The new PPM advection schemes in the MesoNH Jean-Pierre Pinty, Christine Lac, Tomislav Mari ć

1. The new PPM advection schemes in the MesoNHJean-Pierre Pinty, Christine Lac, Tomislav Marić

2. PPM scheme • Eulerian, Piecewise Parabolic Method, introduced by Colella and Woodward in 1984 • implemented and used in many atmospheric sciences and astrophysics applications (Carpenter 1990, Lin 1994, Lin 1996, … , available in WRF) • possible to remove time-step restriction (works when Courant number > 1), Skamarock 2006

3. PPM scheme • finite volume scheme adapted for treating sharp gradients • unique parabola is fit to each grid zone and advected • monotonicity constraints can be applied to parabolas or zone fluxes • no new extremes are generated during advection • total mass conserved

4. New advection schemes in MesoNH • momentum (U, V, W) and meteorological variables • CEN4TH – centered 4th order • meteorological variables (Θ, TKE, Rx, SV) • PPM_00 – unlimited PPM • PPM_01 – monotonic PPM (Colella, Woodward), classic limiter • PPM_02 – monotonic PPM (Skamarock) • different limiter (possible extension to remove time step restriction)

5. Implementing the PPM in MesoNH • PPM algorithm requires forward in time integration, not leap-frog • extension of advection operator to 3D done with time-split scheme as described in Skamarock (2006): • sequential application of 1D algorithm • altering order at each time step (Strang, 1968)

6. 3 4 2 1 Implementing the PPM in MesoNH • advection operator in 3D, x – y – z

7. Implementing the PPM in MesoNH • advection operator in 3D, z – y – x 4 3 1 2

8. 2D test case – trapped waves CTURB = “TKEL” CCLOUD = “KESS” CRAD = “NONE” CTURBDIM = “3DIM” CTURBLEN = “DELT” dx = 250 m dz = 50 – 250 m initial sounding

9. MASDEV 4.6 UVW_ADV = CEN2ND, MET_ADV = FCT2ND U m/s 2500s 2000s 3500s 3000s

10. MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = FCT2ND t = 5000 s W U RC TKE

11. MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_00 t = 5000 s W U RC TKE

12. MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_01 t = 5000 s W U RC TKE

13. MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_02 t = 5000 s W U RC TKE

14. Real case test: Île-de-France squall line • NMODEL = 2 • Δx =10 km and 2.5km • CTURB = ‘TKEL’ • CCLOUD = ‘KESS’ • CRAD = ‘ECMWF’ • CTURBDIM = ‘1DIM’ • CTURBLEN = ‘BL89’

15. MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = CEN4TH 16H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_00 16H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv

16. MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_01 18H: ACPRT+ θv 18H: ACPRT+ θv 16H: INPRT+ θv 16H: INPRT+ θv 17H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: INPRT+ θv 18H:APRT+qv MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_02

17. New schemes - summary • both CEN4TH and PPM schemes are an order of magnitude more accurate than the CEN2ND, FCT2ND and MPDATA • CEN4TH is strongly recommended for momentum advection • PPM schemes for meteorological variables • monotonic PPM_01 or PPM_02 for variables that should remain within initial range

18. Stability and time step • PPM schemes stable up to Courant numbers (2D horizontal advection) • FCT and MPDATA schemes become unstable at much smaller Courant numbers (less than 0.35 for MPDATA)

19. Stability and time step • CEN4TH scheme stable for: • overall stability of the model improved, but still limited by the momentum advection

20. Current and future work • use unlimited PPM_00 scheme for momentum advection

21. scheme for momentum advection: CEN4TH PPM_00

22. Current and future work • fully implement the existing PPM schemes into the new version of the model, MASDEV 4.7 • parallelization

25. Stability of the advection schemes FCT Cx,y=0.25 C = 0.35 PPM_01 Cx,y = 1 C = 1.41 MPDATA Cx,y=0.25 C = 0.35

26. Testing the PPM – cyclogenesis, ω(r) • max Courant number = 0.32 • average Courant number = 0.1

27. Testing the PPM – cyclogenesis, ω(r) PPM_01 FCT

28. Testing the PPM – cyclogenesis, ω(r) PPM_01 MPDATA