  We investigate the statistical recovery of missing physics and turbulent
phenomena in fluid flows using generative machine learning. Here we develop a
two-stage super-resolution method using spectral filtering to restore the
high-wavenumber components of a Kolmogorov flow. We include a rigorous
examination of generated samples through the lens of statistical turbulence. By
extending the prior methods to a combined super-resolution and conditional
high-wavenumber generation, we demonstrate turbulence recovery on a $8\times$
upsampling task, effectively doubling the range of recovered wavenumbers.
