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

1. 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 Morín
Mariano Hortal Reymundo
Area of Modelization
Spanish Meteorological Agency (AEMET)

2. 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.

3. 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
GM
SLSI scheme for
h y d r o s t a t i c
p r i m i t i v e
equations
LAM
SLSI scheme for
non-hydrostatic
p r i m i t i v e
equations

4. 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

5. 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

6. A non-interpolating SL scheme for the continuity equation
Temporal discretization
Non-interpolating semi-Lagrangian semi-implicit time integration scheme (only for the
continuity equation).
Spacial discretization
Spectral 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

7. 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
of the wind
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.F90
Implementation of NI-SLCE

8. 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: fair
Date: 20091015 12UTC to 20091015 12UTC
NHem Extratropics (lat 20.0 to 90.0, lon -180.0 to 180.0)
Anomaly correlation
500hPa geopotential
NISLCE
MR

9. Results

10. Results

11. 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.