The Thermodynamic Energy Equation and WRF1 The governing equations for various vertical coordinates In any coordinate system, the total derivative operator may be written as D = Dt ∂ ∂ ∂t + v · ∇π + π˙ ∂π (1) π where π represents the choice of coordinate system, π˙ = Dπ D t and v = (u, v) is the two-dimensional cartesian wind vector (Laprise 1992). The Advanced Research WRF (ARW) equations use a terrainfollowing hydrostatic-pressure vertical coordinate called η defined by (2) ph − ph − pht η = p − p = pht hs ht µ where ph is the hydrostatic pressure, pht is the hydrostatic pressure at the top of the model, phs is the hydrostatic pressure at the model surface, and µ = phs − pht (3) is the mass per unit area within the column in the model domain (x,y). Using this coordinate system, we can rewrite the total derivative operator defined in (1) as Dt 2 D = ∂ ∂t η + v · ∇η + η˙ ∂ ∂η . (4) WRF flux-form dry thermodynamic equation Following Ooyama (1990), we can write prognostic equations in terms of variables that have conservation properties. For the thermodynamic equation, this is D(µθ) = FΘ (5) Dt where FΘ represents all sources and sinks and µθ represents the mass theta flux. Using (1) we can expand the above as ∂(µθ) + µ v · ∇ηθ + ∂µ = FΘ (6) θ η˙ ∂t ∂η or more η concisely ∂(µθ) + µ u · ∇ η θ = FΘ (7) ∂t η where u = (u, v, w) is the three-dimensional cartesian wind vector. Letting 1 Letting α = 1 the above can be written as ρ pα = RdT (17) which solving for p gives Rd . (15) η The equation of state is p = ρRdT. T: T =θ p po Substituting the above into (18): p= or rearranging Rd / cp −Rd/cp = θpRd/cp po Rd α 2 . (8) (9) (10) + U · ∇ η θ = FΘ . θ is θ=T po Rd/cp (18) (19) .U = µu Θ = µθ Ω = µη˙ we can then write (7) ∂Θ ∂t According to vector identity. p= T α The definition of potential temperature. Solving the above for temperature. (11) η ∇ · (aA ) = a∇ · A + A · ∇a where here we let a = θ and A = U (12) so that ∇ · (θU ) = θ∇ · U + U · ∇θ. (16) where p is the dry air pressure. (20) . Rd is the dry air gas constant. (13) By mass conservation. ρ is the density of air. (14) Substituting the above into (11) the flux-form thermodymamic equation becomes ∂Θ ∂t η r + ∇ ·U θ = FΘ. and T is the actual temperature. p where po is the pressure at 1000 hPa and cp is the specific heat capacity at constant presssure. ∇ · = 0 leaving ∇ · (θU U )=U · ∇θ. θpRd/cp p (21) −Rd/cp o Rd pp−Rd/cp = 1− ⇒p R cpd Rαd = α 3 θp −Rd/cp o −Rd/cp θp o . (22) (23) . α 1 γ (29) Raising both sides to the power γ gives 1 γ 1−γ γ Rd θ rpγ = ⇒p = γ α (30) po d 1−γ R θ p (31) γ α ⇒ p = po o d . then we can note that 1 Rd − and p c Rd = − p c cp − = Rd cp cv = c = cp − cv cv − cp = cp cv − p p c γ (24) (25) − 1 (26) −1 γ 1− γ (27) = cp 1 = 1 = (28) γ c where γ = p v c and cv is the specific heat capacity at constant volume. R θ (32) γ αpo 3 WRF moist thermodynamic equation When including moisture. The coupling of the variables to dry mass is retained and written as .Noting that cp = cv + R. the vertical coordinate from (2) can be written as pdh − pdht = η= pdhs − pdht pdh − (33) pdht µd where the subscript ’d’ now represents that it is for the dry air component and µd represents the mass of dry air in the column. Subsituting (24) and into (23) gives (28) 1−γ R θ γ p = d po . pressure is adjusted to include moisture via the moist potential temperature (not to be confused with virtual potential temperature) by re-writing (32) as p = po where 4 θm "" θ d m R θ γ αp (40) d o 1+ v R qv = θ(1 + 1. Polar strereographic. if we let be a vector representing the cartesian distance on the X model’s plane. qr. snow. qi. say m. qc. we can write changes along the cartesian plane in terms of changes on the Earth’s surface as = d md x (42) X dX ⇒dx= m . In general. and Mercator. The true velocity vector is Substituting the above into (34): . then to obtain the true distance x on the sphere we must make some transformation by using some scale. and any additional mixing ratio class such as graupel for instance. This means that the calculations produced on the model plane must be projected onto the Earth’s sphere. Further. (43) This map factor varies but remains close to unity where it is exactly unity at the true latitude of the map projection.61qv ). cloud liquid water. Rd (41) Map factors The WRF model uses map projections such as the Lambert conformal. the thermodynamic equation (15) is not directly changed. Using this notation. cloud ice. qs.U = µd u Θ = µd θ Ω = µd η˙. etc and (39) are the mass mixing ratios of water vapor. which is called the map factor. rain water. (34) (35) (36) In this method. but instead an additional conservation equation is added to the governing equations to include the mass mixing ratios of water in all of its phases: ∂Qm = FQm (37) + rU · η ∂t ∇qm where Qm = µd q m (38) qm = qv. (48) .u= (44) . dx dt dx U = µ d u = µd dt which using (43) gives (45) µd u U = (46) m and similarly µdη˙ Ω= . (47) m Applying this to the thermodynamic equation (15). the flux-form moist thermodynamic equation with map factors m is ∂Θ 2 ∂(Uθ) +m ∂t ∂x + ∂(V θ) + m ∂(Ωθ) ∂y ∂η = F Θ. there is a smaller timestepping acoustic loop so that the meteorologically insignificant acoustic modes do not limit the RK3 accuracy. FΘ include potential temperature tendencies from radiation. and mass µ. planetary bound.cumulus + FΘ. Perturbations are added to the pressure p. mixing.balanced reference state.5 Perturbation equations The WRF equations are modified so that they are written as perturbations from a hydrostatically. Within the large timestepping Runge-Kutta loop. The forcing terms. inverse density α. the thermodynamic equation uses the following: Θ = Θ − Θt∗ (55) V Ω t∗ = V −V = Ω − Ωt∗. We can write that as FΘ = FΘ. (56) (57) . we will rewrite the thermodynamic equation (48) as Θt + m2 [∂x(Uθ) + ∂y (V θ)] + m∂η (Ωθ) = FΘ (53) where the subscripts t.pbl + FΘ.radiation + FΘ. x. (51) (52) For notational simplicity. 6 Model integration 6. geopotential φ.ping loop. The microphysics contributions to θ are calculated in a non-timesplit technique and are therefore calculated outside the Runge-Kutta timestep. Thus to increase the accuracy. and diffusion. For instance. y and η are the respective partial derivatives. (54) The above is calculated only on the first RK3 step. Thus the thermodynamic equation is left unchanged and is given by (48). The low frequency modes are integrated using a Runge-Kutta third order scheme while the fast (acoustic) modes are integrated over smaller timesteps. cumulus parametrization.mixing/dif fusion. a perturbation form of the governing equations are integrated using smaller acoustic time steps by defining small time step variables that are deviations from the most recent RK3 predictor (denoted by the superscript t*).1 Overview A time-split time integration scheme is used.ary layer (pbl) schemes. This acoustic timestepping is performed in a correction type manner. The Runge-Kutta three step time integration process from θ t to θt+∆t for potential temperature is ∆t t R(Θ) ∗ = Θ + (49) Θ 3 ∆t t R(Θ∗ ) Θ∗∗ = Θ + (50) 2 Θt+∆t = Θt + ∆tR(Θ∗∗ ). θ t∗ + RLarge.mp.θ l (63) in which at the end of the acoustic loop Θτ +∆τ = Θt+∆t.θ +∆τ θ ∗ m∂η (Ω where Θτ +∆τ − Θτ . (60) δτ Θ = ∆τ Just as with the right hand side tendency from the large timestep.mp then the final result is Θt+∆t = Θt+∆t + ∆t ∗ Θt. After the coupled potential temperature Θ is calculated from the above. but NOT within the acoustic integration. Substituting (55)-(57) into (53) and using (58) the acoustic thermodynamic equation is l t∗ τ t δτ Θ + m2 ∂x(U θt∗) + ∂y (V θt∗) + Large. τ (61) Small.θ θ θ +∆τ θ R ∂ (U ∗ (V x ∗ ∗ (Ω ∗ −m Substituting the above into (59) the acoustic tendency is δ τ Θ = Rτ ∗ t∗ Small. then it is adjusted for the microphysics tendencies. .θ 2 = −m [∂x(Uθ) + ∂y (V θ)] − m∂η (Ωθ) + FΘ (58) which is calculated in each of the RK3 substeps. Then from (53): θ Rt∗ Large. The variables in (58) are used as a beginning reference to which the perturbations in (55) . After the scalars in (37) are updated. (64) )=R (59) . The default number of acoustic timesteps for RK3 is 4.(57) are initially built upon entry into the acoustic integration and are not themselves updated within the acoustic loop. the Runge-Kutta loop is then complete.θ + RLarge. If we let the coupled potential temperature tendencies from the microphysics be Θt.Let the right hand side of the tendency from the large timestep which is fixed throughout the acoustic integration be Rt∗ Large. we can write the right hand side tendency from the small timestep using (59) as l 2 τ = t ) + ∂y t ) − m∂η t ).θ (62) which using (60) can be written as Θτ +∆τ = Θτ + ∆τ Rτ ∗ Small. the WRF (ARW) calculation flow procedure for potential temperature is as follows: 1. i. but then only carries their sum Fθ. This is done to prevent the excitation of acoustic waves . 6. ii. you must decouple it from the total dry air mass. End acoustic loop. If RK3 step = 1. then the number of iterations is n = 1. However. If RK3 step = 3. calculate time-split physics: FΘ = θt.cumulus + θt. the total radiation term must be coupled with total dry mass in order to be integrated. 2. s 2 C.radiation may be expanded as Fθ. After this we’ll have Θt+∆t. Begin time step (a) Begin Runge-Kutta three step (RK3) time integration loop (steps 1.radiation into the model integration.lw and FΘ.2 Some notes on calculating tendencies To calculate the tendenices in WRF from (54).lw and Fθ. 2. if you add the total radiation tendency to the WRF Registry (RTHRATEN). After this we’ll have Θ∗∗. if the number of acoustic timesteps is 4 (n = 4).mixing/dif fusion. if the number of acoustic timesteps is 4 (ns = 4). The WRF ARW technically carries the microphysics tendency term when integrating through the timesteps. (b) Update scalar equation using (37). Return to step 1.θ Begin acoustic time step loop (1 → n) using (63). The dry air mass in terms of WRF variables is mut = mu + mub (67) where mu is the perturbation dry air mass and mub is the background dry air mass. For instance. (d) Adjust for the non time-split schemes (only microphysics) using (64).sw (66) where FΘ. the radiation component. Calculate Rt∗ iii. A. B. one must use extreme caution due to the coupling of the tendencies with the total dry air mass. the shortwave and longwave radiation components are NEVER coupled with dry air mass.radiation = FΘ. End time step.lw + FΘ. if the number of acoustic timesteps is 4 (ns = 4). Large. If RK3 step = 2.pbl + θt. so they are not coupled when written out of WRF. WRF calculates the individual components FΘ.To summarize. 3) using (49)-(51). (c) End (RK3) loop.sw within the radiation_driver module.radiation + θt. However. thenthe number of iterations is n = ns = 4. The calculations for radiation are not coupled with total dry mass when passing through this subroutine.sw are the longwave and shortwave radiation components respectively. but removes it in the end. iv. If RK3 step = 1. then the number of iterations is n = ns = 2. Therefore. (65) (= Advection + FΘ) using (58). Fθ. After this we’ll have Θ∗. physics + FΘ. The inclusion of the diabatic forcing tendency estimate used in the RK3 integration is from the previous timestep.1 RK3 timestep Limitations to the timestep used in the RK3 integration arise through the advective Courant number ∆ ( u∆t ). The above term. the tendencies that are computed in the time stepping themselves are coupled with the total dry air mass while variables that are only calculated individually within physics or microphysics schemes are not coupled. WRF subtracts off this tendency with θt+∆t = θt+∆t − ns ∗ ∆t ∗ µd ∗ H_DIABATIC (70) where ns is the number of timesteps.mp (68) Large. is given by (54) and Fθ. Since the ARW integration involves equations in mass-flux form.mp is the microphysics tendency given by FΘ. The suggested scheme is the 5th order scheme.WRF Variable name RTHRATEN RTHRATENLW RTHRATENSW RTHBLTEN RTHCUTEN H_DIABATIC T_TENDF Description θt from total radiation in K/s θt from longwave radiation in K/s θt from shortwave radiation in K/s θt from planetary boundary layer scheme in K/s θt from cumulus scheme in K/s θt from microphysics scheme in K/s θt from physics in K/s Coupled with MUT? Yes No No Yes Yes No Yes Table 1: Summary of potential temperature tendency variable names in WRF ARW model.mp . which is called T_TENDF in WRF.physics. while retaining the balances produced by the physics at the end. which depends on the advection order chosen by the user. Currently options for x the advection scheme are from 2nd order through 6th order. It is necessary to include it here to relieve the acoustic noise that would be produced otherwise. and before updating the microphysics. as they are already divided by the model timestep when written out. it is important that the physics be balanced at the end of the timestep. At the same time. 7 Stability and the CFL condition 7.θ = t 2 R−m ∗ where Fθ. when uncoupled is carried in WRF as variF able H_DIABATIC or H_DIABATIC = µdΘ. No timestep adjustments are necessary.mp = µd θt+∆t θt − (69) ∆t where µd is the total dry air mass. Table 1 gives a summmary of the WRF ARW potential temperature tendency terms that can be written out when added to the Registry. Thus this microphysics term can be thought of in relation to FΘ as rewritting (58) as [∂x(Uθ) + ∂y (V θ)] − m∂η (Ωθ) + FΘ. The only adjustments that need to be accounted for are the couplings with the total dry air mass. However. after the timestep. of which it was shown by Wicker and Skamarock (2002) that the maximum stable 8 . we must therefore scale the Courant number from the 1D case by a √factor of 1 .8198 . Solving the above for the timestep ∆t: ∆tmax ≤ Cr ∆x / ma (3) x (72) u 1. That is. To get a more conservative estimate of the maximum acoustic timestep. we can replace √1 with 1 to get an acoustic timestep constraint of (2) 2 ∆τ < 2 ∆x < ∆x (77) . umax (73) (74) A general guideline for a maximum timestep is: ∆t6 ∗ ∆x "" 1000 (75) where ∆x is the grid spacing in meters. 7. Since the spatial dimension is 3 in the model.Courant number for 1D advection (while using the RK3 timestep scheme) is 1.42. This will allow stability for up to a 136 m/s jet.2 Acoustic timestep The maximum Courant number for the acoustic timestep τ is Cr < cs∆τ 1 <√ ∆x 2 (76) where cs = 300ms−1 is the speed of sound.42 ∆x ≤ √ 3 umax ∆x ≤ 0. 3 Cr u∆t (71) ∆x = √ 3 where Cr is the Courant number as found by Wicker and Skamarock (2002) for 1D advection. the acoustic timestep should be approximately 1/4 of the RK3 timestep. Generally. then the radiation timestep should be 30 minutes.3 Radiation timesteps The recommended timestep is 1 minute per km dx. if horizontal grid resolution is 30 km.1. which means it is updated with the model large (RK3) timestep from Section 7.4 Cumulus and planetary boundary layer scheme timesteps Generally set to zero minutes. 7. For example.cs 150 where ∆x is in meters. 9 . Therefore the default number of acoustic timesteps should be 4. 7.