Semi-Lagrangian
Dynamics in GFS
Sajal K. Kar
Introduction



Over the years, the accuracy of medium-range
forecasts has steadily improved with increasing
resolution at the ECMWF. Introduction of the semiLagrangian (SL) treatment of advection has been
recognized as a contributing factor.
An SL scheme for advection, compared to an Eulerian
scheme, allows larger time steps, thus improving
model efficiency, particularly at high resolutions.
Joe Sela and colleagues developed a semiLagrangian semi-implicit (SLSI) version of the
Eulerian-SI (operational) GFS. The numerical
schemes used broadly follow the ECMWF approach.
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Some details of SL GFS

Hydrostatic shallow-atmosphere primitive equations in
terrain-following s-p hybrid vertical coordinate on
Lorenz grid.
 Prognostic field variables include u, v, Tv, lnps, q, and
a few other tracers.
 Vertical finite-difference scheme designed to conserve
angular momentum and total energy.
 Governing equations are space-time discretized using
the SLSI-SETTLS scheme. SETTLS stands for the
Stable-Extrapolation Two-Time-Level Scheme (Hortal
2002, QJRMS), which will be reviewed later.
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More details of SL GFS
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The ECMWF model employs a quasi-cubic Lagrangepolynomial 3D interpolation for the prognostic fields,
whereas a tri-linear interpolation is used for the other
rhs terms.
 The SL GFS includes options of (i) Hermite-, (ii) quasicubic Lagrange-polynomial 3D interpolations used for
all fields to be interpolated.
 ECMWF model employs finite-elements and SL
employs finite-difference in the vertical.
 Recently, we have added options of a SETTLS based
departure-point scheme and a ‘modified Lagrange’
interpolation scheme that mimics the ECMWF.
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Semi-Lagrangian vs. Eulerian


The SLSI-SETTLS scheme, compared to the Eulerian
time-filtered-leapfrog-SI scheme, is a two time-level
scheme. Thus, the SL model does not need a time
filter and is relatively more efficient.
Overhead of the SL scheme comes from the
departure-point calculations and 3D/2D interpolations
of prognostic variables to the departure points at each
time step. However, reasonably large time steps
allowed by the SL scheme offsets this computational
overhead.

For example, T574 Eulerian GFS, equivalent grid-resolution 27 km,
uses a time step of 180 s. The SL GFS for T1534, with equivalent
grid-resolution of 13 km, can use a time step of 450 s.
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Semi-Lagrangian Explicit SETTLS
Forcedadvectionequation for  :
d
 R( ).
dt
ExplicitSL  SET T LSfor (1)

n 1
A
(1)

1 n 1
n 1 2
 RM  ( RD  RAn ),
( 2)
t
2
A  Arrivalgrid point ;D  Departurenongridpoint .
n
D
Ext rapolation in t ime: RAn 1  2 RAn  RAn 1.
n 1
D
Int erpolation in space: R
 (2 R  R
n
A
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(3a)
n 1
A
D
) .
(3b)
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Details of the SETTLS
.
Backward Trajectory:
A
M
D
t + t
t + t/2
t
SETTLS uses the points, (D, t+t) and (A, t), to
evaluate the R term at M. Also used to locate
departure points.
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Semi-Lagrangian Semi-Implicit
SETTLS
Forced
advectionequationfor  rewrittenas
.
d
 R( )  L()  N ()
(4)
dt
L  Linearpartof R to be treatedwithCN scheme
N  'Nonlinear'partof R to be treatedwithSETTLS
SLSI SETTLS for (4)
An1  Dn 1 n1 n
1
 ( LA  LD )  N Mn1 2  ( LnA1  2 LnM1 2  LnD )  ( LnM1 2  N Mn1 2 )
t
2
2


( LnA1  2 LnM1 2  LnD )  RMn1 2 

( tt L) nM1 2  RMn1 2 ,
2
2
  0  SL ExplicitSETTLS;   1  SLSI SETTLS
 tt L  Semi- implicitadjustmentterm.
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(5)
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Updates on SL GFS
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Recently, a systematic inter-comparison of 4 selected SL options
in the SL-T1148 with the Eu-T574 was carried out in test runs
without cycling. See details of the SL options in
http://www.emc.ncep.noaa.gov/mmb/skar/Kar_GCWMB_2012112
9.pdf and the model inter-comparison scores in
http://www.emc.ncep.noaa.gov/mmb/skar/SLG1134_options2_.pp
tx
Encouraged by the T1148 results and other considerations, the
SL GFS (T1534-L64) is slated for operational implementation in
2014.
Preliminary test runs without cycling of SL-T1534 are being
carried out.
The SL GFS is about to become a part of NEMS.
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500 hPa Hgt AC for SL-T1534
(Courtesy of DaNa Carlis)
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References
IFS Documentation – Cy38r1, 2012: Part III: Dynamics
and numerical procedures, 1-29.
 Sela, J., 2009a: The implementation of the sigma
pressure hybrid coordinate into the GFS. NOAA/NCEP
Office Note 461. 25 pp.
 Sela, J., 2010: The derivation of the sigma pressure
hybrid coordinate semi-Lagrangian model equations
for the GFS. NOAA/NCEP Office Note 462. 31 pp.

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