2 results
Instability and breakdown of a vertical vortex pair in a strongly stratified fluid
- MICHAEL L. WAITE, PIOTR K. SMOLARKIEWICZ
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- Journal:
- Journal of Fluid Mechanics / Volume 606 / 10 July 2008
- Published online by Cambridge University Press:
- 10 July 2008, pp. 239-273
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- Article
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The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Frh < 0.2 and 0.1 < Frz < 10. Here Frh ≡ U/(NR) and Frz ≡ Ukz/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt–Väisälä frequency, and 2π/kz is the vertical wavelength of a small initial perturbation. At early times with Frz < 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Frz > 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R ≈ 0.2/Frz for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 < Frz < 0.8. Short perturbations with Frz > 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds number Re ≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.
14 - Studies of Geophysics
- from SECTION D - FRONTIER FLOWS
- Edited by Fernando F. Grinstein, Len G. Margolin, Los Alamos National Laboratory, William J. Rider, Los Alamos National Laboratory
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- Book:
- Implicit Large Eddy Simulation
- Published online:
- 08 January 2010
- Print publication:
- 30 July 2007, pp 413-438
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Summary
Introduction
Prediction of the Earth's climate andweather is difficult in large part because of the ubiquity of turbulence in the atmosphere and oceans. Geophysical flows evince fluid motions ranging from dissipation scales as small as a fraction of a millimeter to planetary scales of thousands of kilometers. The span in time scales (from a fraction of a second to many years) is equally large. Turbulence in the atmosphere and the oceans is generated by heating and by boundary stresses – just as in engineering flows. However, geophysical flows are further complicated by planetary rotation and density–temperature stratification, which lead to phenomena not commonly found in engineering applications. In particular, rotating stratified fluids can support a variety of inertia-gravity and planetary waves. When the amplitude of such a wave becomes sufficiently large (i.e., comparable to the wavelength), the wave can break, generating a localized burst of turbulence. If one could see the phenomena that occur internally in geophysical flows at any scale, one would be reminded of familiar pictures of white water in a mountain stream or of breaking surf on a beach. The multiphase thermodynamics of atmosphere and oceans – due to ubiquity of water substance and salt, respectively – adds complexity of its own.
Because of the enormous range of scales, direct numerical simulation (DNS) of the Earth's weather and climate is far beyond the reach of current computational technology. Consequently, all numerical simulations truncate the range of resolved scales to one that is tractable on contemporary computational machines. However, retaining the physicality of simulation necessitates modeling the contribution of truncated scales to the resolved range.