Development and application of adjoint sensitivity to potential vorticity and unbalanced flows in a numerical weather prediction model

Abstract

The adjoint of a numerical model provides a tool to measure the sensitivity of a chosen forecast aspect to small model state perturbations in the earlier time. For the response function describing the growth or behavior of extratropical and tropical systems, the sensitivity gradient provides dynamical information about certain aspects of storm development. Previous studies employed Numerical Weather Prediction (NWP) models with either their adjoint or ensemble experiments to investigate dynamical mechanisms contributing to the change in the response function. However, most of them focus on the impact of individual model variable perturbation on the response function. This study is one of the first to derive the sensitivities to three-dimensional Quasi-geostrophic (QG) potential vorticity (PV) and Ertel PV that combine different sensitivity variables and the first to provide a “PV thinking” in the adjoint framework. Same to the invertibility of PV, provided a balance constraint, a balanced model state sensitivity can be obtained from sensitivity to PV. Using balanced sensitivity fields as the adjoint forcing is shown to alleviate the geostrophic adjustment process often seen in the adjoint integration which emanates an unphysical high-frequency wave that contaminates the interpretation of the adjoint analysis result. In addition to the balanced adjoint dynamics, the physical meaning of the deviation from the geostrophic balance, imbalance, is explored under the adjoint framework for the first time. Sensitivities to QGPV, geostrophic imbalance, and Ertel PV all integrate wind and temperature sensitivities. Applying these tools to different cases reveals their individual attributes that help understand the dynamical mechanism of storm intensification. The case study of the March 2020 Atlantic midlatitude cyclone development confirms the general application of sensitivity to QGPV. In the case study of the November 1998 Winter Storm, sensitivity to winds is found to be dominated by the balanced component recovered from sensitivity to QGPV, while the sensitivity to potential temperature is dominated by the unbalanced component recovered from sensitivity to geostrophic imbalance. Further, the collocation between the unbalanced temperature sensitivity and diabatic heating separates the temperature sensitivity into adiabatic and diabatic sources. Balanced and unbalanced initial small perturbations are found to insert a similar amount of impact on the storm development. The ability of sensitivity to Ertel PV to integrate the features of sensitivity to QGPV and sensitivity to geostrophic imbalance suggests the better compatibility of “adjoint nonlinear balance” than the “adjoint geostrophic balance” in the case study of Hurricane Ian (2022).