Interpretations of Adjoint Sensitivities to Quasi-Geostrophic Potential Vorticity and Imbalance

The adjoint of a numerical weather prediction model is a tool that can be used to evaluate the sensitivity of a chosen aspect of a forecast at a specific forecast time (the response function) to small perturbations to the model state at an earlier times. The sensitivity is defined as the gradient of the chose response function with respect to the model state variables. For a response function describing the evolution of a weather system, the sensitivity gradient provides dynamical and thermodynamical insights into the processes involved in the weather system's evolution.

While many extant studies have demonstrated the interpretation of adjoint-derived sensitivity gradients in synoptic case studies, challenges remain in providing physical interpretations of these gradients with respect to model forecast fields like horizontal wind and temperature. It is desirable to distill the substantive implications that each model state sensitivity reveals into a single sensitivity field, like the sensitivity to potential vorticity (PV). In this way, much as the potential vorticity "thinking" has provided a concise framework to study the nearly balanced dynamics of weather systems, a consideration of sensitivities to PV may offer a foundation for a more efficient interpretation of adjoint-derived sensitivity output. Moreover, for a chosen response function, the sensitivity to the "imbalance" (defined as the difference between the two-dimensional non-divergent streamfunction and the geostrophic streamfunction) offers insights into the adjustment processes associated with a forecast evolution leading up to the final time response function value.

In this study, expressions for calculating the sensitivities to PV and to imbalance from adjoint sensitivities to horizontal wind and temperature are presented. Applications of these expressions to WRF adjoint model output are demonstrated and interpreted for a case of western Atlantic explosive cyclogenesis in March 2020.

This study follows in the tradition of Professor Dan Keyser who has been concerned with applying principles and concepts from mathematics and physics to understanding weather systems and their associated processes with the additional goal of refining numerical and conceptual models used in operational weather prediction.