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 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