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An Adjoint Examination of a Nudging Method for Data Assimilation

Jian-Wen BaoCooperative Institute for Research in Environmental Sciences, University of Colorado/NOAA, Environmental Technology Laboratory, Boulder, Colorado

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Ronald M. ErricoNational Center for Atmospheric Research, Boulder, Colorado

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Abstract

A regional adjoint modeling system is modified to determine the sensitivities of data assimilation and forecast results with respect to perturbations of the nudging fields and coefficients. A generalized linear system is used to explain the sensitivities both mathematically and physically. A linearized shallow-water model is utilized to show that the dynamics determining the sensitivities can be well described in terms of the dynamics of geostrophic adjustment, with the added effects of dissipation and nudging terms. The purpose of the study is to reveal the dynamics responsible for the sensitivities of assimilated fields and forecasts to a given observed variable, and thus to gain insight into what kinds of information are most (or least) effectively assimilated by the nudging method.

The results of the adjoint study reveal that the nudging terms contribute significantly to the prognostic tendencies, even if the values of the nudging coefficient are smaller than those commonly used. When either all dynamic fields or only wind fields are nudged, the assimilation result is much more sensitive to the analyzed data at a later time. The sensitivity of the variance of the difference between the assimilation result and the analyzed data at the final time within various bands of horizontal and vertical spatial scales shows that little scale interaction is evident in this study.

The qualitative comparison of the sensitivity results for nudging only wind or temperature or both are apparently well explained by referring to results of a sensitivity analysis for a nudged, linear shallow-water model. The latter results indicate that nudging high-frequency gravity waves toward an analysis that varies on a much slower timescale had little effect on the final assimilation fields, aside from damping. The same was not true for either rotational modes or slowly propagating inertial-gravitational modes. The sensitivity analysis of the shallow-water model also explains why nudging temperature alone does not produce desirable results.

All the results indicate that the advection is being overwhelmed by the nudging even when the value of the nudging coefficient is half as large as commonly used, but geostrophic and dissipative adjustment are acting effectively. For larger values of the nudging coefficients, the effects of advection are diminished more.

Corresponding author address: Dr. Jian-Wen Bao, NOAA/ETL, 325 Broadway, Boulder, CO 80303-3328.

Email: jbao@etl.noaa.gov

Abstract

A regional adjoint modeling system is modified to determine the sensitivities of data assimilation and forecast results with respect to perturbations of the nudging fields and coefficients. A generalized linear system is used to explain the sensitivities both mathematically and physically. A linearized shallow-water model is utilized to show that the dynamics determining the sensitivities can be well described in terms of the dynamics of geostrophic adjustment, with the added effects of dissipation and nudging terms. The purpose of the study is to reveal the dynamics responsible for the sensitivities of assimilated fields and forecasts to a given observed variable, and thus to gain insight into what kinds of information are most (or least) effectively assimilated by the nudging method.

The results of the adjoint study reveal that the nudging terms contribute significantly to the prognostic tendencies, even if the values of the nudging coefficient are smaller than those commonly used. When either all dynamic fields or only wind fields are nudged, the assimilation result is much more sensitive to the analyzed data at a later time. The sensitivity of the variance of the difference between the assimilation result and the analyzed data at the final time within various bands of horizontal and vertical spatial scales shows that little scale interaction is evident in this study.

The qualitative comparison of the sensitivity results for nudging only wind or temperature or both are apparently well explained by referring to results of a sensitivity analysis for a nudged, linear shallow-water model. The latter results indicate that nudging high-frequency gravity waves toward an analysis that varies on a much slower timescale had little effect on the final assimilation fields, aside from damping. The same was not true for either rotational modes or slowly propagating inertial-gravitational modes. The sensitivity analysis of the shallow-water model also explains why nudging temperature alone does not produce desirable results.

All the results indicate that the advection is being overwhelmed by the nudging even when the value of the nudging coefficient is half as large as commonly used, but geostrophic and dissipative adjustment are acting effectively. For larger values of the nudging coefficients, the effects of advection are diminished more.

Corresponding author address: Dr. Jian-Wen Bao, NOAA/ETL, 325 Broadway, Boulder, CO 80303-3328.

Email: jbao@etl.noaa.gov

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