The linear stability analysis of the above equilibrium horizontal magnetic layer (Newcomb, 1961) showed that the necessary and sufficient condition for the onset of the general 3D instability with non-zero wavenumbers (, ) in both horizontal directions parallel and perpendicular to the magnetic field is that
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 and , where , , and 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, 1970; Acheson, 1979; Schmitt and Rosner, 1983). The diffusive effects are shown to alter the stability criteria of Equations (24) and (25) by reducing the term by a factor of (see Acheson, 1979). 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, 1988; Cattaneo et al., 1990; Matthews et al., 1995; Wissink et al., 2000; Fan, 2001a).
Cattaneo and Hughes (1988), Matthews et al. (1995), and Wissink et al. (2000) 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, 1988) 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. (1995) and Wissink et al. (2000) extend the simulations of Cattaneo and Hughes (1988) 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. (2000) 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 (2001a) 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, 2001a) of the growth of the 3D undulatory instability of this horizontal magnetic layer show formation of significantly arched magnetic flux tubes (see Figure 7) whose apices become increasingly buoyant as a result of the diverging flow of plasma from the apices to the troughs.
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The decrease of the field strength at the apex of the arched flux tube as a function of height is found to follow approximately the relation , or, the Alfven speed being constant, which is a significantly slower decrease of with height compared to that for the rise of a horizontal flux tube without any field line stretching, for which case should remain constant. The variation of the apex field strength with height following 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 -loops (see Moreno-Insertis, 1992) 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. (2007) 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 8) which are stress-free and impermeable.
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 9 and 10). The development of isolated flux tube concentrations results from convergent downflows continually driven by the instability (Figure 10). 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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