the new ppm advection schemes in the mesonh jean-pierre pinty, christine lac, tomislav mari ć
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The new PPM advection schemes in the MesoNH
Jean-Pierre Pinty, Christine Lac, Tomislav Marić
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
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
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)
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)
Implementing the PPM in MesoNH
• advection operator in 3D, x – y – z
1 2
3 4
Implementing the PPM in MesoNH
• advection operator in 3D, z – y – x
1
3 4
2
2D test case – trapped waves
CTURB = “TKEL”CCLOUD = “KESS”CRAD = “NONE”CTURBDIM = “3DIM”CTURBLEN = “DELT”dx = 250 mdz = 50 – 250 m
initial sounding
MASDEV 4.6 UVW_ADV = CEN2ND, MET_ADV = FCT2ND
2000 s 2500 s
3000 s3500 s
U m/s
MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = FCT2ND
t = 5000 s
U
TKE
W
RC
MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_00
U
t = 5000 s
TKE
W
RC
MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_01
U
t = 5000 s
TKE
W
RC
MASDEV 4.7 UVW_ADV = CEN4TH, MET_ADV = PPM_02
U
t = 5000 s
TKE
W
RC
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’
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+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv
16H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv
18H:APRT+qv
MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_01 16H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv
MASDEV4_7: ADV_u,v,w = CEN4TH, ADV_θ,rv,TKE = PPM_02 16H: INPRT+ θv 17H: INPRT+ θv 18H: INPRT+ θv 18H: ACPRT+ θv
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
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)
Stability and time step
• CEN4TH scheme stable for:
• overall stability of the model improved, but still limited by the momentum advection
Current and future work
• use unlimited PPM_00 scheme for momentum advection
CEN4TH PPM_00
scheme for momentum advection:
Current and future work
• fully implement the existing PPM schemes into the new version of the model, MASDEV 4.7– parallelization
Stability of the advection schemes
PPM_01Cx,y = 1C = 1.41
FCTCx,y=0.25C = 0.35
MPDATACx,y=0.25C = 0.35
Testing the PPM – cyclogenesis, ω(r)
max Courant number = 0.32
• average Courant number = 0.1
Testing the PPM – cyclogenesis, ω(r)
PPM_01 FCT
Testing the PPM – cyclogenesis, ω(r)
PPM_01 MPDATA
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