Model Error in Filtering Random Compressible Flows Utilizing Noisy Lagrangian Tracers

Nan Chen Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Andrew J. Majda Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York, and Center for Prototype Climate Modeling, NYU Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates

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Abstract

Lagrangian tracers are drifters and floaters that collect real-time information of fluid flows. This paper studies the model error in filtering multiscale random rotating compressible flow fields utilizing noisy Lagrangian tracers. The random flow fields are defined through random amplitudes of Fourier eigenmodes of the rotating shallow-water equations that contain both incompressible geostrophically balanced (GB) flows and rotating compressible gravity waves, where filtering the slow-varying GB flows is of primary concern. Despite the inherent nonlinearity in the observations with mixed GB and gravity modes, there are closed analytical formulas for filtering the underlying flows. Besides the full optimal filter, two practical imperfect filters are proposed. An information-theoretic framework is developed for assessing the model error in the imperfect filters, which can apply to a single realization of the observations. All the filters are comparably skillful in a fast rotation regime (Rossby number ). In a moderate rotation regime (), significant model errors are found in the reduced filter containing only GB forecast model, while the computationally efficient 3D-Var filter with a diagonal covariance matrix remains skillful. First linear then nonlinear coupling of GB and gravity modes is introduced in the random Fourier amplitudes, while linear forecast models are retained to ensure the filter estimates have closed analytical expressions. All the filters remain skillful in the regime. In the regime, the full filter with a linear forecast model has an acceptable filtering skill, while large model errors are shown in the other two imperfect filters.

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Corresponding author address: Nan Chen, Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012. E-mail: chennan@cims.nyu.edu

Abstract

Lagrangian tracers are drifters and floaters that collect real-time information of fluid flows. This paper studies the model error in filtering multiscale random rotating compressible flow fields utilizing noisy Lagrangian tracers. The random flow fields are defined through random amplitudes of Fourier eigenmodes of the rotating shallow-water equations that contain both incompressible geostrophically balanced (GB) flows and rotating compressible gravity waves, where filtering the slow-varying GB flows is of primary concern. Despite the inherent nonlinearity in the observations with mixed GB and gravity modes, there are closed analytical formulas for filtering the underlying flows. Besides the full optimal filter, two practical imperfect filters are proposed. An information-theoretic framework is developed for assessing the model error in the imperfect filters, which can apply to a single realization of the observations. All the filters are comparably skillful in a fast rotation regime (Rossby number ). In a moderate rotation regime (), significant model errors are found in the reduced filter containing only GB forecast model, while the computationally efficient 3D-Var filter with a diagonal covariance matrix remains skillful. First linear then nonlinear coupling of GB and gravity modes is introduced in the random Fourier amplitudes, while linear forecast models are retained to ensure the filter estimates have closed analytical expressions. All the filters remain skillful in the regime. In the regime, the full filter with a linear forecast model has an acceptable filtering skill, while large model errors are shown in the other two imperfect filters.

Denotes Open Access content.

Corresponding author address: Nan Chen, Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012. E-mail: chennan@cims.nyu.edu
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