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Librational forcing of a rapidly rotating fluid-filled cube

Published online by Cambridge University Press:  13 March 2018

Ke Wu ,
Bruno D. Welfert  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Ke Wu
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
*
†Email address for correspondence: juan.m.lopez@asu.edu
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Abstract

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 842 , 10 May 2018 , pp. 469 - 494
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Copyright
© 2018 Cambridge University Press 

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References

Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307–323.Google Scholar
Boisson, J., Lamriben, C., Maas, L. R. M., Cortet, P. P. & Moisy, F. 2012 Inertial waves and modes excited by the libration of a rotating cube. Phys. Fluids 24, 076602.Google Scholar
Cortet, P.-P., Lamriben, C. & Moisy, F. 2010 Viscous spreading of an inertial wave beam in a rotating fluid. Phys. Fluids 22, 086603.Google Scholar
Davidson, P. A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Gutierrez-Castillo, P. & Lopez, J. M. 2017 Differentially rotating split-cylinder flow: responses to weak harmonic forcing in the rapid rotation regime. Phys. Rev. Fluids 2, 084802.Google Scholar
Hugues, S. & Randriamampianina, A. 1998 An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 28, 501–521.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Le Bars, M., Cebron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.CrossRefGoogle Scholar
Lemasquerier, D., Grannan, A. M., Vidal, J., Cébron, D., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Libration-driven flows in ellipsoidal shells. J. Geophys. Res. Planet. 122, 1926–1950.Google Scholar
Lopez, J. M. & Marques, F. 2014 Rapidly rotating cylinder flow with an oscillating sidewall. Phys. Rev. E 89, 013013.Google Scholar
Maas, L. R. M. 2003 On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dyn. Res. 33, 373–401.CrossRefGoogle Scholar
Malkus, W. V. R. 1968 Precession of the Earth as the cause of geomagnetism. Science 160, 259–264.Google Scholar
Noir, J., Hemmerlin, F., Wicht, J., Baca, S. M. & Aurnou, J. M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173, 141–152.Google Scholar
Nurijanyan, S., Bokhove, O. & Maas, L. R. M. 2013 A new semi-analytical solution for inertial waves in a rectangular parallelepiped. Phys. Fluids 25, 126601.Google Scholar
Rao, D. 1966 Free gravitational oscillations in rotating rectangular basins. J. Fluid Mech. 25, 523–555.Google Scholar
Sauret, A., Cébron, D., Le Bars, M. & Le Dizés, S. 2012 Fluid flows in a librating cylinder. Phys. Fluids 24, 026603.CrossRefGoogle Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.Google Scholar
Vanel, J.-M., Peyret, R. & Bontoux, P. 1986 A pseudo-spectral solution of vorticity-stream function equations using the influence matrix technique. In Journal of Numerical Methods for Fluid Dynamics II (ed. Morton, K. W. & Baines, M.), pp. 463–475. Clarendon Press Oxford.Google Scholar
Wood, W. W. 1966 An oscillatory disturbance of rigidly rotating fluid. Proc. R. Soc. Lond.A 293, 181–212.Google Scholar
Wu, C.-C. & Roberts, P. H. 2013 On a dynamo driven topographically by longitudinal libration. Geophys. Astrophys. Fluid Dyn. 107, 20–44.Google Scholar

Wu et al. supplementary movie 1

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=1/6 (right), for libration frequencies 0.6484 (top) and 0.6742 (bottom), over one libration period, both at Ekman number 1e-4.6 and libration amplitude 0.04.

Download Wu et al. supplementary movie 1(Video)
Video 7.9 MB

Wu et al. supplementary movie 2

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.677 (top), and of the eigenmode at eigenfrequency 0.6772 (bottom).

Download Wu et al. supplementary movie 2(Video)
Video 1.6 MB

Wu et al. supplementary movie 3

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699 (top), and of the eigenmode at eigenfrequency 0.6962 (bottom).

Download Wu et al. supplementary movie 3(Video)
Video 1.5 MB

Wu et al. supplementary movie 4

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.705 (top), and of the eigenmode at eigenfrequency 0.7022 (bottom).

Download Wu et al. supplementary movie 4(Video)
Video 1.7 MB

Wu et al. supplementary movie 5

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.708 (top), and of the eigenmode at eigenfrequency 0.7045 (bottom).

Download Wu et al. supplementary movie 5(Video)
Video 1.8 MB

Wu et al. supplementary movie 6

Contours of z-vorticity in the horizontal planes z=0.45, 0.30, 0.15 and 0.0 of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699.

Download Wu et al. supplementary movie 6(Video)
Video 1.8 MB

Wu et al. supplementary movie 7

Contours of x-vorticity in the vertical planes x=0.45, 0.30, 0.15 and 0.0 of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699.

Download Wu et al. supplementary movie 7(Video)
Video 2.1 MB

Wu et al. supplementary movie 8

Contours of x-vorticity in the vertical plane z=0 at the oscillation phase pi/2 of the nonlinear simulations at Ekman number 1e-6 and libration amplitude 1e-6, animating 999 snapshots at libration frequencies 0.001 to 0.999.

Download Wu et al. supplementary movie 8(Video)
Video 21.3 MB