Nonlinear dissipative spherical Alfvén waves in solar coronal holes

The weakly nonlinear dynamics of linearly polarized, spherical Alfvén waves in coronal holes is investigated. An evolutionary equation, combining the effects of spherical stratification, nonlinear steepening and dissipation due to shear viscosity is derived. The equation is a spherical analog of the scalar Cohen-Kulsrud-Burgers equation. Three main stages of the wave evolution are distinguished: geometrical amplification, wave breaking and enhanced dissipation. The wave dissipation is dramatically increased by the nonlinear transfer of energy to smaller scales. The scenario of the nonlinear dissipation is practically independent of viscosity. The dissipation rate is stronger for highest amplitudes, and depends weakly on the wave period and the temperature of the atmosphere. Waves with periods less than 300 s and initial amplitudes about 2-3% of the Alfvén speed at the base of the corona are subject to the nonlinear steepening and dissipation in less than 10 solar radii. For the Alfvén waves with amplitudes less than 25 km s^-1 at the base of the corona, the maximum amplitude of up to 200 km s^-1 is reached at several solar radii. The nonlinear distortion of the wave shape is accompanied by the generation of longitudinal motions and density perturbations.