9.1 Subsurface evolution of active region magnetic fields

If we believe that the magnetic field seen in sunspots and active regions on the solar surface originates from a large-scale, predominantly toroidal magnetic field generated at the base of the solar convection zone, then the process of how active region flux is transported through the convection zone and emerge into the solar atmosphere becomes a crucial link in the solar cycle dynamo puzzle. Theoretical, numerical, and observational studies in the past 3 decades have greatly improved our understanding of the process, but have also raised many new questions. One of the key questions remaining unanswered is the field strength B of the toroidal magnetic field at the base of the convection zone generated by the solar cycle dynamo. Whether B is significantly greater than or comparable to 4 Beq ∼ 10 G, which is the field strength in equipartition with the kinetic energy density of the convective motions, will critically affect the formation, dynamic evolution, and properties of the emerging active region flux tubes. As was described in Section 5.7, for a buoyant flux tube with B ≳ (H ∕a)1∕2B ∼ 3B p eq eq, the magnetic buoyancy force of the flux tube dominates the hydrodynamic force of the strongest downdrafts, and the flux tube can rise unaffected by convection. On the other hand, for Beq ≲ B ≲ 3 × Beq the strong convective downdrafts can overcome the magnetic buoyancy and hence plays a significant role in affecting the dynamics and structure of the emerging Ω-shaped tubes.

9.1.1 Magnetic buoyancy dominated regime

Thin flux tube simulations of emerging flux loops in the solar convective envelope (Section 5.1) suggest that the toroidal magnetic fields at the base of the solar convection zone is in the range of about 3 × 104 to 105 G, significantly higher than the equipartition field strength, in order for the emerging loops to be consistent with the observed properties of solar active regions. In this field strength range, it is found that Coriolis force acting on rising Ω-shaped loops produce several asymmetries between the leading and the following sides of the emerging loops, providing explanations for the observed asymmetries between the leading and the following polarities of bipolar active regions, e.g. the active region tilts described by Joy’s law and the observed asymmetric proper motions of the two polarities of newly developing active regions.

At B ∼ 105 G, toroidal magnetic fields stored in the weakly subadiabatic overshoot region are preferably in an equilibrium state of neutral buoyancy, with the magnetic curvature force balanced by the Coriolis force due to a prograde toroidal flow (Section 3.1). This is true regardless of whether the field is in the state of an extended magnetic layer or isolated flux tubes. Detecting this prograde toroidal flow through helioseismology may be a way to probe and measure the toroidal magnetic fields stored in the tachocline region. Isolated toroidal flux tubes stored in the weakly subadiabatic overshoot region should experience a radiative heating due to the non-zero divergence of the radiative heat flux (Section 3.2). This radiative heating causes a quasi-static upward drift of the toroidal flux tubes. In order to maintain toroidal flux tubes in the overshoot region for a time scale comparable to the solar cycle period, a rather strong subadiabaticity of δ < − 10 −4 is needed, which is significantly more subadiabatic than the values obtained by most of the overshoot models based on the non-local mixing length theory. This strong subadiabaticity may be achieved as a result of some level of suppression of convective motions by the toroidal magnetic flux tubes themselves. Furthermore, a semi-analytical model of convective overshoot (Rempel, 2004) – based on the assumption that the convective energy transport is governed by coherent downflow plumes – shows that the overshoot region can have a subadiabaticity of δ ≲ − 10 −4 if the downflow filling factor at the base of the convection zone is −4 ≲ 10.

Neutrally buoyant toroidal flux tubes stored in the weakly subadiabatic overshoot region are subject to the onset of the undulatory buoyancy instability depending on the field strength and the value of the subadiabaticity (Section 4.1). The toroidal flux tubes become buoyantly unstable and develop Ω-shaped emerging flux loops when the field strength becomes sufficiently large, or when the quasi-static upward drift due to radiative heating brings the tubes out to regions of sufficiently weak subadiabaticity or into the convection zone. A neutrally buoyant equilibrium magnetic layer is also subject to the same type of undulatory buoyancy instability and 3D MHD simulations show that arched buoyant flux tubes form as a result of the non-linear growth of the instability (Section 4.2).

There are several major difficulties associated with explaining active regions as rising flux tubes in the magnetic buoyancy dominated regime (with B ∼ 105 G at the base of the convection zone) as listed in the following. (1) Recent solar cycle dynamo models which take into account the dynamic effects of the Lorentz force from the large-scale mean fields have suggested that the toroidal magnetic field generated at the base of the solar convection zone is ∼ 1.5 × 104 G (Rempel, 2006bJump To The Next Citation Point). (2) The long length scales of the Ω-tubes required for the onset of the magnetic buoyancy instability are too large compared to the observed longitudinal extent of the active regions (see Section 8.3). (3) The high twist rate required to counteract the vorticity generation by the magnetic buoyancy, which tends to break up the rising flux tube, is inconsistent with the observed twists and tilts in the majority of solar active regions on the surface (Sections 5.4 and 5.5). Possible solutions for resolving the above difficulties have been proposed. For (1), for example, the amplification of a toroidal magnetic field by conversion of potential energy associated with the superadiabatic stratification of the convection zone may be a means to reach field strength that is significantly above the equipartition value (Section 7). For (2), a possible solution may be the dynamic disconnection of surface active regions from their parent tubes in the deep interior (Section 8.3). For (3), further 3D simulations of the formation and rise of buoyant flux tubes from toroidal magnetic fields initially in mechanical equilibrium in the presence of solar rotation are necessary. Another possible solution for (3) is the much lower rate of twist of the emerged magnetic field in the solar atmosphere compared to that in the interior portion of the tube as a result of the rapid and extreme stretching of the field in the atmosphere (see Figure 38View Image and the discussion in Section 8.2).

9.1.2 The effects of convection

The range of field strengths of 104 G ≲ B ≲ 3 × 104 G for toroidal flux tubes at the base of the solar convection is an interesting regime where the strong convective downdrafts can overcome the magnetic buoyancy of the tube, and hence play a significant role in affecting the formation, dynamics, and structure of the emerging Ω-tubes (Section 5.7). Some simulations have found that magnetic fields of B ≲ Beq are preferentially transported downward against their magnetic buoyancy out of the turbulent convection zone into the stably stratified overshoot region by compressible penetrative convection (Section 6). This is also the range of field strength for the toroidal magnetic field suggested by the recent dynamic mean field dynamo model for the solar cycle (Rempel, 2006a,b). The rise of flux tubes with these field strengths through the solar convective envelope, under the influence of both the Coriolis force and the turbulent convective flows has not been well studied.

Previous thin flux tubes simulations (Section 5.1) which neglect the effect of convection, have shown that tubes at these field strengths tend to be significantly deflected poleward by the Coriolis force during their rise, and thus have difficulty reproducing the emergence of active regions at low latitudes. They are also found to produce tilt angles that are inconsistent with Joy’s law. Therefore, an important question is whether the the convective flows in the solar convective envelope can modify the dynamics of the rising Ω-tubes such that they emerge with properties that are consistent with the observed properties of solar active regions. At these field strengths, the convective downdrafts are capable of pinning down the flux tube while the broad helical upflows in between the downdrafts can boost the rise and thus possibly produce emerging Ω-loops which are consistent with the properties of the majority of solar active regions.

Some of the major difficulties associated with explaining active regions as rising flux tubes in the magnetic buoyancy dominated regime (with B ∼ 105 G at the base of the convection zone) discussed above may be ameliorated. For such equipartition or weakly super-equipartition field strengths, the length scale of the Ω-tubes may be largely defined by the separations of convective downdrafts extending into the deep convection zone (see e.g. Miesch et al., 2008), which are comparable to the size scale of solar active regions. Furthermore, significant twist of the tube may no longer be required to obtain reasonably cohesive emerging tubes, since the generation of strong vortex tubes by magnetic buoyancy is suppressed by the tension force resulting from the pinning-down of the flux tube at short intervals by the convective downdrafts (Section 5.7). These possibilities need to be examined by self-consistent 3D spherical-shell simulations of rising flux tubes in a rotating solar convective envelope in the presence of convective flows and the associated large-scale mean flows. Important initial steps in this area have been carried out by Jouve and Brun (2009), although the simulations presented so far have considered buoyant toroidal flux tubes whose initial field strengths and fluxes are too large compared to the values expected for active region scale flux tubes.


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