
Square Wave Advection Tests:

22. Runge-Kutta Time Step Constraint
RK3 is limited by the advective Courant number (ut/x) and the user’s choice of advection schemes (2nd through 6th order)
The maximum stable Courant numbers for advection in the RK3 scheme are almost double those in the leapfrog time-integration scheme
Maximum Courant number for 1D advection in RK3
Time Scheme
Spatial order
3rd
4th
5th
6th
Leapfrog
Unstable
0.72
0.62
RK2
0.88
0.30
RK3
1.61
1.26
1.42
1.08

23. Time Step Configuration
For 3 spatial directions, the maximum advective time step should satisfy:
tmax < C x
√3 umax
Assuming a 10 km horizontal grid with a maximum velocity of 100 m/s, 5th-order spatial differencing and a Runge-Kutta 3rd-order time integration scheme (Courant number = 1.42):
tmax < * 10,000
1.732 * 100
< 82 seconds
Use 25% less and round down, so tmax = .75 * 82 = 60 seconds.
Rule of thumb:
For MM5 tmax = 3 x
For WRF-ARW tmax = 6 x

24. Acoustic Time Step Configuration
For the forward-backward scheme used in the WRF-ARW’s 2D explicit acoustic time integration scheme, the maximum acoustic time step should satisfy:
tmax < x
√2 cs
Assuming a 10 km horizontal grid and a sound speed of 300 m/s:
tmax < ,000
1.414 * 300
< 23.5 seconds
In WRF-ARW, the ratio of the advective/acoustic time steps should be an even integer, so round down to tmax = 15 seconds. The number of sound time steps per advective time steps is what is specified in the WRF namelist input file as variable time_step_sound, which in this case would be 60 / 15 = 4.

25. Microphysics
Includes explicitly resolved water vapor, cloud and precipitation processes
Model accommodates any number of mixing-ratio variables
Four-dimensional arrays with 3 spatial indices and one species index
Memory (size of 4th dimension) is allocated depending on the scheme
Carried out at the end of the time-step as an adjustment process, does not provide tendencies
Rationale: condensation adjustment should be at the end of the time step to guarantee that the final saturation balance is accurate for the updated temperature and moisture
Latent heating forcing for potential temperature during dynamical sub-steps (saving the microphysical heating as an approximation for the next time step)
Sedimentation process is accounted for, a smaller time step is allowed to calculate vertical flux of precipitation to prevent instability
Saturation adjustment is also included

26. WRF Microphysics Options
Mixed-phase processes are those that result from the interaction of ice and water particles (e.g. riming that produces graupel or hail)
For grid sizes ≤ 10 km, where updrafts may be resolved, mixed-phase schemes should be used, particularly in convective or icing situations
For coarser grids the added expense of these schemes is not worth it because riming is not likely to be resolved
Scheme
Number of Moisture Variables
Ice-Phase Processes
Mixed-Phase Processes
Kessler 3 N
Purdue Lin 6 Y
WSM3
WSM5 5
WSM6
Eta GCP 2
Thompson 7

27. WRF Planetary Boundary Layer Options
All PBL schemes are 1D, assume a clear separation between subgrid and resolved eddies. Not valid for grid sizes below a few hundred meters, so must use a full 3D local sub-grid turbulence scheme instead. WRF provides a prognostic TKE equation with 1.5 order turbulence closure for this.
Medium Range Forecast Model (MRF) PBL
Counter-gradient flux for heat and moisture in unstable conditions
Enhanced vertical flux coefficients in PBL
PBL height is defined using a critical bulk Richardson number = .5
Yonsei University (YSU) PBL
Uses counter-gradient terms to represent fluxes due to non-local gradients
Explicit treatment of entrainment layer at PBL top, entrainment is proportional to the surface buoyancy flux (supported by studies using large-eddy models)
PBL height is defined using a critical bulk Richardson number = 0, so it is only dependent on the buoyancy profile. This lowers the PBL top compared with MRF.
Mellor-Yamada-Janjic (MYJ) PBL
2.5 order turbulence closure through a full range of atmospheric turbulent regimes
Upper limit imposed on master length scale, which depends on TKE, buoyancy and shear
In the unstable range, upper limit derived from requirement that TKE production be non-singular in the case of growing turbulence
In the stable range, upper limit derived from requirement that the ratio of variance of vertical velocity deviation and TKE cannot be smaller than in the vanishing turbulence regime
TKE production/dissipation differential equation is solved iteratively, with recently improved empirical constants.

28. Current Defects of WRF
Serious deficiencies in PBL parameterizations and land surface models produce biases/errors in the predicted surface and 2-meter temperatures, and PBL height. WRF cannot maintain shallow stable layers.
3D/4D Variational data assimilation and Ensemble Kalman Filtering (EnKF) still under development, EnKF available to community from NCAR as part of the Data Assimilation Research Testbed (DART).
Not clear yet what to do in “convective no-man’s land” – convective parameterizations valid only at horizontal scales > 10 km, but needed to trigger convection at 5-10 km scales.
Multi-species microphysics schemes with more accurate particle size distributions and multiple moments should be developed to rectify errors in the prediction of convective cells.
Heat and momentum exchange coefficients need to be improved for high-wind conditions in order to forecast hurricane intensity. Wind wave and sea spray coupling should also be implemented. Movable, vortex-following 2-way interactive nested grid capability has recently been incorporated into the WRF framework.
Upper atmospheric processes (gravity wave drag and stratospheric physics) need to be improved for coupling with global models.

29. Technical Description of WRF-ARW
For more information on the technical details of the
WRF model: