Empirical constraints on the turbulence in QSO host nebulae from velocity structure function measurements

We present the first empirical constraints on the turbulent velocity field of the diffuse circumgalactic medium around four luminous QSOs at $z\!\approx\!0.5$--1.1. Spatially extended nebulae of $\approx\!50$--100 physical kpc in diameter centered on the QSOs are revealed in [OII]$\lambda\lambda\,3727,3729$ and/or [OIII]$\lambda\,5008$ emission lines in integral field spectroscopic observations obtained using MUSE on the VLT. We measure the second- and third-order velocity structure functions (VSFs) over a range of scales, from $\lesssim\!5$ kpc to $\approx\!20$--50 kpc, to quantify the turbulent energy transfer between different scales in these nebulae. While no constraints on the energy injection and dissipation scales can be obtained from the current data, we show that robust constraints on the power-law slope of the VSFs can be determined after accounting for the effects of atmospheric seeing, spatial smoothing, and large-scale bulk flows. Out of the four QSO nebulae studied, one exhibits VSFs in spectacular agreement with the Kolmogorov law, expected for isotropic, homogeneous, and incompressible turbulent flows. The other three fields exhibit a shallower decline in the VSFs from large to small scales. However, with a limited dynamic range in the spatial scales in seeing-limited data, no constraints can be obtained for the VSF slopes of these three nebulae. For the QSO nebula consistent with the Kolmogorov law, we determine a turbulence energy cascade rate of $\approx\!0.2$ cm$^{2}$ s$^{-3}$. We discuss the implication of the observed VSFs in the context of QSO feeding and feedback in the circumgalactic medium.