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

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  • 1 Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin
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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” () and the sensitivity to potential vorticity (PV) . The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, and , and height (fluid depth), : − (g/f)(). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of .

The sensitivity to PV determines the long-time (t → ∞), steady behavior of the adjoint sensitivity to height, = −(f/H2). 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: = (1/H)/∂y and = (1/H)/∂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.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Michael C. Morgan, mcmorgan@wisc.edu

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” () and the sensitivity to potential vorticity (PV) . The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, and , and height (fluid depth), : − (g/f)(). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of .

The sensitivity to PV determines the long-time (t → ∞), steady behavior of the adjoint sensitivity to height, = −(f/H2). 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: = (1/H)/∂y and = (1/H)/∂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.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Michael C. Morgan, mcmorgan@wisc.edu
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