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Improved time-stepping schemes in weather and climate models

Department of Meteorology

About the Project

"The task of predicting weather and climate may be reduced to the following iterative procedure. First, given the state of the system at any time (the input), use the governing equations to compute the state at a slightly later time (the output). Then, repeat the loop as many times as required, always using the previous output as the next input. This prediction framework presents three challenges that degrade the reliability of the forecast. First, observations containing measurement errors are required to serve as the initial state. Second, the physical processes and their interactions are imperfectly represented in the spatially truncated governing equations. Third, the discrete time stepping is merely an approximation to the exact time-continuous evolution. This project will work on the third of these challenges, which (until the recent work of the supervisors) had received the least attention.

The leapfrog (second-order centred-difference) time-stepping scheme is widely used in weather and climate models, because it is easy to implement, computationally inexpensive, and has low run-time storage requirements. Unfortunately, it admits a spurious computational mode, which is manifest as a growing 2Δt oscillation. For over 40 years, the solution has been to apply the time filter that was developed by Robert (1966) and Asselin (1972). The filter successfully suppresses the computational mode, but it also weakly damps the physical mode and reduces the accuracy to first-order.

Recent work by the supervisors has shown that simple modifications to the filtered leapfrog scheme can increase the amplitude accuracy from first-order to third-order (Williams 2009) and even seventh-order (Williams 2013), without sacrificing the phase accuracy, stability, or computational expense. Furthermore, the modifications are suitable for use in semi-implicit models (Williams 2011). The modified filter has become known as the Robert–Asselin–Williams (RAW) filter. It is now used in many weather and climate models and has significantly increased the skill of medium-range weather forecasts (Amezcua, Kalnay & Williams 2011, Amezcua & Williams 2014). More advanced Runge–Kutta schemes should also be considered (Weller et al. 2013), which enable high-order, stable combinations of implicit and explicit schemes. These have been applied to linear equations but their application to non-linear equations is not understood.

This project will develop and analyse several new possible improvements to the filtered leapfrog scheme, such as combining a second-order filter with a fourth-order filter, and will make comparisons with implicit–explicit Runge–Kutta techniques. Appropriate linearisations will be found in order to apply the implicit part of the scheme. Various analysis techniques will be used to interrogate the stability and accuracy properties, including the derivation of truncated power-series expansions of the complex amplification factors, the derivation of the equivalent partitioned multi-step methods, and the derivation of root locus curves. The schemes will also be tested in numerical integrations of a hierarchy of nonlinear models of varying complexity, ranging from the simple nonlinear pendulum, through the classical Lorenz system, to a comprehensive atmosphere general circulation model, and the impacts on the prediction skill and climatology (i.e., attractor statistics) will be quantified."


"Amezcua, J., Kalnay, E. and Williams, P. D. (2011) The effects of the RAW filter on the climatology and forecast skill of the SPEEDY model. Monthly Weather Review, 139(2), pp 608-619. doi:10.1175/2010MWR3530.1
Amezcua, J. and Williams, P. D. (2014) The composite-tendency RAW filter in semi-implicit integrations. Quarterly Journal of the Royal Meteorological Society, in press. doi:10.1002/qj.2391
Asselin, R. (1972) Frequency filter for time integrations. Monthly Weather Review, 100(6), 487-490. doi:10.1175/1520-0493(1972)1002.3.CO;2
Robert, A. J. (1966) The integration of a low order spectral form of the primitive meteorological equations. Journal of the Meteorological Society of Japan, 44, 237-245.
Weller, H., Lock, S.-J. and Wood, N. (2013) Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations. Journal of Computational Physics, 252. pp 365-381. doi:10.1016/
Williams, P. D. (2009) A proposed modification to the Robert–Asselin time filter. Monthly Weather Review, 137(8), pp 2538-2546. doi:10.1175/2009MWR2724.1
Williams, P. D. (2011) The RAW filter: an improvement to the Robert–Asselin filter in semi-implicit integrations. Monthly Weather Review, 139(6), pp 1996-2007. doi:10.1175/2010MWR3601.1
Williams, P. D. (2013) Achieving seventh-order amplitude accuracy in leapfrog integrations. Monthly Weather Review, 141(9), pp 3037-3051. doi:10.1175/MWR-D-12-00303.1

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