1. The new PPM advection schemes in the MesoNH Jean-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 4 th 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. Implementing the PPM in MesoNH advection operator in 3D, x – y – z 1 2 34

7. Implementing the PPM in MesoNH advection operator in 3D, z – y – x 1 3 4 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 2000 s 2500 s 3000 s 3500 s U m/s

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

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

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

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

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 MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_00 16H: INPRT+ θv17H: INPRT+ θv18H: INPRT+ θv18H: ACPRT+ θv 16H: INPRT+ θv17H: INPRT+ θv18H: INPRT+ θv18H: ACPRT+ θv

16. 18H:APRT+qv MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_01 16H: INPRT+ θv17H: INPRT+ θv18H: INPRT+ θv 18H: ACPRT+ θv MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_02 16H: INPRT+ θv17H: INPRT+ θv18H: INPRT+ θv 18H: ACPRT+ θv

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. CEN4THPPM_00 scheme for momentum advection:

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 PPM_01 Cx,y = 1 C = 1.41 FCT Cx,y=0.25 C = 0.35 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_01FCT

28. Testing the PPM – cyclogenesis, ω(r) PPM_01MPDATA