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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 Beq, where Beq is the field strength that is in equipartition with the kinetic energy density of the convective motions: 2 2 B eq∕8π = ρvc∕2. If we use the results from the mixing length models of the solar convection zone for the convective flow speed vc, then we find that in the deep convection zone Beq is on the order of 104G. 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., 1990Spruit, 1997). Hence it should be noted that the Beq 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 ∼ 104 G eq as the field strength in equipartition with convection in this review.

Assuming that in the deep solar convection zone the magnetic field strength for flux tubes responsible for active region formation is at least 104 G, and given that the amount of flux observed in solar active regions ranges from ∼ 1020 Mx to 1022Mx (see Zwaan, 1987), then one finds that the cross-sectional sizes of the flux tubes are small in comparison to other spatial scales of variation, e.g. the pressure scale height. For an isolated magnetic flux tube that is thin in the sense that its cross-sectional radius a is negligible compared to both the scale height of the ambient unmagnetized fluid and any scales of variation along the tube, the dynamics of the flux tube may be simplified with the thin flux tube approximation (see Spruit, 1981Jump To The Next Citation PointLongcope and Klapper, 1997Jump To The Next Citation Point) which corresponds to the lowest order in an expansion of MHD in powers of a∕L, where L represents any of the large length scales of variation. Under the thin flux tube approximation, all physical quantities of the tube, such as position, velocity, field strength, pressure, density, etc. are assumed to be averages over the tube cross-section and they vary spatially only along the tube. Furthermore, because of the much shorter sound crossing time over the tube diameter compared to the other relevant dynamic time scales, an instantaneous pressure balance is assumed between the tube and the ambient unmagnetized fluid:

B2- p + 8π = pe (1 )
where p is the tube internal gas pressure, B is the tube field strength, and pe is the pressure of the external fluid. Applying the above assumptions to the ideal MHD momentum equation, Spruit (1981Jump To The Next Citation Point) derived the equation of motion of a thin untwisted magnetic flux tube embedded in a field-free fluid. Taking into account the differential rotation of the Sun, Ωe (r) = Ωe(r)ˆz, the equation of motion for the thin flux tube in a rotating reference frame of angular velocity Ω = Ωˆz is (Ferriz-Mas and Schüssler, 1993Jump To The Next Citation PointCaligari et al., 1995Jump To The Next Citation Point)
dv 2 2 ρ dt-= 2ρ(v × Ω ) + ρ(Ω − Ω e)ϖ ˆϖ + (ρ − ρe)geff ( 2 ) 2 (2 ) + ˆl ∂- B-- + B--k − CD ρe|(vrel)⊥|(vrel)⊥-, ∂s 8π 4π π (Φ∕B )1∕2
where
geff = g + Ω2eϖ ˆϖ, (3 ) (vrel)⊥ = [v − (Ωe − Ω ) × r]⊥. (4 )
In the above, r, v, B, p, ρ, denote the position vector, velocity, magnetic field strength, plasma pressure and density of a Lagrangian tube element respectively, each of which is a function of time t and the arc-length s measured along the tube, ρe(r) denotes the external density at the position r of the tube element, ˆz is the unit vector pointing in the direction of the solar rotation axis, ϖˆ denotes the unit vector perpendicular to and pointing away from the rotation axis at the location of the tube element and ϖ denotes the distance to the rotation axis, ˆl ≡ ∂r∕∂s is the unit vector tangential to the flux tube, k ≡ ∂2r∕∂s2 is the tubes curvature vector, the subscript ⊥ denotes the vector component perpendicular to the local tube axis, g is the gravitational acceleration, and CD is the drag coefficient. The drag term (the last term on the right hand side of the equation of motion (2View Equation)) is added to approximate the opposing force experienced by the flux tube as it moves relative to the ambient fluid. The term is derived based on the case of incompressible flows pass a rigid cylinder under high Reynolds number conditions, in which a turbulent wake develops behind the cylinder, creating a pressure difference between the up- and down-stream sides and hence a drag force on the cylinder (see Batchelor, 1967).

If one considers only the solid body rotation of the Sun, then the Equations (2View Equation), (3View Equation), and (4View Equation) can be simplified by letting Ωe = 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 (2View Equation), 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 (2View Equation) differently, with the term ρ(dv ∕dt)⊥ on the left-hand-side of the perpendicular component of the equation being replaced by (ρ + ρe)(dv∕dt)⊥ (see Spruit, 1981Jump To The Next Citation Point). 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, 1992Fan et al., 1994Jump To The Next Citation PointMoreno-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., 1995Jump To The Next Citation Point1998Jump To The Next Citation PointFan and Fisher, 1996Jump To The Next Citation Point).

Equations (1View Equation) and (2View Equation) are to be complemented by the following equations to completely describe the dynamic evolution of a thin untwisted magnetic flux tube:

( ) [ ] d- B- B- ∂(v-⋅ˆl) dt ρ = ρ ∂s − v ⋅ k , (5 ) 1dρ-= -1-dp-− ∇ad- dQ-, (6 ) ρdt γp dt p dt ρRT p = -----, (7 ) μ
where ∇ad ≡ (∂ lnT ∕∂ ln p)s. Equation (5View Equation) describes the evolution of the tube magnetic field and is derived from the ideal MHD induction equation (Spruit, 1981). Equation (6View Equation) is the energy equation for the thin flux tube (Fan and Fisher, 1996Jump To The Next Citation Point), in which dQ ∕dt corresponds to the volumetric heating rate of the flux tube by non-adiabatic effects, e.g. by radiative diffusion (Section 3.2). Equation (7View Equation) is simply the equation of state for an ideal gas. Thus the five Equations (1View Equation), (2View Equation), (5View Equation), (6View Equation), and (7View Equation) completely determine the evolution of the five dependent variables v (t,s), B (t,s), p(t,s), ρ (t,s), and T (t,s) for each Lagrangian tube element of the thin flux tube.

Spruit’s original formulation for the dynamics of a thin isolated magnetic flux tube as described above assumes that the tube consists of untwisted flux B = B ˆl. Longcope and Klapper (1997Jump To The Next Citation Point) extend the above model to include the description of a weak twist of the flux tube, assuming that the field lines twist about the axis at a rate q whose magnitude is 2π∕Lw, where Lw is the distance along the tube axis over which the field lines wind by one full rotation and |qa | ≪ 1. Thus in addition to the axial component of the field B, there is also an azimuthal field component in each tube cross-section, which to lowest order in qa is given by Bθ = qr⊥B, where r⊥ denotes the distance to the tube axis. An extra degree of freedom for the motion of the tube element – the spin of the tube cross-section about the axis – is also introduced, whose rate is denoted by ω (angle per unit time). By considering the kinematics of a twisted ribbon with one edge corresponding to the tube axis and the other edge corresponding to a twisted field line of the tube, Longcope and Klapper (1997Jump To The Next Citation Point) derived an equation that describes the evolution of the twist q in response to the motion of the tube:

ˆ dq-= − dln-δsq + ∂-ω + (ˆl × k ) ⋅ dl, (8 ) dt dt ∂s dt
where δs denotes the length of a Lagrangian tube element. The first term on the right-hand-side describes the effect of stretching on q: Stretching the tube reduces the rate of twist q. The second term is simply the change of q resulting from the gradient of the spin along the tube. The last term is related to the conservation of total magnetic helicity which, for the thin flux tube structure, can be decomposed into a twist component corresponding to the twist of the field lines about the axis, and a writhe component corresponding to the “helicalness” of the axis (see discussion in Longcope and Klapper, 1997Jump To The Next Citation Point). It describes how the writhing motion of the tube axis can induce twist of the opposite sense in the tube.

Furthermore, by integrating the stresses over the surface of a tube segment, Longcope and Klapper (1997Jump To The Next Citation Point) evaluated the forces experienced by the tube segment. They found that for a weakly twisted (|qa| ≪ 1) thin tube (|a∂s| ≪ 1), the equation of motion of the tube axis differs very little from that for an untwisted tube – the leading order term in the difference is 2 O [qa ∂s] (see also Ferriz-Mas and Schüssler, 1990). Thus the equation of motion (2View Equation) applies also to a weakly twisted thin flux tube. By further evaluating the torques exerted on a tube segment, Longcope and Klapper (1997Jump To The Next Citation Point) also derived an equation for the evolution of the spin ω:

dω 2da ∂q ---= − ----ω + v2a--, (9 ) dt a dt ∂s
where √ ---- va = B ∕ 4πρ is the Alfvén speed. The first term on the right hand side simply describes the decrease of spin due to the expansion of the tube cross-section as a result of the tendency to conserve angular momentum. The second term, in combination with the second term on the right hand side of Equation (8View Equation), describes the propagation of torsional Alfvén waves along the tube.

The two new Equations (8View Equation) and (9View Equation) – derived by Longcope and Klapper (1997Jump To The Next Citation Point) – together with the earlier Equations (1View Equation), (2View Equation), (5View Equation), (6View Equation), and (7View Equation) 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 (1View Equation), (2View Equation), (5View Equation), (6View Equation), and (7View Equation), and then apply the resulting motion of the tube axis to Equations (8View Equation) and (9View Equation) 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 (8View Equation)) and the torsional Alfvén waves (2nd term on the right-hand-side of Equation (8View Equation)). Twist can also be generated due to writhing motion of the tube axis (last term on the right-hand-side of Equation (8View Equation)), as required by the conservation of total helicity.


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