Cellular convection with finite amplitude in a rotating fluid

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.

Oscillatory instability and fluid patterns in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

“…Only the frequency f (1) 101 exists for T a ≥ 2700, which then increases monotonically with T a. Figure 5 (b) shows the variation of frequencies f (2) 101 and f (2) 011 with T a. They also remain initially equal, and then become unequal at T a ≥ 1115.…”

“…Rubio, Lopez & Marques 27 showed interesting effects of the centrifugal force even for small values (∼ 10 −2 ) of the Froude number F r, which is a ratio of the centrifugal force to the force of buoyancy. The Coriolis force is known to break the mirror symmetry of the convective flow patterns 2,6 even at small rotation rates. Chandrasekhar 3 analyzed the effects of Coriolis force on the onset of convection in a Rayleigh-Bénard system.…”

Localized rotating convection with no-slip boundary conditions

“…r ), where r is a reference temperature and > 0. This physical configuration has been studied for the wealth of instabilities it exhibits 3,20,21 and for the small amplitude but complex dynamics that result. [22][23][24] We restrict attention to two-dimensional convection described by the dimensionless equations, written in the rotating frame,…”

Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence

“…The functional form of the zonal velocity components differs from the corresponding expressions in the previous studies [7][8][9][10][11][14][15][16] because of the fact that the velocity divergence is nonzero. Thus, we seek the expression of u in the form:…”

“…A typical three-dimensional basic state is cyclonic vorticity with rising motion due to surface convergence and anticyclonic vorticity as well as sinking motion due to surface divergence. Exact solutions of the Navier-Stokes equations for a three-dimensional vortex have been discovered [7][8][9][10][11]; of particular interest is Sullivan's two-cell vortex solution, because the flow not only spirals in toward the axis and out along it, but it also has a region of reverse flow near the axis.…”