Recent magnetic studies of Mars suggest that (1) it possessed a periodically reversing magnetic field for the first ~500 Myr of its existence and (2) plate tectonics may have been operating during this time. On Earth the geodynamo is thought to occur because of convection in the outer core. This paper estimates the amount of heat the Martian core can conduct in the absence of convection. It uses parameterized, variable-viscosity thermal evolution models to show that the core heat flux increases if the planet's surface heat flux is increased above the value required to eliminate instantaneous radiogenic heat production. Conversely, a sudden reduction in surface heat flux causes the mantle to heat up and the core heat flux to become negative. Thus, if plate tectonics, or some other process causing high surface heat flux, was occurring on early Mars, it is likely to have caused convection in the core and hence generated a magnetic field. Conversely, a reduction in surface heat flux would probably have caused the core to stop convecting and shut off the magnetic field. There is thus an important link between surface processes and core magnetism, which may also be relevant to planets such as Earth and Venus. ¿ 2000 American Geophysical Union