Alfvén continuum modes around an X-point with a guide field

Context. Oscillations around X-points are important for the local heating of the coronal plasma by magnetic reconnection and for the generation of quasi-periodic pulsations in solar flares. The processes of phase mixing and resonant absorption are of particular interest in this context. Aims. The aim of this paper is to find the Alfvén continuum eigenmodes around an X-point with a guide field, as these modes are particularly important for phase mixing and resonant absorption. Methods. We studied the linear oscillations in the neighborhood of an X-point with the use of flux coordinates. Reduced equations that describe continuum oscillations were used to determine the Alfvén continuum modes both with and without a guide field in the direction normal to the X-point plane. Results. We determined Alfvén continuum mode solutions in the form of analytical solutions for the special case without a guide field. We also derived numerical solutions in the more general case with a guide field under the assumption of pressureless plasma.