E-mail: edward.bolton@yale.edu

Computational and Theoretical Plane Layer Convection

Thermal convection in horizontal plane layers is widely regarded as a classic example of hydrodynamic instability. In the companion papers Busse and Bolton [1984] and Bolton and Busse [1985], we examined the instabilities of nonlinear convection rolls with stress-free boundary conditions. Both the numerical and analytical studies indicated that, except at infinite Prandtl number, the convection rolls with the critical wavenumber are unstable to the skewed-varicose instability. We used a perturbation expansion technique for the analytical paper, and a Galerkin technique for the numerical paper.

In another paper [Fauve, Bolton, and Brachet 1987], we developed phase equations for the oscillatory instability of both rigid and free-slip horizontal-layer convection. This was the first paper in which the coefficients of phase equations were calculated from nonlinear simulations. We found that the oscillatory instability should consist of traveling waves, rather than standing waves. In addition, we predicted that, for convection of mercury, the oscillatory instability should have a subcritical onset for low wavenumbers.

In a second numerical study [Bolton, Busse, and Clever 1986], we examined the stability of convection rolls with rigid boundary conditions at intermediate Prandtl numbers. A family of oscillatory "blob" instabilities were discovered to limit the region of stable rolls at high Rayleigh numbers and low wavenumbers. We were also successful in characterizing the transition from the knot to the cross-roll instability.

Nonlinear convection in plane layers may take on a variety of planforms when the viscosity is temperature dependent. For the square planform case, we have developed neutral curves and small-amplitude results, as well as a fully nonlinear code [Bolton and Ribe, 1989, abst.].

Busse, F.H. and E.W. Bolton (1984) Instabilities of convection rolls with stress-free boundaries near threshold, J. Fluid Mech., 146, 115-125.

Bolton, E.W. and F.H. Busse (1985) Stability of convection rolls in a layer with stress-free boundaries, J. Fluid Mech., 150, 487-498.

Bolton, E.W., F.H. Busse and R.M. Clever (1986) Oscillatory instabilities of convection rolls at intermediate Prandtl numbers, J. Fluid Mech., 164, 469-485.

Fauve, S., E.W. Bolton and M.E. Brachet (1987) Nonlinear oscillatory convection: A quantitative phase dynamics approach, Physica, 29D, 202-214.

E.W. Bolton and N.M. Ribe (1989) Square-cell convection in a fluid with temperature dependent viscosity, EOS, Trans of Am. Geoph. Union, 70, p. 1333.