  We investigate quantum turbulence in a two-dimensional trapped supersolid and
demonstrate that both the wave and vortex turbulence involve triple rather than
dual cascades, as in a superfluid. Because of the presence of a second gapless
mode associated with translation symmetry breaking, a new $k^{-13/3}$ scaling
law is predicted to occur in the wave turbulence. Simultaneous fast
vortex-antivortex creation and annihilation in the interior of the oscillating
supersolid results in a $k^{-1}$ scaling law in the vortex turbulence.
Numerical simulations based on the Gross-Pitaevskii equation confirmed the
predictions.
