Wave field and zonal flow of a librating disk

Abstract

In this work, we provide a viscous solution of the wave field generated by librating a disk (harmonic oscillation of the rotation rate) in a stably stratified rotating fluid. The zonal flow (mean flow correction) generated by the nonlinear interaction of the wave field is also calculated in the weakly nonlinear framework. We focus on the low dissipative limit relevant for geophysical applications and for which the wave field and the zonal flow exhibit generic features (Ekman scaling, universal structures, etc.). General expressions are obtained which depend on the disk radius a * , the libration frequency ω * , the rotation rate Ω * of the frame, the buoyancy frequency N * of the fluid, its kinematic diffusion ν * and its thermal diffusivity κ * . When the libration frequency is in the inertia-gravity frequency interval (min(Ω * , N * ) < ω * < max(Ω * , N * )), the presence of conical internal shear layers is observed in which the spatial structures of the harmonic response and of the mean flow correction are provided. At the point of focus of these internal shear layers on the rotation axis, the largest amplitudes are obtained: the angular velocity of the harmonic response and the mean flow correction are found to be O(εE −1/3 ) and (ε 2 E −2/3 ) respectively, where ε is the libration amplitude and E = ν * /(Ω * a * 2 ) is the Ekman number. We show that the solution in the internal shear layers and in the focus region is at leading order the same as that generated by an oscillating source of axial flow localized at the edge of the disk (oscillating Dirac ring source).

Near-field internal wave beams in two dimensions

“…Adapting the results of Tilgner (2000), Bardakov etal. (2007), Davis & Llewellyn Smith (2010) and Le Dizès (2015) for the oscillations of a horizontal circular disc to the line source , with the Dirac delta function, we obtain or, equivalently, At first glance the two wave structures, shown in figure 3( a , b ), appear incompatible with each other. In particular, the spectrum of the line source leaves the normal wavenumber arbitrary, thereby allowing to become infinitely large and preventing the pole displacements (2.15) and (2.26) from remaining small, however small and can be.…”

“…The determination of this model for an actual forcing mechanism, typically the oscillations of a plate, requires the calculation of the boundary layer, which then predates the calculation of the waves. In three dimensions the boundary layer has been obtained by Davis & Llewellyn Smith (2010) and Le Dizès (2015) by solving the no-slip boundary-value problem for oscillating horizontal discs.…”

“…By contrast, for thin forcing, the evolution of the waves is set by the distance normal to the forcing. This can be seen in the inviscid calculations of Oser (1957), Reynolds (1962), Martin & Llewellyn Smith (2011, 2012 b ) and Davis (2012) for a horizontal disc, Hurley (1969) for an inclined plate and Llewellyn Smith & Young (2003) for a vertical plate, or in the viscous calculations of Kistovich & Chashechkin (1999 a , b ) for a two-dimensional inclined plate, Vasil'ev & Chashechkin (2003, 2006 a , b , 2012) for a three-dimensional inclined plate, Tilgner (2000), Bardakov, Vasil'ev & Chashechkin (2007), Davis & Llewellyn Smith (2010), Le Dizès (2015) and Le Dizès & Le Bars (2017) for a horizontal disc, Maurer etal. (2017) and Boury, Peacock & Odier (2019) for a horizontal wave generator and Beckebanze, Raja & Maas (2019) for a vertical wave generator.…”

Internal shear layers from librating objects

“…The two velocity components v and v φ (aligned along with L j and the azimut, respectively) are found to be identical albeit a phase factor. Around the line L 1 , Le Dizès (2015) Figure 3. Contour of the azimuthal velocity v φ in the (r,z) plane of the librating disk in the plane obtained from the numerical simulation at t = 501.5π/ω for E = 10 −5 , ε = 10 −4 , ω = √ 2.…”

Axisymmetric inertial modes in a spherical shell at low Ekman numbers

“…This behaviour also emerges in the analysis of shear layers produced by a librating disc, which reflect on the axis (e.g. Le Dizès 2015).…”