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 $zsimeq0$ 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.
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