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This paper deals with the spatial structure of zero azimuthal wave number ULF oscillations in a1-D inhomogeneous multi-component plasma when a finite ion gyrofrequency is taken into account. Such oscillations mayoccur in the terrestrial magnetosphere as Pc1-3 waves or in the magnetosphere of the planet Mercury. The wave fieldwas found to have a sharp peak on some magnetic surfaces, an analogy of the Alfv\ue9n (field line) resonance inone-fluid MHD theory. The resonance can only take place for waves with frequencies in the intervalsω<ωch or Ω0<ω< ωcp, where ωch and ωcp are heavy and light ionsgyrofrequencies, and Ω0 is a kind of hybrid frequency. Contrary to ordinary Alfv\ue9n resonance, the waveresonance under consideration takes place even at the zero azimuthal wave number. The radial component of the waveelectric field has a pole-type singularity, while the azimuthal component is finite but has a branching pointsingularity on the resonance surface. The later singularity can disappear at some frequencies. In the regionadjacent to the resonant surface the mode is standing across the magnetic shells

Axisymmetric Alfvén resonances in a multi-component plasma at finite ion gyrofrequency