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 crosssection. The first of such calculations was done in fact much earlier by Schüssler (1979) and later, simulations of higher numerical resolutions have been performed (MorenoInsertis and Emonet, 1996; Longcope et al., 1996; Fan et al., 1998a; Emonet and MorenoInsertis, 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 crosssection, 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 crosssection 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).


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 selfconsistently 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 crosssection, preventing it from breaking up into two vortex rolls. The 2D models (MorenoInsertis and Emonet, 1996; Longcope et al., 1996; Fan et al., 1998a; Emonet and MorenoInsertis, 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 nonuniform 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 threedimensional effect (Abbett et al., 2000). By further including the effect of solar rotation using a local fplane 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 18).

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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