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)
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4. In This Talk
Describe a semi-Lagrangian trajectory Laplace Transform scheme
Compare with semi-implicit schemes in a shallow water model
Show benefits when orography is added
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5. 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)
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9. We consider the transform of the general equation
L is a linear operator and N a nonlinear vector function.
The Laplace transform is
The initial value is X^0 and we have held the nonlinear term at its initial value N^0.
We apply the inversion operator at time t = Δt to get the filtered state at this time:
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11. Evaluating the contour integral
The inversion using L∗ requires the complex integration in (3), around the circle C∗.
To apply the filter in practice, we replace C∗ by the
N-sided polygon C∗N to reduce the integration to a summation.
The length of each edge is Δs_n and the midpoints are labelled s_n for n = 1, 2, ,N.
The Figure below shows the case with N = 8.
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13. We can now define the numerical operator used for the
modified inversion as
We truncate the exponential expansion at N and introduce a correction term to ensure that the inversion is exact for low-order polynomials.
Then the final form of the numerical filtering inversion integral is
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14. LT Filtering Integration Scheme
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15. Phase Error Analysis Relative Phase Change: R = (numerical) / (actual)
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16. Relative phase errors for the semi-implicit (SI) and LT methods, for Kelvin waves with m = 1 and m = 5.
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17. Kelvin wave with zonal wave number m = 5. Hourly height at 0. 0◦E, 0
Kelvin wave with zonal wave number m = 5. Hourly height at 0.0◦E, 0.9◦N over 10 hours with τc = 3 hours T63 spectral resolution.
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18. Semi-Lagrangian Laplace Transform
Define the LT along
a trajectory
Then
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19. 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
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20. Shallow Water Equations
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22. 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
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23. Orographic Resonance Analysis
Linear analysis of orographically forced stationary waves
Numerical simulations with shallow water SLLT
Results consistently show benefits of SLLT scheme
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24. Linear Analysis: (Numerical)/(Analytic)
Analytic solution vanishes
Spurious numerical resonance
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25. Linear Analysis: (Numerical)/(Analytic)
Analytic solution vanishes
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26. 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
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32. Efficiency SLSI 27 minutes SLLT 45 SLLT (symm) 35
Symmetry in the LT inversion
Timing for T119, dt = 600s, 24 hour forecasts:
SLSI
27 minutes
SLLT 45
SLLT (symm) 35
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33. Conclusions
Shallow water model using a semi-Lagrangian Laplace Transform method
Advantages over a semi-implicit method
Accurate phase speed
Stability
No orographic resonance
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34. 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,
Lindberg K., Alexeev V.A. (2000): A Study of the Spurious Orographic Resonance in Semi-Implicit Semi-Lagrangian Models. Monthly Weather Review, 128,
Lynch P. (1985): Initialization using Laplace Transforms. Quart. J. R. Met. Soc., 111,
Lynch P. (1986): Initialization of a Barotropic Limited-Area Model Using the Laplace Transform Technique. Monthly Weather Review, 113,
Lynch P. (1991): Filtering Integration Schemes Based on the Laplace and Z Transforms. Monthly Weather Review, 119,
Ritchie H., Tanguay M. (1996): A Comparison of Spatially Averaged Eulerian and Semi-Lagrangian Treatments of Mountains. Monthly Weather Review, 124,
Van Isacker J., Struylaert W (1985): Numerical Forecasting Using Laplace Transforms. Royal Belgian Meteorological Institute Publications Serie A, 115
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