An experimental investigation of convection in a rotating sphere subject to time varying thermal boundary conditions

Chamberlain, J. A.; Carrigan, Charles R.

doi:10.1080/03091928608245897

Cited by 42 publications

(8 citation statements)

References 18 publications

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“…More recently, Guervilly, Cardin & Schaeffer (2019) examined the convective length scales in a rotating sphere using both quasi-geostrophic and fully three-dimensional methods in the regime of very low Ekman number and small Prandtl number, albeit with a weak thermal forcing (around the onset of convection). Convection was studied experimentally in the full sphere by Chamberlain & Carrigan (1986) using centrifugal gravity, though practical considerations precluded the exploration of very supercritical Rayleigh numbers: in that study, Ra 2.5Ra c . The highly supercritical regime of convection in a whole sphere is of great relevance to the dynamics of the Earth's core prior to inner core nucleation and to some fully convective stars, yet the problem remains largely unexplored.…”

Section: Introductionmentioning

confidence: 99%

Large-scale vortices and zonal flows in spherical rotating convection

Lin

Jackson

2021

J. Fluid Mech.

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Section: Introductionmentioning

confidence: 99%

Large-scale vortices and zonal flows in spherical rotating convection

Lin

Jackson

2021

J. Fluid Mech.

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“…As well as our investigations of the cases with stress-free boundaries, we also consider the solutions in the case of rigid boundaries at which we apply a no-slip boundary condition. This is, of course, the true situation in the Earth's context as well as for experiments (Busse & Carrigan 1976;Carrigan & Busse 1983;Chamberlain & Carrigan 1986;Hart, Glatzmaier & Toomre 1986;Cordero & Busse 1992). In that case, an Ekman layer forms at the boundary outside which the mainstream solution on the short E 1/3 azimuthal length scale is influenced at O(E 1/6 ) by Ekman-layer suction, as noted by Zhang & Jones (1993).…”

Section: Introductionmentioning

confidence: 85%

“…The former 'internal heating' case is motivated by the situation for a complete sphere (r i = 0) for which the uniform heat source distribution leads to the temperature gradient −β r, where β ≡ 2(T i − T o )/r 2 o (see also Chamberlain & Carrigan 1986). In the latter 'differential heating' case, there are no internal heat sources and the temperature gradient is simply maintained by the temperature difference between the inner and outer boundaries (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

The onset of thermal convection in rotating spherical shells

et al. 2004

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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.

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“…On the other hand, in liquid metals where P 1 the thermal Ekman number E κ = E/P becomes the relevant control parameter. The description of the convective motions as barotropic thermal Rossby waves has been validated in water experiments, both at low forcings (Busse & Carrigan 1976;Carrigan & Busse 1983 for the onset of convection with E > 1.7 × 10 −5 , Chamberlain & Carrigan 1986;Cordero & Busse 1992) and strong forcings (Cardin & Olson 1992 where E > 4 × 10 −6 and R up to 50 times critical), R = γ α T d 4 /κν being the Rayleigh number with γ the gravity gradient, α the thermal expansion coefficient of the fluid and T the temperature difference between the inner and the outer boundaries. Cardin & Olson (1994) -see also Sumita & Olson (2000 who used a hemispherical shell -concentrated on the evolution of the heat flux as convection develops from onset by comparing experiments performed in water with quasi-geostrophic numerical simulations.…”

Section: Definitionmentioning

confidence: 99%

Experimental and numerical studies of convection in a rapidly rotating spherical shell

et al. 2007

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Thermal convection in a rapidly rotating spherical shell is investigated experimentally and numerically. The experiments are performed in water (Prandtl number P=7) and in gallium (P=0.025), at Rayleigh numbers R up to 80 times the critical value in water (up to 6 times critical in gallium) and at Ekman numbers E∼10−6. The measurements of fluid velocities by ultrasonic Doppler velocimetry are quantitatively compared with quasi-geostrophic numerical simulations incorporating a varying β-effect and boundary friction (Ekman pumping). In water, unsteady multiple zonal jets, weaker in amplitude than the non-axisymmetric flow, are experimentally observed and numerically reproduced at moderate forcings (R/Rc<40). In this regime, zonal flows and vortices share the same length scale. Gallium experiments and strongly supercritical convection experiments in water correspond to another regime. In these turbulent flows, the zonal motion amplitude U dominates the non-axisymmetric motion amplitude Ũ. As a result of the reverse cascade of kinetic energy, the characteristic Rhines length scale $\ell_{\beta} \,{\sim}\, \sqrt{\overline{U}/\beta}$ of zonal jets emerges, and the boundary friction becomes the main brake on the growth of the zonal flow. A scaling law U ∼ Ũ4/3 is then derived and verified both numerically and experimentally.

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An experimental investigation of convection in a rotating sphere subject to time varying thermal boundary conditions

Cited by 42 publications

References 18 publications

Large-scale vortices and zonal flows in spherical rotating convection

Large-scale vortices and zonal flows in spherical rotating convection

The onset of thermal convection in rotating spherical shells

Experimental and numerical studies of convection in a rapidly rotating spherical shell

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