Simple Systems with Anomalous Dissipation and Energy Cascade

Abstract

We analyze a class of dynamical systems of the type $\dot a_n(t) = c_{n-1} a_{n-1}(t) - c_n a_{n+1}(t) + f_n(t), n \epsilon {{\mathbb{N}}}, a_0=0,$ where f _n (t) is a forcing term with $f_n(t)\not = 0$ only for $n\le n_\star < \infty$ and the coupling coefficients c _n satisfy a condition ensuring the formal conservation of energy $\frac12\sum_n |a_n(t)|^2$ . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f _n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c _n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely $\mathbb{E} |a_n|^2$ scales as $n^{-\alpha}$ as n→∞. Here the exponents α depend on the coupling coefficients c _n and $\mathbb{E}$ denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.