  We demonstrate that systems with a parameter-controlled inverse cascade can
exhibit critical behavior for which at the critical value of the control
parameter the inverse cascade stops. In the vicinity of such a critical point
standard phenomenological estimates for the energy balance will fail since the
energy flux towards large length scales becomes zero. We demonstrate these
concepts using the computationally tractable model of two-dimensional
magneto-hydrodynamics in a periodic box. In the absence of any external
magnetic forcing the system reduces to hydrodynamic fluid turbulence with an
inverse energy cascade. In the presence of strong magnetic forcing the system
behaves as 2D magneto-hydrodynamic turbulence with forward energy cascade. As
the amplitude of the magnetic forcing is varied a critical value is met for
which the energy flux towards the large scales becomes zero. Close to this
point the energy flux scales as a power law with the departure from the
critical point and the normalized amplitude of the fluctuations diverges.
Similar behavior is observed for the flux of the square vector potential for
which no inverse flux is observed for weak magnetic forcing, while a finite
inverse flux is observed for magnetic forcing above the critical point. We
conjecture that this behavior is generic for systems of variable inverse
cascade.
