  We predict the emergence of turbulent scaling in the quench dynamics of the
two-dimensional Heisenberg model for a wide range of initial conditions and
model parameters. In the isotropic Heisenberg model, we find that the spin-spin
correlation function exhibits universal scaling consistent with a turbulent
energy cascade. When the spin rotational symmetry is broken by an easy-plane
exchange anisotropy, we find a dual cascade of energy and an emergent conserved
charge associated to transverse magnetization fluctuations. The scaling
exponents are estimated analytically and agree with numerical simulations using
phase space methods. We also define the space of initial conditions (as a
function of energy, magnetization, and spin number $S$) that lead to a
turbulent cascade. The universal character of the cascade, insensitive to
microscopic details or initial conditions, suggests that turbulence in spin
systems can be broadly realized in cold atom and solid-state experiments.

Statistical Mechanics