Using Jacobian sensitivities to assess a linearization of the relaxed Arakawa–Schubert convection scheme

Authors

  • D. Holdaway,

    Corresponding author
    1. Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, MD, USA
    2. Goddard Earth Sciences Technology and Research, Universities Space Research Association, MD, USA
    • Correspondence to: D. Holdaway, Code 610.1, Goddard Space Flight Center, Greenbelt, MD 20771, USA. E-mail: dan.holdaway@nasa.gov

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  • R. Errico

    1. Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, MD, USA
    2. Goddard Earth Sciences Technology and Research, Morgan State University, MD, USA
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Abstract

The inclusion of linearized moist physics can increase the accuracy of 4D-Var data assimilation and adjoint-based sensitivity analysis. Moist processes such as convection can exhibit nonlinear behaviour. As a result, representation of these processes in a linear way requires much care; a straightforward linearization may yield a poor approximation to the behaviour of perturbations of interest and could contain numerical instability. Here, an extensive numerical study of the Jacobian of the relaxed Arakawa–Schubert (RAS) convection scheme is shown. A Jacobian based on perturbations at individual model levels can be used to understand the physical behaviour of the RAS scheme, predict how sensitive that behaviour is to the prognostic variables and determine the stability of a linearization of the scheme. The linearity of the scheme is also considered by making structured perturbations, constructed from the principle components of the model variables. Based on the behaviour of the Jacobian operator and the results when using structured perturbations, a suitable method for linearizing the RAS scheme is determined. For deep, strong convection, the structures of the RAS Jacobian are reasonably simple, the rate at which finite-amplitude estimates of the structures change with respect to input perturbations is small and the eigenmodes of the Jacobian are not prohibitively unstable. For deep convection, an exact linearization is therefore suitable. For shallow convection, the RAS scheme can be more sensitive to the input prognostic variables due to the faster time-scales and proximity to switches. Linearization of the RAS therefore requires some simplifications to smooth the behaviour for shallow convection. It is noted that the physical understanding of the scheme gained from examining the Jacobian provides a useful tool to the developers of nonlinear physical parametrizations.

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