Thermoconvective waves in a rotating fluid.

Abstract

Propagation of thermoconvective waves (TCW) in an unbounded incompressible viscous heat-conducting fluid is investigated when the fluid rotates about a horizontal axis with uniform angular velocity \ensuremath{\Omega} and is subjected to a constant temperature gradient \ensuremath{\beta} perpendicular to the axis of rotation. A novel result of the analysis is that, unlike the nonrotating case, undamped TCW can propagate (with \ensuremath{\beta}>0 such that there is no thermal instability) if 2${P}^{2}$S(1+P)P\ensuremath{\gamma}2S, where P=\ensuremath{\nu}/\ensuremath{\kappa}, S=4(\ensuremath{\nu}/${g}^{2}$${)}^{2/3}$${\ensuremath{\Omega}}^{2}$, and \ensuremath{\gamma}=(${\ensuremath{\nu}}^{2}$/g${)}^{1/3}$\ensuremath{\alpha}\ensuremath{\beta}. Here, \ensuremath{\nu}, \ensuremath{\kappa}, \ensuremath{\alpha}, and g denote the kinematic viscosity, the thermal diffusivity, the fluid volume-expansion coefficient, and gravitational acceleration, respectively. It is also shown that, unlike the nonrotating case where weakly damped low-frequency TCW can propagate when the fluid is heated from below, such waves in a rotating fluid are possible if \ensuremath{\beta}0 and P\ensuremath{\gamma}S. It turns out that when a layer of finite thickness d rotates about a horizontal axis and is subjected to a constant temperature gradient \ensuremath{\beta} perpendicular to the rotation axis, weakly damped low-frequency TCW can propagate without violating the condition of thermal stability if 8/3${\mathrm{\ensuremath{\pi}}}^{4}$\ensuremath{\le}R-${\mathit{S}}_{2}$27/4${\mathrm{\ensuremath{\pi}}}^{4}$, where R=g\ensuremath{\alpha}\ensuremath{\beta}${\mathit{d}}^{4}$/\ensuremath{\kappa}\ensuremath{\nu} and ${\mathit{S}}_{1}$=4${\mathrm{\ensuremath{\Omega}}}^{2}$${\mathit{d}}^{4}$/${\ensuremath{\nu}}^{2}$.