Most of the earlier calculations (see Choudhuri and Gilman, 1987; Choudhuri, 1989; D’Silva and Choudhuri, 1993; Fan et al., 1993, 1994) considered initially buoyant toroidal flux tubes by assuming that they are in temperature equilibrium with the external plasma. Various types of initial undulatory displacements are imposed on the buoyant tube so that portions of the tube will remain anchored within the stably stratified overshoot layer and other portions of the tube are displaced into the unstable convection zone which subsequently develop into emerging -shaped loops.
Later calculations (see Schüssler et al., 1994; Caligari et al., 1995, 1998; Fan and Gong, 2000) considered more physically self-consistent initial conditions where the initial toroidal flux ring is in the state of mechanical equilibrium. In this state the buoyancy force is zero (neutrally buoyant) and the magnetic curvature force is balanced by the Coriolis force resulting from a prograde toroidal motion of the tube plasma. It is argued that this mechanical equilibrium state is the preferred state for the long-term storage of a toroidal magnetic field in the stably stratified overshoot region (Section 3.1). In these simulations, the development of the emerging -loops is obtained naturally by the non-linear, adiabatic growth of the undulatory buoyancy instability associated with the initial equilibrium toroidal flux rings (Section 4.1). As a result there is far less degree of freedom in specifying the initial perturbations. The eruption pattern needs not be prescribed in an ad hoc fashion but is self-consistently determined by the growth of the instability once the initial field strength, latitude, and the subadiabaticity at the depth of the tube are given. For example Caligari et al. (1995) modeled emerging loops developed due to the undulatory buoyancy instability of initial toroidal flux tubes located at different depths near the base of their model solar convection zone which includes a consistently calculated overshoot layer according to the non-local mixing-length treatment. They choose values of initial field strengths and latitudes that lie along the contours of constant instability growth times of 100 days and 300 days in the instability diagrams (see Figure 6), given the subadiabaticity at the depth of the initial tubes. The tubes are then perturbed with a small undulatory displacement which consists of a random superposition of Fourier modes with azimuthal order ranging from through , and the resulting eruption pattern is determined naturally by the growth of the instability.
On the other hand, non-adiabatic effects may also be important in the destabilization process. It has been discussed in Section 3.2 that isolated magnetic flux tubes with internally suppressed convective transport experience a net heating due to the non-zero divergence of radiative heat flux in the weakly subadiabatically stratified overshoot region and also in the lower solar convection zone. The radiative heating causes a quasi-static upward drift of the toroidal flux tube with a drift velocity . Thus the time scale for a toroidal flux tube to drift out of the stable overshoot region may not be long compared to the growth time of its undulatory buoyancy instability. For example if the subadiabaticity is , the time scale for the flux tube to drift across the depth of the overshoot region is about 20 days, smaller than the growth times ( 100 – 300 days) of the most unstable modes for tubes of a field as shown in Figure 6. Therefore radiative heating may play an important role in destabilizing the toroidal flux tubes. The quasi-static upward drift due to radiative heating can speed-up the development of emerging -loops (especially for weaker flux tubes) by bringing the tube out of the inner part of the overshoot region of stronger subadiabaticity, where the tube is stable or the instability growth is very slow, to the outer overshoot region of weaker subadiabaticity or even into the convection zone, where the growth of the undulatory buoyancy instability occurs at a much shorter time scale.
A possible scenario in which the effect of radiative heating helps to induce the formation of -shaped emerging loops has been investigated by Fan and Fisher (1996). In this scenario, the initial neutrally buoyant toroidal flux tube is not exactly uniform, and lies at non-uniform depths with some portions of the tube lying at slightly shallower depths in the overshoot region. Radiative heating and quasi-static upward drift of this non-uniform flux tube bring the upward protruding portions of the tube first into the unstably stratified convection zone. These portions can become buoyantly unstable (if the growth of buoyancy overcomes the growth of tension) and rise dynamically as emerging loops. In this case the non-uniform flux tube remains close to a mechanical equilibrium state during the initial quasi-static rise through the overshoot region. The emerging loop develops gradually as a result of radiative heating and the subsequent buoyancy instability of the outer portion of the tube entering the convection zone.
In the following subsections we review the major findings and conclusions that have been drawn from the various thin flux tube simulations of emerging flux loops.
Axisymmetric simulations of the buoyant rise of toroidal flux rings in a rotating solar convective envelope by Choudhuri and Gilman (1987) first demonstrate the significant influence of the Coriolis force on the rising trajectories. The basic effect is that the Coriolis force acting on the radial outward motion of the flux tube (or the tendency for the rising tube to conserve angular momentum) drives a retrograde motion of the tube plasma. This retrograde motion then induces a Coriolis force directed towards the Sun’s rotation axis which acts to deflect the trajectory of the rising tube poleward. The amount of poleward deflection by the Coriolis force depends on the initial field strength of the emerging tube, being larger for flux tubes with weaker initial field. For flux tubes with an equipartition field strength of , the effect of the Coriolis force is so dominating that it deflects the rising tubes to emerge at latitudes poleward of the sunspot zones even though the flux tubes start out from low latitudes at the bottom of the convective envelope. In order for the rising trajectory of the flux ring to be close to radial so that the emerging latitudes are within the observed sunspot latitudes, the field strength of the toroidal flux ring at the bottom of the solar convection zone needs to be . This basic result is confirmed by later simulations of non-axisymmetric, -shaped emerging loops rising through the solar convective envelope (see Choudhuri, 1989; D’Silva and Choudhuri, 1993; Fan et al., 1993; Schüssler et al., 1994; Caligari et al., 1995, 1998; Fan and Fisher, 1996).
Simulations by Caligari et al. (1995) of emerging loops developed self-consistently due to the undulatory buoyancy instability show that, for tubes with initial field strength , the trajectories of the emerging loops are primarily radial with poleward deflection no greater than . For tubes with initial field strength exceeding , the poleward deflection of the emerging loops remain reasonably small (no greater than about ). However, for a tube with equipartition field strength of , the rising trajectory of the emerging loop is deflected poleward by about . Such an amount of poleward deflection is too great to explain the observed low latitudes of active region emergence. Furthermore, it is found that with such a weak initial field the field strength of the emerging loop falls below equipartition with convection throughout most of the convection zone. Such emerging loops are expected to be subjected to strong deformation by turbulent convection and may not be consistent with the observed well defined order of solar active regions.
Fan and Fisher (1996) modeled emerging loops that develop as a result of radiative heating of non-uniform flux tubes in the overshoot region. The results on the poleward deflection of the emerging loops as a function of the initial field strength are very similar to that found in Caligari et al. (1995). Figure 8 shows the latitude of loop emergence as a function of the initial latitude at the base of the solar convection zone. It can be seen that tubes of emerge essentially radially with very small poleward deflection (), and for tubes with , the poleward deflections remain reasonably small so that the emerging latitudes are within the observed sunspot zones.
A well-known property of the solar active regions is the so called Joy’s law of active region tilts. The averaged orientation of bipolar active regions on the solar surface is not exactly toroidal but is slightly tilted away from the east-west direction, with the leading polarity (the polarity leading in the direction of rotation) being slightly closer to the equator than the following polarity. The mean tilt angle is a function of latitude, being approximately (Wang and Sheeley Jr, 1989, 1991; Howard, 1991a,b; Fisher et al., 1995).
Using thin flux tube simulations of the non-axisymmetric eruption of buoyant -loops in a rotating solar convective envelope, D’Silva and Choudhuri (1993) are the first to show that the active region tilts as described by Joy’s law can be explained by Coriolis forces acting on the flux loops. As the emerging loop rises, there is a relative expanding motion of the mass elements at the summit of the loop. The Coriolis force induced by this diverging, expanding motion at the summit is to tilt the summit clockwise (counter-clockwise) for loops in the northern (southern) hemisphere as viewed from the top, so that the leading side from the summit is tilted equatorward relative to the following side. Since the component of the Coriolis force that drives this tilting has a dependence, the resulting tilt angle at the apex is approximately .
Caligari et al. (1995) studied tilt angles of emerging loops developed self-consistently due to the undulatory buoyancy instability of flux tubes located at the bottom as well as just above the top of their model overshoot region, with selected values of initial field strengths and latitudes lying along contours of constant instability growth times (100 days and 300 days). The resulting tilt angles at the apex of the emerging loops (see Figure 9) produced by these sets of unstable tubes (whose field strengths are within the range of to ) show good agreement with the observed tilt angles for sunspot groups measured by Howard (1991b).
Similar results of loop tilt angles (see Figure 10) are found by Fan and Fisher (1996) who considered formation of emerging loops by gradual radiative heating of non-uniform toroidal flux tubes initially in mechanical equilibrium in the overshoot region.
Thin flux tube simulations also show that the tilt angles of the emerging loops tend to be smaller for tubes with smaller total flux or radius (see D’Silva and Choudhuri, 1993; Fan and Fisher, 1996). This is because smaller tubes are more influenced by the drag force, whose direct effect is to oppose the tilting motion of the loops. This prediction on the dependence of the tilt angle on active region flux is confirmed by Fisher et al. (1995), who studied the relation between the tilt angle and the mean separation distance between the leader and follower spots of a sunspot group. Wang and Sheeley Jr (1989) and Howard (1992) have shown that the mean polarity separation distance of an active region is a good proxy for the total magnetic flux in the region. Fisher et al. (1995) found that for a fixed latitude, the tilt angle of sunspot groups decreases with decreasing polarity separation, and hence decreasing total flux, consistent with the results from the thin flux tube calculations. This agreement adds support to the explanation that the Coriolis force acting on rising flux loops is the main cause of active region tilts and argues against the suggestion by Babcock (1961) that the active region tilt angles simply reflect the orientation of the underlying toroidal magnetic field stretched out by the latitudinal differential rotation.
Using Mount Wilson sunspot group data, Fisher et al. (1995) further studied the dispersion or scatter of spot-group tilts away from the mean tilt behavior described by Joy’s law. First they found that the magnitude of the tilt dispersion is significantly greater than the level expected from measurement errors, suggesting a solar origin of the tilt angle scatter. Furthermore, they found that the root-mean-square tilt scatter decreases with increasing polarity separation (or total flux) and does not vary with latitude (in contrast to the latitudinal dependence of the mean tilts). This result is consistent with the picture that scatter of active region tilts away from the mean Joy’s law behavior results from buffeting of emerging loops by convective motions during their rise through the solar convection zone (Longcope and Fisher, 1996).
Recent non-linear simulations of the two-dimensional MHD tachocline (Cally et al., 2003) show that bands of toroidal magnetic fields in the solar tachocline may become tipped relative to the azimuthal direction by an amount that is within at sunspot latitudes due to the non-linear evolution of the 2D global joint instability of differential rotation and toroidal magnetic fields. This tipping may either enhance or reduce the observed tilt in bipolar active regions depending on from which part of the tipped band the emerging loops develop. Thus the basic consequence of the possible tipping of the toroidal magnetic fields in the tachocline is to contribute to the spread of the tilts of bipolar active regions.
An intriguing property of solar active regions is the asymmetry in morphology between the leading and following polarities. The leading polarity of an active region tends to be in the form of large sunspots, whereas the following polarity tends to appear more dispersed and fragmented; moreover, the leading spots tend to be longer lived than the following. Fan et al. (1993) offered an explanation for the origin of this asymmetry. In their thin flux tube simulations of the non-axisymmetric eruption of buoyant -loops through a rotating model solar convective envelope, they found that an asymmetry in the magnetic field strength develops between the leading and following legs of an emerging loop, with the field strength of the leading leg being about 2 times that of the following leg. The field strength asymmetry develops because the Coriolis force, or the tendency for the tube plasma to conserve angular momentum, drives a counter-rotating flow of plasma along the emerging loop, which, in conjunction with the diverging flow of plasma from the apex to the troughs, gives rise to an effective asymmetric stretching of the two legs of the loop with a greater stretching and hence a stronger field strength in the leading leg. Fan et al. (1993) argued that the stronger field along the leading leg of the emerging loop makes it less subject to deformation by the turbulent convection and therefore explains the more coherent and less fragmented appearance of the leading polarity of an active region.
However, the calculations by Caligari et al. (1995) and Caligari et al. (1998) show that the field strength asymmetry found in Fan et al. (1993) depends on the choices of the initial conditions. They found that if one uses the more self-consistent mechanical equilibrium state (instead of the buoyant, temperature equilibrium state used in Fan et al. (1993)) for the initial toroidal tube, for which there exists an initial prograde toroidal motion, the subsequent differential stretching of the emerging loop is quite different. It is found that a consistently stronger field along the leading leg of the emerging loop only occurs for cases with relatively weak initial field strengths (). For stronger fields (), which seem to fit the observed properties of solar active regions better in all the other aspects, e.g. the latitude of flux emergence and the tilt angles, the field strength asymmetry becomes very small and even reverses its sense in the upper half of the solar convection zone.
Further work by Fan and Fisher (1996) examined the field strength asymmetry for emerging flux loops that form as a result of gradual radiative heating of non-uniform flux tubes initially in mechanical equilibrium.
Another asymmetry in the emerging loop generated by the effect of the Coriolis force is the asymmetry in the east-west inclinations of the two sides of the loop. This asymmetry is first shown in the thin flux tube calculations of Moreno-Insertis et al. (1994) and Caligari et al. (1995) who modeled emerging loops that develop self-consistently as a result of the buoyancy instability of toroidal magnetic flux tubes initially in mechanical equilibrium. Moreno-Insertis et al. (1994) and Caligari et al. (1995) found that as the emerging loop rises, the Coriolis force, or the tendency for the tube to conserve angular momentum, drives a counter-rotating motion of the tube plasma, which causes the summit of the loop to move retrograde relative to the valleys, resulting in an asymmetry in the inclinations of the two legs of the loop with the leading leg being inclined more horizontally with respect to the surface than the following leg. This asymmetry in inclination can be clearly seen in Figure 12, which shows a view from the north pole of an asymmetric emerging loop obtained from a simulation by Caligari et al. (1995).
The observational consequences of this geometric asymmetry are discussed in Moreno-Insertis et al. (1994) and Caligari et al. (1995). The emergence of such an eastward inclined loop is expected to produce apparent asymmetric east-west proper motions of the two polarities of the emerging region, with a more rapid motion of the leading polarity spots away from the emerging region compared to the motion of the following polarity spots. Such asymmetric proper motions are observed in young active regions and sunspot groups (see Chou and Wang, 1987; van Driel-Gesztelyi and Petrovay, 1990; Petrovay et al., 1990). Furthermore, the asymmetry in the inclination of the emerging loop may also explain the observation that the magnetic inversion line in bipolar regions is statistically nearer to the main following spot than to the main proceeding one (van Driel-Gesztelyi and Petrovay, 1990; Petrovay et al., 1990).
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