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

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.

Inclusion of Linearized Moist Physics in NASA’s Goddard Earth Observing System Data Assimilation Tools

“…The findings of Holdaway and Errico (2013) motivate an approach that involves the exact linearization of the RAS scheme, with only minor simplifications to the algorithm. These simplifications are related to the way certain constants are chosen and have a minimal effect on the overall behavior of the scheme.…”

“…When attempting to represent moist physics in the linear model, a number of options exist, as outlined by Holdaway and Errico (2013). Ideally, an exact linearization of the schemes would be implemented.…”

“…To estimate the nonlinearity in the RAS scheme, Holdaway and Errico (2013) performed an extensive study examining Jacobian sensitivities. Despite the potential nonlinearity in the RAS formulation, they found that a large degree of linearity exists, especially for midto deep convection.…”

“…Large-scale condensation is modeled using the scheme developed by Bacmeister et al (2006). Holdaway and Errico (2013) examined linearity and stability in the RAS convection scheme by studying Jacobian sensitivities. Holdaway and Errico (2013) examined linearity and stability in the RAS convection scheme by studying Jacobian sensitivities.…”

Identifying a Barotropic Growth Mechanism in East Pacific Tropical Cyclogenesis Using Adjoint-Derived Sensitivity Gradients

“…The effect of moist physics was not considered in this study for the sake of maintaining realistic results in the adjoint model simulations. Sophisticated adjoint models now exist that can explicitly handle the physics of parameterized moist convection, including the WRF, COAMPS (e.g., Doyle et al 2012), and GEOS-5 (Holdaway and Errico 2014), and the inclusion of these moisture physics and their interaction with the barotropic growth mechanism in east Pacific tropical cyclogenesis is an area of further study. The results shown here omit the impact of moisture where moisture likely plays a significant role, and previous results using a similar technique with a moist adjoint have found moisture perturbations to be dominant (Doyle et al 2012).…”