Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

A comprehensive study of the classical linear spin-down of a constant density viscous fluid (kinematic viscosity \nu) rotating rapidly (angular velocity \Omega) inside an axisymmetric cylindrical container (radius L, height H) with rigid boundaries, that follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=\nu/H^2\Omega \ll 1$, was provided by Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}\Omega^{-1})$ time-scale. On the longer lateral viscous diffusion time-scale $O(L^2/\nu)$, the QG-flow responds to the $E^{1/3}$-side-wall shear-layers. In our variant the side-wall and top boundaries are stress-free; a setup 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) side-wall. This is evinced by the QG-angular velocity, \omega*, evolution on the O(L^2/\nu) time-scale: Spatially, \omega* increases (decreases) outwards from the axis for a slippery (rigid) side-wall; temporally, the long-time ($\gg L^2/\nu)$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery side-wall case, the $E^{1/2} \times E^{1/2}$ corner region that forms at the side-wall 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 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.Comment: 33 pages, 10 figure

On the equatorial Ekman layer

This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit E≪1, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the E1/2 Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder C, is geostrophic. Stewartson identified a complicated nested layer structure on C, which comprises relatively thick quasigeostrophic E2/7- (inside C) and E1/4E1/4- (outside C) layers. They embed a thinner ageostrophic E1/3 shear layer (on C), which merges with the inner sphere Ekman layer to form the E2/5-equatorial Ekman layer of axial length E1/5. Under appropriate scaling, this E2/5-layer problem may be formulated, correct to leading order, independent of E. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate z) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite z to account for the far-field behaviour. Adopting z−1 as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.F.M. and E.D. have been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01. F.M. acknowledges a PhD mobility grant from Institut de Physique du Globe de Paris. A.M.S. visited ENS, Paris (19–25 October 2014), while F.M. and E.D. visited the School of Mathematics and Statistics, Newcastle University (respectively, 7–25 September 2015 and 25–30 November 2015); the authors wish to thank their respective host institutions for their hospitality and support

Kinematic dynamo action in large magnetic Reynolds number flows driven by shear and convection

Copyright © 2001 Cambridge University Press. Published version reproduced with the permission of the publisher.A numerical investigation is presented of kinematic dynamo action in a dynamically driven fluid flow. The model isolates basic dynamo processes relevant to field generation in the Solar tachocline. The horizontal plane layer geometry adopted is chosen as the local representation of a differentially rotating spherical fluid shell at co-latitude ϑ; the unit vectors x^, y^ and z^ point east, north and vertically upwards respectively. Relative to axes moving easterly with the local bulk motion of the fluid the rotation vector Ω lies in the (y,z)-plane inclined at an angle ϑ to the z-axis, while the base of the layer moves with constant velocity in the x-direction. An Ekman layer is formed on the lower boundary characterized by a strong localized spiralling shear flow. This basic state is destabilized by a convective instability through uniform heating at the base of the layer, or by a purely hydrodynamic instability of the Ekman layer shear flow. The onset of instability is characterized by a horizontal wave vector inclined at some angle ε to the x-axis. Such motion is two-dimensional, dependent only on two spatial coordinates together with time. It is supposed that this two-dimensionality persists into the various fully nonlinear regimes in which we study large magnetic Reynolds number kinematic dynamo action. When the Ekman layer flow is destabilized hydrodynamically, the fluid flow that results is steady in an appropriately chosen moving frame, and takes the form of a row of cat's eyes. Kinematic magnetic field growth is characterized by modes of two types. One is akin to the Ponomarenko dynamo mechanism and located close to some closed stream surface; the other appears to be associated with stagnation points and heteroclinic separatrices. When the Ekman layer flow is destabilized thermally, the well-developed convective instability far from onset is characterized by a flow that is intrinsically time-dependent in the sense that it is unsteady in any moving frame. The magnetic field is concentrated in magnetic sheets situated around the convective cells in regions where chaotic particle paths are likely to exist; evidence for fast dynamo action is obtained. The presence of the Ekman layer close to the bottom boundary breaks the up-down symmetry of the layer and localizes the magnetic field near the lower boundary

The onset of thermal convection in Ekman–Couette shear flow with oblique rotation

Copyright © 2003 Cambridge University Press. Published version reproduced with the permission of the publisher.The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Ω, which is inclined at an angle ϑ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U0 normal to the plane containing the rotation vector and gravity g (i.e. U0 || g × Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ2, the Reynolds number Re (based on U0), the Prandtl number Pr and the angle ϑ. The basic equilibrium state consists of a linear temperature profile and an Ekman–Couette flow, both dependent only on the vertical coordinate z. Our linear stability study involves determining the critical Rayleigh number Rac as a function of τ and Re for representative values of ϑ and Pr. Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical ϑ = 0 and τ >> 1, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy Ra ≠ 0 mode competition occurs. On increasing τ from zero, with fixed large Re, we identify four types of mode: a convective mode stabilized by the strong shear for moderate τ, hydrodynamic type I and II modes either assisted (Ra > 0) or suppressed (Ra < 0) by buoyancy forces at numerically large τ, and a convective mode for very large τ that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation ϑ _= 0, the symmetry associated with U0 ↔ −U0 for the vertical rotation is broken and so the cases of positive and negative Re exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case ϑ = π/4. Though the overall features of the stability results are broadly similar to the case of vertical rotation , their detailed structure possesses a surprising variety of subtle differences

Non-local effects in the mean-field disc dynamo. II. Numerical and asymptotic solutions

The thin-disc global asymptotics are discussed for axisymmetric mean-field dynamos with vacuum boundary conditions allowing for non-local terms arising from a finite radial component of the mean magnetic field at the disc surface. This leads to an integro-differential operator in the equation for the radial distribution of the mean magnetic field strength, $Q(r)$ in the disc plane at a distance $r$ from its centre; an asymptotic form of its solution at large distances from the dynamo active region is obtained. Numerical solutions of the integro-differential equation confirm that the non-local effects act similarly to an enhanced magnetic diffusion. This leads to a wider radial distribution of the eigensolution and faster propagation of magnetic fronts, compared to solutions with the radial surface field neglected. Another result of non-local effects is a slowly decaying algebraic tail of the eigenfunctions outside the dynamo active region, $Q(r)\sim r^{-4}$, which is shown to persist in nonlinear solutions where $\alpha$-quenching is included. The non-local nature of the solutions can affect the radial profile of the regular magnetic field in spiral galaxies and accretion discs at large distances from the centre.Comment: Revised version, as accepted; Geophys. Astrophys. Fluid Dyna

The publications of P.H. Roberts in year order from PAUL HARRY ROBERTS. 13 September 1929 — 17 November 2022

Paul Roberts was a physicist and applied mathematician. He made important and often pioneering contributions in diverse research areas, but mainly with a fluid dynamic theme, which include rotating fluids, geophysical and astrophysical flows and superfluids. The cornerstone was magnetohydrodynamics with particular application to the geodynamo. His most notable achievement was the first three-dimensional self-consistent numerical geodynamo model, which built on the equations he had derived previously governing the Earth's core dynamics. His considerations included the role of thermal and compositional convection, and the inner core boundary layer (a mixed phase region). In addition to his remarkable illustration of magnetic field reversals, he applied his model to core–mantle coupling and variations in the length of the day. Paul investigated superfluid liquid helium on two fronts. He had a lifelong interest in Bose–Einstein condensates on the microscale governed by the nonlinear Schrodinger equation. His early work on the robust derivation of the mean-field Hall–Vinen–Bekharevic–Khalatnikov equations was far ahead of its time. Their importance and relevance was not appreciated until much later

Bénard convection in a slowly rotating penny shaped cylinder subject to constant heat flux boundary conditions

International audienceWe consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity Ω 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 Ω 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, σ. As σ 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 σ, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction ψ derived from the asymptotics. With ψ 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, Davidson & Dormy (J. Fluid Mech.,vol. 812, 2017, pp. 890-904)

Bénard convection in a slowly rotating penny shaped cylinder subject to constant heat flux boundary conditions

International audienceWe consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity Ω 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 Ω 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, σ. As σ 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 σ, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction ψ derived from the asymptotics. With ψ 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, Davidson & Dormy (J. Fluid Mech.,vol. 812, 2017, pp. 890-904)

Nonlinear solutions of the amplitude equations governing fluid flow in rotating spherical geometries

We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo