CONVECTION IN A SPHERICALLY SYMMETRICAL ACCRETION FLOW

We develop a simple model of convection in a fluid spherically accreti ng on to a black hole. The model is based on a generalization of the s tandard mixing length theory of convection and is applicable if the co nvective velocities are either larger or smaller than the accretion ve locity - as long as both are significantly lower than the speed of sou nd. The change in the accretion velocity with radius leads to a radial stretching of the inflowing fluid which impedes convection. Thus, in order for convection to occur, the specific entropy has to grow inward s at a rate higher than some threshold value [dS/dr < (dS/dr)(threshol d) < 0], which modifies the familiar Schwarzschild criterion for conve ction. However, this minimum rate is of such a small magnitude that ev en, for instance, optically thick, gas-pressure-dominated accretion fl ows, as discussed by Flammang, would necessarily involve convection in their subsonic portions, although the convective luminosity would be small compared with the total luminosity. We illustrate our model thro ugh its application to accretion on to a small primordial black hole a t the centre of a Sun-like star.