Relationships between the Limit of Predictability and Initial Error
in the Uncoupled and Coupled Lorenz Models |
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| Authors: | DING Ruiqiang and LI Jianping |
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| Institution: | State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics,
Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029;State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics,
Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029 |
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| Abstract: | In this study, the relationship between the limit of predictability and
initial error was investigated using two simple chaotic systems: the Lorenz
model, which possesses a single characteristic time scale, and the coupled
Lorenz model, which possesses two different characteristic time scales. The
limit of predictability is defined here as the time at which the error
reaches 95% of its saturation level; nonlinear behaviors of the error
growth are therefore involved in the definition of the limit of
predictability. Our results show that the logarithmic function performs well
in describing the relationship between the limit of predictability and
initial error in both models, although the coefficients in the logarithmic
function were not constant across the examined range of initial errors.
Compared with the Lorenz model, in the coupled Lorenz model---in which the
slow dynamics and the fast dynamics interact with each other---there is a
more complex relationship between the limit of predictability and initial
error. The limit of predictability of the Lorenz model is unbounded as the
initial error becomes infinitesimally small; therefore, the limit of
predictability of the Lorenz model may be extended by reducing the amplitude
of the initial error. In contrast, if there exists a fixed initial error in
the fast dynamics of the coupled Lorenz model, the slow dynamics has an
intrinsic finite limit of predictability that cannot be extended by reducing
the amplitude of the initial error in the slow dynamics, and vice versa. The
findings reported here reveal the possible existence of an intrinsic finite
limit of predictability in a coupled system that possesses many scales of
time or motion. |
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| Keywords: | limit of predictability initial error Lorenz model coupled Lorenz model |
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