On the Dynamics of Adjustment in the f-Plane Shallow Water Adjoint System

Abstract

Analytic results and numerical experimentation reveal that a “backward” integration of the adjoint of the shallow water system linearized about a basic state at rest on an f plane is characterized by a radiation of gravity wave–like structures and the emergence of a steady adjoint state. The earlier adjoint states are linked to the prescribed adjoint state (i.e., the adjoint forcing) through the locally conserved dynamical adjoint variables of the shallow water system: the sensitivity to “balanced height” ([Formula: see text]) and the sensitivity to potential vorticity (PV) [Formula: see text]. The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, [Formula: see text] and [Formula: see text], and height (fluid depth), [Formula: see text]: [Formula: see text]− ( g/ f)([Formula: see text]). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of [Formula: see text]. The sensitivity to PV determines the long-time ( t → ∞), steady behavior of the adjoint sensitivity to height, [Formula: see text]= −( f/ H2)[Formula: see text]. 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 nondivergent sensitivities to the flow that resemble geostrophic balance: [Formula: see text]= (1/ H)[Formula: see text]/∂ y and [Formula: see text]= (1/ H)[Formula: see text]/∂ x. The process by which this long-time, nondivergent, adjoint state emerges is termed adjoint adjustment. For the system considered, sensitivities to the ageostrophic and irrotational components of the flow vanish for the adjusted state near the prescribed adjoint forcing.