Resonant absorption of shear Alfv¿n waves is thought to be a likely candidate to explain heating of the solar corona and acceleration of the solar wind. A difficulty with the theory is that the absorption process is slow. It takes a long time ~S1/3&tgr;A, where S≫1 is the magnetic Reynolds number, and &pgr;A a typical Alfv¿n wave propagation time, for wave fronts to steepen sufficiently for absorption to occur. Moreover, heating occurs in a very thin layer whose width scales as S-1/3. A faster absorption mechanism is nonresonant absorption by compressional viscosity, in a curved magnetic field. Heating is nonresonant and is not localized to a narrow layer. The effect could be quite important where the solar coronal magnetic field is strongly curved, in the chromosphere. It could also be important on open field lines in the upper corona, where the compressional viscosity is large. It might imply that a significant part of outgoing Alfv¿n waves are absorbed in the corona. ¿American Geophysical Union 1991