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

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

PPM schemeEulerian, Piecewise Parabolic Method, introduced by Colella and Woodward in 1984implemented 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

PPM schemefinite volume scheme adapted for treating sharp gradientsunique parabola is fit to each grid zone and advectedmonotonicity constraints can be applied to parabolas or zone fluxesno new extremes are generated during advectiontotal mass conserved

New advection schemes in MesoNHmomentum (U, V, W) and meteorological variablesCEN4TH centered 4th ordermeteorological variables (, TKE, Rx, SV)PPM_00 unlimited PPMPPM_01 monotonic PPM (Colella, Woodward), classic limiterPPM_02 monotonic PPM (Skamarock)different limiter (possible extension to remove time step restriction)

Implementing the PPM in MesoNHPPM 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 algorithmaltering order at each time step (Strang, 1968)

Implementing the PPM in MesoNHadvection operator in 3D, x y z

Implementing the PPM in MesoNHadvection operator in 3D, z y x 1342

2D test case trapped wavesCTURB = TKELCCLOUD = KESSCRAD = NONECTURBDIM = 3DIMCTURBLEN = DELTdx = 250 mdz = 50 250 minitial sounding

MASDEV 4.6 UVW_ADV = CEN2ND, MET_ADV = FCT2ND2000 s2500 s3000 s3500 sU m/s

MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = FCT2NDt = 5000 sUTKEWRC

MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_00Ut = 5000 sTKEWRC

MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_01Ut = 5000 sTKEWRC

MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_02Ut = 5000 sTKEWRC

Real case test: le-de-France squall lineNMODEL = 2x =10 km and 2.5kmCTURB = TKELCCLOUD = KESSCRAD = ECMWFCTURBDIM = 1DIMCTURBLEN = BL89

MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_,rv,TKE = CEN4TH MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_,rv,TKE = PPM_00 16H: INPRT+ v17H: INPRT+ v18H: INPRT+ v18H: ACPRT+ v16H: INPRT+ v17H: INPRT+ v18H: INPRT+ v18H: ACPRT+ v

18H:APRT+qvMASDEV4_7: ADV_u,v,w = CEN4TH, ADV_,rv,TKE = PPM_01 MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_,rv,TKE = PPM_02

New schemes - summaryboth CEN4TH and PPM schemes are an order of magnitude more accurate than the CEN2ND, FCT2ND and MPDATACEN4TH is strongly recommended for momentum advectionPPM schemes for meteorological variablesmonotonic PPM_01 or PPM_02 for variables that should remain within initial range

Stability and time stepPPM 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)

Stability and time stepCEN4TH scheme stable for:overall stability of the model improved, but still limited by the momentum advection

Current and future workuse unlimited PPM_00 scheme for momentum advection

CEN4THPPM_00scheme for momentum advection:

Current and future workfully implement the existing PPM schemes into the new version of the model, MASDEV 4.7parallelization

Stability of the advection schemesPPM_01Cx,y = 1C = 1.41FCTCx,y=0.25C = 0.35MPDATACx,y=0.25C = 0.35

Testing the PPM cyclogenesis, (r)max Courant number = 0.32average Courant number = 0.1

Testing the PPM cyclogenesis, (r)PPM_01FCT

Testing the PPM cyclogenesis, (r)PPM_01MPDATA