The absorption of power is studied with fluid and gyrokinetic plasma models, when Alfven resonances provide for a weak damping in a partially standing wave field. Examples chosen in slab and toroidal geometry show that the fluid predictions based on resonance absorption are generally very different from the Landau damping of mode-converted slow waves. They, in particular, suggest that the continuum damping of toroidal Alfven eigenmodes (TAE) and the power deposition profiles obtained in the ion-cyclotron range of frequencies (ICRF) using fluid plasma models can be very misleading. (C) 1998 American Institute of Physics. [S1070-664X(98)01111-2].