Maximal Stochastic Transport in the Lorenz Equations

TL;DR

This paper extends the background method to calculate stochastic upper bounds for heat transport in the Lorenz equations, revealing how noise influences bounds differently in chaotic and non-chaotic regimes and highlighting the coupling with unstable periodic orbits.

Contribution

It introduces a novel extension of the background method to stochastic Lorenz equations and analyzes the complex effects of noise on transport bounds across different dynamical regimes.

Findings

Stochastic upper bounds are larger than deterministic ones.

Bounds increase monotonically with noise below chaos transition.

In chaotic regimes, bounds vary non-monotonically with noise amplitude.

Abstract

We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-B\'enard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of \citet{Doering15}, but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.