On the edge of an inverse cascade

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 this using the computationally tractable model of two-dimensional (2D) magnetohydrodynamics 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 magnetohydrodynamic 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.

Figure 1

Normalized energy dissipation in the large scales ${\epsilon }_{{}_E}^{-}/{\epsilon }_{{}_E}$ and normalized square vector potential dissipation ${\epsilon }_{{}_A}^{-}/{\epsilon }_{{}_A}$ as a function of $\mu$ for different values of $k_{f}L$ (as indicated by the table in the figure). Inset: A zoom of the same data close to the critical point. The red lines show the fitting curves ${({\mu }_c-\mu )}^{{\gamma }_{{}_E}}$ and ${(\mu -{\mu }_c)}^{{\gamma }_{{}_A}}$.

Figure 2

The energy flux ${\Pi }_{{}_E}\left(k\right)$ (top panel) and the flux of the square vector potential ${\Pi }_{{}_A}\left(k\right)$ (lower panel) for different values of $\mu$ around the critical point and $k_{f}L=16$.

Figure 3

Inset: The instantaneous energy flux ${\Pi }_{{}_E}(t,k)$ with cyan lines for various times $t$ and the time averaged energy flux ${\Pi }_{{}_E}\left(k\right)$ with the dark blue line for $\mu =0.21$ and $k_{f}L=16$. The main figure shows the variance of the energy flux ${\sigma }_{{}_E}={\langle {[{\Pi }_{{}_E}(t,k)-{\Pi }_{{}_E}\left(k\right)]}^2\rangle }^{1/2}$ and the variance of the square vector potential flux ${\sigma }_{{}_A}$ normalized by the large scale dissipations. The fluxes were evaluated at $k$ slightly smaller than the forcing wave number indicated by the vertical dashed line in the inset.

Figure 4

Kinetic energy spectra (top panel) and magnetic energy spectra (bottom panel) for $k_{f}L=64$ and different values of $\mu$ varying from 0.21 to 0.26.