3 results
Convectively driven shear and decreased heat flux
- David Goluskin, Hans Johnston, Glenn R. Flierl, Edward A. Spiegel
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- Journal:
- Journal of Fluid Mechanics / Volume 759 / 25 November 2014
- Published online by Cambridge University Press:
- 31 October 2014, pp. 360-385
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We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ($\mathit{Pr}$) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ($\mathit{Ra}$) sufficiently, and we explore the resulting convection for $\mathit{Ra}$ up to $10^{10}$. When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $\mathit{Ra}\rightarrow \infty$. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $\mathit{Ra}$. When the large-scale shear is present with $\mathit{Pr}\lesssim 2$, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $\mathit{Ra}$ for $\mathit{Pr}=1$. When the shear is present with $\mathit{Pr}\gtrsim 3$, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $\mathit{Ra}$, but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to $\mathit{Ra}^{0.077}$ when $\mathit{Pr}=3$ and to $\mathit{Ra}^{0.19}$ when $\mathit{Pr}=10$. Analogies with tokamak plasmas are described.
2 - Reflections on the solar tachocline
- Edited by D. W. Hughes, University of Leeds, R. Rosner, University of Chicago, N. O. Weiss, University of Cambridge
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- Book:
- The Solar Tachocline
- Published online:
- 21 August 2009
- Print publication:
- 31 May 2007, pp 31-50
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Summary
Solar activity takes place in narrow bands of latitude that move like solitary waves from mid-latitudes toward the solar equator. This behaviour points to the existence of a thin layer in the Sun that may serve as a waveguide. With its grand minima, the cycle is intermittent in a way that does not occur in the simplest chaos models. To be useful as a primitive model of the cycle, a differential equation should be of high enough order to display such strong intermittency. These and other features of solar fluid dynamics led to the adumbration of an intermediate shear layer between the convection zone and the radiative core. This layer, like the weather layers in planetary atmospheres, produces coherent structures – sunspots and perhaps vortices. Similar layers may play a role in stellar activity in cool stars other than the Sun and perhaps even in hot stars if their atmospheres are turbulent.
The maculate Sun
Rotation and turbulence in stars are significant for an understanding of stellar evolution and for the fluid dynamics of accretion discs. We can watch these processes most closely in our own Solar System. Observations of the Sun, the giant planets and the earth reveal coherent structures whose study has been one of the most exciting adventures in the mathematical science of the twentieth century. (At a meeting in the Newton Institute, we ought to recall this.)
26 - Continuum equations for stellar dynamics
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- By Edward A. Spiegel, Department of Astronomy Columbia University, New York, NY 10027, USA, Jean-luc Thiffeault, Department of Applied Physics and Applied Mathematics Columbia University, New York, NY 10027, USA
- Edited by Michael J. Thompson, Imperial College of Science, Technology and Medicine, London, Jørgen Christensen-Dalsgaard, Aarhus Universitet, Denmark
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- Book:
- Stellar Astrophysical Fluid Dynamics
- Published online:
- 11 November 2009
- Print publication:
- 01 May 2003, pp 377-392
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Summary
The description of a stellar system as a continuous fluid represents a convenient first approximation to stellar dynamics, and its derivation from the kinetic theory is standard. The challenge lies in providing adequate closure approximations for the higher-order moments of the phase-space density function that appear in the fluid dynamical equations. Such closure approximations may be found using representations of the phase-space density as embodied in the kinetic theory. In the classic approach of Chapman and Enskog, one is led to the Navier–Stokes equations, which are known to be inaccurate when the mean free paths of particles are long, as they are in many stellar systems. To improve on the fluid description, we derive here a modified closure relation using a Fokker–Planck collision operator. To illustrate the nature of our approximation, we apply it to the study of gravitational instability. The instability proceeds in a qualitative manner as given by the Navier–Stokes equations but, in our description, the damped modes are considerably closer to marginality, especially at small scales.
A kinetic equation
If we have a system of N stars, with N very large, and wish to study its large-scale dynamics, we have to choose the level of detail we can profitably treat. Even if we could know the positions and velocities of all N stars for all times, we would be mainly interested in the global properties that are implied by this information.