### 3.2 Effect of radiative heating

Storage of a strong super-equipartition field of 10^{5} G at the base of the solar convection zone requires a
state of mechanical equilibrium since convective motion is not strong enough to counteract the
magnetic stress (Section 5.7). For isolated flux tubes stored in the weakly subadiabatic overshoot
layer, the mechanical equilibrium corresponds to a neutrally buoyant state with a lower internal
temperature (Section 3.1). Therefore flux tubes will be heated by radiative diffusion due to the mean
temperature difference between the tube and the surrounding field-free plasma (see Parker, 1979; van
Ballegooijen, 1982). Moreover, it is not adequate to just consider this zeroth order contribution due to the
mean temperature difference in evaluating the radiative heat exchange between the flux tube and its
surroundings. Due to the convective heat transport, the temperature gradient in the overshoot region and
the lower convection zone is very close to being adiabatic, deviating significantly from that of a radiative
equilibrium, and hence there is a non-zero divergence of radiative heat flux (see Spruit, 1974; van
Ballegooijen, 1982). Thus an isolated magnetic flux tube with internally suppressed convective transport
should also experience a net heating due to this non-zero divergence of radiative heat flux,
provided that the radiative diffusion is approximately unaffected within the flux tube (Fan and
Fisher, 1996; Moreno-Insertis et al., 2002; Rempel, 2003). In the limit of a thin flux tube, the rate
of radiative heating (per unit volume) experienced by the tube is estimated to be (Fan and
Fisher, 1996)
where is the unperturbed radiative energy flux, is the unperturbed radiative conductivity,
is the first zero of the Bessel function , is the tube radius, is the mean temperature of the
flux tube, and is the corresponding unperturbed temperature at the location of the tube. Under the
conditions prevailing near the base of the solar convection zone and for flux tubes that are responsible for
active region formation, the first term due to the non-vanishing divergence of the radiative
heat flux is found in general to dominate the second term. In the overshoot region, it can be
shown that for these flux tubes the time scale for the heating to significantly increase their
buoyancy from an initial neutrally buoyant state is long compared to the dynamic time scale
characterized by the Brunt–Väisälä frequency. Thus the radiative heating is found to cause a
quasi-static rise of the toroidal flux tubes, during which the tubes remain close to being neutrally
buoyant. The upward drift velocity is estimated to be which does not depend
sensitively on the field strength of the flux tube (Fan and Fisher, 1996; Rempel, 2003). This implies
that maintaining toroidal flux tubes in the overshoot region for a period comparable to the
solar cycle time scale requires a strong subadiabaticity of , which is significantly
more subadiabatic than the values obtained by most of the overshoot models based on the
non-local mixing length theory (see van Ballegooijen, 1982; Schmitt et al., 1984; Skaley and
Stix, 1991).
On the other hand if the spatial filling factor of the toroidal flux tubes is large, or if the toroidal
magnetic field is stored in the form of an extended magnetic layer, then the suppression of convective
motion by the magnetic field is expected to alter the overall temperature stratification in the overshoot
region. Rempel (2003) performed a 1D thermal diffusion calculation to model the change of the mean
temperature stratification in the overshoot region when convective heat transport is being significantly
suppressed. It is found that a reduction of the convective heat conductivity by a factor of 100 leads to the
establishment of a new thermal equilibrium of significantly more stable temperature stratification
with in a time scale of a few months. Thus as the toroidal magnetic field is being
amplified by the solar dynamo process, it may improve the conditions for its own storage by
reducing the convective energy transport and increasing the subadiabaticity in the overshoot
region.