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We address the problem of evaluating the power spectrum of the velocity fieldof the ICM using only information on the plasma density fluctuations, which canbe measured today by Chandra and XMM-Newton observatories. We argue that forrelaxed clusters there is a linear relation between the rms density andvelocity fluctuations across a range of scales, from the largest ones, wheremotions are dominated by buoyancy, down to small, turbulent scales:$(\delta\rho_k/\rho)^2 = \eta_1^2 (V_{1,k}/c_s)^2$, where $\delta\rho_k/\rho$is the spectral amplitude of the density perturbations at wave number $k$,$V_{1,k}^2=V_k^2/3$ is the mean square component of the velocity field, $c_s$is the sound speed, and $\eta_1$ is a dimensionless constant of order unity.Using cosmological simulations of relaxed galaxy clusters, we calibrate thisrelation and find $\eta_1\approx 1 \pm 0.3$. We argue that this value is set atlarge scales by buoyancy physics, while at small scales the density andvelocity power spectra are proportional because the former are a passive scalaradvected by the latter. This opens an interesting possibility to use gasdensity power spectra as a proxy for the velocity power spectra in relaxedclusters, across a wide range of scales.

The relation between gas density and velocity power spectra in galaxy clusters: qualitative treatment and cosmological simulations