6 results
The onset of strongly localized thermal convection in rotating spherical shells
- Andrew P. Bassom, Andrew M. Soward, Sergey V. Starchenko
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- Journal:
- Journal of Fluid Mechanics / Volume 689 / 25 December 2011
- Published online by Cambridge University Press:
- 16 November 2011, pp. 376-416
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A Boussinesq fluid of kinematic velocity and thermal diffusivity is confined within a rapidly rotating shell with inner and outer sphere boundary radii and , respectively. The boundaries of the shell corotate at angular velocity and a continuously varying stratification profile is applied which is unstable in and stable in . When , the unstable zone attached to the inner boundary is thin. As in previous small Ekman number studies, convection at the onset of instability takes on the familiar ‘cartridge belt’ structure, which is localized within a narrow layer adjacent to, but outside, the cylinder tangent to the inner sphere at its equator (Dormy et al. J. Fluid Mech., 2004, vol. 501, pp. 43–70), with estimated radial width of order . The azimuthally propagating convective columns, described by the cartridge belt, reside entirely within the unstable layer when , and extend from the equatorial plane an axial distance along the tangent cylinder as far as its intersection with the neutrally stable spherical surface . We investigate the eigensolutions of the ordinary differential equation governing the axial structure of the cartridge belt both numerically for moderate-to-small values of the stratification parameter and analytically when . At the lowest order of the expansion in powers of , the eigenmodes resemble those for classical plane layer convection, being either steady (exchange of stabilities) or, for small Prandtl number , oscillatory (overstability) with a frequency . At the next order, the axial variation of the basic state removes any plane layer degeneracies. First, the exchange of stabilities modes oscillate at a low frequency causing the short axial columns to propagate as a wave with a small angular velocity, termed slow modes. Second, the magnitudes of both the Rayleigh number and frequency of the two overstable modes, termed fast modes, split. When the slow modes that exist at large azimuthal wavenumbers make a continuous transition to the preferred fast modes at small . At all values of the critical Rayleigh number corresponds to a mode exhibiting prograde propagation, whether it be a fast or slow mode. This feature is shared by the uniform classical convective shell models, as well as Busse’s celebrated annulus model. None of them possess any stable stratification and typically are prone to easily excitable Rossby or inertial modes of convection at small . By way of contrast these structures cannot exist in our model for small due to the viscous damping in the outer thick stable region.
The influence of surface topography on rotating convection
- Peter I. Bell, Andrew M. Soward
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- Journal:
- Journal of Fluid Mechanics / Volume 313 / 25 April 1996
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- 26 April 2006, pp. 147-180
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Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.
On finite-amplitude subcritical instability in narrow-gap spherical Couette flow
- ANDREW P. BASSOM, ANDREW M. SOWARD
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- Journal:
- Journal of Fluid Mechanics / Volume 499 / 25 January 2004
- Published online by Cambridge University Press:
- 27 January 2004, pp. 277-314
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We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii $R_1$ and $R_2$$({>}R_1)$ in the narrow-gap limit, $\varepsilon\,{\equiv}\,(R_2-R_1)/R_1\,{\ll}\,1$, caused by rotating them both about a common axis with distinct angular velocities $\Omega_1$ and $\Omega_2$ respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number $T_{\rm crit}$ which, remarkably, exceeds the local value $T_c$ obtained by approximating the spheres as cylinders in the vicinity of the equator even as $\varepsilon\,{\downarrow}\,0$. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with $\delta\,{\approx}\,(\Omega_1\,{-}\,\Omega_2)/\Omega_1\,{=}\,\OR(\varepsilon^{1/2})$ for which $T_{\rm crit}\,{=}\,T_c\,{+}\,\OR(\delta^2)\,{+}\,\OR(\varepsilon)$. In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent $\OR(\varepsilon^{1/2}R_1)$ gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions.
Here we switch attention to the case when $\varepsilon^{1/2}\,{\ll}\,\delta\,{\le}\, 1$. We show that for Taylor numbers $T\,{=}\,T_c+\OR((\delta\varepsilon)^{2/3})$ there exists a locally unstable region of width $\OR((\delta\varepsilon)^{1/3}R_1)$ within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale $\OR((\varepsilon^{2}/\delta)^{1/3}R_1)$. The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that $T\to T_c$ as $\varepsilon\,{\downarrow}\,0$.
Two particular limits of our theory are examined. In the first, $\varepsilon^{1/2}\,{\ll}\,\delta\,{\ll}\,1$, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of $\delta/\varepsilon^{1/2}$. The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region.
Our second special case $\delta\,{=}\,1$ relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width $\OR(\varepsilon^{1/3}R_1)$ on its associated time scale, which is $\OR(\varepsilon^{-1/3})$ longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.
Kinematic dynamo action in large magnetic Reynolds number flows driven by shear and convection
- YANNICK PONTY, ANDREW D. GILBERT, ANDREW M. SOWARD
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- Journal:
- Journal of Fluid Mechanics / Volume 435 / 25 May 2001
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- 22 June 2001, pp. 261-287
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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 [xcirc ], ŷ and zˆ 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 ∈ 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.
Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell
- ANDREW M. SOWARD, RAINER HOLLERBACH
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- Journal:
- Journal of Fluid Mechanics / Volume 408 / 10 April 2000
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- 10 April 2000, pp. 239-274
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Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.
In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.
The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.
The onset of thermal convection in a rapidly rotating sphere
- CHRIS A. JONES, ANDREW M. SOWARD, ALI I. MUSSA
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- Journal:
- Journal of Fluid Mechanics / Volume 405 / 25 February 2000
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- 25 February 2000, pp. 157-179
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The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than the true global value by an O(1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex s-plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the global criterion on a related two-parameter family of models, which includes the spherical convection problem for particular O(1) values of his parameters. Since he used one of them as the basis of a small-parameter expansion, his results are necessarily approximate for our problem.
Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Yano's relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the leading-order value of the critical Rayleigh number together with its first-order correction for various values of the Prandtl number.
Numerical solutions of the relevant PDEs have also been found, for values of the Ekman number down to 10−6; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.