The saturation of the VSI in protoplanetary disks via parametric instability

The vertical shear instability (VSI) is a robust and potentially important phenomenon in irradiated protoplanetary disks (PPDs), yet the mechanism by which it saturates remains poorly understood. Global simulations suggest that the non-linear evolution of the VSI is dominated by radially propagating inertial wavetrains (called `body modes'), but these are known to be susceptible to a parametric instability. In this paper, we propose that the global VSI saturates via this secondary instability, which initiates a redistribution of energy from the large scales to smaller-scale inertial waves, and finally into a turbulent cascade. We present an analytic theory of the instability in a simple idealised model that captures the main physical and mathematical details of the problem. In addition, we conduct numerical simulations with the SNOOPY code to consolidate the theory. Once the parametric instability prevails, the VSI is likely far more disordered and incoherent than current global simulations suggest. We also argue that it is challenging to capture parametric instability in global simulations unless the radial resolution is very fine, possibly $\sim 300$ grid cells per scale height in radius.