Leaky and non-leaky kink oscillations of magnetic flux tubes

Abstract

We study the normal and leaky kink oscillations of a thin magnetic tube embedded in a surrounding magnetized plasma. We adopt the cold plasma (zero plasma $\beta$) approximation, appropriate for typical solar coronal conditions. Using the Laplace transform, we solve the initial value problem to determine the motions of the flux tube. The asymptotic description of this motion is determined by the properties of a dispersion function $D(\omega)$, an infinitely valued function of the complex frequency $\omega$. Two cases arise. When the plasma density $\rho_{\rm i}$ inside the tube exceeds the density $\rho_{\rm e}$ of the surrounding plasma, the asymptotic behaviour is given by the normal mode that is associated with two zeros of $D(\omega)$. These zeros are real and symmetric with respect to the imaginary axis, and are situated on the principal sheet of the Riemann surface of $D(\omega)$. The asymptotics given by the normal mode is valid for times much longer than the period of tube oscillation, and is homogeneous with respect to the radial distance $r$. In the second case, where $\rho_{\rm i} < \rho_{\rm e}$, there are no zeros of $D(\omega)$ on the principal sheet. The asymptotics of the solution of the initial value problem is then given by a leaky mode of the tube kink oscillations. This mode is associated with two zeros of $D(\omega)$ on two different, non-principal, sheets of the Riemann surface of $D(\omega)$ attached to the principal sheet. The asymptotics given by the leaky mode is intermediate, i.e. valid for times much longer than the period of oscillations but smaller than (or of the order of) the damping time due to energy leakage. The asymptotics given by the leaky mode is inhomogeneous with respect to $r$. We also consider other solutions associated with the zeros of $D(\omega)$ on non-principal sheets, and argue that in the coronal loop case where $\rho_{\rm i} > \rho_{\rm e}$ these solutions have no physical meaning. In the opposite case, where $\rho_{\rm i} < \rho_{\rm e}$, only the two solutions associated with the two zeros of $D(\omega)$ lying on two different non-principal sheets attached to the principal sheet are physically meaningful; they determine the leaky mode. All other solutions of this type are devoid of physical meaning. On the basis of our results, we conclude that there is no radial wave energy leakage from a dense coronal loop oscillating in the kink mode; other mechanisms must be sought to explain the observed oscillation damping.