The helical kink instability of isolated, twisted magnetic flux tubes

To understand the dynamics of twisted active region flux tubes below the solar photosphere, we investigate the linear kink stability of isolated, twisted tubes of magnetic flux. We apply linearized equations of MHD to a cylindrical magnetic equilibrium (screw pinch), but with significant differences from earlier work. The magnetic field vanishes outside a radius r = R where it is confined by the higher pressure of the unmagnetized plasma. The outside boundary of the tube is free to move, displacing the unmagnetized plasma as it does so. We concentrate on equilibria where all field lines have the same helical pitch: B-theta/rB(z) = q = const. The main results are as follows. 1. These equilibria are stable, provided that the field line pitch does not exceed a threshold; q less than or equal to q(cr) for stability. The threshold is q(cr) = (alpha)(1/2), where alpha is the r(2) coefficient in the series expansion of the equilibrium axial magnetic field (B-z) about the tube axis (r = 0): B-z(r) = B-0(1 - alpha r(2) +...). When this criterion is violated, there are unstable eigemnodes, xi proportional to e'((0+kz)). The most unstable of these have a helical pitch k which is near (but not equal to) the field line pitch q. 2. For weakly twisted tubes (qR much less than 1) we derive growth rates and unstable eigenfunctions analytically. For strongly twisted tubes (qR greater than or similar to 1), we find growth rates and unstable eigenfunctions numerically. 3. The maximum growth rate and range of unstable wavenumbers for a strongly twisted tube can be predicted qualitatively by using the analytical results from the weakly twisted case, The maximum growth rate in that case is given by omega(max) = nu(A) R(q(2) - q(cr)(2))/3.83, where nu(A) is the axial Alfven speed. The range of unstable wavenumbers is (-q - Delta k/2) < k < (-q + Delta k/2), where Delta k = 4qR(q(2) - q(cr)(2))(1/2)/3.83. 4. The kink instability we find consists mainly of internal motions. Helical translations of the entire tube are stable. 5. We argue that an emerging, twisted magnetic flux loop will tend to have a uniform q along its length. The increase in the tube radius R as it rises results in a decreasing value of q(cr). This means that the apex of the flux loop will become kink unstable before the rest of the tube. 6. Our results suggest that most twisted flux tubes rising through the convection zone will be stable to kinking. Those few tubes which are kink unstable, and which presumably become knotted or kinked active regions upon emergence, only become kink unstable some time after they have begun rising through the convection zone.