Circuit Bounds on Stochastic Transport in the Lorenz Equations

TL;DR

This paper introduces an experimental circuit model of the Lorenz equations to study stochastic transport bounds in turbulent convection, revealing bifurcation phenomena and reentrant behavior of transport with noise.

Contribution

It presents a novel experimental circuit analogue of the Lorenz equations to investigate stochastic bounds and bifurcations in turbulent convection models.

Findings

Reentrant behavior of transport as a function of noise amplitude.

Bifurcation phenomena influenced by offsets in the circuit.

Comparison of circuit output with theoretical stochastic Lorenz bounds.

Abstract

In turbulent Rayleigh-B\'enard convection one seeks the relationship between the heat transport, captured by the Nusselt number, and the temperature drop across the convecting layer, captured by Rayleigh number. In experiments, one measures the Nusselt number for a given Rayleigh number, and the question of how close that value is to the maximal transport is a key prediction of variational fluid mechanics in the form of an upper bound. The Lorenz equations have traditionally been studied as a simplified model of turbulent Rayleigh-B\'enard convection, and hence it is natural to investigate their upper bounds, which has previously been done numerically and analytically, but they are not as easily accessible in an experimental context. Here we describe a specially built circuit that is the experimental analogue of the Lorenz equations and compare its output to the recently determined upper bounds of the stochastic Lorenz equations \cite{AWSUB:2016}. The circuit is substantially more efficient than computational solutions, and hence we can more easily examine the system. Because of offsets that appear naturally in the circuit, we are motivated to study unique bifurcation phenomena that arise as a result. Namely, for a given Rayleigh number, we find a reentrant behavior of the transport on noise amplitude and this varies with Rayleigh number passing from the homoclinic to the Hopf bifurcation.