Article Type:
Research Article

The Role of Time Step Size in Numerical Stability of Tangent Linear Models

Jiang Zhu Meteorological Research Institute, Ibaraki, Japan, and Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

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Masafumi Kamachi Meteorological Research Institute, Ibaraki, Japan

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Print Publication:
01 May 2000
DOI:
https://doi.org/10.1175/1520-0493(2000)128<1562:TROTSS>2.0.CO;2
Page(s):
1562–1572
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Abstract

It is found that some stable time-integration schemes for some nonlinear models do not guarantee stable integrations of the associated tangent linear models with the same time step size. These problems usually occur when the nonlinear models describe vertical diffusion processes and are numerically implemented by semi-implicit time-integration schemes that are unconditionally stable. The direct linearization procedure performed on such numerical schemes of nonlinear models can be interpreted as some conditionally stable numerical schemes of the underlying linearized equations.

Numerical experiments using a simple, illustrative model and a realistic ocean mixed layer model and their tangent linear models showed instabilities in the tangent linear models. Several methods are tried to reduce the nonphysical noise caused by the numerical instabilities. This study suggests that reducing time step size can give good results compared to some other methods that either are not accurate enough or change too much from the original nonlinear model.

Corresponding author address: Dr. Jiang Zhu, Meteorological Research Institute, 1-1 Nagamine, Tsukuba, 305-0052 Ibaraki, Japan.

Email: jzhu@mri-jma.go.jp

Abstract

It is found that some stable time-integration schemes for some nonlinear models do not guarantee stable integrations of the associated tangent linear models with the same time step size. These problems usually occur when the nonlinear models describe vertical diffusion processes and are numerically implemented by semi-implicit time-integration schemes that are unconditionally stable. The direct linearization procedure performed on such numerical schemes of nonlinear models can be interpreted as some conditionally stable numerical schemes of the underlying linearized equations.

Numerical experiments using a simple, illustrative model and a realistic ocean mixed layer model and their tangent linear models showed instabilities in the tangent linear models. Several methods are tried to reduce the nonphysical noise caused by the numerical instabilities. This study suggests that reducing time step size can give good results compared to some other methods that either are not accurate enough or change too much from the original nonlinear model.

Corresponding author address: Dr. Jiang Zhu, Meteorological Research Institute, 1-1 Nagamine, Tsukuba, 305-0052 Ibaraki, Japan.

Email: jzhu@mri-jma.go.jp

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