  Context: In the context of the solar coronal heating problem, one possible
explanation for the high coronal temperature is the release of energy by
magnetohydrodynamic (MHD) waves. The energy transfer is believed to be
possible, among others, by the development of the Kelvin-Helmholtz instability
(KHI) in coronal loops. Aims: Our aim is to determine if standing slow waves in
solar atmospheric structures such as coronal loops, and also prominence
threads, sunspots, and pores, can trigger the KHI due to the oscillating shear
flow at the structure's boundary. Methods: We used linearized nonstationary MHD
to work out an analytical model in a cartesian reference frame. The model
describes a compressible plasma near a discontinuous interface separating two
regions of homogeneous plasma, each harboring an oscillating velocity field
with a constant amplitude which is parallel to the background magnetic field
and aligned with the interface. The obtained analytical results were then used
to determine the stability of said interface, both in coronal and photospheric
conditions. Results: We find that the stability of the interface is determined
by a Mathieu equation. In function of the parameters of this equation, the
interface can either be stable or unstable. For coronal as well as photospheric
conditions, we find that the interface is stable with respect to the KHI.
Theoretically, it can, however, be unstable with respect to a parametric
resonance instability, although it seems physically unlikely. We conclude that,
in this simplified setup, a standing slow wave does not trigger the KHI without
the involvement of additional physical processes.
