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A Semi-Lagrangian Laplace Transform Filtering Integration Scheme

A Semi-Lagrangian Laplace Transform Filtering Integration Scheme. Colm Clancy and Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin. Aim. To develop a time-stepping scheme that filters

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A Semi-Lagrangian Laplace Transform Filtering Integration Scheme

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  1. A Semi-Lagrangian Laplace Transform Filtering Integration Scheme Colm Clancy and Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin

  2. Aim • To develop a time-stepping scheme that filters high-frequency noise based on Laplace Transform theory • First used by Lynch (1985). Further work in Lynch (1986), (1991) and Van Isacker & Struylaert (1985), (1986) PDEs On The Sphere 2010

  3. In This Talk • Describe a semi-Lagrangian trajectory Laplace Transform scheme • Compare with semi-implicit schemes in shallow water model. and show benefits when orography is added: Stability No orographic resonance PDEs On The Sphere 2010

  4. LT Filtering Integration Scheme • At each time-step, solve for the Laplace Transform of the prognostic variables • Alter the inversion so as to remove high-frequency components (numerically) PDEs On The Sphere 2010

  5. LT Filtering Integration Scheme PDEs On The Sphere 2010

  6. Phase Error Analysis Relative Phase Change: R = (numerical) / (actual) PDEs On The Sphere 2010

  7. PDEs On The Sphere 2010

  8. Semi-Lagrangian Laplace Transform • Define the LT along a trajectory • Then PDEs On The Sphere 2010

  9. Semi-Lagrangian Laplace Transform SLLT • Based on spectral SWEmodel (John Drake, ORNL) • Compared with semi-Lagrangian semi-implicit SLSI • Stability not dependent on reference geopotential PDEs On The Sphere 2010

  10. Shallow Water Equations PDEs On The Sphere 2010

  11. Orographic Resonance • Spurious resonance from coupling semi-Lagrangian and semi-implicit methods [reviewed in Lindberg & Alexeev (2000)] • LT method has benefits over semi-implicit schemes • Motivates investigating orographic resonance in SLLT model PDEs On The Sphere 2010

  12. Orographic Resonance Analysis • Linear analysis of orographically forced stationary waves • Numerical simulations with shallow water SLLT • Results consistently show benefits of SLLT scheme PDEs On The Sphere 2010

  13. Linear Analysis: (Numerical)/(Analytic) Analytic solution vanishes Spurious numerical resonance PDEs On The Sphere 2010

  14. Linear Analysis: (Numerical)/(Analytic) Analytic solution vanishes PDEs On The Sphere 2010

  15. Test Case with 500hPa Data • Initial data: ERA-40 analysis of 12 UTC 12th February 1979 • Used by Ritchie & Tanguay (1996) and Li & Bates (1996) • Running at T119 resolution PDEs On The Sphere 2010

  16. PDEs On The Sphere 2010

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  20. Efficiency • Symmetry in the LT inversion • Relative overhead of SLLT method, compared to SLSI: • Reduces with increasing resolution PDEs On The Sphere 2010

  21. Conclusions • Shallow water model using a semi-Lagrangian Laplace Transform method • Advantages over a semi-implicit method • Accurate phase speed • Stability • No orographic resonance PDEs On The Sphere 2010

  22. PDEs On The Sphere 2010

  23. References • Li Y., Bates J.R. (1996): A study of the behaviour of semi-Lagrangian models in the presence of orography. Quart. J. R. Met. Soc., 122, 1675-1700 • Lindberg K., Alexeev V.A. (2000): A Study of the Spurious Orographic Resonance in Semi-Implicit Semi-Lagrangian Models.Monthly Weather Review, 128, 1982-1989 • Lynch P. (1985): Initialization using Laplace Transforms. Quart. J. R. Met. Soc., 111, 243-258 • Lynch P. (1986): Initialization of a Barotropic Limited-Area Model Using the Laplace Transform Technique. Monthly Weather Review, 113, 1338-1344 • Lynch P. (1991): Filtering Integration Schemes Based on the Laplace and Z Transforms. Monthly Weather Review, 119, 653-666 • Ritchie H., Tanguay M. (1996): A Comparison of Spatially Averaged Eulerian and Semi-Lagrangian Treatments of Mountains. Monthly Weather Review, 124, 167-181 • Van Isacker J., Struylaert W (1985): Numerical Forecasting Using Laplace Transforms. Royal Belgian Meteorological Institute Publications Serie A, 115 PDEs On The Sphere 2010

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