Global variable-resolution semi-Lagrangian model SL-AV: current status and further developments Mikhail Tolstykh Institute of Numerical Mathematics, Russian Academy of Sciences and Hydrometeorological Research Center of Russia

SL-AV model (semi-Lagrangian absolute vorticity) Shallow water constant-resolution version demonstrated the accuracy of a spectral model for most complicated tests from the standard test set (JCP 2002 v. 179, ) 3D constant-resolution version (Russian Meteorology and Hydrology, 2001, N4) passed quasioperational tests at RHMC 3D dynamical core passed Held-Suarez test

SL-AV model (constant resolution version) Accepted by Roshydromet comission 27/01/06 (forecast of upper-air fields and MSLP) Precipitation forecasts are on trials since 01/07/06 Suppression of spurious orographic resonance (Nov. 2005) PBL parameterization with “interactive mixing length” (PBL height is calculated following Ayotte-Piriou-Geleyn-Tudor) ISBA parameterization and assimilation scheme close to enter

Features of dynamics Semi-Lagrangian scheme – SETTLS (Hortal, QJRMS 2003) Semi-implicit scheme – follows (Bates et al, MWR 1993) but with trapezoidal rather than midpoint rule in hydrostatic equation 4th-order differencing formulae (compact and explicit) for horizontal derivatives Direct FFT solvers for semi-implicit scheme, U-V reconstruction, and 4th order horizontal diffusion

Held-Suarez test of 3D dynamical core ( 2 degrees lat/lon resolution, 20 levels )

Parallel implementation for version 0.225ºх0.18ºх28

2d, u cmp, u 2d, v cmp, v d, u cmp, u 2d, v cmp, v 1.E E E E E E

Extension to the case of variable resolution in latitude  Discrete coordinate transformation (given as a sequence of local map factors), subject to smoothness and ratio constraints. This requires very moderate changes in the constant resolution code (introduction of map factors in computation of gradients, semi-implicit scheme etc) and also allows to preserve all compact differencing and its properties intact.  Some changes in the semi-Lagrangian advection - interpolations and search of trajectories on a variable mesh.

Monthly mean skill score S1 H500 of 24 and 48h forecasts. Dec Aug UTC, Europe (verification against analyses)

Monthly mean RMS errors of 24h and 48h T850 forecasts dec aug UTC, Europe. (verification against analyses)

Preliminary evaluation of precipitation forecasts over Central European part of Russia during 1/07-24/09/2006 Models compared: Two versions of ММ5 running at Hydrometcentre with 18 km resolution (MM5-1) and Moscow Hydrometeobureau with 15 km res. (MM5-2), and SL-AV VR model (30 km over Russia). MM5 used NCEP analyses, SL-AV VR used interpolated analyses of Hydrometcentre OI data assimilation for constant –resolution SL-AV model Period is too short to make conclusions

Recent development work Design of reduced grid Implementation of linear finite-element scheme for integration of hydrostatics equation 2D nonhydrostatic version

Idea:The accuracy of the SLscheme substantially depends on the interpolation procedure A reduced grid for the SL-AV global model (R. Yu. Fadeev)

n rel is the relative reduction of the total number of nodes with respect to the regular grid Reduced grid for the SL-AV global model (R. Yu. Fadeev)

The normalized r. m. s. error of the numerical solution with respect to analytical solution numerical solution obtained on the regular grid Solid body rotation test: n is the number of rotations Williamson D. L. et al. - J. Comput. Phys., vol. 102, pp Reduced grid for the SL-AV global model

n is the number of rotations Smooth deformational flow Reduced grid for the SL-AV global model The normalized r. m. s. error of the numerical solution with respect to analytical solution numerical solution obtained on the regular grid Doswell S. A. - J. Atmos. Sci., 1984, vol. 41, pp Nair R., et. al. - Mon. Wea. Rev., 2002, vol. 130, pp

Reduced grid in SL-AV Currently is in the implementation stage. As a first step, will touch only calculations of parameterizations and semi-Lagrangian advection (including calculations of variables to be interpolated).

Av. mean error of geopotential forecasts vs. time for FD scheme (left), linear FE scheme (right) (Aug. 2005, 12 UTC, Southern Hemisphere)

Av. RMS error of geopotential forecasts vs. time for FD scheme (left), linear FE scheme (right) (Aug. 2005, 12 UTC, Southern Hemisphere)

2D nonhydrostatic dynamical core Based on SL-AV dynamics approaches (vorticity- divergence formulation on the unstaggered grid, high- order finite differences) and NH HIRLAM core developed by R.Room et al. Room’s approach is semi-implicit semi-Lagrangian, does not contain triple nonlinear terms, however has many simplifications in equations. It is planned to implement first 3D version with Room’s approach and the modify it (drop at least some of simplifications)

2D mountain waves: Agnesi hill 250m height, halfwidth 2.5 km; U=30 m/s; Dx=530m, 101 levels, dt=30s :  -velocity (left), temperature departure from constant reference profile (right)

SL-AV VR nearest future development Implementation of “quasi-assimilation” Increase of resolution (horizontal to km over Russia, vertical to at least 41 levels) Implementation of 3D nonhydrostatic version with reduced grid