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We study the statistical properties of turbulence driven by structureformation in a massive merging galaxy cluster at z=0. The development ofturbulence is ensured as the largest eddy turnover time is much shorter thanthe Hubble time independent of mass and redshift. We achieve a large dynamicrange of spatial scales through a novel Eulerian refinement strategy where thecluster volume is refined with progressively finer uniform nested grids duringgravitational collapse. This provides an unprecedented resolution of 7.3 h^{-1}kpc across the virial volume. The probability density functions of variousvelocity derived quantities exhibit the features characteristic of fullydeveloped compressible turbulence observed in dedicated periodic-boxsimulations. Shocks generate only 60% of the total vorticity within \rvir/3region and 40% beyond that. We compute second and third order, longitudinal andtransverse, structure functions for both solenoidal and compressionalcomponents, in the cluster core, virial region and beyond. The structurefunctions exhibit a well defined inertial range. The injection scale iscomparable to the virial radius but increases towards the outskirts. Within\rvir/3, the spectral slope of the solenoidal component is close toKolmogorov's, but for the compressional component is substantially steeper andclose to Burgers'; the flow is mostly solenoidal and statistically rigorouslyconsistent with fully developed, homogeneous and isotropic turbulence. Smallscale anisotropy appears due to numerical artifact. Towards the virial region,the flow becomes compressional, the structure functions flatter and modestgenuine anisotropy appear. In comparison, mesh adaptivity based on Lagrangianrefinement and the same finest resolution, leads to lack of turbulent power onsmall scale and excess thereof on large scales, and unreliable density weightedstructure functions.

The Matryoshka Run: Eulerian Refinement Strategy to Study Statistics of Turbulence in Virialized Cosmic Structures