  We use the Matryoshka run to study the time dependent statistics of
structure-formation driven turbulence in the intracluster medium of a
10$^{15}M_\odot$ galaxy cluster. We investigate the turbulent cascade in the
inner Mpc for both compressional and incompressible velocity components. The
flow maintains approximate conditions of fully developed turbulence, with
departures thereof settling in about an eddy-turnover-time. Turbulent velocity
dispersion remains above $700$ km s$^{-1}$ even at low mass accretion rate,
with the fraction of compressional energy between 10% and 40%. Normalisation
and slope of compressional turbulence is susceptible to large variations on
short time scales, unlike the incompressible counterpart. A major merger occurs
around redshift $z\simeq0$ and is accompanied by a long period of enhanced
turbulence, ascribed to temporal clustering of mass accretion related to
spatial clustering of matter. We test models of stochastic acceleration by
compressional modes for the origin of diffuse radio emission in galaxy
clusters. The turbulence simulation model constrains an important unknown of
this complex problem and brings forth its dependence on the elusive
micro-physics of the intracluster plasma. In particular, the specifics of the
plasma collisionality and the dissipation physics of weak shocks affect the
cascade of compressional modes with strong impact on the acceleration rates. In
this context radio halos emerge as complex phenomena in which a hierarchy of
processes acting on progressively smaller scales are at work. Stochastic
acceleration by compressional modes implies statistical correlation of radio
power and spectral index with merging cores distance, both testable in
principle with radio surveys.
