Jun Zhang, Stephen Childress, and Albert Libchaber

**In an experiment set up and run at Rockefeller University,
Zhang and
Libchaber studied the these non-Boussinesq features of
convection in a 20cm cube of glycerol, at Rayleigh numbers
10 ^{6}-10^{9} and Prandtl numbers 10^{2}-10^{3}. In connection with this
work we have developed a nonlinear thermal boundary layer model to
explain the scaling observed in this and other experiments, and to
predict the bulk temperature of the fluid. The letter deviates from
the arithmetic mean of the wall temperatures, because of the
differing structure of the top and bottom thermal layers.**

**We have also examined the mean velocity near within the
thermal layers, and compared this with the theoretical model. In
all respects the model agrees qualitatively with the experimental
findings, thus explaining the non-Boussinesq effects as principally
a result of the nonlinear effect of variable viscosity on the
thermal boundary layer equation.**

**In an ongoing extension of this work, we have set up in the
WetLab a second generation of the glycerol experiment, and
developed a theoretical model of the large-scale flow developed in
the experiment. In our model, the mean flow is viewed as an
instability of well-established upward and downward thermal
plumes.**

**[1] J. Zhang, S. Childress, and A. Libchaber,
Non-Boussinesq effect: thermal convection with broken symmetry,
Phys. Fluids {\bf 9}, (4), 1034 (1997).**

**[2] J. Zhang, S. Childress, and A. Libchaber,
Non-Boussinesq effect: asymmetric velocity profiles in thermal
convection. Submitted to Physics of Fluids as a brief communication
(1997).**

**Abstract for ref [1]:
We investigate large Rayleigh number (10 ^{6}-10^{9}) and large Prandtl
number (10^{2}-10^{3}) thermal convection in glycerol in as aspect
ratio one cubic cell. The kinematic viscosity of the fluid strongly
depends upon the temperature. The symmetry between the top and
bottom boundary layers is thus broken, the so-called non-Boussinesq
regime. In a previous paper Wu and Libchaber have proposed that in
such a state the two thermal boundary layers adjust their length
scales so that the mean hot and cold temperature fluctuations are
equal in the center of the cell. We confirm this equality. A
simplified two-dimensional model for the mean center temperature
based on an equation for the thermal boundary layer is presented
and compared with the experimental results. The conclusion is that
the central temperature adjusts itself so that the heat fluxes from
the boundary layers are equal, temperature fluctuations at the
center symmetrical, at a cost of very different temperature drops
and Rayleigh number for each boundary.**

**Abstract for ref [2]:
In thermal convection at high Rayleigh numbers, in the hard
turbulent regime, a large scale flow is present. When the viscosity
of the fluid strongly depends on temperature, the top-bottom
symmetry is broken. In addition to the asymmetric temperature
profile across the convection cell, the velocity profiles near the
plate boundaries show dramatic differences from the symmetric case.
We report here that the second derivative of the velocity profiles
are of opposite signs in the thermal sub-layers, through
measurements derived from the power spectrum of temperature
time-series. As a result, the stress rate applied at the plates is
maintained constant within a factor of 3, while the viscosity
changes by a factor of 53i, in qualitative agreement with previous
theory.**