  A three-dimensional MHD model for the propagation and dissipation of Alfven
waves in a coronal loop is developed. The model includes the lower atmospheres
at the two ends of the loop. The waves originate on small spatial scales (less
than 100 km) inside the kilogauss flux elements in the photosphere. The model
describes the nonlinear interactions between Alfven waves using the reduced MHD
approximation. The increase of Alfven speed with height in the chromosphere and
transition region (TR) causes strong wave reflection, which leads to
counter-propagating waves and turbulence in the photospheric and chromospheric
parts of the flux tube. Part of the wave energy is transmitted through the TR
and produces turbulence in the corona. We find that the hot coronal loops
typically found in active regions can be explained in terms of Alfven wave
turbulence, provided the small-scale footpoint motions have velocities of 1-2
km/s and time scales of 60-200 s. The heating rate per unit volume in the
chromosphere is 2 to 3 orders of magnitude larger than that in the corona. We
construct a series of models with different values of the model parameters, and
find that the coronal heating rate increases with coronal field strength and
decreases with loop length. We conclude that coronal loops and the underlying
chromosphere may both be heated by Alfvenic turbulence.
