### 5.7 Buoyant flux tubes in a 3D stratified convective velocity field

#### 5.7.1 General considerations

To understand how active region flux tubes emerge through the solar convection zone, it is certainly important to understand how 3D convective flows in the solar convection zone affect the rise and the structure of buoyant flux tubes. The well-defined order of solar active regions as described by the Hale polarity rule suggests that the emerging flux tubes are not subject to strong deformation by the turbulent convection. One can thus consider the following two simplified order-of-magnitude estimates. First, the magnetic buoyancy of the flux tube should probably dominate the downward hydrodynamic force from the convective downflows:

where is the field strength at which the magnetic energy density is in equipartition with the kinetic energy density of the convective downdrafts, is the flow speed of the downdrafts, is the local pressure scale height, is the tube radius, and is the aerodynamic drag coefficient which is of order unity. In Equation (29) we have used the aerodynamic drag force as an estimate for the magnitude of the hydrodynamic forces. The estimate (29) leads to the condition that the field strength of the flux tube needs to be significantly higher than the equipartition field strength by a factor of . For flux tubes responsible for active region formation, near the bottom of the solar convection zone. Thus we call the “magnetic buoyancy dominated regime”, and expect to be the regime where convective downdrafts become dominant. A second consideration is that the magnetic tension force resulting from bending the flux tube by the convective flows should also dominate the hydrodynamic force due to convection:
where denotes the size scales of the convective flows. This leads to a condition for the magnetic field strength very similar to Equation (29) if we consider the largest convective cell scale for to be comparable to the pressure scale height. Similar results have also been found in Cline (2003).

#### 5.7.2 Simulations in a local Cartesian geometry without rotation

Fan et al. (2003) carried out direct 3D MHD simulations of the evolution of a buoyant magnetic flux tube in a stratified convective velocity field. The basic result is illustrated in Figure 30.

They first computed a 3D convective velocity field in a superadiabatically stratified fluid, until the convection reaches a statistical steady state. The resulting velocity field (see top-left image in Figure 30) shows the typical features of overturning convection in a stratified fluid as found in many previous investigations. The surface layer displays a cellular pattern with patches of upflow region surrounded by narrow downflow lanes. In the bulk of the convecting domain, the downflows are concentrated into narrow filamentary plumes, some of which extend all the way across the domain, while the upflows are significantly broader and are of smaller velocity amplitude in comparison to the downdrafts. A uniformly buoyant, twisted horizontal magnetic flux tube having an entropy that is equal to the entropy at the base of the domain is inserted into the convecting box (see top-right image in Figure 30). In the case where the field strength of the tube is in equipartition to the kinetic energy density of the strongest downdraft (left column in the bottom panel of Figure 30), the magnetic buoyancy for this flux tube is weaker than the hydrodynamic force resulting from the convective downflows and the evolution of the tube depends sensitively on the local condition of the convective flows. Despite being buoyant, the portions of the tube in the paths of downdrafts are pushed downward and pinned down to the bottom, while the rise speed of sections within upflow regions is significantly boosted. The pinned-down flux is then further distorted and transported laterally by the horizontal diverging flow at the bottom. On the other hand in the case where the tube field strength is 10 times the equipartition value (right column in the bottom panel of Figure 30), the horizontal flux tube rises under its uniform buoyancy, nearly unaffected by the convection. In this case the horizontal flux tube is sufficiently twisted so that it does not break up into two vortex tubes.

In case , it is found that the random north-south tilting of the flux tube caused by convection is of the amplitude  30°, which is greater than the r.m.s. scatter of the active region tilts away from Joy’s law for large active regions (10°), but is not beyond the r.m.s. tilt scatter for small active regions (30°) (see Fisher et al., 1995). This indicates that the distortion of flux tubes of equipartition field strength by the convective flows during their buoyant rise through the solar convection zone is probably too large to be consistent with the observational constraint of tilt dispersion for large solar active regions. Furthermore it should be noted that the realistic convective flows in the solar convection zone is probably far more turbulent than that computed in Fan et al. (2003), containing flows of scales significantly smaller than the cross-section of the flux tube. Hence the flux tube distortion found in the simulations of Fan et al. (2003) is most likely a lower limit.

The distances between the major downflow plumes are also an important factor in determining the fate of the buoyant flux tubes of equipartition field strength. In Fan et al. (2003), the distance between neighboring downflow plumes can be as large as about , hence allowing the portion of the buoyant tube between the plumes to rise up to the top of the domain. If on the other hand, the distances between the downflow plumes are below , then the tension force for the tube between the pinned-down points will exceed the magnetic buoyancy force () and the entire flux tube will be prevented from emerging to the surface.

#### 5.7.3 Global rotating spherical-shell simulations

Jouve and Brun (2009) have carried out the first set of global anelastic MHD simulations of the buoyant rise of an initially toroidal flux ring in a rotating, fully convective spherical shell, possessing self-consistently generated mean flows such as meridional circulations and differential rotation, representative of the conditions of the solar convective envelope (see e.g. review by Miesch, 2005). They inserted into the fully developed convecting envelope a buoyant toroidal flux ring with different initial field strengths, twist rates, and initial latitudes, to study how the flux tube rises in the presence of convection and the associated mean flows, and how the dynamic evolution depends on the above initial parameters.

It is found that the magnetic field strength corresponding to the value that is in equipartition with the kinetic energy of the strongest downflows is rather high, . The initial field strength of the toroidal flux ring considered in the simulations are all significantly greater than , being , , and . Thus, except for the case with , all of the other cases simulated are essentially in the magnetic buoyancy dominated regime (with as discussed in Section 5.7.1). As a result, the simulations recovered many of the findings obtained from previous simulations in the absence of convective flows. These include the dependence of the poleward deflection of the tube on the initial tube field strength (e.g. Choudhuri and Gilman, 1987Fan, 2008), the critical dependence on the initial twist for the cohesion of the buoyantly rising flux tube (e.g. Emonet and Moreno-Insertis, 1998Abbett et al., 2000), and the dependence of the tilt angle of the emerging tube on the initial twist (Section 5.5 and  Fan, 2008). Due to the relatively high magnetic diffusivity in the code, flux tubes with a very large initial field strength (ranging from 1.5 × 105 G to 6 × 105 G) and a large radius, corresponding to a total flux on the order of a few times 1023 Mx, significantly greater than the typical active region fluxes, are considered, such that the rise times of the flux tubes are the diffusive time scale of about 14.5 days. Because most of the cases considered are essentially in the magnetic buoyancy dominated regime, the rising toroidal flux tube only develops rather moderate undulations by the influence of the convective flows (see Figure 32), and tubes with undulations extending the depth of the convection zone are not found.

It is also found that flux tubes introduced at lower latitudes (e.g. at 15°) have difficulty reaching the top of the domain (even with a strong initial field strength of ), and the authors attributed the cause of this to the differential rotation. For the weakest field strength case (with ), it is found that portions of the toroidal ring are pinned down by the convective downdrafts, and eventually the tube loses its buoyancy due to magnetic diffusion and is unable to rise to the top (see top panel of Figure 32).

Clearly simulations with a reduced magnetic diffusion are necessary to model the evolution of rising flux tubes in more realistic parameter regimes. Specifically, it is important to model cases with a weaker initial field strength () and thus smaller magnetic buoyancy to study whether large-undulation -shaped emerging tubes with properties consistent with solar active regions can develop (see more discussions in Section 9.1). Note also that the twist of the initial toroidal flux rings introduced in the northern hemisphere (see Figure 1 of Jouve and Brun, 2009) in the simulations is right-handed or positive, which is opposite to the observed preferred sign of twist (left-handed) for active regions in the northern hemisphere (e.g. Pevtsov et al., 2001). With a magnitude of the twist that is just above the critical twist needed for the buoyant flux tube to rise cohesively, the orientation of the tilt angle of the final emerging tube is found to be largely determined by the sign of the twist rather than by the Coriolis force (Fan, 2008). Thus the sign of the tilt angle of the final emerging tube in the simulations may reverse (i.e. becomes opposite to the direction of active region tilts) if toroidal flux rings with a negative or left-handed initial twist were introduced in the northern hemisphere.