A non-interpolating semi-Lagrangian scheme for the continuity equation of the ECMWF forecast

model.

Project 4: Formation in technical used in the dynamics of the numerical weather models.

Tomás Morales MorínMariano Hortal Reymundo

Area of Modelization

Spanish Meteorological Agency (AEMET)

Outline

• Conclusions of previous works.

• A non-interpolating semi-Lagrangian scheme for the continuity equation (NI-SLCE).

• Implementation of NI-SLCE in semi-Lagrangian semi-implicit (SLSI) scheme of the ECMWF Forecast Model.

• New conclusions.

IFS

Spectral Spacial

Correction

SL Interpolation Correction

HARMONIE

2-D and 3-D interpolation

scheme

Non-Interpolating

semi-Lagrangian scheme

Conservation in Continuity equation

Conservation of NOx, Ozone, Humidity equation without sink and source terms

GMSLSI scheme for h y d r o s t a t i c p r i m i t i v e equations

LAMSLSI scheme for non-hydrostat ic p r i m i t i v e equations

Why should we use a NI-SLCE scheme to calculate the value of the fields at the departure point of the semi-Lagrangian trajectory?

Conclusions of previous works

SL scheme

NI-SL scheme

dX

dt= RHS XA,+ = XD,0 +RHSD,0

XA,+ =

⇢X � d

�X

�x

�⇤,0+RHS⇤,0

�X

�t+ (v

0+ v⇤)

�X

�x= RHS

D⇤X

Dt+ v

0 �X

�x= RHS

• •• •dd

0

d⇤v = v

0+ v⇤

NI-SLCE

A non-interpolating SL scheme for the continuity equation

Temporal discretizationNon-interpolating semi-Lagrangian semi-implicit time integration scheme (only for the continuity equation).

Spacial discretizationSpectral in the horizontal (spherical harmonics).

Pseudo-spectral (finite-element) vertical representation (cubic B-spline).

LnpA,+sup = {Lnpsup � d⇥Lnpsup}⇤,0 + {N.L.+ L.}⇤,0 + {L.}A,+

LnpA,+sup + {L.}A,+ = {Lnpsup � d⇥Lnpsup}⇤,0 + {N.L.+ L.}⇤,0

A B C

21

NI-SLCE

Stepo.F90 / scan2h.F90 / scan2mdm.F90

gp_model.F90

3. Dynamics 4. ECMWF Radiation 5. SL Interpolation

cpg.F90

cpg_dyn.F90

lacdyn.F90

3. Computation o f t he w ind c o m p o n e n t s necessary for SL trajectory

4. Computations of the 3-D eq. RHS terms

5. Computations of the 2-D eq. RHS terms

lavent.F90 lattex.F90 lattes.F90

call_sl.F90

3. SL trajectory research weight and interpolation grid calculation

5. Interpolations

lapinea.F90

larmes.F90

.

..

.

lapineb.F90

larcinb.F90

OpenMP

laiddi.F90 laitli.F90

binislce.F90 trinislce.F90

Bi-dimensional 1 2 - p o i n t interpolations (in horizontal)

T r i l i n e a r interpolations for one variable

/module/ptrslb1.F90 /setup/suslb.F90Implementation of NI-SLCE

Results

40

50

60

70

80

90

100

%

0 1 2 3 4 5 6 7 8 9 10

Forecast Day

es oper 12UTC | Mean method: fairDate: 20091015 12UTC to 20091015 12UTCNHem Extratropics (lat 20.0 to 90.0, lon -180.0 to 180.0)

Anomaly correlation500hPa geopotential

NISLCE

MR

Results

Results

New conclusions

Conclusions

• The NI-SLCE in the horizontal scheme does not improve the conservation of the mass in the continuity equation.

• It is better to continue using interpolation methods to calculate the value of the field at the departure point of the semi-Lagrangian trajectory.