Tracking Atmospheric Instabilities with the Kalman Filter. Part 1: Methodology and One-Layer Resultst

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  • 1 Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California
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Abstract

Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper we implement a KF for a two-dimensional shallow-water model, with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

A model-error covariance matrix for the KF simulations is constructed based on the hypothesis that an ensemble of slow modes dominates the errors. In the 1-L case, the system is stable for a meridionally constant basic flow. Assuming equipartition of energy in the construction of the model-error covariance matrix has a deleterious effect on the process of data assimilation in both the stable and unstable cases. Estimation errors are found to be smaller for a model-error spectrum that decays exponentially with wavenumber than an equipartition spectrum. Then the model-error covariance matrix for the 2-L model is also obtained using a decaying-energy spectrum.

The barotropically unstable 1-L case is studied for a basic velocity profile that has a cosine-square shape. Given this linear instability, forecast errors grow exponentially when no observations are present. The KF keeps the errors bounded, even when very few observations are available. The best placement of a single observation is determined in this simple situation and shown to be where the instability is strongest. The 2-L case and a comparison with the performance of a currently operational data assimilation scheme will appear in Part II.

Abstract

Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper we implement a KF for a two-dimensional shallow-water model, with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

A model-error covariance matrix for the KF simulations is constructed based on the hypothesis that an ensemble of slow modes dominates the errors. In the 1-L case, the system is stable for a meridionally constant basic flow. Assuming equipartition of energy in the construction of the model-error covariance matrix has a deleterious effect on the process of data assimilation in both the stable and unstable cases. Estimation errors are found to be smaller for a model-error spectrum that decays exponentially with wavenumber than an equipartition spectrum. Then the model-error covariance matrix for the 2-L model is also obtained using a decaying-energy spectrum.

The barotropically unstable 1-L case is studied for a basic velocity profile that has a cosine-square shape. Given this linear instability, forecast errors grow exponentially when no observations are present. The KF keeps the errors bounded, even when very few observations are available. The best placement of a single observation is determined in this simple situation and shown to be where the instability is strongest. The 2-L case and a comparison with the performance of a currently operational data assimilation scheme will appear in Part II.

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