Magnetic Fields in the Solar Convection Zone

2.1 The thin flux tube model

The well-defined order of the solar active regions (see description of the observational properties in Section 1) suggests that their precursors at the base of the solar convection zone should have a field strength that is at least B_eq, where B_eq is the field strength that is in equipartition with the kinetic energy density of the convective motions: \(B_{{\rm eq}}^{2}/8\pi=\rho v_{c}^{2}/2\). If we use the results from the mixing length models of the solar convection zone for the convective flow speed v_c, then we find that in the deep convection zone B_eq is on the order of 10⁴ G. In the past two decades, direct 3D numerical simulations have led to a new picture for solar convection that is non-local, driven by the concentrated downflow plumes formed by radiative cooling at the surface layer, and with extreme asymmetry between the upward and downward flows (see reviews by Spruit et al., 1990; Spruit, 1997). Hence it should be noted that the B_eq derived based on the local mixing length description of solar convection may not really reflect the intensity of the convective flows in the deep solar convection zone. With this caution in mind, we nevertheless refer to B_eq ∼ 10⁴ G as the field strength in equipartition with convection in this review.

If one considers only the solid body rotation of the Sun, then the Equations (2), (3), and (4) can be simplified by letting Ω_e = Ω. Calculations using the thin flux tube model (see Section 5.1) have shown that the effect of the Coriolis force 2ρ(v × Ω) acting on emerging flux loops can lead to east-west asymmetries in the loops that explain several well-known properties of solar active regions.

Note that in the equation of motion (2), the effect of the “enhanced inertia” caused by the back-reaction of the fluid to the relative motion of the flux tube is completely ignored. This effect has sometimes been incorporated by treating the inertia for the different components of Equation (2) differently, with the term ρ(dv/dt)_⊥ on the left-hand-side of the perpendicular component of the equation being replaced by (ρ + ρ_e)(dv/dt)_⊥ (see Spruit, 1981). This simple treatment is problematic for curved tubes and the proper ways to treat the back-reaction of the fluid are controversial in the literature (Cheng, 1992; Fan et al., 1994; Moreno-Insertis et al., 1996). Since the enhanced inertial effect is only significant during the impulsive acceleration phases of the tube motion, which occur rarely in the thin flux tube calculations of emerging flux tubes, and the results obtained do not depend significantly on this effect, many later calculations have taken the approach of simply ignoring it (see Caligari et al., 1995, 1998; Fan and Fisher, 1996).

The two new Equations (8) and (9) — derived by Longcope and Klapper (1997) — together with the earlier Equations (1), (2), (5), (6), and (7) provide a description for the dynamics of a weakly twisted thin flux tube. Note that the two new equations are decoupled from and do not have any feedback on the solutions for the dependent variables described by the earlier equations. One can first solve for the motion of the tube axis using Equations (1), (2), (5), (6), and (7), and then apply the resulting motion of the tube axis to Equations (8) and (9) to determine the evolution of the twist of the tube. If the tube is initially twisted, then the twist q can propagate and re-distribute along the tube as a result of stretching (1st term on the right-hand-side of Equation (8)) and the torsional Alfvén waves (2nd term on the right-hand-side of Equation (8)). Twist can also be generated due to writhing motion of the tube axis (last term on the right-hand-side of Equation (8)), as required by the conservation of total helicity.

2.2 MHD simulations

The thin flux tube (TFT) model described above is physically intuitive and computationally tractable. It provides a description of the dynamic motion of the tube axis in a three-dimensional space, taking into account large scale effects such as the curvature of the convective envelope and the Coriolis force due to solar rotation. The Lagrangian treatment of each tube segment in the TFT model allows for preserving perfectly the frozen-in condition of the tube plasma. Thus there is no magnetic diffusion in the TFT model. However, the TFT model ignores variations within each tube cross-section. It is only applicable when the flux tube radius is thin (Section 2.1) and the tube remains a cohesive object (Section 5.3). Clearly, to complete the picture, direct MHD calculations that resolve the tube cross-section and its interaction with the surrounding fluid are needed. On the other hand, direct MHD simulations that discretize the spatial domain are subject to numerical diffusion. The need to adequately resolve the flux tube — so that numerical diffusion does not have a significant impact on the dynamical processes of interest (e.g. the variation of magnetic buoyancy) — severely limits the spatial extent of the domain that can be modeled. So far the MHD simulations cannot address the kinds of large scale dynamical effects that have been studied by the TFT model (Section 5.1). Thus the TFT model and the resolved MHD simulations complement each other.

For the bulk of the solar convection zone, the fluid stratification is very close to being adiabatic with δ ≪ 1, where δ ≡ ∇ − ∇_ad is the non-dimensional superadiabaticity with ∇ = d ln T/d ln p and ∇_ad = (d ln T/d ln p)_ad denoting the actual and the adiabatic logarithmic temperature gradient of the fluid respectively, and the convective flow speed v_c is expected to be much smaller than the sound speed c_s: v_c/c_s ∼ δ^1/2 ≪ 1 (see Schwarzschild, 1958; Lantz, 1991). Furthermore, the plasma β defined as the ratio of the thermal pressure to the magnetic pressure (β ≡ p/(B²/8π)) is expected to be very high (β ≫ 1) in the deep convection zone. For example for flux tubes with field strengths of order 10⁵ G, which is significantly super-equipartition compared to the kinetic energy density of convection, the plasma β is of order 10⁵. Under these conditions, a very useful computational approach for modeling subsonic magnetohydrodynamic processes in a pressure dominated plasma is the well-known anelastic approximation (see Gough, 1969; Gilman and Glatzmaier, 1981; Glatzmaier, 1984; Lantz and Fan, 1999). The main feature of the anelastic approximation is that it filters out the sound waves so that the time step of numerical integration is not limited by the stringent acoustic time scale which is much smaller than the relevant dynamic time scales of interest as determined by the flow velocity and the Alfvén speed.

Fully compressible MHD simulations have also been applied to study the dynamic evolution of a magnetic field in the deep solar convection zone using non-solar but reasonably large β values such as β ∼ 10 to 1000. In several cases comparisons have been made between fully compressible simulations using large plasma β and the corresponding anelastic MHD simulations, and good agreement was found between the results (see Fan et al., 1998a; Rempel, 2002). Near the top of the solar convection zone, neither the TFT model nor the anelastic approximation are applicable because the active region flux tubes are no longer thin (Moreno-Insertis, 1992) and the velocity field is no longer subsonic. Fully compressible MHD simulations are necessary for modeling flux emergence near the surface (Section 8).

3.1 The mechanical equilibria for an isolated toroidal flux tube or an extended magnetic layer

The Hale’s polarity rule of solar active regions indicates a subsurface magnetic field that is highly organized, of predominantly toroidal direction, and with sufficiently strong field strength (super-equipartition compared to the kinetic energy density of convection) such that it is not subjected to strong deformation by convective motions. It is argued that the weakly subadiabatically stratified overshoot layer at the base of the solar convection zone is the most likely site for the storage of such a strong coherent toroidal magnetic field against buoyant loss for time scales comparable to the solar cycle period (see Parker, 1979; van Ballegooijen, 1982).

It is not clear if the toroidal magnetic field is in the state of isolated flux tubes or stored in the form of a more diffuse magnetic layer. Moreno-Insertis et al. (1992) have considered the mechanical equilibrium of isolated toroidal magnetic flux tubes (flux rings) in a subadiabatic layer using the thin flux tube approximation (Section 2.1). The forces experienced by an isolated toroidal flux ring at the base of the convection zone is illustrated in Figure 5(a). The condition of total pressure balance (1) and the presence of a magnetic pressure inside the flux tube require a lower gas pressure inside the flux tube compared to the outside. Thus either the density or the temperature inside the flux tube needs to be lower. If the flux tube is in thermal equilibrium with the surrounding, then the density inside needs to be lower and the flux tube is buoyant. The buoyancy force associated with a magnetic flux tube in thermal equilibrium with its surrounding is often called the magnetic buoyancy (Parker, 1975). It can be seen in Figure 5(a) that a radially directed buoyancy force has a component that is parallel to the rotation axis, which cannot be balanced by any other forces associated with the toroidal flux ring. Thus for the toroidal flux ring to be in mechanical equilibrium, the tube needs to be in a neutrally buoyant state with vanishing buoyancy force, and with the magnetic curvature force pointing towards the rotation axis being balanced by a Coriolis force produced by a faster rotational speed of the flux ring (see Figure 5(a)). Such a neutrally buoyant flux ring (with equal density between inside and outside) then requires a lower internal temperature than the surrounding plasma to satisfy the total pressure balance. If one starts with a toroidal flux ring that is initially in thermal equilibrium with the surrounding and rotates at the same ambient angular velocity, then the flux ring will move radially outward due to its buoyancy and latitudinally poleward due to the unbalanced poleward component of the tension force. As a result of its motion, the flux ring will lose buoyancy due to the subadiabatic stratification and attain a larger internal rotation rate with respect to the ambient field-free plasma due to the conservation of angular momentum, evolving towards a mechanical equilibrium configuration. The flux ring will undergo superposed buoyancy and inertial oscillations around this mechanical equilibrium state. It is found that the oscillations can be contained within the stably stratified overshoot layer and also within a latitudinal range of Δθ ≲ 20° to be consistent with the active region belt, if the field strength of the toroidal flux ring B ≲ 10⁵ G and the subadiabaticity of the overshoot layer is sufficiently strong with δ ≡ ∇ − ∇_ad ≲ −10⁻⁵, where ∇ ≡ d ln T/d ln P is the logarithmic temperature gradient and ∇_ad is ∇ for an adiabatically stratified atmosphere. Flux rings with significantly larger field strength cannot be kept within the low latitude zones of the overshoot region.

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Rempel et al. (2000) considered the mechanical equilibrium of a layer of an axisymmetric toroidal magnetic field of 10⁵ G in a subadiabatically stratified region near the bottom of the solar convection zone in full spherical geometry. In this case, as illustrated in Figure 5(b), a latitudinal pressure gradient can be built up, allowing for force balance between a non-vanishing buoyancy force, the magnetic curvature force, and the pressure gradient without requiring a prograde toroidal flow. Thus a wider range of equilibria can exist. Rempel et al. (2000) found that under the condition of a strong subadiabatic stratification such as the radiative interior with δ ∼ − 0.1, the magnetic layer tends to establish a mechanical equilibrium where a latitudinal pressure gradient is built up to balance the poleward component of the magnetic tension, and where the net radial component of the buoyancy and magnetic tension forces is efficiently balanced by the strong subadiabaticity. The magnetic layer reaches this equilibrium solution in a time scale short compared to the time required for a prograde toroidal flow to set up for the Coriolis force to be significant. For this type of equilibrium where a latitudinal pressure gradient is playing a dominant role in balancing the poleward component of the magnetic curvature force, there is significant relative density perturbation (≫ 1/β) in the magnetic layer compared to the background stratification. On the other hand, under the condition of a very weak subadiabatic stratification such as that in the overshoot layer near the bottom of the convection zone with δ ∼ −10⁻⁵, the magnetic layer tends to evolve towards a mechanical equilibrium which resembles that of an isolated toroidal flux ring, where the relative density perturbation is small (≪C 1/β), and the magnetic curvature force is balanced by the Coriolis force induced by a prograde toroidal flow in the magnetic layer. Thus regardless of whether the field is in the state of an extended magnetic layer or isolated flux tubes, a 10⁵ G toroidal magnetic field stored in the weakly subadiabatically stratified overshoot region is preferably in a mechanical equilibrium with small relative density perturbation and with a prograde toroidal flow whose Coriolis force balances the magnetic tension. The prograde toroidal flow necessary for the equilibrium of the 10⁵ G toroidal field is about 200 ms⁻¹, which is approximately 10% of the mean rotation rate of the Sun. Thus one may expect significant changes in the differential rotation in the overshoot region during the solar cycle as the toroidal field is being amplified (Rempel et al., 2000). Detecting these toroidal flows and their temporal variation in the overshoot layer via helioseismic techniques is a means by which we can probe and measure the toroidal magnetic field generated by the solar cycle dynamo.

3.2 Effect of radiative heating

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 δ ∼ −10⁻⁴ 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.

4.1 The buoyancy instability of isolated toroidal magnetic flux tubes

By linearizing the thin flux tube dynamic equations (1), (2), (5), (6), and (7), the stability of neutrally buoyant toroidal magnetic flux tubes to isentropic perturbations have been studied (see Spruit and van Ballegooijen, 1982a,b; Ferriz-Mas and Schüssler, 1993, 1995).

The general problem of the linear stability of a thin toroidal flux ring in mechanical equilibrium in a differentially rotating spherical convection zone at arbitrary latitudes has been studied in detail by Ferriz-Mas and Schüssler (1993, 1995). For general non-axisymmetric perturbations, a sixth-order dispersion relation is obtained from the linearized thin flux tube equations. It is not possible to obtain analytical stability criteria. The dispersion relation is solved numerically to find instability and the growth rates of the unstable modes. The regions of instability in the (B₀, λ₀) plane (with B₀ being the magnetic field strength of the flux ring and λ₀ being the equilibrium latitude), under the conditions representative of the overshoot layer at the base of the solar convection zone are shown in Figure 6 (from Caligari et al., 1995). The basic parameters that determine the stability of an equilibrium toroidal flux ring are its field strength and the subadiabaticity of the external stratification. In the case δ ≡ ∇ − ∇_ad = −2.6 × 10⁻⁶ (upper panel of Figure 6), unstable modes with reasonably short growth times (less than about a year) only begin to appear at sunspot latitudes for B₀ ≳ 1.2 × 10⁵ G. These unstable modes are of m = 1 and 2. In case of a weaker subadiabaticity, δ ≡ ∇ − ∇_ad = −1.9 × 10⁻⁷ (lower panel of Figure 6), reasonably fast growing modes (growth time less than a year) begin to appear at sunspot latitudes for B₀ ≳ 5 × 10⁴ G, and the most unstable modes are of m = 1 and 2. These results suggest that toroidal magnetic fields stored in the overshoot layer at the base of the solar convection zone do not become unstable until their field strength becomes significantly greater than the equipartition value of 10⁴ G.

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Thin flux tube simulations of the non-linear growth of the non-axisymmetric instabilities of initially toroidal flux tubes and the emergence of Ω-shaped flux loops through the solar convective envelope will be discussed in Section 5.1.

4.2 Breakup of an equilibrium magnetic layer and formation of buoyant flux tubes

The above discussion on the buoyancy instabilities considers ideal adiabatic perturbations. It should be noted that the role of finite diffusion is not always stabilizing. In the solar interior, it is expected that η ≪ K and ν ≪ K, where η, ν, and K denote the magnetic diffusivity, the kinematic viscosity, and the thermal diffusivity respectively. Under these circumstances, it is shown that thermal diffusion can be destabilizing (see Gilman, 1970; Acheson, 1979; Schmitt and Rosner, 1983). The diffusive effects are shown to alter the stability criteria of Equations (24) and (25) by reducing the term ds/dz by a factor of η/K (see Acheson, 1979). In other words, efficient heat exchange can significantly “erode away” the stabilizing effect of a subadiabatic stratification. This process is called the doubly diffusive instabilities.

Direct multi-dimensional MHD simulations have been carried out to study the break-up of a horizontal magnetic layer by the non-linear evolution of the buoyancy instabilities and the formation of buoyant magnetic flux tubes (see Cattaneo and Hughes, 1988; Cattaneo et al., 1990; Matthews et al., 1995; Wissink et al., 2000; Fan, 2001a).

Cattaneo and Hughes (1988), Matthews et al. (1995), and Wissink et al. (2000) have carried out a series of 2D and 3D compressible MHD simulations where they considered an initial horizontal magnetic layer that supports a top-heavy density gradient, i.e. an equilibrium with a lower density magnetic layer supporting a denser plasma on top of it. It is found that for this equilibrium configuration, the most unstable modes are the Rayleigh-Taylor type 2D interchange modes. Two-dimensional simulations of the non-linear growth of the interchange modes (Cattaneo and Hughes, 1988) found that the formation of buoyant flux tubes is accompanied by the development of strong vortices whose interactions rapidly destroy the coherence of the flux tubes. In the non-linear regime, the evolution is dominated by vortex interactions which act to prevent the rise of the buoyant magnetic field. Matthews et al. (1995) and Wissink et al. (2000) extend the simulations of Cattaneo and Hughes (1988) to 3D allowing variations in the direction of the initial magnetic field. They discovered that the flux tubes formed by the initial growth of the 2D interchange modes subsequently become unstable to a 3D undulatory motion in the non-linear regime due to the interaction between neighboring counter-rotating vortex tubes, and consequently the flux tubes become arched. Matthews et al. (1995) and Wissink et al. (2000) pointed out that this secondary undulatory instability found in the simulations is of similar nature as the undulatory instability of a pair of counter-rotating (non-magnetic) line vortices investigated by Crow (1970). Wissink et al. (2000) further considered the effect of the Coriolis force due to solar rotation using a local f-plane approximation, and found that the principle effect of the Coriolis force is to suppress the instability. Further 2D simulations have also been carried out by Cattaneo et al. (1990) where they introduced a variation of the magnetic field direction with height into the previously unidirectional magnetic layer of Cattaneo and Hughes (1988). The growth of the interchange instability of such a sheared magnetic layer results in the formation of twisted, buoyant flux tubes which are able to inhibit the development of vortex tubes and rise cohesively.

On the other hand, Fan (2001a) has considered a different initial equilibrium state for a horizontal unidirectional magnetic layer, where the density stratification remains unchanged from that of an adiabatically stratified polytrope, but the temperature and the gas pressure are lowered in the magnetic layer to satisfy the hydrostatic condition. For such a neutrally buoyant state with no density change inside the magnetic layer, the 2D interchange instability is completely suppressed and only 3D undulatory modes (with non-zero wavenumbers in the field direction) are unstable. The strong toroidal magnetic field stored in the weakly subadiabatic overshoot region below the bottom of the convection zone is likely to be close to such a neutrally buoyant mechanical equilibrium state (see Section 3.1). Anelastic MHD simulations (Fan, 2001a) of the growth of the 3D undulatory instability of this horizontal magnetic layer show formation of significantly arched magnetic flux tubes (see Figure 7) whose apices become increasingly buoyant as a result of the diverging flow of plasma from the apices to the troughs. The decrease of the field strength B at the apex of the arched flux tube as a function of height is found to follow approximately the relation \(B/\sqrt \rho = {\rm{constant}}\), or, the Alfvén speed being constant, which is a significantly slower decrease of B with height compared to that for the rise of a horizontal flux tube without any field line stretching, for which case B/ρ should remain constant. The variation of the apex field strength with height following \(B/\sqrt \rho = {\rm{constant}}\) found in the 3D MHD simulations of the arched flux tubes is in good agreement with the results of the thin flux tube models of emerging Ω-loops (see Moreno-Insertis, 1992) during their rise through the lower half of the solar convective envelope where the stratification is very close to being adiabatic as is assumed in the 3D simulations.

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5.1 Results from thin flux tube simulations of emerging loops

Beginning with the seminal work of Moreno-Insertis (1986) and Choudhuri and Gilman (1987), a large body of numerical simulations solving the thin flux tube dynamic equations (1), (2), (5), (6), and (7) — or various simplified versions of them — have been carried out to model the evolution of emerging magnetic flux tubes in the solar convective envelope (see Choudhuri, 1989; D’Silva and Choudhuri, 1993; Fan et al., 1993, 1994; Schüssler et al., 1994; Caligari et al., 1995; Fan and Fisher, 1996; Caligari et al., 1998; Fan and Gong, 2000). The results of these numerical calculations have contributed greatly to our understanding of the basic properties of solar active regions and provided constraints on the field strengths of the toroidal magnetic fields at the base of the solar convection zone.

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 Ω-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 Ω-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 m = 1 through m = 5, 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 ∼ 10⁻³|δ|⁻¹ cms⁻¹. 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 δ is ∼ −10⁻⁶, 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 ∼ 10⁵ G 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 Ω-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 Ω-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.

5.1.1 Latitude of flux emergence

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 B ∼ 10⁴ G, 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 ∼ 10⁵ G. This basic result is confirmed by later simulations of non-axisymmetric, Ω-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 ≳ 10⁵ G, the trajectories of the emerging loops are primarily radial with poleward deflection no greater than 3°. For tubes with initial field strength exceeding 4 × 10⁴ G, the poleward deflection of the emerging loops remain reasonably small (no greater than about 6^°). However, for a tube with equipartition field strength of 10⁴ G, the rising trajectory of the emerging loop is deflected poleward by about 20°. 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 10⁵ G emerge essentially radially with very small poleward deflection (< 3°), and for tubes with B ≳ 3 × 10⁴ G, the poleward deflections remain reasonably small so that the emerging latitudes are within the observed sunspot zones.

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5.1.2 Active region tilts

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 ∝ sin(latitude) (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 Ω-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 sin(latitude) dependence, the resulting tilt angle at the apex is approximately ∝ sin(latitude).

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 4 × 10⁴ G to 1.5 × 10⁵ G) show good agreement with the observed tilt angles for sunspot groups measured by Howard (1991b). They also found that loops formed from toroidal flux tubes with an equipartition field strength of 10⁴ G develop a tilt angle (−9°) of the wrong sign at the loop apex.

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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. It is found that emerging loops with initial field strength of the range 4 × 10⁴ G to 10⁵ G show tilt angles that are in good agreement with the observed tilt angles of sunspot groups. Tilts of the wrong sign or direction begin to appear for tubes with initial field strength ≲ 3 × 10⁴ G.

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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 +/−10° 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.

5.1.3 Morphological asymmetries of 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 Ω-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 (∼ 10⁴ G). For stronger fields (∼ 10⁵ G), 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. They find (see Figure 11) significantly stronger fields in the leading leg compared to the following in the lower solar convection zone for all initial field strengths, but nearly equal field strengths of the two legs in the upper convection zone for stronger initial fields (B ≳ 6 × 10⁴ G). Clearly, further investigations, and perhaps fully resolved three-dimensional MHD simulations of emerging flux loops, are needed to understand the origin of the observed morphological asymmetries of the two polarities of solar active regions.

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5.1.4 Geometrical asymmetry of emerging loops and the asymmetric proper motions of active regions

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). Caligari et al. (1998) and Fan and Fisher (1996) show that the geometrical asymmetry is a robust result for models of emerging loops that originate from initial flux tubes in mechanical equilibrium. Loops with initial field strength ranging from 3 × 10⁴ to 10⁵ G consistently show this asymmetry.

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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).

5.2 Hemispheric trend of the twist in solar active regions

Vector magnetic field observations of active regions on the photosphere have revealed that on average the solar active regions have a small but statistically significant mean twist that is left-handed in the northern hemisphere and right-handed in the southern hemisphere (see Pevtsov et al., 1995, 2001, 2003). What is being measured is the quantity α ≡ 〈J _z /B _z 〉, the ratio of the vertical electric current over the vertical magnetic field averaged over the active region. When plotted as a function of latitude, the measured α for individual solar active regions show considerable scatter, but there is clearly a statistically significant trend for negative (positive) α in the northern (southern) hemisphere (see Figure 13). A linear least squares fit to the data of α as a function of latitude (Figure 13a) found that α = −2.7 × 10⁻¹° θ_deg m⁻¹, where θ_deg is latitude in degrees, and that the r.m.s. scatter of a from the linear fit is Δα = 1.28 × 10⁻⁸ m⁻¹ (Longcope et al., 1998). The observed systematic a in solar active regions may reflect a systematic field line twist in the subsurface emerging flux tubes.

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If the measured α values are a direct consequence of the emergence of twisted magnetic flux tubes from the interior, then it would imply subsurface emerging tubes with a field line twist of q = α/2 (Longcope and Klapper, 1997), where q denotes the angular rate of field line rotation about the axis over a unit axial distance along the tube. Several subsurface mechanisms for producing twist in emerging flux tubes have been proposed (see e.g. the review by Petrovay et al., 2006). 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 (Gilman and Charbonneau, 1999), or it may be acquired during the rise of the flux tubes through the solar convection zone (Longcope et al., 1998; Choudhuri, 2003; Choudhuri et al., 2004; Chatterjee et al., 2006).

Longcope et al. (1998) explain the origin of the observed twist in emerging active region flux tubes as a result of buffeting by the helical turbulence in the solar convection zone during the rise of the tubes. Applying the dynamic model of a weakly twisted thin flux tube (Section 2.1), Longcope et al. (1998) modeled the rise of a nearly straight, initially untwisted tube, buffeted by a random velocity field representative of the turbulent convection in the solar convection zone, which has a nonzero kinetic helicity due to the effect of solar rotation. The kinetic helicity causes helical distortion of the tube axis, which in turn leads to a net twist of the field lines about the axis in the opposite sense within the tube as a consequence of the conservation of magnetic helicity. This process is termed the Σ-effect by Longcope et al. (1998). Quantitative model calculations of Longcope et al. (1998) show that the Σ-effect can explain the hemispheric sign, magnitude, latitude variation, and the r.m.s. dispersion of the observed a of solar active regions. Furthermore, because the aerodynamic drag force acting on flux tubes by the convective flows has an a⁻² dependence, where a is the tube radius, this model predicts that the resulting twist generated should also have an a⁻² dependence, suggesting a systematic trend for greater twist in smaller flux tubes.

As discussed in Section 5.1.2, Ω-shaped emerging loops are themselves acted upon by the Coriolis force, developing a “tilt” of the loop. This helical deformation of the tube axis will then also induce a twist of the field lines of the opposite sense within the tube as a consequence of conservation of magnetic helicity. Calculations based on the weakly twisted thin flux tube model show that this twist generated by the large scale tilting of the emerging Ω-loop resulting from the Coriolis force has the right hemispheric sign and latitude dependence, but is of too small a magnitude to account for the observed twist in solar active regions (Longcope and Klapper, 1997; Fan and Gong, 2000).

Another interesting and natural explanation for the origin of twist in emerging flux tubes is the accretion of the background mean poloidal field onto the rising flux tube as it traverse through the solar convection zone (Choudhuri, 2003; Choudhuri et al., 2004; Chatterjee et al., 2006). In a Babcock-Leighton type dynamo, the dispersal of solar active regions with a slight mean tilt angle at the surface generates a mean poloidal magnetic field. The mean tilt angle of solar active regions is produced by the Coriolis force acting on the rising flux tubes (see Section 5.1.2). In the northern hemisphere, when a toroidal flux tube rises into a poloidal field that has been created due to the tilt of the same type of flux tubes emerged earlier, the poloidal field gets wrapped around the flux tube will produce a left-handed twist for the tube. This is illustrated in Figure 14. Using a circulation-dominated (or flux-transport) Babcock-Leighton type mean-field dynamo model, Choudhuri et al. (2004) did a rough estimate of the twist acquired by an emerging flux tube rising through the solar convection zone. Figure 15 shows the resulting butterfly diagram indicating the sign of α of the emerging regions as a function of latitude and time.

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It is found that at the beginning of a solar cycle, there is a short duration where the sign of α is opposite to the preferred sign for the hemisphere. This is because of the phase relation between the toroidal and poloidal magnetic fields produced by this types of solar dynamo models. At the beginning of a cycle, the mean poloidal magnetic field in the convection zone is still dominated by that generated by the emerging flux tubes of the previous cycle, and toroidal flux tubes of the new cycle emerging into this poloidal field gives rise to a right-handed (left-handed) twist of the tube in the northern (southern) hemisphere. However, for the rest of the cycle starting from the solar maximum, the poloidal magnetic field changes sign and the twist for the emerging tubes becomes consistent with the hemispheric preference. For the whole cycle, it is found that about 67% of the emerging regions have a sign of α consistent with the hemispheric rule. The rough estimate also shows that the magnitude of the α values produced by poloidal flux accretion is consistent with the observed values, and that there is an a⁻² dependence on the radius a of the emerging tube, i.e. smaller sunspots should have greater α values. This prediction is also made by the Σ-effect mechanism.

Given the frozen-in condition of the magnetic field, it is expected that the accreted poloidal flux be confined in a sheath at the outer periphery of the rising tube, and that in order to produce a twist within the tube, some form of turbulent diffusion needs to be invoked (Chatterjee et al., 2006). By solving the induction equation in a co-moving Lagrangian frame following the rising flux tube and using several simplifying assumptions, Chatterjee et al. (2006) modeled the evolution of the magnetic field in the rising tube cross-section as a result of poloidal flux accretion and penetration due to a field strength dependent turbulent diffusivity. They found that with plausible choices of assumptions and parameter values an α value comparable to the observations is obtained.

When a buoyant magnetic flux tube form at the base of the solar convection zone from the dynamo generated (pre-dominantly) toroidal magnetic field, it should already obtain an initial twist due to the weak poloidal mean field contained in the magnetic layer. This initial twist will then be further augmented or altered due to poloidal flux accretion and also due to the Σ-effect as the tube rises through the solar convection zone. MHD simulations of the formation and rise of buoyant magnetic flux tubes directly incorporating the mean field profiles from dynamo models (for both the fields at the base and in the bulk of the convection zone) as the initial state is necessary to quantify the initial twist and contribution from poloidal field accretion. Such simulations should be done for dynamo mean fields at different phases of the cycle to access the cycle variation of the twist in the emerging flux tubes.

Observationally, looking for any systematic variations of the the α value (or twist) of solar active regions with the solar cycle phase is helpful for identifying the main mechanisms for the origin of the twist. So far results in this area have been inconclusive (Bao et al., 2000; Pevtsov et al., 2001). On the other hand, observational studies of the correlation between active region twist (as measured by α) and tilt angles have revealed interesting results (Holder et al., 2004; Tian et al., 2005; Nandy, 2006). The Σ-effect predicts that the twist being generated in the tube is uncorrelated to the local tilt of the tube at the apex (Longcope et al., 1998). However, due to the Coriolis force, active region Ω-tubes acquire a mean tilt that has a well defined latitudinal dependence as described by the Joy’s law. The mean twist generated by the Σ-effect also has a latitudinal dependence that is consistent with the observed hemispheric rule. Thus, due to the mutual dependence of their mean values on latitude there should be a correlation between the tilt angle and twist of solar active regions and the correlation is expected to be positive if one assigns negative (positive) sign to a clockwise (counter-clockwise) tilt. However, Holder et al. (2004) found a statistically significant negative correlation between the twist and tilt for the 368 bipolar active regions studied, opposite to that expected from the mutual dependence on latitude of mean twist and mean tilt of active regions. Removing the effect of the mutual dependence of the mean tilt and mean twist on latitude (by either determining the correlation at a fixed latitude, or by subtracting off the fitted mean tilt and mean twist at the corresponding latitude), they found that the negative correlation is enhanced. Furthermore, it is found that the negative correlation is mainly contributed by those active regions (174 out of 368 regions) that deviate significantly from Joy’s law (by > 6σ), while regions that obey Joy’s law to within 6σ show no significant correlation between their twist and tilt. A separate study by Tian et al. (2005) found that a sample of 104 complex δ-configuration active regions, more than half of which have tilts that are opposite to the direction prescribed by Joy’s law, show a significant negative correlation between their twist and tilt (after correcting for their definition of the sign of tilt which is opposite to the definition used in Holder et al. (2004)). The results of Holder et al. (2004) and Tian et al. (2005) both indicate that there is a significant population (about one half in the case of Holder et al. (2004)) of solar active regions whose twist/tilt properties cannot be explained by the Σ-effect together with the effect of the Coriolis force alone. These active regions are consistent with the situation where the buoyant flux tube form at the base of the solar convection zone has acquired an initial twist, such that as it rises upward due to buoyancy into an Ω-tube, it develops a writhe that is of the same sense as the initial twist of the tube. In the extreme case, the twist can be so large that the flux tube becomes kink unstable (see section 5.4). The resulting tilt at the apex due to the writhe has the negative correlation with the twist as described in the above observations.

5.3 On the minimum twist needed for maintaining cohesion of rising flux tubes in the solar convection zone

As described in Section 5.1, simulations based on the thin flux tube approximation have revealed many interesting results with regard to the global dynamics of active region emerging flux loops in the solar convective envelope, which provide explanations for several basic observed properties of solar active regions. However, one major question ignored by the thin flux tube model is how a flux tube remains a discrete and cohesive object as it moves in the solar convection zone. The manner in which solar active regions emerge on the photosphere suggests that they are coherent flux bundles rising through the solar convection zone and reaching the photosphere in a reasonably cohesive fashion. To address this question, 3D MHD models that fully resolve the rising flux tubes are needed.

As a natural first step, 2D MHD simulations have been carried out to model buoyantly rising, infinitely long horizontal magnetic flux tubes in a stratified layer representing the solar convection zone, focusing on the dynamic evolution of the tube cross-section. The first of such calculations was done in fact much earlier by Schüssler (1979) and later, simulations of higher numerical resolutions have been performed (Moreno-Insertis and Emonet, 1996; Longcope et al., 1996; Fan et al., 1998a; Emonet and Moreno-Insertis, 1998). The basic result from these 2D models of buoyant horizontal flux tubes is that due to the vorticity generation by the buoyancy gradient across the flux tube cross-section, if the tube is untwisted, it quickly splits into a pair of vortex tubes of opposite circulations, which move apart horizontally and cease to rise. If on the other hand, the flux tube is sufficiently twisted such that the magnetic tension of the azimuthal field can effectively suppress the vorticity generation by the buoyancy force, then most of the flux in the initial tube is found to rise in the form of a rigid body whose rise velocity follows the prediction by the thin flux tube approximation. The result described above is illustrated in Figure 16 which shows a comparison of the evolution of the tube cross-section between the case where the buoyant horizontal tube is untwisted (upper panels) and a case where the twist of the tube is just above the minimum value needed for the tube to rise cohesively (lower panels). This minimum twist needed for tube cohesion can be estimated by considering a balance between the magnetic tension force from the azimuthal field and the magnetic buoyancy force. For a flux tube near thermal equilibrium whose buoyancy |Δρ/ρ| ∼ 1/Β, where ΔΡ = Ρ − Ρ_e denotes the density difference between the inside and the outside of the tube and β ≡ p/(B²/8π) denotes the ratio of the gas pressure over the magnetic pressure, such an estimate (Moreno-Insertis and Emonet, 1996) yields the condition that the pitch angle ψ of the tube field lines on average needs to reach a value of order

$$\tan \Psi \equiv {{{B_\phi }} \over {{B_z}}} \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {\left({{a \over {{H_p}}}} \right)^{1/2}}.$$

(26)

In (26)B _z and denote the axial and azimuthal field of the horizontal tube respectively, a is the characteristic radius of the tube, and H _p is the local pressure scale height. The above result on the minimum twist can also be expressed in terms of the rate of field line rotation about the axis per unit length along the tube q. For a uniformly twisted flux tube, B _φ = qr B _z , where r is the radial distance to the tube axis. Then q needs to reach a value of order (Longcope et al., 1999)

$$q \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} {\left({{1 \over {{H_p}a}}} \right)^{1/2}}$$

(27)

for the flux tube to maintain cohesion during its rise. Note that the conditions given by Equations (26) and (27) and also the 2D simulations described in this section all assume buoyant flux tubes with initial buoyancy |Δρ/ρ| ∼ 1/β. For tubes with lower level of buoyancy, the necessary twist is smaller with tan ψ and q both ∝ |Δρ/ρ|^1/2 (see Emonet and Moreno-Insertis, 1998).

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Longcope et al. (1999) pointed out that the amount of twist given by Equation (27) is about an order of magnitude too big compared to the twist deduced from vector magnetic field observations of solar active regions on the photosphere. They assumed that the averaged α ≡ J _z /B _z (the ratio of the vertical electric current over the vertical magnetic field) measured in an active region on the photosphere directly reflects the twist in the subsurface emerging tube, i.e. q = α/2 (Longcope and Klapper, 1997; Longcope et al., 1998). If this is true then it seems that the measured twists in solar active regions directly contradict the condition for the cohesive rise of a horizontal flux tube with buoyancy as large as |Aρ/ρ| ∼ 1/β.

More recently, 3D simulations of Ω-shaped arched flux tubes have been carried out (Abbett et al., 2000, 2001; Fan, 2001a). Fan (2001a) performed 3D simulations of arched flux tubes which form from an initially neutrally buoyant horizontal magnetic layer as a result of its undulatory buoyancy instability (see Section 4.2 and Figure 7). It is found that without any initial twist the flux tubes that form rise through a distance of about one density scale height included in the simulation domain without breaking up. This significantly improved cohesion of the 3D arched flux tubes compared to the previous 2D models of buoyant horizontal tubes is not only due to the additional tension force made available by the 3D nature of the arched flux tubes, but also due largely to the absence of an initial buoyancy and a slower initial rise (Fan, 2001a). With a neutrally buoyant initial state, both the buoyancy force and the magnetic tension force grow self-consistently from zero as the flux tube arches. The vorticity source term produced by the growing magnetic tension as a result of bending and braiding the field lines is found to be able to effectively counteract the vorticity generation by the growing buoyancy force in the apex cross-section, preventing it from breaking up into two vortex rolls. The 2D models (Moreno-Insertis and Emonet, 1996; Longcope et al., 1996; Fan et al., 1998a; Emonet and Moreno-Insertis, 1998) on the other hand considered an initially buoyant flux tube for which there is an impulsive initial generation of vorticity by the buoyancy force. A significant initial twist is thus required to suppress this initial vorticity generation. Therefore the absence of an initial vorticity generation by buoyancy, and the subsequent magnetic tension force resulting from bending and braiding the field lines allow the arched tube with no net twist in Fan (2001a) to rise over a significantly greater distance without disruption.

Abbett et al. (2000) performed 3D simulations where an initial horizontal flux tube is prescribed with a non-uniform buoyancy distribution along the tube such that it rises into an Ω-shaped loop. As discussed above, due to the prescribed buoyancy in the initial horizontal tube, there is an impulsive initial generation of vorticity by the buoyancy force which breaks up the apex of the rising Ω-loop if there is no initial twist. However the separation of the two vortex fragments at the apex is reduced due to the three-dimensional effect (Abbett et al., 2000). By further including the effect of solar rotation using a local f-plane approximation, Abbett et al. (2001) found that the influence of the Coriolis force significantly suppresses the degree of fragmentation at the apex of the Ω-loop (Figure 18). They also found that the Coriolis force causes the emerging loop to become asymmetric about the apex, with the leading side (leading in the direction of rotation) having a shallower angle with respect to the horizontal direction compared to the following (Figure 18), consistent with the geometric asymmetry found in the thin flux tube calculations (Section 5.1.4).

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Another interesting possibility is suggested by the 3D simulations of Dorch and Nordlund (1998), who showed that a random or chaotic twist with an amplitude similar to that given by Equation (26) or (27) in the flux tube can ensure that the tube rises cohesively. Such a random twist may not be detected in the photosphere measurement of active region twists which is determined by taking some forms of average of the quantity α = J _z /B _z over the active region.

5.4 The rise of kink unstable magnetic flux tubes and the origin of delta-sunspots

Most of the solar active regions are observed to have very small twists, with an averaged value of α ≡ 〈J _z /B _z 〉 (the averaged ratio of the vertical electric current over the vertical magnetic field) measured to be on the order of 0.01 Mm⁻¹ (see Pevtsov et al., 1995, 2001). However there is a small but important subset of active regions, called the δ-sunspots, which are observed to be highly twisted with a reaching a few times 0.1 Mm⁻¹ (Leka et al., 1996), and to have unusual polarity orientations that are sometimes reversed from Hale’s polarity rule (see Zirin and Tanaka, 1973; Zirin, 1988; Tanaka, 1991). These δ-sunspots are compact structures where umbrae of opposite polarity are contained within a common penumbra. They are found to be the most flare productive active regions (see Zirin, 1988). Through careful analysis of the evolution of flare-active δ-sunspot groups, Tanaka (1991) proposed a model of an emerging twisted flux rope with kinked or knotted geometry to explain the observed evolution of these regions.

The non-linear evolution of the kink instability of twisted magnetic flux tubes in a high-β plasma has been investigated by 3D MHD simulations (Linton et al., 1998, 1999; Fan et al., 1998b, 1999). Fan et al. (1998b, 1999) modeled the rise of a kink unstable flux tube through an adiabatically stratified model solar convection zone. In the case where the initial twist of the tube is significantly supercritical such that the e-folding growth time of the most unstable kink mode is smaller than the rise time scale, they found sharp bending of the flux tube as a result of the non-linear evolution of the kink instability. During the onset of the kink instability, the magnetic energy decreases while the magnetic helicity is approximately conserved. The writhing of the flux tube also significantly increases the axial field strength and hence enhances the buoyancy of the flux tube. The flux tube rises and arches upward at the portion where the kink concentrates, with a rotation of the tube orientation at the apex that exceeds 90° (see Figure 19). The emergence of this kinked flux tube can give rise to a compact magnetic bipole with polarity order inverted from the Hale polarity rule (Figure 20) as often seen in δ-sunspots. The conservation of magnetic helicity requires that the writhing of the tube due to the kink instability is of the same sense as the twist of the field lines. Hence for a kinked emerging tube, the rotation or tilt of the magnetic bipole from the east-west polarity orientation defined by the Hale’s polarity rule should be related to the twist of the tube. The rotation or tilt should be clockwise (counterclockwise) for right-hand-twisted (left-hand-twisted) flux tubes. This tilt-twist relation can be used as a means to test the model of kinked flux tubes as the origin of δ-sunspots (Tanaka, 1991; Leka et al., 1994, 1996; Lópezs Fuentes et al., 2003). Observations have found with both consistent and opposing cases (Leka et al., 1996; López Fuentes et al., 2003). A recent study (Tian et al., 2005) which includes a large sample (104) of complex δ-configuration active regions shows that 65–67% of these δ-regions have the same sign of twist and writhe, supporting the model of kinked flux tubes.

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5.5 Buoyant flux tubes in a 3D stratified convective velocity field

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 21. 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 21) 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 21). 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 21), 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 21), 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.

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In case B = B_eq, 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 5H _p , 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 ∼ 2H _p , then the tension force for the tube between the pinned-down points will exceed the magnetic buoyancy force (∼ B²/8πH _p ) and the entire flux tube will be prevented from emerging to the surface.

8.1 Evolution in the top layer of the solar convection zone

A complete 3D MHD model of emerging active region flux tubes, extending from the base of the solar convection zone up into to the visible solar atmosphere, is not yet possible. Thin flux tube calculations of rising flux tubes in the solar convection zone usually extend from the base of the solar convection zone up to roughly 20–30 Mm below the surface, at which depths the validity of the thin flux tube approximation breaks down since the diameter of the flux tube begins to exceed the local pressure scale height (Moreno-Insertis, 1992). The domain of fully 3D MHD simulations of rising magnetic flux tubes in a model convection zone typically contains a vertical stratification of up to 3 density scale heights (see Abbett et al., 2000, 2001; Fan et al., 2003), which corresponds to the density stratification over the range from the bottom of the solar convection zone to roughly 36 Mm below the photosphere. Because of the rapid decrease of the various scale heights in the top layer of the solar convection zone which demands increasing numerical resolution, it is not yet feasible to perform 3D MHD simulations that extend from the bottom of the convection zone all the way to the photosphere. Furthermore, there is an increased complexity in the physics of the top layer of the solar convection zone. The thermodynamics of the plasma is complicated by ionization effects and the radiative exchange is expected to play an important role in the heat transport (see Lantz, 1991; Stein and Nordlund, 2000). The anelastic approximation breaks down because the plasma flow speed is no longer slow compared to the sound speed. Fully compressible 3D MHD simulations of magneto-convection and emerging magnetic flux in the top few Mm layer of the solar convection zone and the overlying photospheric layer, incorporating realistic physics such as partial ionization of the dominant constituents and radiative transfer, have been carried out (see review by Stein and Nordlund, 2000, and the references therein). These simulations have produced results that can be directly compared with high resolution photospheric observations of the solar granulations and have found the formation of intense magnetic flux tubes of kG field strength with size scales up to that of a magnetic pore. However realistic simulations of the evolution of emerging active region scale flux tubes in the top ∼ 20 Mm layer of the solar convection zone are still beyond the reach of current numerical modeling capabilities. It remains an open question how the top of the emerging Ω-tube intensifies to form sunspots with kG field strength and β ∼ 1 at the photosphere. The intensification of small scale magnetic flux tubes to kG field strength at the photosphere can be explained by the process of “convective collapse” which results from the convective instability of plasma inside the vertical thin flux tubes in the top few hundred kilometers of the solar convection zone (see Parker, 1978; Spruit, 1979; Spruit and Zweibel, 1979). This process may also be responsible for the intensification of sunspot scale flux tubes, but there have been no realistic model calculations.

8.2 Flux emergence into the solar atmosphere

A fundamental and challenging question concerning the emergence of solar active regions is how buoyant magnetic fields from the convectively unstable interior emerge dynamically into the stably stratified, rarefied solar atmosphere and corona. Pioneering work in this area was carried out by Shibata and collaborators (see Shibata et al., 1989) who showed that the magnetic buoyancy instability is a mechanism through which magnetic flux reaching the photosphere can expand dynamically into the stably stratified solar atmosphere. Many 2D and 3D simulations of flux emergence into the atmosphere have since been carried out (see e.g. Magara, 2001; Murray et al., 2006, and the references therein).

During the run-away expansion of the magnetic flux tube into the solar atmosphere, strong diverging downflows are found along the emerged field lines, forming shock fronts just above the photosphere (e.g. Fan, 2001b; Magara and Longcope, 2003; Archontis et al., 2004). Due to the magnetic tension force associated with the twisted field lines, a shear flow pattern develops on the photosphere with the plasma on the two sides of the polarity inversion line moving oppositely in the east-west direction (Manchester IV, 2001; Fan, 2001b; Magara and Longcope, 2003; Manchester IV et al., 2004; Archontis et al., 2004). The effect of the tension force driven shear flow is to transport the axial magnetic flux upward into the expanding portion of the flux tube in the solar atmosphere (Manchester IV et al., 2004). Newly developing active regions on the photosphere often exhibit such a shear flow pattern (Zirin, 1988; Strous et al., 1996). The two polarities of the emerging region on the photosphere initially emerge as north-south oriented, and over time they appear to shear along the polarity inversion line and separate in the east-west direction (e.g. Fan, 2001b; Magara and Longcope, 2003; Manchester IV et al., 2004; Archontis et al., 2004; Magara, 2006). A detailed study of both the apparent and plasma motions in the photosphere resulting from the emergence of a twisted flux tube (with different degree of twist for the initial subsurface flux tube) is carried out by Magara (2006). It is found that the apparent rotational motion exhibited by the two polarities are opposite to the direction of the actual rotational motion of the plasma.

The simulations also suggest that it is difficult for a twisted flux tube to rise bodily into the corona as a whole due to the heavy plasma that is trapped at the bottom concave (or U-shaped) portions of the winding field lines. Independent simulations by Fan (2001b) and Archontis et al. (2004), considering nearly identical initial conditions both find that that as the upper parts of the winding field lines of the twisted flux tube expand into the atmosphere and the corona, the axial field line of the tube settles to an equilibrium not much above the photosphere, and the lower U-shaped parts of the field lines largely remain trapped below the photosphere. On the other hand, Manchester IV et al. (2004) considered the case where the length of the emerging portion of the tube is shortened to a half of that in Fan (2001b), such that the field lines only perform about one turn about the axis over the emerging segment. In this case it is found that after the growth of the magnetic buoyancy instability which causes the expansion of the flux tube into the solar atmosphere, a secondary eruptive phase sets in where the upper emerged part of the flux tube begins to “pinch off” from the lower mass-laden part via magnetic reconnection, forming a new flux rope with a new central axial field line that rises well into the corona. Manchester IV et al. (2004) explains that the cause of the eruptive phase is the shear flow driven by the magnetic tension force, which continually transport axial flux and magnetic energy to the upper expanded portion of the magnetic field, driving its eruption. It is found that a sigmoid shaped current sheet develops in the chromospheric heights during the eruptive phase. Simulations by Magara and Longcope (2001), Magara (2004) Magara et al. (2005), and Magara (2006) also found formation of current concentration of similar sigmoid morphology in the chromospheric heights (see e.g. Figure 25), and the ascent of a flux rope structure into the atmosphere with an O-point of the magnetic field in the central cross-section reaching the base of the corona. It is found that the emerged field lines with footpoints rooted in regions of high electric current density in the chromosphere display an inverse-S shape for a left-hand-twisted emerging tube, in agreement with the X-ray sigmoid morphology and the sense of active region twist in the northern hemisphere (see Figure 25 and Figure 4).

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The effects of the presence of a simple horizontal coronal magnetic field and the interaction and reconnection of the pre-existing coronal field with the emerging flux rope have been investigated in a series of work (Archontis et al., 2004, 2005, 2006; Galsgaard et al., 2005). It is found that the dynamic interaction and the rate of magnetic reconnection depends sensitively on the relative orientations between the upcoming emerging flux and the pre-existing coronal magnetic field. When the two flux systems coming into contact have a relative angle above 90°, the simulations show immediate and substantial magnetic reconnection, producing collimated high-speed and high-temperature jets from the reconnection site. As a result of the reconnection, most of the flux in the subsurface flux rope end up connecting to the coronal magnetic field. The interaction with the pre-existing coronal field tends to slow down the rise speed of the upper-boundary of the emerging flux tube into the corona, compared to the case where the flux tube emerges into a field-free atmosphere (Archontis et al., 2005). Isobe et al. (2005) performed high resolution 3D simulations of emerging flux and its reconnection with pre-existing magnetic field in the corona. In this calculation the emerging flux develops from an initial horizontal magnetic flux sheet (with uni-direction field) situated in the top of the convection zone. Due to the growth of the three-dimensional magnetic buoyancy instability of modes with high wave number in the direction perpendicular to the field (or magnetic Rayleigh-Taylor instability), undulating flux bundles with fine spacial scales rise into the corona and reconnect with the pre-existing coronal magnetic field in a spatially intermittent way. This results in the formation of filamentary structure resembling the observed arch filament systems in emerging flux regions (EFRs). The spatially intermittent reconnection and heating also explain the coexistence of many hot and cold loops and the jets being ejected from the loop footpoints in EFRS observed in EUV by the TRACE satellite.

Due to the need to resolve the photosphere pressure scale height (∼ 150 km), the 3D calculations of twisted emerging flux tubes described above are done for a domain size of up to a few tens of Mm, the size of a small active region. The implicit assumption is that the basic dynamical behavior of the smaller emerging flux tubes modeled in these simulations is representative of that of the larger active region scale emerging tubes. Furthermore, adiabatic evolution of an ideal gas is assumed in these simulations, which is not appropriate for the atmosphere layers considered. The adiabatic expansion of the tube plasma emerging into the atmosphere resulting in unphysically low temperatures. Radiative heat exchange, thermal conduction, and the still uncertain sources of coronal heating all play an important role in the thermal energy evolution of the plasma in emerging flux regions.

8.3 Post-emergence evolution of subsurface fields

This is so far a largely unexplored area with mostly speculations and very few quantitative calculations. One outstanding question is whether and to what extent sunspots and active region fields at the surface remain connected to the subsurface flux tubes in the deep solar convection zone. If the surface fields remain connected to the subsurface flux tubes, then based on the shape of the Ω-loop as shown in Figure 12, it seems difficult to understand why the separation of the two polarities of a solar active region stops at the scales of ≲ 100 Mm, without continuously increasing to at least the footpoint separation of the Ω-loop at the base of the convection zone, which is ∼ 1000 Mm! Due to the angles between the legs of the Ω-loop and the surface, the magnetic tension of the legs is expected to be pulling the two poles of the bipolar region apart until the legs become vertical. However there is also an attracting force between the two poles due to the emerged magnetic field above the photosphere. If we assume that the field above the photosphere is potential, then the attracting force of the two poles due to the field above can be estimated by considering the attraction between two opposite electric charges of ±q, where q = Φ/2π and Φ is the total magnetic flux of the active region. This yields (Φ/2π)²/d² for the attracting force, where d is the separation between the poles. On the other hand, the lateral force from the subsurface field that acts to pull the two poles apart can be estimated by integrating over the area of each pole the Maxwell stress exerted by the subsurface field from below. This leads to an estimated lateral force of \((B_{n}^{2}/4\pi){\rm ctg}\theta\Delta S\), where B _n and ΔS are the photosphere normal field strength and area of each magnetic pole respectively (note that we have B _n ΔS = Φ), and θ is the angle between the slanted subsurface tube and the surface. Balancing the forces from above and below the photosphere (taking a representative value of θ = 45°), we find that it is possible for the two poles to reach a lateral force balance if πd² = ΔS. This suggests that the two poles of the bipolar region need to be very close, with the polarity separation no greater than the radius of each pole, in order for the attracting force from above to balance the force that drives the separation from below. Furthermore it appears that the force balance is an unstable equilibrium because of the 1/d² dependence of the attracting force which weakens in response to an increase in d.

A first quantitative calculation of the above process of “dynamic disconnection” has been carried out by Schüssler and Rempel (2005), using a 1-D self-similar vertical flux tube model whose top end has reached the photosphere. The model computes the quasi-static evolution of the flux tube under the effects of radiative cooling, convective energy transport, and a pressure buildup by a prescribed inflow at the bottom of the tube due to the high entropy tube plasma flowing upward to establish hydrostatic equilibrium along the field. The calculation shows that after emergence, the radiative losses near the surface drives an inward propagating cooling front accompanied by a downflow, which leads to a decrease of the gas pressure and an intensification of the magnetic field in the surface layers. In the mean time, the convergence of the radiative cooling driven downflow and the buoyancy driven upflow results in an increase of the gas pressure and a weakening of the magnetic field in the tube at a depth of a few Mms. It is found that for a reasonable range of the upflow speed, the magnetic field weakens to fall below the local equipartition value (i.e. dynamic disconnection takes place) at a depth between 2 and 6 Mm, in a time scale of up to 3 days. This is consistent with the time scale over which the evolution of a newly emerging active region changes from an “active phase” of growth with increasing polarity separation to a “passive phase” which can be well represented by transport by near surface flows and supergranular diffusion.