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

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

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.

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)

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

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

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:

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

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

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!