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

Abstract

The unsteady Ekman problem involves finding the response of the near-surface currents to wind stress forcing under linear 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 horizontal momentum equations, rendered linear by the assumption of horizontally uniform motion, 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.

Frequency dependence of near-surface oceanic kinetic energy from drifter observations and global high-resolution models

“…When such a cubic profile is applied to the wind-driven Ekman momentum equation for the surface boundary layer, this results in a frequency-dependent shear which is minimum at the inertial frequency and increases away from the inertial frequency (Elipot, 2006). This theoretical framework is useful to understand how the locally wind-driven component of oceanic currents (Lilly & Elipot, 2021), from the inertial frequency to the low-frequency motions, can be sheared in the upper 15 m of the ocean, as seen here in models and observations. In addition, upper-ocean stratification modulates the ultimate penetration depth of wind momentum (Large & Crawford, 1995;Crawford & Large, 1996;Elipot & Gille, 2009;Dohan & Davis, 2011;Lilly & Elipot, 2021).…”

“…This theoretical framework is useful to understand how the locally wind-driven component of oceanic currents (Lilly & Elipot, 2021), from the inertial frequency to the low-frequency motions, can be sheared in the upper 15 m of the ocean, as seen here in models and observations. In addition, upper-ocean stratification modulates the ultimate penetration depth of wind momentum (Large & Crawford, 1995;Crawford & Large, 1996;Elipot & Gille, 2009;Dohan & Davis, 2011;Lilly & Elipot, 2021). The reported values of the 0 m to 15 m KE ratio in the drifters and the models are therefore consistent with these expectations.…”

“…In addition, low-frequency motions may also have a substantial "surface quasi-geostrophic" component (Lapeyre & Klein, 2006;LaCasce & Wang, 2015), which is surface intensified and may therefore contribute to the weaker motions at 15 m depth relative to the surface. However, the simplest explanation for the vertical structure seen at low-frequencies (Figure 4c) is that it is due to Ekman flows, which exhibit substantial variation over short vertical scales (Elipot & Gille, 2009;Lilly & Elipot, 2021).…”

“…As such, undrogued drifter observations likely exhibit larger downwind velocity errors but these are yet to be comprehensively distinguished from real oceanic processes. For example, locally wind-driven velocities at the surface are more energetic than at 15-m depth because of vertical shear at a broad range of frequencies through Ekman dynamics (Elipot & Gille, 2009;Lilly & Elipot, 2021), or through surface gravity wave processes and their associated Stokes drift (Polton et al, 2005). Yet, undrogued drifters qualitatively capture the same KE features as drogued drifters (Yu et al, 2019).…”

Near‐Surface Oceanic Kinetic Energy Distributions From Drifter Observations and Numerical Models

“…This shear is minimum at the inertial frequency and increases away from the inertial frequency (Elipot, 2006). This theoretical framework is useful for understanding how the locally wind-driven component of oceanic currents (J. M. Lilly & Elipot, 2021), ranging from the inertial frequency to low-frequency Ekman motions, can be sheared in the upper 15 m of the ocean. In addition, upper-ocean stratification modulates the ultimate penetration depth of wind momentum (Large & Crawford, 1995;Crawford & Large, 1996;Elipot & Gille, 2009;Dohan & Davis, 2011; J. M. Lilly & Elipot, 2021).…”