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. PDEs On The Sphere 2010 24th August 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) Aim

3. PDEs On The Sphere 2010 24th August 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

4. PDEs On The Sphere 2010 24th August 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)

5. PDEs On The Sphere 2010 24th August LT Filtering Integration Scheme

6. PDEs On The Sphere 2010 24th August Phase Error Analysis Relative Phase Change: R = (numerical) / (actual)

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8. PDEs On The Sphere 2010 24th August Define the LT along a trajectory Then Semi-Lagrangian Laplace Transform

9. PDEs On The Sphere 2010 24th August Based on spectral SWEmodel (John Drake, ORNL) Compared with semi-Lagrangian semi-implicit SLSI Stability not dependent on reference geopotential Semi-Lagrangian Laplace Transform SLLT

10. PDEs On The Sphere 2010 24th August Shallow Water Equations

11. PDEs On The Sphere 2010 24th August 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 Orographic Resonance

12. PDEs On The Sphere 2010 24th August Linear analysis of orographically forced stationary waves Numerical simulations with shallow water SLLT Results consistently show benefits of SLLT scheme Orographic Resonance Analysis

13. PDEs On The Sphere 2010 24th August Linear Analysis: (Numerical)/(Analytic) Spurious numerical resonance Analytic solution vanishes

14. PDEs On The Sphere 2010 24th August Linear Analysis: (Numerical)/(Analytic) Analytic solution vanishes

15. PDEs On The Sphere 2010 24th August Test Case with 500hPa Data Initial data: ERA-40 analysis of 12 UTC 12 th February 1979 Used by Ritchie & Tanguay (1996) and Li & Bates (1996) Running at T119 resolution

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20. PDEs On The Sphere 2010 24th August Efficiency Symmetry in the LT inversion Relative overhead of SLLT method, compared to SLSI: Reduces with increasing resolution T42 ~50% T119 ~30%

21. PDEs On The Sphere 2010 24th August Shallow water model using a semi-Lagrangian Laplace Transform method Advantages over a semi-implicit method  Accurate phase speed  Stability  No orographic resonance Conclusions

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23. PDEs On The Sphere 2010 24th August 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