We present the $0.6<z<2.6$ evolution of the ionized gas velocity dispersion in 175 star-forming disk galaxies based on data from the full KMOS$^{rm 3D}$ integral field spectroscopic survey. In a forward-modelling Bayesian framework including instrumental effects and beam-smearing, we fit simultaneously the observed galaxy velocity and velocity dispersion along the kinematic major axis to derive the intrinsic velocity dispersion $sigma_0$. We find a reduction of the average intrinsic velocity dispersion of disk galaxies as a function of cosmic time, from $sigma_0sim45$ km s$^{-1}$ at $zsim2.3$ to $sigma_0sim30$ km s$^{-1}$ at $zsim0.9$. There is substantial intrinsic scatter ($sigma_{sigma_0, {rm int}}approx10$ km s$^{-1}$) around the best-fit $sigma_0-z$-relation beyond what can be accounted for from the typical measurement uncertainties ($deltasigma_0approx12$ km s$^{-1}$), independent of other identifiable galaxy parameters. This potentially suggests a dynamic mechanism such as minor mergers or variation in accretion being responsible for the scatter. Putting our data into the broader literature context, we find that ionized and atomic+molecular velocity dispersions evolve similarly with redshift, with the ionized gas dispersion being $sim10-15$ km s$^{-1}$ higher on average. We investigate the physical driver of the on average elevated velocity dispersions at higher redshift, and find that our galaxies are at most marginally Toomre-stable, suggesting that their turbulent velocities are powered by gravitational instabilities, while stellar feedback as a driver alone is insufficient. This picture is supported through comparison with a state-of-the-art analytical model of galaxy evolution.
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