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, 1982a,b)
where is the wavenumber along the tube of the undulatory perturbation, is the local pressure scale height, is the ratio of the plasma pressure divided by the magnetic pressure of the flux tube, is the superadiabaticity, and is the ratio of the specific heats. If all values of are allowed, then the condition for the presence of instability is Note that is a singular limit. For perturbations with which do not involve bending the field lines, the condition for instability becomes (Spruit and van Ballegooijen, 1982a) which is a significantly more stringent condition than (18), even more stringent than the convective instability for a fieldfree fluid (). Thus the undulatory instability (with ) is of a very different nature and is easier to develop than the instability associated with uniform upanddown motions of the entire flux tube. The undulatory instability can develop even in a convectively stable stratification with as long as the field strength of the flux tube is sufficiently strong (i.e. is of sufficiently small amplitude) such that is smaller than . In the regime of where only the undulatory modes with 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, 1982a).Including the curvature effect of spherical geometry, but still ignoring solar rotation, Spruit and van Ballegooijen (1982a,b) 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 (FerrizMas and Schüssler, 1993). For motions within the equatorial plane, the conditions for instabilities are (Spruit and van Ballegooijen, 1982a,b)
where is the ratio of the pressure scale height over the radius of the bottom of the solar convection zone, (having integer values ) denotes the azimuthal order of the undulatory mode of the closed toroidal flux ring, i.e. the wavenumber , is a parameter that describes the variation of the gravitational acceleration: . Near the base of the solar convection zone, , . Thus conditions (20) show that it is possible for modes to become unstable in the weakly subadiabatic overshoot region, and that the instabilities of modes require less stringent conditions than the instability of mode. Since Equation (20) 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 FerrizMas and Schüssler (1993, 1995). For general nonaxisymmetric perturbations, a sixthorder 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 plane (with being the magnetic field strength of the flux ring and being the equilibrium latitude), under the conditions representative of the overshoot layer at the base of the solar convection zone are shown in Figure 6 (from Caligari et al., 1995).

Thin flux tube simulations of the nonlinear growth of the nonaxisymmetric instabilities of initially toroidal flux tubes and the emergence of shaped flux loops through the solar convective envelope will be discussed in Section 5.1.
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