Vertical motions in an intense magnetic flux tube

Abstract

The nature of convective instability in a slender magnetic flux tube is explored. A sufficient condition for stability is derived for the case of anarbitrary temperature profile in the external medium. The discussion allows for the possibility of a temperature difference between the interior and exterior of the tube. Special cases of our sufficiency condition reduce to Schwarzschild's criterion and its generalisation by Gough and Tayler (1966).

The distribution of stable and unstable eigenvalues, for the particular case of alinear temperature profile, is discussed in detail.

For a tube ofinfinite depth, with a uniform temperature gradientΛ′ ₀ inside the tube equal to that in the ambient medium, anecessary and sufficient condition for convective stability to occur inside the tube is

$$\frac{4}{{\gamma (\Lambda _0^{ \prime } )^2 }}\left( {\frac{{\gamma - 1}}{\gamma } + \Lambda _0^{ \prime } } \right)\left( {\frac{\gamma }{2} + \frac{{c_0^2 }}{{\upsilon _A^2 }}} \right) + \left( {1 + \frac{1}{{2\Lambda _0^{ \prime } }}} \right)^2 > 0,$$

wherec ₀ andυ _A are the sound and Alfvén speeds inside the tube, andγ is the ratio of specific heats. Thestable modes form acontinuous spectrum; the unstable modes arediscrete and infinite in number.

In a tube offinite depthd ^* with a linear temperature profile, a necessary and sufficient condition for convective stability is derived. There exists a critical depthd such that tubes of depthd are stable if and only ifd <d ^*. In a finite tube, only discrete eigenvalues occur.

The critical depthd ^* is determined for a wide range of conditions, and the results applied to the Sun. Under the assumptions of the present model, intense flux tubes are convectively stable if sufficiently shallow (with depths 1−2 × 10³ km or less). Tubes that extend deeper into the solar convection zone are potentially (convectively) unstable, but may be stabilised for sufficiently strong magnetic fields (typically greater than about a kilogauss). The observed downdraftsinside intense flux tubes, if a manifestation of convective instability, are thus likely to be atransient phenomenon in which the field inside the tube is further intensified until hydrostatic equilibrium obtains.Convective instability in a flux tube is thus a possible means of achieving kilogauss field strengths.