4.2 Breakup of an equilibrium magnetic layer and formation of buoyant flux tubes

UpdateJump To The Next Update Information It is possible that the toroidal magnetic field stored at the base of the convection zone is in the form of an extended magnetic layer, instead of individual magnetic flux tubes for which the thin flux tube approximation can be applied. The classic problem of the buoyancy instability of a horizontal magnetic field B = B (z)ˆx in a plane-parallel, gravitationally stratified atmosphere with a constant gravity − gˆz, pressure p (z ), and density ρ(z), in hydrostatic equilibrium,
( ) d B2 --- p + --- = − ρg, (21 ) dz 8π
has been studied by many authors in a broad range of astrophysics contexts including

The linear stability analysis of the above equilibrium horizontal magnetic layer (Newcomb, 1961Jump To The Next Citation Point) showed that the necessary and sufficient condition for the onset of the general 3D instability with non-zero wavenumbers (kx ⁄= 0, ky ⁄= 0) in both horizontal directions parallel and perpendicular to the magnetic field is that

dρ ρ2g ---> − ---, (22 ) dz γp
is satisfied somewhere in the stratified fluid. On the other hand the necessary and sufficient condition for instability of the purely interchange modes (with kx = 0 and ky ⁄= 0) is that
2 dρ-> − ----ρ-g----. (23 ) dz γp + B2 ∕4π
is satisfied somewhere in the fluid – a more stringent condition than (22View Equation). Note in Equations (22View Equation) and (23View Equation), p and ρ are the plasma pressure and density in the presence of the magnetic field. Hence the effect of the magnetic field on the instability criteria is implicitly included. As shown by Thomas and Nye (1975Jump To The Next Citation Point) and Acheson (1979Jump To The Next Citation Point), the instability conditions (22View Equation) and (23View Equation) can be alternatively written as
2 va-dlnB--< − -1 ds- (24 ) c2s dz cp dz
for instability of general 3D undulatory modes and
[ ( ) ] v2a-d-- B- 1-ds- c2 dz ln ρ < − c dz (25 ) s p
for instability of purely 2D interchange modes, where va is the Alfvén speed, cs is the sound speed, cp is the specific heat under constant pressure, and ds∕dz is the actual entropy gradient in the presence of the magnetic field. The development of these buoyancy instabilities is driven by the gravitational potential energy that is made available by the magnetic pressure support. For example, the magnetic pressure gradient can “puff-up” the density stratification in the atmosphere, making it decrease less steeply with height (causing condition (22View Equation) to be met), or even making it top heavy. This raises the gravitational potential energy and makes the atmosphere unstable. In another situation, the presence of the magnetic pressure can support a layer of cooler plasma with locally reduced temperature embedded in an otherwise stably stratified fluid. This can also cause the instability condition (22View Equation) to be met locally in the magnetic layer. In this case the pressure scale height within the cooler magnetic layer is smaller, and upon bending the field lines, plasma will flow from the crests to the troughs to establish hydrostatic equilibrium, thereby releasing gravitational potential energy and driving the instability. This situation is very similar to the buoyancy instability associated with the neutrally buoyant magnetic flux tubes discussed in Section 4.1.

The above discussion on the buoyancy instabilities considers ideal adiabatic perturbations. It should be noted that the role of finite diffusion is not always stabilizing. In the solar interior, it is expected that η ≪ K and ν ≪ K, where η, ν, and K denote the magnetic diffusivity, the kinematic viscosity, and the thermal diffusivity respectively. Under these circumstances, it is shown that thermal diffusion can be destabilizing (see Gilman, 1970Acheson, 1979Jump To The Next Citation PointSchmitt and Rosner, 1983). The diffusive effects are shown to alter the stability criteria of Equations (24View Equation) and (25View Equation) by reducing the term ds ∕dz by a factor of η∕K (see Acheson, 1979Jump To The Next Citation Point). In other words, efficient heat exchange can significantly “erode away” the stabilizing effect of a subadiabatic stratification. This process is an example of the double-diffusive instabilities.

Direct multi-dimensional MHD simulations have been carried out to study the break-up of a horizontal magnetic layer by the non-linear evolution of the buoyancy instabilities and the formation of buoyant magnetic flux tubes (see Cattaneo and Hughes, 1988Jump To The Next Citation PointCattaneo et al., 1990Jump To The Next Citation PointMatthews et al., 1995Jump To The Next Citation PointWissink et al., 2000Jump To The Next Citation PointFan, 2001aJump To The Next Citation Point).

Cattaneo and Hughes (1988Jump To The Next Citation Point), Matthews et al. (1995Jump To The Next Citation Point), and Wissink et al. (2000Jump To The Next Citation Point) have carried out a series of 2D and 3D compressible MHD simulations where they considered an initial horizontal magnetic layer that supports a top-heavy density gradient, i.e. an equilibrium with a lower density magnetic layer supporting a denser plasma on top of it. It is found that for this equilibrium configuration, the most unstable modes are the Rayleigh–Taylor type 2D interchange modes. Two-dimensional simulations of the non-linear growth of the interchange modes (Cattaneo and Hughes, 1988Jump To The Next Citation Point) found that the formation of buoyant flux tubes is accompanied by the development of strong vortices whose interactions rapidly destroy the coherence of the flux tubes. In the non-linear regime, the evolution is dominated by vortex interactions which act to prevent the rise of the buoyant magnetic field. Matthews et al. (1995Jump To The Next Citation Point) and Wissink et al. (2000Jump To The Next Citation Point) extend the simulations of Cattaneo and Hughes (1988Jump To The Next Citation Point) to 3D allowing variations in the direction of the initial magnetic field. They discovered that the flux tubes formed by the initial growth of the 2D interchange modes subsequently become unstable to a 3D undulatory motion in the non-linear regime due to the interaction between neighboring counter-rotating vortex tubes, and consequently the flux tubes become arched. Matthews et al. (1995) and Wissink et al. (2000Jump To The Next Citation Point) pointed out that this secondary undulatory instability found in the simulations is of similar nature as the undulatory instability of a pair of counter-rotating (non-magnetic) line vortices investigated by Crow (1970). Wissink et al. (2000) further considered the effect of the Coriolis force due to solar rotation using a local f-plane approximation, and found that the principle effect of the Coriolis force is to suppress the instability. Further 2D simulations have also been carried out by Cattaneo et al. (1990) where they introduced a variation of the magnetic field direction with height into the previously unidirectional magnetic layer of Cattaneo and Hughes (1988). The growth of the interchange instability of such a sheared magnetic layer results in the formation of twisted, buoyant flux tubes which are able to inhibit the development of vortex tubes and rise cohesively.

On the other hand, Fan (2001aJump To The Next Citation Point) has considered a different initial equilibrium state for a horizontal unidirectional magnetic layer, where the density stratification remains unchanged from that of an adiabatically stratified polytrope, but the temperature and the gas pressure are lowered in the magnetic layer to satisfy the hydrostatic condition. For such a neutrally buoyant state with no density change inside the magnetic layer, the 2D interchange instability is completely suppressed and only 3D undulatory modes (with non-zero wavenumbers in the field direction) are unstable. The strong toroidal magnetic field stored in the weakly subadiabatic overshoot region below the bottom of the convection zone is likely to be close to such a neutrally buoyant mechanical equilibrium state (see Section 3.1). Anelastic MHD simulations (Fan, 2001aJump To The Next Citation Point) of the growth of the 3D undulatory instability of this horizontal magnetic layer show formation of significantly arched magnetic flux tubes (see Figure 7Watch/download Movie) whose apices become increasingly buoyant as a result of the diverging flow of plasma from the apices to the troughs.

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Figure 7: mpg-Movie (250 KB) The formation of arched flux tubes as a result of the non-linear growth of the undulatory buoyancy instability of a neutrally buoyant equilibrium magnetic layer perturbed by a localized velocity field. From Fan (2001aJump To The Next Citation Point). The images show the volume rendering of the absolute magnetic field strength |B |. Only one half of the wave length of the undulating flux tubes is shown, and the left and right columns of images show, respectively, the 3D evolution as viewed from two different angles.

The decrease of the field strength B at the apex of the arched flux tube as a function of height is found to follow approximately the relation √ -- B ∕ ρ = constant, or, the Alfven speed being constant, which is a significantly slower decrease of B with height compared to that for the rise of a horizontal flux tube without any field line stretching, for which case B ∕ρ should remain constant. The variation of the apex field strength with height following √ -- B ∕ ρ = constant found in the 3D MHD simulations of the arched flux tubes is in good agreement with the results of the thin flux tube models of emerging Ω-loops (see Moreno-Insertis, 1992Jump To The Next Citation Point) during their rise through the lower half of the solar convective envelope where the stratification is very close to being adiabatic as is assumed in the 3D simulations.

Kersalé et al. (2007Jump To The Next Citation Point) studied the nonlinear 3D evolution of the magnetic buoyancy instability resulting from a smoothly stratified horizontal magnetic field, and with the instability continually driven via the boundary conditions. They considered the case where the prescribed magnetic pressure gradient is such that the equilibrium is unstable to the 3D modes but stable to 2D interchange modes. One important distinction of this work compared to many of the previous studies is that the instability is continually driven through imposing a fixed magnetic pressure gradient at the top and bottom boundaries (Figure 8View Image) which are stress-free and impermeable.

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Figure 8: Horizontal average of the magnetic field B x as a function depth for the initial state (dotted line), at a later time when the instability saturates (dashed line), and in the final steady state (solid line). The magnetic pressure gradient is maintained at the top and bottom boundaries during the non-linear evolution of the magnetic buoyancy instability. From Kersalé et al. (2007Jump To The Next Citation Point). Figure reproduced by permission of the AAS.

The initial growth of the instabilities from a random perturbation results in the formation of arched flux tubes. In the non-linear stage, the system is found to establish a modulated periodic state where discrete flux tube concentrations with field strength significantly stronger than the initial mean field form periodically as modulated traveling waves (see Figures 9View Image and 10View Image). The development of isolated flux tube concentrations results from convergent downflows continually driven by the instability (Figure 10View Image). This result provides an interesting mechanism for the formation of strong active region flux tubes from dynamo generated large scale field at the base of the convection zone.

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Figure 9: Evolution of the kinetic energy density. The system eventually establishes a modulated periodic state with two disparate time scales. From Kersalé et al. (2007Jump To The Next Citation Point). Figure reproduced by permission of the AAS.
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Figure 10: Space-time plots for fixed values of x and z of the magnetic energy density (left), the transverse horizontal (y) velocity (middle), and the vertical velocity (right), with the horizontal axis being the y-axis and vertical axis denoting the time. From Kersalé et al. (2007). Figure reproduced by permission of the AAS.

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