Turbulence and conduction can dramatically affect the evolution of baryons in the universe ; current constraints are however rare and uncertain .
Using 3D high-resolution hydrodynamic simulations , tracking both electrons and ions , we study the effects of turbulence and conduction in the hot intracluster medium .
We show how the power spectrum of the gas density perturbations ( \delta = \delta \rho / \rho ) can accurately constrain both processes .
The characteristic amplitude of density perturbations is linearly related to the strength of turbulence , i.e .
the 3D Mach number , as A ( k ) _ { \delta, max } = c M , where c \simeq 0.25 for injection scale of 500 kpc .
The slope of A _ { \delta } ( k ) in turn reflects the level of diffusion , dominated by conduction .
In a non-conductive medium , subsonic stirring motions advect density with a similar nearly Kolmogorov cascade , E _ { \delta } ( k ) \propto k ^ { -5 / 3 } .
Increasing conduction ( parametrized via the magnetic suppression f = 10 ^ { -3 } \rightarrow 1 ) progressively steepens the spectrum towards the Burgers-like regime , E _ { \delta } ( k ) \propto k ^ { -2 } .
The slope is only weakly dependent on M .
The turbulent Prandtl number defines the dynamic similarity of the flow ; at scales where P _ { t } \equiv t _ { cond } / t _ { turb } < 100 , the power spectrum develops a significant decay , i.e .
conduction stifles turbulent regeneration .
The transition is gentle for highly suppressed conduction , f \leq 10 ^ { -3 } , while sharp in the opposite regime .
For strong conductivity ( f \geq 0.1 ) , P _ { t } \sim 100 occurs on spatial scales larger than the injection scale , globally damping density perturbations by a factor of 2 - 4 , from large to small scales .
The velocity spectrum is instead not much affected by conduction .
The f \geq 0.1 regime should also affect the appearance of X-ray images , in which Kelvin-Helmholtz and Rayleigh-Taylor rolls and filaments are washed out .
In a stratified system , perturbations are characterized by a mixture of modes : weak/strong turbulence induces higher isobaric/adiabatic fluctuations , while conduction forces both modes towards the intermediate isothermal regime .
We provide a general analytic fit which is applied to new deep Chandra observations of Coma cluster .
The observed spectrum is best consistent with strongly suppressed effective isotropic conduction , f \simeq 10 ^ { -3 } , and mild subsonic turbulence , M \simeq 0.45 ( assuming injection scale at \sim 250 kpc ) .
The latter implies E _ { turb } \simeq 0.11 E _ { th } , in agreement with cosmological simulations and line-broadening observations .
The low conductivity corroborates the survival of sharp features in the ICM ( cold fronts , filaments , bubbles ) , and indicates that cooling flows may not be balanced by conduction .