Global variable-resolution semi-Lagrangian model

SL-AV: current status and further developments

Mikhail TolstykhInstitute 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, 180-200)• 3D constant-resolution version (Russian

Meteorology and Hydrology, 2001, N4) passed quasioperational tests at RHMC

• 3D dynamical core passed Held-Suarez test

Versions

• Constant resolution version – 0.9x0.72 degrees (lon x lat), 28 sigma levels (1.40625x1.125 for seasonal forecasts)

• Variable resolution version – 0.5625° lon, lat resolution varying between ~30 and 70 km, 28 levels

• Possibility to use configuration with rotated pole in future (‘advected’ Coriolis term)

For all versions parameterizations from Meteo-France ARPEGE/IFS model with minor modifications

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

Unstaggered grid

• Does not need multiple trajectories for SL• Does not need additional interpolations between U, V

and T points (hence it is easier to apply high-ordercompact finite differences).

But: U-V formulation is bad in this case (properties ofRossby and gravity waves propagation).

Vorticity-divergence formulation- is known to have better waves propagation properties than U-V formulation with B or C grid.

Requires fast and accurate solver to reconstruct U-V fromvorticity and divergence.

O(h³) –accurate solver based on FFT in longitude andcompact differences in latitude (Hybrid approach).

Normalized L2 error for u and vas functions of horizontal resolution:

left – cross-polar flow, right – Rossby-Haurwitz wave.2d – second-order algorithm, cmp – compact schemes

2d, ucmp, u2d, vcmp, v

5.

2.

5.

2.

5.

2.

5.

1.0 1.5 2.0 2.5

2d, ucmp, u2d, vcmp, v

1.E-082.5.

1.E-072.5.

1.E-062.5.

1.E-052.5.

1.E-042.5.

1.E-032

1.0 1.5 2.0 2.5

Extension to the case of variableresolution 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 2004 - Aug 2005. 12 UTC, Europe (verification against

analyses)

S1 на 48 ч.

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

1 2 3 4 5 6 7 8 9

SMA

SLM

SLMV

S1 на 24 ч.

0,00

0,05

0,10

0,15

0,20

0,25

0,30

1 2 3 4 5 6 7 8 9

SMA

SLM

SLMV

Monthly mean RMS errors of 24h and 48h T850 forecasts dec 2004 - aug 2005. 12 UTC, Europe. (verification against

analyses)

Т850, 48 h

0,0

0,5

1,0

1,5

2,0

2,5

1 2 3 4 5 6 7 8 9

RM

SE

, К

SMA

SLM

SLMV

,

Т850, 24 h

0,0

0,5

1,0

1,5

2,0

2,5

1 2 3 4 5 6 7 8 9

SMA

SLM

SLMV

Monthly mean RMS errors of 24h and 48h T850 forecasts dec 2004 - aug 2005. 12 UTC, Europe. (verification against

analyses)

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

1 2 3 4 5 6 7 8 9

SMA

SLM

SLMV

H500, 72h Н500, 24 h

0,0

0,5

1,0

1,5

2,0

2,5

1 2 3 4 5 6 7 8 9

SMA

SLM

SLMV

8,0

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

Proportion correct. Central Russia

0,0

10,0

20,0

30,0

40,0

50,0

60,0

70,0

80,0

90,0

12 24 36 48

hours

MM5-1

MM5-2

SLAV-VR

Heidke skill score (HSS), Central Russia

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

1 2 3 4

hour

MM5-1

MM5-2

SLAV-VR

Recent development work

• Design of reduced grid • Implementation of linear finite-element

scheme for integration of hydrostatics equation

• 2D nonhydrostatic version

Problem in the variable-resolution version of the model

– clustering of grid points near the poles due to convergence of meridians on the latitude-longitude grid, especially in the high-resolution case. This drawback leads to problems in use of parallel iterative solvers, calculation of grid-point humidity convergence neededfor deep convection parameterization, and also to expenses oncalculation in "wasted" grid points. The situation aggravateswhen one uses the variable resolution in latitude. It is necessary to work on reduced grid implementation

A reduced grid for the SL-AV global model(R. Yu. Fadeev)

Idea:The accuracy of the SL scheme substantially depends on the interpolation procedure

nrel 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)

Reduced grid for the SL-AV global model

The normalized r. m. s. error of the numerical solution with respect toanalytical solution numerical solution obtained on the regular grid

Solid body rotation test:Solid body rotation test:

n is the number of rotations

Williamson D. L. et al. - J. Comput. Phys., vol. 102, pp. 211-224.

Reduced grid for the SL-AV global model

n is the number of rotations

Smooth deformational flowSmooth deformational flow

The normalized r. m. s. error of the numerical solution with respect toanalytical solution numerical solution obtained on the regular grid

Doswell S. A. - J. Atmos. Sci., 1984, vol. 41, pp. 1242-1248.Nair R., et. al. - Mon. Wea. Rev., 2002, vol. 130, pp. 649-667.

Reduced grid in SL-AV

• Currently is in the implementation stage. • As a first step, will touch only calculations

of parameterizations and semi-Lagrangianadvection (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, halfwidth2.5 km; U=30 m/s; Dx=530m, 101 levels, dt=30s :

ω-velocity (left), temperature departure from constant reference profile (right)

SL-AV VR nearest future development

• Implementation of “quasi-assimilation”• Increase of resolution (horizontal to

12-15 km over Russia, vertical to at least 41 levels)

• Implementation of 3D nonhydrostaticversion with reduced grid