  Recent numerical simulations have demonstrated that transverse coronal loop
oscillations are susceptible to the Kelvin-Helmholtz (KH) instability due to
the counter-streaming motions at the loop boundary. We present the first
analytical model of this phenomenon. The region at the loop boundary where the
shearing motions are greatest is treated as a straight interface separating
time-periodic counter-streaming flows. In order to consider a twisted tube, the
magnetic field at one side of the interface is inclined. We show that the
evolution of the displacement at the interface is governed by Mathieu's
equation and we use this equation to study the stability of the interface. We
prove that the interface is always unstable, and that, under certain
conditions, the magnetic shear may reduce the instability growth rate. The
result, that the magnetic shear cannot stabilise the interface, explains the
numerically found fact that the magnetic twist does not prevent the onset of
the KH instability at the boundary of an oscillating magnetic tube. We also
introduce the notion of the loop $\sigma$-stability. We say that a
transversally oscillating loop is $\sigma$-stable if the KH instability growth
time is larger than the damping time of the kink oscillation. We show that even
relatively weakly twisted loops are $\sigma$-stable.

Solar and Stellar Astrophysics