  In the framework of compressional magnetohydrodynamics (MHD), we numerically
studied the commonly accepted presumption that the Alfv\'enic turbulence is
generated by the collisions between counter-propagating Alfv\'en waves (AWs).
In the conditions typical for the low-beta solar corona and inner solar wind,
we launched two counter-propagating AWs in the three-dimensional simulation box
and analyzed polarization and spectral properties of perturbations generated
before and after AW collisions. The observed post-collisional perturbations
have different polarizations and smaller cross-field scales than the original
waves, which supports theoretical scenarios with direct turbulent cascades.
However, contrary to theoretical expectations, the spectral transport is
strongly suppressed at the scales satisfying the classic critical balance of
incompressional MHD. Instead, a modified critical balance can be established by
colliding AWs with significantly shorter perpendicular scales. We discuss
consequences of these effects for the turbulence dynamics and turbulent heating
of compressional plasmas. In particular, solar coronal loops can be heated by
the strong turbulent cascade if the characteristic widths of the loop
substructures are more than ten times smaller than the loop width. The revealed
new properties of AW collisions have to be incorporated in the theoretical
models of AW turbulence and related applications.
