5.4 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, 1996Jump To The Next Citation PointLongcope et al., 1996Jump To The Next Citation PointFan et al., 1998aJump To The Next Citation PointEmonet and Moreno-Insertis, 1998Jump To The Next Citation Point). 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 22View Image 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).

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Figure 22: Upper panel: Evolution of a buoyant horizontal flux tube with purely longitudinal magnetic field. Lower panel: Buoyant rise of a twisted horizontal flux tube with twist that is just above the minimum value given by Equation (26View Equation). The color indicates the longitudinal field strength and the arrows describe the velocity field. From Fan et al. (1998aJump To The Next Citation Point). (For a corresponding movie showing the evolution of the tube for the untwisted and the twisted cases refer to Figure 23Watch/download Movie.)

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Figure 23: mpg-Movie (196 KB) The evolution of a rising flux tube. From Fan et al. (1998aJump To The Next Citation Point). For a detailed description see Figure 22View Image.

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 Δ ρ ≡ ρ − ρe 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, 1996Jump To The Next Citation Point) yields the condition that the pitch angle Ψ of the tube field lines on average needs to reach a value of order

( ) B ϕ a 1∕2 tanΨ ≡ --- ≳ --- . (26 ) Bz Hp
In Equation (26View Equation), Bz and B ϕ denote the axial and azimuthal field of the horizontal tube respectively, a is the characteristic radius of the tube, and Hp 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 = qrBz, where r is the radial distance to the tube axis. Then q needs to reach a value of order (Longcope et al., 1999Jump To The Next Citation Point)
( 1 )1∕2 q ≳ ---- (27 ) Hpa
for the flux tube to maintain cohesion during its rise. Note that the conditions given by Equations (26View Equation) and (27View Equation) and also the 2D simulations described in this section all assume buoyant flux tubes with initial buoyancy |Δ ρ∕ρ| ∼ 1∕β. For tubes with lower level of buoyancy, the necessary twist is smaller with tan Ψ and q both ∝ |Δ ρ∕ρ |1∕2 (see Emonet and Moreno-Insertis, 1998Jump To The Next Citation Point).

Longcope et al. (1999) pointed out that the amount of twist given by Equation (27View Equation) 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 ∕B z 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, 1997Longcope 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 |Δ ρ∕ρ| ∼ 1∕β.

More recently, 3D simulations of Ω-shaped arched flux tubes have been carried out (Abbett et al., 2000Jump To The Next Citation Point2001Jump To The Next Citation PointFan, 2001aJump To The Next Citation Point). Fan (2001aJump To The Next Citation Point) 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 7Watch/download Movie). 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, 2001aJump To The Next Citation Point). 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, 1996Longcope et al., 1996Fan et al., 1998aEmonet and Moreno-Insertis, 1998Jump To The Next Citation Point) 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. (2000Jump To The Next Citation Point) 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., 2000Jump To The Next Citation Point). By further including the effect of solar rotation using a local f-plane approximation, Abbett et al. (2001Jump To The Next Citation Point) found that the influence of the Coriolis force significantly suppresses the degree of fragmentation at the apex of the Ω-loop (Figure 24View Image).

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Figure 24: The rise of a buoyant Ω-loop with an initial field strength B = 105 G in a rotating model solar convection zone at a local latitude of 15° (from Abbett et al. (2001Jump To The Next Citation Point)). The Ω-loop rises cohesively even though it is untwisted. The loop develops an asymmetric shape with the leading side (leading in the direction of rotation) having a shallower angle relative to the horizontal direction compared to the following side.

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 24View Image), 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 (26View Equation) or (27View Equation) 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 ∕B z z over the active region.


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