Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem

Version 1 : Received: 18 November 2020 / Approved: 20 November 2020 / Online: 20 November 2020 (09:32:45 CET)

How to cite: Lilly, J.M.; Elipot, S. A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem. Preprints 2020, 2020110529 (doi: 10.20944/preprints202011.0529.v1). Lilly, J.M.; Elipot, S. A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem. Preprints 2020, 2020110529 (doi: 10.20944/preprints202011.0529.v1).

Abstract

The unsteady Ekman problem involves finding the response of the near-surface currents to wind stress forcing under linearized dynamics. Its solution can be conveniently framed in the frequency domain in terms of a quantity that is known as the transfer function, the Fourier transform of the impulse response function. In this paper, a theoretical investigation of a fairly general transfer function form is undertaken with the goal of paving the way for future observational studies. Building on earlier work, we consider in detail the transfer function arising from a linearly-varying profile of the vertical eddy viscosity, subject to a no-slip lower boundary condition at a finite depth. The linearized horizontal momentum equations are shown to transform to a modified Bessel’s equation for the transfer function. Two self-similarities, or rescalings that each effectively eliminate one independent variable, are identified, enabling the dependence of the transfer function on its parameters to be more readily assessed. A systematic investigation of asymptotic behaviors of the transfer function is then undertaken, yielding expressions appropriate for eighteen different regimes, and unifying the results from numerous earlier studies. A solution to a numerical overflow problem that arises in the computation of the transfer function is also found. All numerical code associated with this paper is distributed freely for use by the community.

Subject Areas

Ekman currents; ocean surface currents; wind stress forcing; transfer function; wind-driven response

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