How vortices and shocks provide for a flux loop in two-dimensional
  compressible turbulence
  Large-scale turbulence in fluid layers and other quasi-two-dimensional
compressible systems consists of planar vortices and waves. Separately, wave
turbulence usually produces a direct energy cascade, while solenoidal planar
turbulence transports energy to large scales by an inverse cascade. Here, we
consider turbulence at finite Mach numbers when the interaction between
acoustic waves and vortices is substantial. We employ solenoidal pumping at
intermediate scales and show how both direct and inverse energy cascades are
formed starting from the pumping scale. We show that there is an inverse
cascade of kinetic energy up to a scale $\ell$, where a typical velocity
reaches the speed of sound; this creates shock waves, which provide for a
compensating direct cascade. When the system size is less than $\ell$, the
steady state contains a system-size pair of long-living condensate vortices
connected by a system of shocks. Thus turbulence in fluid layers processes
energy via a loop: Most energy first goes to large scales via vortices and is
then transported by waves to small-scale dissipation.
