  We use MHD simulations to detect the nonlinear effects of torsional Alfv\'en
wave propagation in a potential magnetic field with exponentially divergent
field lines, embedded in a stratified solar corona. In Paper I we considered
solutions to the linearised governing equations torsional Alfv\'en wave
propagation and showed, using a finite difference solver we developed named
WiggleWave, that in certain scenarios wave damping is stronger than what would
be predicted by our analytic solutions. In this paper we consider whether
damping would be further enhanced by the presence of nonlinear effects. We
begin by deriving the nonlinear governing equations for torsional Alfv\'en wave
propagation and identifying the terms that cause coupling to magnetosonic
perturbations. We then compare simulation outputs from an MHD solver called
Lare3d, which solves the full set of nonlinear MHD equations, to the outputs
from WiggleWave to detect nonlinear effects such as: the excitation of
magnetosonic waves by the Alfv\'en wave, self-interaction of the Alfv\'en wave
through coupling to the induced magnetosonic waves, and the formation of shock
waves higher in the atmosphere caused by the steepening of these compressive
perturbations. We suggest that the presence of these nonlinear effects in the
solar corona would lead to Alfv\'en wave heating that exceeds the expectation
from the phase mixing alone.
