1. Overview of WRF ARW Thermodynamic Equation
Steven Cavallo
University of Washington
Group Meeting
August 12, 2005

2. Outline & Overview Why I love theta The WRF thermodynamic equation
Model integration
Calculation flow

3. Why I so deeply care about theta
Recall from the Ertel PV equation
so that in the absence of friction, in order for PV to be conserved the following must be true:
The Advanced Research WRF (ARW) calculates this in flux-form:
where d is the total dry air mass in a column and F are the forcing tendencies including radiation, pbl, cumulus, microphysics, mixing, and diffusion effects.

4. WRF (ARW) system
The total derivative operator in any coordinate system (say ) may be written as (Laprise 1992):
WRF (ARW) uses a terrain following sigma coordinate system they call  and use the following couplings:
so that the thermodynamic equation becomes (after adding the map factors)

5. WRF Integration
The large timestep integration is done using a third order Runge-Kutta 3-step scheme (RK3):
where the *’s represent intermediate timesteps, and R() = t are the tendencies.
Solving for the large timestep tendency from the thermodynamic equation and calling it then
where

6. WRF Integration (cont’d)
However within the large time step, smaller time steps must be taken due to the high frequency (but meteorologically insignificant) acoustic modes. This is done by defining the following perturbations:
where the superscript t* which represents the corresponding values from the latest RK3 predictor.
After substituting these into the thermodynamic equation:
where  is the acoustic timestep and

7. WRF Integration (cont’d)
Finally, the next time step can then be written as
where the large time step tendency term is updated at each RK3 substep and the small time step tendency term is updated in each acoustic iteration.
The forcings, F are only updated on the first RK3 substep.
Microphysics tendencies are calculated as an adjustment of  outside the RK3 loop:

8. WRF (ARW) time stepping summary
Begin time step
Begin Runge-Kutta (RK3) time stepping loop (substeps 1,2, and 3).
If RK3 substep = 1, then calculate physics tendencies (F)..
Calculate
Begin acoustic time stepping loop (1  n).
If RK3 substep = 1, then there are n = 1 iterations. After this we get *.
If RK3 substep = 2, then there are n=2 iterations. After this we get **.
If RK3 substep = 3, then there are n=4 iterations. After this we get t+t.
End acoustic loop.
Update scalars (mixing ratios).
End RK3 loop
Adjust  for microphysics tendencies.
End time step

9. Coupled with total dry mass?
Writing out the tendencies
To add any physics or microphysics tendency to the WRF output, simply adjust the variable so that is has the correct write code in the registry (see registry documentation page). HOWEVER, since only the variables that are themselves part of the time step are coupled with mass, this can make it quite tricky.
The following is a table listing the WRF variable name, description, and whether it is coupled with the total dry air mass:
WRF variable name
Description
Coupled with total dry mass?
RTHRATEN
t from total radiation in K s-1
Yes
RTHRATENLW
t from long wave radiation in K s-1 No
RTHRATENSW
t from short wave radiation in K s-1
RTHBLTEN
t from planetary boundary layer scheme in K s-1
RTHCUTEN
t from cumulus scheme in K s-1
H_DIABATIC
t from microphysics (not physics!) scheme in K s-1
T_TENDF
t from physics (not microphysics!) in K s-1

10. Quick summary
Ertel’s PV equation tells us that creation and destruction of PV can arise from diabatic heating.
The ARW calculates the diabatic heating tendencies in a “right hand side” calculation from the thermodynamic energy equation.
Physics tendencies include effects of shortwave and longwave radiation, the pbl, cumulus, and mixing and diffusion. These are calculated at beginning of first RK3 substep.
Microphysics tendencies are calculated and used to make an adjustment or correction to the potential temperature outside of the RK3 time stepping loop.
Physics and microphysics tendencies can easily be added to output from WRF registry. Must be careful about the mass couplings though when doing so!