Evidence for “adjustment” in the Integration of Adjoints of Numerical Weather Prediction Models

Results from analytic work and numerical experimentation that reveal that a “backward” integration of the adjoint of the shallow water system linearized about a basic state at rest on an f-plane from an arbitrary final adjoint state is characterized by a radiation of gravity wave-like structures and the emergence of a steady adjoint state are presented. The earlier adjoint states are linked to the prescribed, final adjoint state (adjoint forcing) through the locally conserved dynamical adjoint variables of the shallow water system: the sensitivity to “balanced height” and sensitivity to PV. The sensitivity to balanced height is determined from the adjoint sensitivity forcings to the wind components and height (fluid depth). The sensitivity to PV is diagnosed from an inversion of an elliptic operator relating the sensitivity to PV to the distribution of balanced height.

The sensitivity to PV determines the long-time, steady behavior of the adjoint model sensitivity to height. In the vicinity of the initial adjoint forcing, the long-time, steady-state behavior of the adjoint system (linearized about a state at rest) is characterized by non-divergent sensitivities to the flow that resemble geostrophic balance. Sensitivities to the ageostrophic components of the flow vanish as the backward integration proceeds. The process by which this long-time, non-divergent, adjoint state emerges is termed adjoint adjustment.

The outcomes of the first steps in a line of inquiry to determine the extensibility of the above results are presented.  These include comparisons between adjoint-derived forecast sensitivities with ensemble sensitivities computed using the WRF forward and adjoint models to determine evidence for the emergence of a balanced adjoint state and conservation of the sensitivity to PV. Implications of these results for four-dimensional variational data assimilation and synoptic case studies are presented.