Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-18T23:42:57.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular convection with finite amplitude in a rotating fluid

Published online by Cambridge University Press:  28 March 2006

G. Veronis
G. Veronis
Affiliation:
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
Get access Rights & Permissions [Opens in a new window]

Abstract

When a rotating layer of fluid is heated uniformly from below and cooled from above, the onset of instability is inhibited by the rotation. The first part of this paper treats the stability problem as it was considered by Chandrasekhar (1953), but with particular emphasis on the physical interpretation of the results. It is shown that the time-dependent (overstable) motions occur because they can reduce the stabilizing effect of rotation. It is also shown that the boundary of a steady convection cell is distorted by the rotation in such a way that the wave length of the cell measured along the distorted boundary is equal to the wavelength of the non-rotating cell. This conservation of cellular wavelength is traced to the constancy of horizontal vorticity in the rotating and non-rotating systems. In the finite-amplitude investigation the analysis, which is pivoted about the linear stability problem, indicates that the fluid can become unstable to finite-amplitude disturbances before it becomes unstable to infinitesimal perturbations. The finite-amplitude motions generate a non-linear vorticity which tends to counteract the vorticity generated by the imposed constraint of rotation. Under experimental conditions the two fluids, mercury and air, which are considered in this paper, will not exhibit this finite amplitude instability. However, a fluid with a sufficiently small Prandtl number will become unstable to finite-amplitude perturbations. The special role of viscosity as an energy releasing mechanism in this problem and in the Orr-Sommerfeld problem suggests that the occurrence of a finite-amplitude instability depends on this dual role of viscosity (i.e. as an energy releasing mechanism as well as the more familiar dissipative mechanism). The relative stability criterion developed by Malkus & Veronis (1958) is used to determine the preferred type of cellular motions which can occur in the fluid. This preferred motion is a function of the Prandtl number and the Taylor number. In the case of air it is shown that overstable square cells become preferred in finite amplitude, even though steady convective motions occur at a lower Rayleigh number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 5 , Issue 3 , April 1959 , pp. 401 - 435
Copyright
© 1959 Cambridge University Press

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

Boussinesq, J. 1903 Theorie Analytique de la Chaleur, 2, 172. Paris: Gauthier-Villars.
Chandrasekhar, S. 1952 On the inhibition of convection by a magnetic field. Phil. Mag. (7), 43, 501.Google Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated below and subject to coriolis forces. Proc. Roy. Soc. A, 217, 306.Google Scholar
Christopherson, D. G. 1940 Quart. J. Math. 11, 63.
Fultz, D. & Nakagawa, Y. 1955 Proc. Roy. Soc. A, 231, 198.
Gor'kov, L. P. 1957 J. Exptl. Theor. Phys. (U.S.S.R.), 33, 402.
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225.Google Scholar
Meksyn, D. & Stuart, J. T. 1951 Proc. Roy. Soc. A, 208, 517.
Nakagawa, Y. & Frenzen, P. 1955 A theoretical and experimental study of cellular convection in rotating fluids. Tellus, 7, 1.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1.Google Scholar