An Adjoint Method for Obtaining the Most Rapidly Growing Perturbation to Oceanic Flows

Brian F. Farrell Department of Earth and Planetary Physics, Division of Applied Sciences, Harvard University, Cambridge, Massachusetts

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Andrew M. Moore Department of Earth and Planetary Physics, Division of Applied Sciences, Harvard University, Cambridge, Massachusetts

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Abstract

This work explores the formation and growth of waves on oceanic flows using a quasigeostrophic model. In particular, we consider flow regimes consisting of zonal oceanic jets, similar in fact to the westward extension of the Gulf Stream. Traditionally, the formation of waves has been ascribed to exponentially unstable modes, but rather than adopt this paradigm, we seek the most rapidly growing perturbation without restriction or its structure to normal-mode form. Optimal excitations are determined using the adjoint of the quasigeostrophic dynamic equations, and the perturbations found by this method are shown to grow mote rapidly than the unstable mode.

Applications of the theory presented here include determination of a tight upper bound on perturbation growth rate, a constructive method for finding the most disruptive disturbance to a given flow, and a method for determining the relative predictability of flows. From the form of the most rapidly growing perturbation, resolution requirements for numerical models can be determined.

Abstract

This work explores the formation and growth of waves on oceanic flows using a quasigeostrophic model. In particular, we consider flow regimes consisting of zonal oceanic jets, similar in fact to the westward extension of the Gulf Stream. Traditionally, the formation of waves has been ascribed to exponentially unstable modes, but rather than adopt this paradigm, we seek the most rapidly growing perturbation without restriction or its structure to normal-mode form. Optimal excitations are determined using the adjoint of the quasigeostrophic dynamic equations, and the perturbations found by this method are shown to grow mote rapidly than the unstable mode.

Applications of the theory presented here include determination of a tight upper bound on perturbation growth rate, a constructive method for finding the most disruptive disturbance to a given flow, and a method for determining the relative predictability of flows. From the form of the most rapidly growing perturbation, resolution requirements for numerical models can be determined.

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