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 22 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 , where denotes the density difference between the inside and the outside of the tube and 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 of the tube field lines on average needs to reach a value of order

In Equation (26), and denote the axial and azimuthal field of the horizontal tube respectively, is the characteristic radius of the tube, and 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 . For a uniformly twisted flux tube, , where is the radial distance to the tube axis. Then needs to reach a value of order (Longcope et al., 1999) 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 . For tubes with lower level of buoyancy, the necessary twist is smaller with and both (see Emonet and Moreno-Insertis, 1998).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 (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. (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 .

More recently, 3D simulations of -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 -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 -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 -loop (Figure 24).

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 24), consistent with the geometric asymmetry found in the thin flux tube calculations (Section 5.1.4).

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 over the active region.

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