Summarizing the studies reviewed above, the following basic conclusions can be drawn:
- 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 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 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 recent 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 if the downflow
filling factor at the base of the convection zone is .
- 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 -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).
- Thin flux tube simulations of emerging flux loops in the solar convective envelope (Section 5.1)
show that several asymmetries between the leading and the following sides of the emerging
loops develop due to the effects of the Coriolis force. These asymmetries provide explanations
for the observed active region tilts described by Joy’s law and the observed asymmetric proper
motions of the two polarities of newly developing active regions. Results from these thin flux
tube simulations also suggest that the field strength for the toroidal magnetic fields at the base
of the solar convection zone is in the range of about to , significantly higher
than the equipartition field strength, in order for the emerging loops to be consistent with the
observed properties of solar active regions.
- Vector magnetic field observations on the photosphere show that on average solar active regions
have a small but statistically significant mean twist that is left-handed sense in the northern
hemisphere and right-handed sense in the southern hemisphere. The twist may be due to the
current helicity in the dynamo generated toroidal magnetic field, from which buoyant flux tubes
form at the base of the convection zone, and it may be acquired during the rise of the flux
tubes through the convection zone as a result of buffeting by the helical convective motions
(called the -effect) and also through the accretion of the mean poloidal magnetic field in the
convection zone onto the rising flux tube (Section 5.2). Observational studies of the correlation
between active region twist and tilt angles indicate that the -effect cannot be the only
source for the twist (Section 5.2).
- Two-dimensional MHD simulations of the rise of buoyant horizontal flux tubes show that a
minimum twist is needed for the tube to rise cohesively (Section 5.3). This minimum twist
depends on the initial buoyancy of the tube. For a flux tube initially in thermal equilibrium
with its surrounding, the minimum twist needed is found to be an order of magnitude too large
compared to the mean twist measured in the majority of solar active regions. Three-dimensional
simulations of arched rising tubes that develop self-consistently from zero initial buoyancy
indicate that the necessary twist needed for tube cohesion may be significantly reduced
(Section 5.3). On the other hand, the rise of highly twisted, kink unstable magnetic flux tubes
can produce kinked or knotted emerging tubes which may explain the origin of the unusual
class of flare productive active regions called -sunspots which are observed to be highly
twisted and often show polarity order inverted from the Hale polarity rule (Section 5.4).
- In order for the magnetic buoyancy force of an emerging flux tube to dominate the
hydrodynamic force from the strong downflows in a 3D stratified convective velocity field, the
field strength of the flux tube needs to be greater than , where is the
pressure scale height, is the tube radius, and is the field in equipartition with the
kinetic energy density of the strong downdrafts. For emerging active region flux tubes in the
deep convection zone . For a buoyant flux tube of equipartition field strength,
the tube is pinned down at the locations of strong downdrafts although sections of the tube
between downdrafts may still emerge (Section 5.5).
- Magnetic fields of are found to be 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 turbulent pumping mechanism
is complementary for storage of toroidal magnetic fields in the overshoot layer. It also provides
an effective mechanism for transporting the weak poloidal mean field from the convection zone
into the tachocline layer which is important for the working of the interface mean field dynamo
models. However the magnetic buoyancy of significantly super-equipartition toroidal fields of
e.g. cannot be balanced by convective “pummeling” (Section 5.5).
- Toroidal fields of significantly super-equipartition field strengths of to
inferred from simulations of rising flux tubes are difficult to generate dynamically. However,
because of the entropy gradient in the solar convection zone, the formation of emerging loops
with equipartition initial field strengths are found to lead to a loss of pressure confinement
or “explosion” in the middle of the convection zone. The “explosion” draws outflow of high
entropy plasma from the tube, and hence intensifies the submerged flux tubes to significantly
super-equipartition field strengths (Section 7).
- The evolution of active region scale flux tubes in the top layer of the solar convection zone, the
process of sunspot formation at the surface, and the post-emergence evolution of subsurface flux
tubes are not well understood (Section 8). It is suggested that active region magnetic flux on
the photosphere becomes “dynamically disconnected” from its roots in the deep convection zone
soon after the rising flux tube reaches the photosphere (see Section 8.3. With regard to flux
emergence into the solar atmosphere, it has been shown that the magnetic buoyancy instability
is a mechanism through which magnetic flux reaching the photosphere can expand dynamically
into the stably stratified solar atmosphere. Recent 3D MHD simulations of magnetic flux
emergence into the solar atmosphere and reconnection of emerging flux with a pre-existing
coronal field are able to reproduce many observed features in newly emerging active regions.
The observed X-ray sigmoids in the corona may be caused by the emergence of twisted magnetic
flux tubes into the corona. (Section 8.2).