3.2 Effect of radiative heating

Storage of a strong super-equipartition field of 105 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, 1979Jump To The Next Citation Pointvan Ballegooijen, 1982Jump To The Next Citation Point). 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, 1974van Ballegooijen, 1982Jump To The Next Citation Point). 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, 1996Jump To The Next Citation PointMoreno-Insertis et al., 2002Rempel, 2003Jump To The Next Citation Point). 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, 1996Jump To The Next Citation Point)
dQ -- -- --- = − ∇ ⋅ (Frad) − κ (x21∕a2)(T − T e) dt
where Frad is the unperturbed radiative energy flux, κ is the unperturbed radiative conductivity, x1 is the first zero of the Bessel function J0(x ), a is the tube radius, -- T is the mean temperature of the flux tube, and -- Te 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 −3 −1 −1 ∼ 10 |δ| cm s which does not depend sensitively on the field strength of the flux tube (Fan and Fisher, 1996Jump To The Next Citation PointRempel, 2003Jump To The Next Citation Point). 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 δ < − 10− 4, 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, 1982Schmitt et al., 1984Skaley 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 −4 δ ∼ − 10 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.


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