4.1 The buoyancy instability of isolated toroidal magnetic flux tubes

By linearizing the thin flux tube dynamic equations (1View Equation), (2View Equation), (5View Equation), (6View Equation), and (7View Equation), the stability of neutrally buoyant toroidal magnetic flux tubes to isentropic perturbations have been studied (see Spruit and van Ballegooijen, 1982aJump To The Next Citation Point,bJump To The Next Citation PointFerriz-Mas and Schüssler, 1993Jump To The Next Citation Point1995Jump To The Next Citation Point).

In the simplified case of a horizontal neutrally buoyant flux tube in a plane parallel atmosphere, ignoring the effects of curvature and solar rotation, the necessary and sufficient condition for instability is (Spruit and van Ballegooijen, 1982aJump To The Next Citation Point,bJump To The Next Citation Point)

2 2 -β∕2-- k Hp < 1 + β(1∕γ + β δ), (17 )
where k is the wavenumber along the tube of the undulatory perturbation, Hp is the local pressure scale height, β ≡ p∕(B2 ∕8π ) is the ratio of the plasma pressure divided by the magnetic pressure of the flux tube, δ = ∇ − ∇ad is the superadiabaticity, and γ is the ratio of the specific heats. If all values of k are allowed, then the condition for the presence of instability is
β δ > − 1∕γ. (18 )
Note that k → 0 is a singular limit. For perturbations with k = 0 which do not involve bending the field lines, the condition for instability becomes (Spruit and van Ballegooijen, 1982aJump To The Next Citation Point)
( ) βδ > 2- -1 − 1- ∼ 0.12 (19 ) γ γ 2
which is a significantly more stringent condition than (18View Equation), even more stringent than the convective instability for a field-free fluid (δ > 0). Thus the undulatory instability (with k ⁄= 0) is of a very different nature and is easier to develop than the instability associated with uniform up-and-down motions of the entire flux tube. The undulatory instability can develop even in a convectively stable stratification with δ < 0 as long as the field strength of the flux tube is sufficiently strong (i.e. β is of sufficiently small amplitude) such that |βδ| is smaller than 1∕γ. In the regime of − 1∕γ < βδ < (2∕γ )(1∕ γ − 1∕2) where only the undulatory modes with k ⁄= 0 are unstable, a longitudinal flow from the crests to the troughs of the undulation is essential for driving the instability. Since the flux tube has a lower internal temperature and hence a smaller pressure scale height inside, upon bending the tube, matter will flow from the crests to the troughs to establish hydrostatic equilibrium along the field. This increases the buoyancy of the crests and destabilizes the tube (Spruit and van Ballegooijen, 1982aJump To The Next Citation Point).

Including the curvature effect of spherical geometry, but still ignoring solar rotation, Spruit and van Ballegooijen (1982aJump To The Next Citation Point,bJump To The Next Citation Point) have also studied the special case of a toroidal flux ring in mechanical equilibrium within the equatorial plane. Since the Coriolis force due to solar rotation is ignored, the flux ring in the equatorial plane needs to be slightly buoyant to balance the inward tension force. For latitudinal motions out of the equatorial plane, the axisymmetric component is unstable, which corresponds to the poleward slip of the tube as a whole. But this instability can be suppressed when the Coriolis force is included (Ferriz-Mas and Schüssler, 1993Jump To The Next Citation Point). For motions within the equatorial plane, the conditions for instabilities are (Spruit and van Ballegooijen, 1982a,b)

1- 2 2 2βδ > (m − 3 − s)f + 2f∕γ − 1∕(2γ) (m ≥ 1), 1 1( 1 1) (20 ) -β δ > f2(1 − s) − 2f∕γ + -- --− -- (m = 0) 2 γ γ 2
where f ≡ H ∕r p 0 is the ratio of the pressure scale height over the radius of the bottom of the solar convection zone, m (having integer values 0,1,...) denotes the azimuthal order of the undulatory mode of the closed toroidal flux ring, i.e. the wavenumber k = m ∕r0, s is a parameter that describes the variation of the gravitational acceleration: g ∝ rs. Near the base of the solar convection zone, f ∼ 0.1, s ∼ − 2. Thus conditions (20View Equation) show that it is possible for m = 0,1, 2,3,4 modes to become unstable in the weakly subadiabatic overshoot region, and that the instabilities of m = 1, 2,3 modes require less stringent conditions than the instability of m = 0 mode. Since Equation (20View Equation) is derived for the singular case of an equilibrium toroidal ring in the equatorial plane, its applicability is very limited.

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 (19931995). 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 6View Image (from Caligari et al., 1995Jump To The Next Citation Point).

View Image

Figure 6: Upper panel: Regions of unstable toroidal flux tubes in the (B0, λ0)-plane (with B0 being the magnetic field strength of the flux tubes and λ0 being the equilibrium latitude). The subadiabaticity at the location of the toroidal flux tubes is assumed to be δ ≡ ∇ − ∇ad = − 2.6 × 10− 6. The white area corresponds to a stable region while the shaded regions indicate instability. The degree of shading signifies the azimuthal wavenumber of the most unstable mode. The contours correspond to lines of constant growth time of the instability. Thicker lines are drawn for growth times of 100 days and 300 days. Lower panel: Same as the upper panel except that the subadiabaticity at the location of the toroidal tubes is δ ≡ ∇ − ∇ad = − 1.9 × 10−7. From Caligari et al. (1995Jump To The Next Citation Point).

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 6View Image), 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, −7 δ ≡ ∇ − ∇ad = − 1.9 × 10 (lower panel of Figure 6View Image), 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.

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.


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