  Insight into the problem of two-dimensional turbulence can be obtained by an
analogy with a heat conduction network. It allows the identification of an
entropy function associated to the enstrophy dissipation and that fluctuates
around a positive (mean) value. While the corresponding enstrophy network is
highly nonlocal, the direction of the enstrophy current follows from the Second
Law of Thermodynamics. An essential parameter is the ratio $T_k = \gamma_k
/(\nu k^2)$ of the intensity of driving $\gamma_k>0$ as a function of
wavenumber $k$, to the dissipation strength $\nu k^2$, where $\nu$ is the
viscosity. The enstrophy current flows from higher to lower values of $T_k$,
similar to a heat current from higher to lower temperature. Our probabilistic
analysis of the enstrophy dissipation and the analogy with heat conduction thus
complements and visualizes the more traditional spectral arguments for the
direct enstrophy cascade. We also show a fluctuation symmetry in the
distribution of the total entropy production which relates the probabilities of
direct and inverse enstrophy cascades.
