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A Semi-Implicit Modification to the Lorenz N-Cycle Scheme and Its Application for Integration of Meteorological Equations

Daisuke HottaJapan Meteorological Agency, Tokyo, Japan, and University of Maryland, College Park, College Park, Maryland

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Eugenia KalnayUniversity of Maryland, College Park, College Park, Maryland

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Paul UllrichDepartment of Land, Air and Water Resources, University of California, Davis, Davis, California

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Abstract

The Lorenz N-cycle is an economical time integration scheme that requires only one function evaluation per time step and a minimal memory footprint, but yet possesses a high order of accuracy. Despite these advantages, it has remained less commonly used in meteorological applications, partly because of its lack of semi-implicit formulation. In this paper, a novel semi-implicit modification to the Lorenz N-cycle is proposed. The advantage of the proposed new scheme is that it preserves the economical memory use of the original explicit scheme. Unlike the traditional Robert–Asselin (RA) filtered semi-implicit leapfrog scheme whose formal accuracy is only of first order, the new scheme has second-order accuracy if it adopts the Crank–Nicolson scheme for the implicit part. A linear stability analysis based on a univariate split-frequency oscillation equation suggests that the 4-cycle is more stable than other choices of N. Numerical experiments performed using the dynamical core of the Simplified Parameterizations Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model under the framework of the Jablonowski–Williamson baroclinic wave test case confirms that the new scheme in fact has second-order accuracy and is more accurate than the traditional RA-filtered leapfrog scheme. The experiments also give evidence for Lorenz’s claim that the explicit 4-cycle scheme can be improved by running its two “isomeric” versions in alternating sequences. Unlike the explicit scheme, however, the proposed semi-implicit scheme is not improved by alternation of the two versions.

Corresponding author address: Daisuke Hotta, Numerical Prediction Division, Japan Meteorological Agency, 1-3-4 Otemachi, Chiyoda-ku, Tokyo 100-8122, Japan. E-mail: hotta.daisuke@met.kishou.go.jp

Abstract

The Lorenz N-cycle is an economical time integration scheme that requires only one function evaluation per time step and a minimal memory footprint, but yet possesses a high order of accuracy. Despite these advantages, it has remained less commonly used in meteorological applications, partly because of its lack of semi-implicit formulation. In this paper, a novel semi-implicit modification to the Lorenz N-cycle is proposed. The advantage of the proposed new scheme is that it preserves the economical memory use of the original explicit scheme. Unlike the traditional Robert–Asselin (RA) filtered semi-implicit leapfrog scheme whose formal accuracy is only of first order, the new scheme has second-order accuracy if it adopts the Crank–Nicolson scheme for the implicit part. A linear stability analysis based on a univariate split-frequency oscillation equation suggests that the 4-cycle is more stable than other choices of N. Numerical experiments performed using the dynamical core of the Simplified Parameterizations Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model under the framework of the Jablonowski–Williamson baroclinic wave test case confirms that the new scheme in fact has second-order accuracy and is more accurate than the traditional RA-filtered leapfrog scheme. The experiments also give evidence for Lorenz’s claim that the explicit 4-cycle scheme can be improved by running its two “isomeric” versions in alternating sequences. Unlike the explicit scheme, however, the proposed semi-implicit scheme is not improved by alternation of the two versions.

Corresponding author address: Daisuke Hotta, Numerical Prediction Division, Japan Meteorological Agency, 1-3-4 Otemachi, Chiyoda-ku, Tokyo 100-8122, Japan. E-mail: hotta.daisuke@met.kishou.go.jp
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