In an earlier paper [Phys. Fluids 31, 2065 (1988)], a numerical simulation of a passive scalar field convected by a frozen velocity field, i.e., a velocity field with an infinite correlation time, was performed. In this paper, a simulation of a passive scalar field convected by a velocity field which is rapidly stirred at all scales of motion, i.e., a velocity field with near zero correlation time, is performed. For an energy spectrum of the form E(k)∝k−5/3, the temperature spectrum G(k) is found to obey G(k)∝k−11/3 when conductive effects are dominant. A theoretical model is proposed which obtains the above result by representing the transfer of scalar variance by an eddy conductivity, whose correlation time is limited by the correlation time of the velocity field.