So long Kolmogorov: the forward and backward turbulence cascades in a supernovae-driven, multiphase interstellar medium

The interstellar medium (ISM) of disk galaxies is turbulent, and yet the fundamental nature of ISM turbulence, the energy cascade, is not understood in detail. In this study, we use high-resolution simulations of a hydrodynamical, gravitationally stratified, supernova (SNe)-driven, multiphase ISM to probe the nature of a galactic turbulence cascade. Through the use of velocity flux transfer functions split into interactions between compressible $\mathbf{u}_c$ and incompressible $\mathbf{u}_s$ modes, we show that there exists a large-to-small-scale cascade in both $\mathbf{u}_c$ and $\mathbf{u}_s$ when mediated by an additional $\mathbf{u}_s$ mode. But the $\mathbf{u}_s$ cascade is highly non-local. Moreover, there is a $\mathbf{u}_c$ mediated component of the $\mathbf{u}_s$ cascade that proceeds in the opposite direction -- an inverse cascade from small-to-large scales. The cascade feeds flux into scales well beyond the scale height, energizing the winds and fueling the direct cascades. Both the strongly non-local and the inverse $\mathbf{u}_s$ cascades happen on scales that have a power law $\mathbf{u}_s$ energy spectrum, highlighting how degenerate the spectrum is to the true underlying physical processes. We directly show that the inverse cascade comes from $\mathbf{u}_s$ modes interacting with expanding SNe remnants (SNRs) and that $\mathbf{u}_s$ modes are generated to leading order via baroclinic, highly corrugated cooling layers between warm $(T\lesssim 10^4\,\rm{K})$ and hot $(T\gg10^4\,\rm{K})$ gas in these SNRs. Finally, we outline a complete phenomenology for SNe-driven turbulence in a galactic disk, estimate a $10^{-16}\,\rm{G}$ Biermann field generated from SNR cooling layers, and highlight the strong deviations that SNe-driven turbulence has from the conventional Kolmogorov model.