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Recent numerical simulations have demonstrated that transverse coronal looposcillations are susceptible to the Kelvin-Helmholtz (KH) instability due tothe counter-streaming motions at the loop boundary. We present the firstanalytical model of this phenomenon. The region at the loop boundary where theshearing motions are greatest is treated as a straight interface separatingtime-periodic counter-streaming flows. In order to consider a twisted tube, themagnetic field at one side of the interface is inclined. We show that theevolution of the displacement at the interface is governed by Mathieu'sequation and we use this equation to study the stability of the interface. Weprove that the interface is always unstable, and that, under certainconditions, the magnetic shear may reduce the instability growth rate. Theresult, that the magnetic shear cannot stabilise the interface, explains thenumerically found fact that the magnetic twist does not prevent the onset ofthe KH instability at the boundary of an oscillating magnetic tube. We alsointroduce the notion of the loop $\sigma$-stability. We say that atransversally oscillating loop is $\sigma$-stable if the KH instability growthtime is larger than the damping time of the kink oscillation. We show that evenrelatively weakly twisted loops are $\sigma$-stable.

An Analytical Model of the Kelvin-Helmholtz Instability of Transverse Coronal Loop Oscillations