2 results
The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers
- Alan C. Newell, Thierry Passot, Mohammad Souli
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- Journal:
- Journal of Fluid Mechanics / Volume 220 / November 1990
- Published online by Cambridge University Press:
- 26 April 2006, pp. 187-252
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We derive the phase diffusion and mean drift equations for the Oberbeck–Boussinesq equations in large-aspect-ratio containers. We are able to recover all the long-wave instability boundaries (Eckhaus, zigzag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. Moreover, the development of the skew-varicose instability can be followed and it becomes clear how the mean drift field conspires to enhance the necking of phase contours necessary for the production of dislocation pairs. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker & Gollub at Prandtl number 2.5, and by Steinberg, Ahlers & Cannell at Prandtl number 6.1. We find a new instability, the focus instability, which causes circular target patterns to destabilize and which, at sufficiently large Rayleigh numbers, may play a major role in the onset of time dependence. Further, we predict the values of the Rayleigh number at which the time-dependent but spatially ordered patterns will become spatially disordered. The key difficulty in obtaining these equations is the fact that the phase diffusion equation appears as a solvability condition at order ε (the inverse aspect ratio) whereas the mean drift equation is the solvability condition at order ε2. Therefore, we had to use extremely robust inversion methods to solve the singular equations at order ε and the techniques we use should prove to be invaluable in a wide range of similar situations. Finally, we discuss the introduction of the amplitude as an active order parameter near pattern defects, such as dislocations and foci.
Turbulence as an Organizing Agent in the ISM
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- By Enrique Vázquez-Semadeni, Instituto de Astronomía, UNAM, Apdo. Postal 70-264, México D. F. 04510, MEXICO, Thierry Passot, Observatoire de la Côte d'Azur, B.P. 4229, 06304, Nice Cedex 4, FRANCE
- Edited by Jose Franco, Universidad Nacional Autónoma de México, Alberto Carraminana, Instituto Nacional de Astrofisica, Optica y Electronica, Tonantzintla, Mexico
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- Book:
- Interstellar Turbulence
- Published online:
- 04 August 2010
- Print publication:
- 28 May 1999, pp 223-231
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Summary
We discuss HD and MHD compressible turbulence as a cloud-forming and cloud-structuring mechanism in the ISM. Results from a numerical model of the turbulent ISM at large scales suggest that the phase-like appearance of the medium, the typical values of the densities and magnetic field strengths in the intercloud medium, as well as the velocity dispersion-size scaling relation in clouds may be understood as consequences of the interstellar turbulence. However, the density-size relation appears to only hold for the densest clouds, suggesting that low-column density clouds, which are hardest to observe, are turbulent transients. We then explore some properties of highly compressible polytropic turbulence, in one and several dimensions, applicable to molecular cloud scales. At low values of the polytropic index γ, turbulence may induce the gravitational collapse of otherwise linearly stable clouds, except if they are magnetically subcritical. The nature of the density fluctuations in the high Mach-number limit depends on γ. In the isothermal (γ = 1) case, the dispersion of In (ρ) scales like the turbulent Mach number. The latter case is singular with a lognormal density pdf, while power-law tails develop at high (resp. low) densities for γ < 1 (resp. γ > 1). As a consequence, density fluctuations originating from Burgers turbulence are similar to those of the polytropic case only at high density when γ « 1 and M » 1.
Introduction
One of the main features of turbulence is its multi-scale nature (e.g., Scalo 1987; Lesieur 1990).