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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'' ((eta) over cap (b)) and the sensitivity to potential vorticity (PV)(q) over cap . The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, (u) over cap and (nu) over cap, and height (fluid depth), (eta) over cap : (eta) over cap (b) (eta) over cap (g/f)(partial derivative(nu) over capx - partial derivative(u) over cap partial derivative y). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of (eta) over cap (b).

The sensitivity to PV determines the long-time (t -> infinity), steady behavior of the adjoint sensitivity to height, (eta) over cap (t) (-> -infinity) = -(f/H-2)(q) over cap. 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: (u) over cap (t) (-> -infinity) = (1/H)partial derivative(q) over cap/partial derivative y and (nu) over cap (t) (-> -infinity) (1/H)partial derivative(q) over cap/partial derivative x. The process by which this longtime, 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.