Linear advection characteristics of a variable resolution global spectral method on the sphere

Transcription

1 Linear advection characteristics of a variable resolution global spectral method on the sphere S. Janakiraman Seasonal Prediction of Indian Monsoon group, Centre for Development of Advanced Computing, Bangalore,India. 25th September 2012 PDEs on the Sphere , Isaac Newton Institute,Cambridge,UK.

2 Outline Background Variable resolution global spectral method High resolution Tropical Belt Transformation Properties Linear advection experiments on the sphere Eulerian formulation Solid body rotation experiments - results Summary and Conclusions References

3 Variable resolution global spectral method spherical harmonics basis - global method Extension of the Schmidt's transformation approach [Schmidt(1977)]. Resolves the tropical belt with ner resolution Resolution decreases as we move towards the poles

4 High resolution Tropical Belt Transformation A smooth coordinate transformation : (λ, φ) (λ, φ ) given by λ =lλ [ ( π φ =2 arctan tan l 4 + φ )] π 2 2 ( Notice that λ [0, 2πl) and φ π 2, π ) for λ [0, 2π) ( 2 and φ π 2, π ) 2 (λ, φ ) refers to a unique point on the sphere through the coordinate function σ : (cos η 1 (λ ) cos(φ ), sin η 1 (λ ) cos η 2 (φ ), sin η 2 (φ )) [ ( )] η 1 (λ ) = λ l and η 2(φ ) = 2 arctan tan 1/l π 4 + φ are the 2 inverse coordinate transformations.

5 Reparametrisation HTBT is a conformal coordinate transformation on the sphere Not conformal at the poles. Equal distribution of points in the (λ, φ ) plane such that λ = φ provides variable concentration of points through σ.

6 Variable resolution through HTBT Figure: Orthographic projection of the variable resolution grid obtained

7 HTBT spectral method Transformed spherical harmonics B m n (λ, φ ) are used expand the functions on the sphere. B m n (λ, φ ) = Y m n (λ, φ ) (Relationship with standard spherical harmonics). FFT and Gaussian quadrature are used evaluate the spectral coecients. Detailed description provided in [Janakiraman et al.(2012)janakiraman, Nanjundiah, and Murthy].

8 Linear advection equation The advective form of the equation is used in the study h t = V h Gaussian hill is dened by h(λ, φ) = a0 exp ( [ b0 (x x c ) 2 + (y y c ) 2 + (z z c ) 2]) a 0 = 6000, b0 = 10 and R 2 R is the radius. The advection eqn is discretised in the spectral space of (λ, φ ).

9 Time-integration 4th order Runge-Kutta scheme is used for time-integration. Time-step size t = 900s. No ltering, No diusion used in this study. To understand the basic nature of the method with respect to transport properties.

10 Advection over the tropical belt

11 Advection over the tropical belt

12 Advection over the tropical belt

13 Advection over the high-latitude

14 Advection over the high-latitude

15 Advection over the high-latitude

16 Advection across latitudes(α = π/4)

17 Advection across latitudes(α = π/4)

18 Advection across latitudes(α = π/4)

19 Advection over poles (α = π/2)

20 Advection over poles (α = π/2)

21 Advection over poles (α = π/2) The resolution at the pole is coarsest possible achieved. Gaussian hill undergoes severe truncation due to coarse resolution. This is a case of representation error cuased by inadequate resolution [Naughton et al.(1996)naughton, Courtier, and Bourke].

22 Advection from low to high res. region

23 Advection from a low to high res. region

24 Advection from low res. region

25 Advection near the pole

26 Advection near the pole

27 Advection near the pole

28 Summary Linear advection characteristics of the variable resolution global spectral method described through solid-body rotation experiment. Advection over the tropical belt is near dispersion free and accurate. Even the advection along a higher latitude also shows good transport characteristics. The transport from high resolution region to low resolution shows some errors due to change in the resolution. The major limitation is the advection over the pole. The feature advected over the pole show severe truncation errors. We can infer that it is due to the very coarse resolution at the poles. Gaussian hill advected from a region close to pole does not show much errors.

29 Conclusions Variable resolution global spectral method shows good advection characteristics for the tropical region. Also, the advection characteristics along the zonal direction is good. Thus it is suitable for resolving the tropical dynamics etc. A major limitation is the poor advection characteristics for the advection over the poles. So this method not suitable for studies that require accurately resolving the cross-polar ows. Further work is needed to improve this aspect of polar advection. Higher order adaptive lters [Boyd(2001)] are promising approach here.

30 Acknowledgements Prof. Ravi S Nanjundiah, Centre for Atmospheric and Oavel suppoceanic Sciences, Indian Institute of Science, Bangalore. (Joint work). Isaac Newton Institute for supporting stay and travel. National Board of Higher Mathematics, Government of India for partial travel support.

31 Thank you for the attention For any queries and comments, you can mail to : jraman@cdac.in

32 F. Schmidt, Variable ne mesh in spectral global models, Beiträge zur Physik der Atmosphäre 50 (12) (1977) S. Janakiraman, R. S. Nanjundiah, A. V. Murthy, A novel variable resolution global spectral method on the sphere, Journal of Computational Physics 231 (7) (2012) , ISSN , doi: /j.jcp , URL S M. Naughton, P. Courtier, W. Bourke, Representation errors in various grid and spectral truncations for a symmetric feature on the sphere, Quarterly Journal of the Royal Meteorological Society 122 (529) (1996) J. P. Boyd, Chebyshev and Fourier spectral methods, Courier Dover Publications, 2001.