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On the inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder: a reduced model

Published online by Cambridge University Press:  06 February 2020

L. Oruba Open the ORCID record for L. Oruba [Opens in a new window] ,
A. M. Soward Open the ORCID record for A. M. Soward [Opens in a new window]  and
E. Dormy Open the ORCID record for E. Dormy [Opens in a new window]
L. Oruba*
Affiliation:
Laboratoire Atmosphères Milieux Observations Spatiales (LATMOS/IPSL), Sorbonne Université, UVSQ, CNRS, Paris, France
A. M. Soward*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon TyneNE1 7RU, UK
E. Dormy*
Affiliation:
Département de Mathématiques et Applications, UMR-8553, École Normale Supérieure, CNRS, PSL University, 75005Paris, France
*
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
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Abstract

In a previous paper, Oruba et al. (J. Fluid Mech., vol. 818, 2017, pp. 205–240) considered the ‘primary’ quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity $\unicode[STIX]{x1D708}$ that occurs during linear spin-down in a cylindrical container of radius $r^{\dagger }=L$ and height $z^{\dagger }=H$, rotating rapidly (angular velocity $\unicode[STIX]{x1D6FA}$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Here, Direct numerical simulation at large $L=10H$ and Ekman numbers $E=\unicode[STIX]{x1D708}/H^{2}\unicode[STIX]{x1D6FA}$ in the range $=10^{-3}{-}10^{-7}$ reveals inertial wave activity on the spin-down time scale $E^{-1/2}\unicode[STIX]{x1D6FA}^{-1}$. Our analytic study, based on $E\ll 1$, builds on the results of Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404) for an infinite plane layer $L\rightarrow \infty$. In addition to QG spin-down, they identify a ‘secondary’ set of quasi-maximum frequency $\unicode[STIX]{x1D714}^{\dagger }\rightarrow 2\unicode[STIX]{x1D6FA}$ (MF) inertial waves, which is a manifestation of the transient Ekman layer, decaying algebraically $\propto 1/\surd \,t^{\dagger }$. Here, we acknowledge that the blocking of the meridional parts of both the primary-QG and the secondary-MF spin-down flows by the lateral boundary $r^{\dagger }=L$ provides a trigger for other inertial waves. As we only investigate the response to the primary QG-trigger, we call the model ‘reduced’ and for that only inertial waves with frequencies $\unicode[STIX]{x1D714}^{\dagger }<2\unicode[STIX]{x1D6FA}$ are triggered. We explain the ensuing organised inertial wave structure via an analytic study of the thin disc limit $L\gg H$ restricted to the region $L-r^{\dagger }=O(H)$ far from the axis, where we make a Cartesian approximation of the cylindrical geometry. Other than identifying a small scale fan structure emanating from the corner $[r^{\dagger },z^{\dagger }]=[L,0]$, we show that inertial waves, on the gap length scale $H$, radiated (wave energy flux) away from the outer boundary $r^{\dagger }=L$ (but propagating with a phase velocity towards it) reach a distance determined by the mode with the fastest group velocity.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A9
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Oruba et al. supplementary movie 1

{\it Movie 1}\\ This movie shows the $\chi$-contours in the case $\Ek=10^{-3}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}\chi_{\tDNS}$ (colour scale from -0.3 to 0.3); (b) the filtered DNS, with the geostrophic flow substracted $\chi_{\tFNS}$ (colour scale from -0.1 to 0.1); (c) the analytic solutions $\chi_{\tIW}$ (colour scale from -0.1 to 0.1).\\

Download Oruba et al. supplementary movie 1(Video)
Video 3.1 MB

Oruba et al. supplementary movie 2

{\it Movie 2}\\ This movie shows the $v$-contours in the case $\Ek=10^{-3}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}v_{\tDNS}$ (colour scale from -30 to 30); (b) the filtered DNS, with the geostrophic flow substracted $v_{\tFNS}$ (colour scale from -0.5 to 0.5); (c) the analytic solutions $v_{\tIW}$ (colour scale from -0.5 to 0.5).\\

Download Oruba et al. supplementary movie 2(Video)
Video 2.8 MB

Oruba et al. supplementary movie 3

{\it Movie 3}\\ This movie shows the $\chi$-contours in the case $\Ek=10^{-5}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}\chi_{\tDNS}$ (colour scale from -0.3 to 0.3); (b) the filtered DNS, with the geostrophic flow substracted $\chi_{\tFNS}$ (colour scale from -0.1 to 0.1); (c) the analytic solutions $\chi_{\tIW}$ (colour scale from -0.1 to 0.1).\\

Download Oruba et al. supplementary movie 3(Video)
Video 4.5 MB

Oruba et al. supplementary movie 4

{\it Movie 4}\\ This movie shows the $v$-contours in the case $\Ek=10^{-5}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}v_{\tDNS}$ (colour scale from -30 to 30); (b) the filtered DNS, with the geostrophic flow substracted $v_{\tFNS}$ (colour scale from -0.5 to 0.5); (c) the analytic solutions $v_{\tIW}$ (colour scale from -0.5 to 0.5).\\

Download Oruba et al. supplementary movie 4(Video)
Video 3 MB

Oruba et al. supplementary movie 5

{\it Movie 5}\\ This movie shows the $\chi$-contours in the case $\Ek=10^{-7}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}\chi_{DNS}$ (colour scale from -0.3 to 0.3); (b) the filtered DNS, with the geostrophic flow substracted $\chi_{FNS}$ (colour scale from -0.1 to 0.1); (c) the analytic solutions $\chi_{IW}$ (colour scale from -0.1 to 0.1).\\

Download Oruba et al. supplementary movie 5(Video)
Video 4.4 MB

Oruba et al. supplementary movie 6

{\it Movie 6}\\ This movie shows the $v$-contours in the case $\Ek=10^{-7}$, as a function of time: (a) the direct numerical simulations $\Ek^{-1/2}v_{\tDNS}$ (colour scale from -30 to 30); (b) the filtered DNS, with the geostrophic flow substracted $v_{\tFNS}$ (colour scale from -0.5 to 0.5); (c) the analytic solutions $v_{\tIW}$ (colour scale from -0.5 to 0.5).\\

Download Oruba et al. supplementary movie 6(Video)
Video 2.9 MB

Oruba et al. supplementary movie 7

{\it Movie 7}\\ This movie shows the meridional speed $E^{-1/2}\sqrt{u^2+w^2}=r^{-1}\bigl|\bfnabla(r\chi)\bigr|$, as a function of time. Respectively, the panels show results for: ($a$)--($c$) the filtered DNS $\chi_\tFNS$, when ($a$) $E=10^{-3}$, ($b$) $E=10^{-5}$, ($c$) $E=10^{-7}$; ($d$) the analytic $\chi^{\rm{wave}}$ in the limit $E\downarrow 0$; ($e$) the full DNS $E^{-1/2}\chi_\tDNS$ when $E=10^{-7}$ (colour scale from $0$ to $3$).

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Video 907.9 KB
Supplementary material: PDF

Oruba et al. supplementary material

Supplementary Data

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