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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. NEMS/GFS Modeling Summer School 2 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. NEMS/GFS Modeling Summer School 3 More details of SL GFS 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. NEMS/GFS Modeling Summer School 4 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. NEMS/GFS Modeling Summer School 5 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 NEMS/GFS Modeling Summer School (3a) n 1 A D ) . (3b) 6 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. NEMS/GFS Modeling Summer School 7 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) An1 Dn 1 n1 n 1 ( LA LD ) N Mn1 2 ( LnA1 2 LnM1 2 LnD ) ( LnM1 2 N Mn1 2 ) t 2 2 ( LnA1 2 LnM1 2 LnD ) RMn1 2 ( tt L) nM1 2 RMn1 2 , 2 2 0 SL ExplicitSETTLS; 1 SLSI SETTLS tt L Semi- implicitadjustmentterm. NEMS/GFS Modeling Summer School (5) 8 Updates on SL GFS 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. NEMS/GFS Modeling Summer School 9 500 hPa Hgt AC for SL-T1534 (Courtesy of DaNa Carlis) NEMS/GFS Modeling Summer School 10 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. NEMS/GFS Modeling Summer School 11