  This article establishes explicit non-asymptotic ergodic bounds in the
renormalized Wasserstein-Kantorovich-Rubinstein (WKR) distance for a viscous
energy shell lattice model of turbulence with random energy injection. The
system under consideration is driven either by a Brownian motion, a symmetric
$\alpha$-stable L\'evy process, a stationary Gaussian or $\alpha$-stable
Ornstein-Uhlenbeck process, or by a general L\'evy process with second moments.
The obtained non-asymptotic bounds establish asymptotically abrupt
thermalization. The analysis is based on the explicit representation of the
solution of the system in terms of convolutions of Bessel functions.
