Global variable-resolution semi-Lagrangian vorticity-

divergence NWP model

Mikhail Tolstykh

Institute of Numerical Mathematics

Russian Academy of Sciences, and Russian Hydrometeorological

Research Centre

Moscow Russia

2

Unstaggered grid is used to avoid expensivemultidimensional high-order reinterpolations.Hence – vorticity-divergence formulation

Currently 1.125/1.40625 degrees lat/lon, 28sigma levels

Possibility to use configuration with rotatedpole

2 time-level scheme dt=36 min, ‘advected’Coriolis term

4th order compact differences for discretizationof derivatives in non-advective terms,including SI scheme and U-V reconstruction

Direct FFT solvers for semi-implicit scheme,U-V reconstruction, and 4th order horizontaldiffusion

Parameterizations from operational Meteo-France ARPEGE/IFS model with some minormodifications (in deep convection closure,shallow convection, and stratiformprecipitation scheme)

3

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-order compactfinite differences).

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

Hence 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).

4

5

7

Momentum equations are used only to obtainthe discrete divergence equation!

Divergence equation is more difficult todiscretize and integrate stably with large timesteps (because of metric terms)

than

to take divergence from RHSs of momentumequations (known departure and arrival pointterms).

(the same as in spectral SL models :-))

Price: one more 3D interpolation in the SL part

8

Distinct features of discretization

Discretization of the hydrostatic equation:Trapezoidal instead of midpoint rule

Orographic resonance treatment: Eulerianspatially averaged + first-order uncentering inall equations except for absolute vorticityequation (hence undisturbed Rossby waves)(Y. Li and J.R.Bates, QJ 96)

Interpolations in semi-Lagrangian advection:Akima in vertical in the PBL

9

Results of forecasts using ECMWF data for 1996

AveragedNORD20 RMS

errors

12 forecastsstarting on 15th

of each month1.5x1.5 deg and 1.40625x1.125

deg versions

10

Results of data assimilation experiments using RHMC OI

(M.Tsyroulnikov,

A.Bagrov,

R.Zaripov)08-28/02/2000

First guess geopotential errors vs radiosondes

(red lines – 1.40625x1.125 deg model,blue lines -

1.5x1.5 deg model,

full line - RMS, dashed line - bias)

P, hPа

0-10 10 20 30 40 50 60 70 80

1000

925

850

700

500

400

300

250

200

150

100

70

50

30

20

10

м

11

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0

1000

925

850

700

500

400

300

250

200

150

100

70

50

30

20

P, гПа

K0 1 2 3 4 5 6 7 8

1000

925

850

700

500

400

300

250

200

150

100

70

50

30

20

10P, гПа

м/с

First guess RMS errors for temperature and wind

12

NORD20 geopotential errors vs analyses

33 forecasts,10-25/02/00, for00 and 12 UTC

Blue line – 24h,red line –72h.

Full line – RMS,dashed line -

bias0-10 10 20 30 40 50 60 70 80 90

1000

925

850

700

500

400

300

250

200

150

100

70

50

30

20

10P, гПа

м

13

Extension to the case of variable resolution in latitude

Discrete coordinate transfromation (given in the differential way, as a sequence of local map factors). This requires very moderate changes in the constant resolution code (introduction of map factors in computation of gradients, SI etc) and also allows to preserve all compact differencing and its properties intact.

Only one sphere is used everywhere. Some changes in the semi-Lagrangian

advection - interpolations and search of trajectories on a variable mesh.

14

Example of the latitude partition used in experiments.The lowest resolution should be at least 250 km.

15

16

17

18

2d, ucmp, u2d, vcmp, v

5.

1.E-06

2.

5.

1.E-05

2.

5.

1.E-04

2.

5.

1.E-03

1.0 1.5 2.0 2.5

Normalized L2 error, polar flow

UV

Deg1.5

2

3

57

1e-04

1.52

3

57

1e-03

1.0 1.5 2.0 2.5

Reconstruction of U-V field: analytic cross-polar flow

Constant and variable resolution

19

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

Normalized L2 error, RG wave

UV

Deg1.5

2

3

57

1e-04

1.52

3

57

1e-03

1.0 1.5 2.0 2.5

Reconstruction of U-V field: analytic RG-4 wave

Constant and variable resolution Normalized global RMS error

20

21

22

Test 2

2.5 2 1.5

x 10-4

Hours0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 24 48 72 96 120

Shallow water model on the sphere

Standard test set: case 2 (rotated zonalgeostrophic flow)

Constant and variable resolutionNormalized global RMS height error

128x80256x160256x160-1.1384x240

Hours5

7

1e-04

1.5

2

3

45

7

1e-03

1.5

2

3

24 48 72 96 120

23

Standard test set: case 3(similar to previous, but the wind field is nonzero

in a limited region) Constant and variable resolutionNormalized global RMS height error

Test 3

2.52 1.5

x 10-4

Hours0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

0 24 48 72 96 120

128x80256x160256x160-1.1384x240

Hours3

5

7

1e-04

1.52

3

5

7

1e-03

1.52

24 48 72 96 120

24

Standard test set: case 6(Rossby-Haurwitz wave N 4)

Constant and variable resolutionNormalized global (and high-res area) RMS height error

Test 6

2.52 1.5

x 10-3

Days0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0. 5. 10.

256x160 Glob256x160-1.1 Glob256x160 Hires256x160-1.1 Hires

x 10-3

Days0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

5 10

25

Test 7a

2.52 1.5

x 10-3

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0 24 48 72 96 120

Standard test set: case 7a (“Real” data 21/12/78)

Constant and variable resolutionNormalized global (and high-res area) RMS height error

1.075 Glob1.1 Glob1.075 Hires1.1 Hires

x 10-3

Hours1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

24 48 72 96 120

26

Test 7b

2.52 1.5

x 10-3

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 24 48 72 96 120

Standard test set: case 7b (“Real” data 16/01/79)

Constant and variable resolutionNormalized global (and hi-res area) RMS height error

1.075 Glob1.1 Glob1.075 Hires1.1 Hires

x 10-3

Hours0.0

1.0

2.0

3.0

4.0

5.0

24 48 72 96 120

27

Test 7c

2.52 1.5

x 10-3

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

0 24 48 72 96 120

1.075 Glob1.1 Glob1.075 Hires1.1 Hires

x 10-3

Hours

1.00

2.00

3.00

4.00

5.00

24 48 72 96 120

Standard test set: case 7c(“Real” data 09/01/79)

Constant and variable resolutionNormalized global (and high-res area) RMS height

error

28

Conclusions

The variable resolution version of the 2Dmodel is capable to produce accurateforecasts in the high-resolution zone for 3-4days range. Beyond thisrange, the solution degrades rapidly.

The extension to the 3D case isstraightforward and is being implementedcurrently.

It is necessary to implement a reduced grid