Simple Systems with Anomalous Dissipation and Energy Cascade

TL;DR

This paper studies linear shell models with stochastic forcing that, despite formal energy conservation, exhibit anomalous dissipation and energy cascades, providing insights into turbulence phenomena through exactly solvable models.

Contribution

It demonstrates that simple, formally conservative shell models can support dissipative solutions and steady states, illustrating mechanisms of anomalous dissipation and energy cascade.

Findings

Models support dissipative solutions despite formal conservation.

Energy spectra follow a power-law decay with mode index.

Mechanism of energy cascade resembles turbulence dynamics.

Abstract

We analyze a class of linear shell models subject to stochastic forcing in finitely many degrees of freedom. The unforced systems considered formally conserve energy. 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 is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients. The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes ($a_n$) with higher $n$; this is responsible for solutions with interesting energy spectra, namely $\EE |a_n|^2$ scales as $n^{-\alpha}$ as $n\to\infty$. Here the exponents $\alpha$ depend on the coupling coefficients $c_n$ and $\EE$ 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.