  Numerical simulations have revealed a new type of turbulence of
unidirectional waves in a plasma that is perpendicularly structured (Magyar et
al. 2017), named uniturbulence. For this new type of turbulence, the transverse
structuring modifies the upward propagating wave to have both Elsasser
variables, leading to the well-known perpendicular cascade. In this paper, we
study an analytical description of the non-linear evolution of kink waves in a
cylindrical flux tube, which are prone to uniturbulence. We show that they lead
to a non-linear cascade for both propagating and standing waves. We calculate
explicit expressions for the wave pressure and energy cascade rate. The
computed damping rate {\tau}/P depends on the density contrast of the flux tube
and the background plasma and is inversely proportional to the amplitude of the
kink wave. The dependence on the density contrast shows that it plays a role
especially in the lower solar corona. These expressions may be added in Alfven
wave driven models of the solar atmosphere (such as AWSOM, van der Holst et al.
2014), modifying it to UAWSOM (Uniturbulence and Alfven Wave Solar Model).
