10 results
Bénard convection in a slowly rotating penny-shaped cylinder subject to constant heat flux boundary conditions
- A.M. Soward, L. Oruba, E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 951 / 25 November 2022
- Published online by Cambridge University Press:
- 02 November 2022, A5
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We consider axisymmetric Boussinesq convection in a shallow cylinder of radius $L$ and depth $H (\ll L)$, which rotates with angular velocity $\varOmega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\varOmega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number $\sigma$. As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity $v$ by DNS. Our ‘hybrid’ methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba et al. (J. Fluid Mech., vol. 812, 2017, pp. 890–904).
Inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder
- L. Oruba, A.M. Soward, E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 915 / 25 May 2021
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- 15 March 2021, A53
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In an earlier 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 $\nu$ that occurs during linear spin-down in a cylindrical container of radius $L$ and height $H$, rotating rapidly (angular velocity $\varOmega$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Direct numerical simulation (DNS) of the linear system at large $L= 10 H$ and Ekman number $E\leqslant \nu /H^2\varOmega =10^{-3}$ by Oruba et al. (J. Fluid Mech., vol. 888, 2020, p. 44) reveals significant inertial wave activity on the spin-down time scale. That analytic study, for $E\ll 1$, builds on the results of Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404) for an infinite plane layer $L\to \infty$. At large but finite distance from the symmetry axis, the meridional (QG-)flow, that causes the QG-spin-down, is blocked by the lateral boundary, which provides the primary QG-trigger for inertial wave generation. For the laterally unbounded layer, Greenspan and Howard identified, in addition to the QG-flow, inertial waves of maximum frequency (MF) $2\varOmega$, which are a manifestation of the transient Ekman layer. The blocking of these additional MF-waves by the lateral boundary provides an extra trigger that complements the QG-triggered inertial waves. Here we obtain analytic results for the full wave activity caused by the combined trigger ($\text {QG}+\text {MF}$) that faithfully capture their true character.
On the inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder: a reduced model
- L. Oruba, A. M. Soward, E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 888 / 10 April 2020
- Published online by Cambridge University Press:
- 06 February 2020, A9
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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.
Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions
- L. Oruba, A. M. Soward, E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 818 / 10 May 2017
- Published online by Cambridge University Press:
- 30 March 2017, pp. 205-240
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A comprehensive study of the classical linear spin-down of a constant-density viscous fluid (kinematic viscosity $\unicode[STIX]{x1D708}$) rotating rapidly (angular velocity $\unicode[STIX]{x1D6FA}$) inside an axisymmetric cylindrical container (radius $L$, height $H$) with rigid boundaries, which follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=\unicode[STIX]{x1D708}/H^{2}\unicode[STIX]{x1D6FA}\ll 1$, was provided by Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404). For that problem $E^{1/2}$ Ekman layers form quickly, triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}\unicode[STIX]{x1D6FA}^{-1})$ time scale. On the longer lateral viscous diffusion time scale $O(L^{2}/\unicode[STIX]{x1D708})$, the QG flow responds to the $E^{1/3}$ sidewall shear layers. In our variant, the sidewall and top boundaries are stress-free, a set-up motivated by the study of isolated atmospheric structures such as tropical cyclones or tornadoes. Relative to the unbounded plane layer case, spin-down is reduced (enhanced) by the presence of a slippery (rigid) sidewall. This is evidenced by the QG angular velocity, $\unicode[STIX]{x1D714}^{\star }$, evolution on the $O(L^{2}/\unicode[STIX]{x1D708})$ time scale: spatially, $\unicode[STIX]{x1D714}^{\star }$ increases (decreases) outwards from the axis for a slippery (rigid) sidewall; temporally, the long-time $(\gg L^{2}/\unicode[STIX]{x1D708})$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery sidewall case, the $E^{1/2}\times E^{1/2}$ corner region that forms at the sidewall intersection with the rigid base is responsible for a $\ln E$ singularity within the $E^{1/3}$ layer, causing our asymptotics to apply only at values of $E$ far smaller than can be reached by our direct numerical simulation (DNS) of the linear equations governing the entire spin-down process. Instead, we solve the $E^{1/3}$ boundary layer equations for given $E$ numerically. Our hybrid asymptotic–numerical approach yields results in excellent agreement with our DNS.
Eye formation in rotating convection
- L. Oruba, P. A. Davidson, E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 812 / 10 February 2017
- Published online by Cambridge University Press:
- 06 January 2017, pp. 890-904
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We consider rotating convection in a shallow, cylindrical domain. We examine the conditions under which the resulting vortex develops an eye at its core; that is, a region where the poloidal flow reverses and the angular momentum is low. For simplicity, we restrict ourselves to steady, axisymmetric flows in a Boussinesq fluid. Our numerical experiments show that, in such systems, an eye forms as a passive response to the development of a so-called eyewall, a conical annulus of intense, negative azimuthal vorticity that can form near the axis and separates the eye from the primary vortex. We also observe that the vorticity in the eyewall comes from the lower boundary layer, and relies on the fact the poloidal flow strips negative vorticity out of the boundary layer and carries it up into the fluid above as it turns upward near the axis. This process is effective only if the Reynolds number is sufficiently high for the advection of vorticity to dominate over diffusion. Finally we observe that, in the vicinity of the eye and the eyewall, the buoyancy and Coriolis forces are negligible, and so although these forces are crucial to driving and shaping the primary vortex, they play no direct role in eye formation in a Boussinesq fluid.
Stability and bifurcation of planetary dynamo models
- E. Dormy
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- Journal:
- Journal of Fluid Mechanics / Volume 688 / 10 December 2011
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- 23 November 2011, pp. 1-4
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Rapidly rotating dynamos, relevant to the origin of the Earth magnetic field, are difficult to model owing to the extreme parameter regimes that occur in their dynamics. Numerical models alone fail to approach the correct regime. However, progress can be achieved by combining numerical and analytical methods. This can offer a better understanding of the variety of behaviour observed near the onset of dynamo action, as seen in the recent study of Sreenivasan & Jones (J. Fluid Mech., this issue, vol. 688, 2011, pp. 5–30).
Shear-layers in magnetohydrodynamic spherical Couette flow with conducting walls
- A. M. SOWARD, E. DORMY
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- Journal:
- Journal of Fluid Mechanics / Volume 645 / 25 February 2010
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- 02 February 2010, pp. 145-185
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We consider the steady axisymmetric motion of an electrically conducting fluid contained within a spherical shell and permeated by a centred axial dipole magnetic field, which is strong as measured by the Hartmann number M. Slow axisymmetric motion is driven by rotating the inner boundary relative to the stationary outer boundary. For M ≫ 1, viscous effects are only important in Hartmann boundary layers adjacent to the inner and outer boundaries and a free shear-layer on the magnetic field line that is tangent to the outer boundary on the equatorial plane of symmetry. We measure the ability to leak electric current into the solid boundaries by the size of their relative conductance ɛ. Since the Hartmann layers are sustained by the electric current flow along them, the current inflow from the fluid mainstream needed to feed them increases in concert with the relative conductance, because of the increasing fraction ℒ of the current inflow leaked directly into the solids. Therefore the nature of the flow is sensitive to the relative sizes of ɛ−1 and M.
The current work extends an earlier study of the case of a conducting inner boundary and an insulating outer boundary with conductance ɛo = 0 (Dormy, Jault & Soward, J. Fluid Mech., vol. 452, 2002, pp. 263–291) to other values of the outer boundary conductance. Firstly, analytic results are presented for the case of perfectly conducting inner and outer boundaries, which predict super-rotation rates Ωmax of order M1/2 in the free shear-layer. Successful comparisons are made with numerical results for both perfectly and finitely conducting boundaries. Secondly, in the case of a finitely conducting outer boundary our analytic results show that Ωmax is O(M1/2) for ɛo−1 ≪ 1 ≪ M3/4, O(ɛo2/3M1/2) for 1 ≪ ɛo−1 ≪ M3/4 and O(1) for 1 ≪ M3/4 ≪ ɛo−1. On increasing ɛo−1 from zero, substantial electric current leakage into the outer boundary, ℒo ≈ 1, occurs for ɛo−1 ≪ M3/4 with the shear-layer possessing the character appropriate to a perfectly conducting outer boundary. When ɛo−1 = O(M3/4) the current leakage is blocked near the equator, and the nature of the shear-layer changes. So, when M3/4 ≪ ɛo−1, the shear-layer has the character appropriate to an insulating outer boundary. More precisely, over the range M3/4 ≪ ɛo−1 ≪ M the blockage spreads outwards, reaching the pole when ɛo−1 = O(M). For M ≪ ɛo−1 current flow into the outer boundary is completely blocked, ℒo ≪ 1.
Ekman layers near wavy boundaries
- D. GÉRARD-VARET, E. DORMY
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- Journal:
- Journal of Fluid Mechanics / Volume 565 / 25 October 2006
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- 28 September 2006, pp. 115-134
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We investigate the effect of boundary roughness on the dynamical properties of the flow in laminar Ekman boundary layers. The study considers wavy boundaries having both horizontal wavelength and vertical extent comparable in size with the boundary layer width. In the case of flat boundaries, Ekman layers are known to be active, i.e. to affect significantly the dynamics of the mainstream flow. We show how the layer modelling needs to be modified to account for such wavy boundaries. In particular, nonlinear terms enter the laminar description. This model can be linearized in the limit of small Reynolds numbers. The resulting equations are studied using both asymptotic expansions and full numerical simulations. We find that small-scale roughness significantly alters energy dissipation in the boundary layer. This can result in either a reduction or an increase of dissipation, depending on, in particular, the orientation of the mainstream flow with respect to boundary modulation. Agreement is obtained between theoretical and computational results.
The onset of thermal convection in rotating spherical shells
- E. DORMY, A. M. SOWARD, C. A. JONES, D. JAULT, P. CARDIN
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- Journal:
- Journal of Fluid Mechanics / Volume 501 / 25 February 2004
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- 23 February 2004, pp. 43-70
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The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number $E$ is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius $r_o$ and have short azimuthal length scale $O(E^{1/3}r_o)$. They also confirmed the localization of the convection about some cylinder radius $s\,{=}\,s_M$ roughly $r_o/2$. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius $r_i$ is small (certainly less than $s_M$), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell $r_i\,{=}\,0.35\,r_o$. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale $O(E^{2/9}r_o)$. The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell $r_i\,{=}\,0.65\,r_o$ and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as $10^{-7}$. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case $r_i\,{=}\,0.35\,r_o$. In every case mentioned, we also apply rigid boundary conditions and determine the $O(E^{1/6})$ corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.
A super-rotating shear layer in magnetohydrodynamic spherical Couette flow
- E. DORMY, D. JAULT, A. M. SOWARD
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- Journal:
- Journal of Fluid Mechanics / Volume 452 / 10 February 2002
- Published online by Cambridge University Press:
- 15 February 2002, pp. 263-291
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We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M−2/3 ([Lt ]M−1/2) and latitudinal extent M−1/3. We construct its analytic solution and so determine an inward displacement width O(M−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 104. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.