Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T07:16:57.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-layered diffusive convection. Part 1. Spontaneous layer formation

Published online by Cambridge University Press:  26 March 2010

TAKASHI NOGUCHI  and
HIROSHI NIINO
TAKASHI NOGUCHI*
Affiliation:
Ocean Research Institute, The University of Tokyo, Nakano, Tokyo 164-8639, Japan
HIROSHI NIINO
Affiliation:
Ocean Research Institute, The University of Tokyo, Nakano, Tokyo 164-8639, Japan
*
†Present address: Department of Aeronautics and Astronautics, Kyoto University, Yoshida-Honmachi, Sakyo, Kyoto 606-8501, Japan. Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Diffusive convection in an infinite two-dimensional fluid with linear vertical gradients of temperature and salinity is studied numerically and analytically. When the density gradient ratio exceeds a critical value above which diffusive convection grows according to the linear stability analysis, spontaneous layer formation is found to occur. At the first stage nearly steady oscillating motions, the horizontal scale of which is of the order of the buoyancy boundary layer scale δ, arise. After several tens of the oscillation cycle, a transition to the second stage occurs in which overturning convective motions develop and well-mixed regions are formed. The convective motions resemble Rayleigh–Bénard convection at a high Rayleigh number. The well-mixed regions are gradually organized into horizontal layers, a typical thickness of which is of the order of δ. A detailed analysis of the nonlinear process during the layer formation reveals that four modes are responsible for the layer formation: The first mode is the linear fastest-growing mode with wavenumber vector (k0, 0). The second mode with (k0, m0) is weakly growing. The third mode with (0, m0) is dissipating, and the fourth mode is its higher harmonic having (0, 2m0). It is shown that a truncated spectral model with the four modes well reproduces the results of the full numerical simulation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 651 , 25 May 2010 , pp. 443 - 464
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahlers, G. & Lücke, M. 1987 Some properties of an eight-mode Lorenz model for convection in binary fluids. Phys. Rev. A 35, 470–473.CrossRefGoogle ScholarPubMed
Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.CrossRefGoogle Scholar
Fernando, H. J. S. 1987 The formation of a layered structure when a stable salinity gradient is heated from below. J. Fluid Mech. 182, 525–541.CrossRefGoogle Scholar
Hoare, R. A. 1966 Problems of heat transfer in Lake Vanda, a density stratified Antarctic lake. Nature 210, 787–789.CrossRefGoogle Scholar
Huppert, H. E. & Linden, P. F. 1979 On heating a stable salinity gradient from below. J. Fluid Mech. 95, 431–464.CrossRefGoogle Scholar
Huppert, H. E. & Turner, J. S. 1981 A laboratory model of a replenished magma chamber. Earth Planet. Sci. Lett. 54, 144–172.CrossRefGoogle Scholar
Linden, P. F. 1976 The formation and destruction of fine-structure by double diffusive processes. Deep-Sea Res. 23, 895–908.Google Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82, 1397–1412.CrossRefGoogle Scholar
Neal, V. T., Neshyba, S. & Denner, W. 1969 Thermal stratification in the Arctic ocean. Science 166, 373–374.CrossRefGoogle ScholarPubMed
Neshyba, S., Neal, V. T. & Denner, W. 1971 Temperature and conductivity measurements under upper Ice Island T-3. J. Geophys. Res. 76, 8107–8120.CrossRefGoogle Scholar
Noguchi, T. & Niino, H. 2010 Multi-layered diffusive convection. Part 2. Dynamics of layer evolution. J. Fluid Mech. doi:10.1017/S0022112010994160.CrossRefGoogle Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Blackie.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa Publishers.Google Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35, 209–218.CrossRefGoogle Scholar
Swallow, J. C. & Crease, J. 1965 Hot salty water at the bottom of the Red Sea. Nature 205, 165–166.CrossRefGoogle Scholar
Tait, R. I. & Howe, M. R. 1968 Some observation of thermohaline stratification in the deep ocean. Deep-Sea Res. 15, 275–280.Google Scholar
Turner, J. S. 1965 The coupled turbulent transports of salt and heat across a sharp density interface. Intl J. Heat Mass Transfer 8, 759–767.CrossRefGoogle Scholar
Turner, J. S. 1968 The behaviour of a stable salinity gradient heated from below. J. Fluid Mech. 33, 183–200.CrossRefGoogle Scholar
Veronis, G. 1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1–17.Google Scholar
Veronis, G. 1968 Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.CrossRefGoogle Scholar