Circuit bounds on stochastic transport in the Lorenz equations

Abstract

In turbulent Rayleigh-Bénard 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-Bénard 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 [1]. 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.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

“…Hence, the leading-order analysis is unchanged, which means that periodic orbits bifurcate/persist as for b = 0, although their symmetry properties may broken. In particular, this applies to the Lorenz models with offsets from Weady (2018), Palmer (1998) for which one can also show that the transport is maximized in an equilibrium (Ovsyannikov 2022). Nonzero B generally requires j ≥ 1 for a regular limit in which the right-hand side of the equation for ω becomes independent of ω and vanishes for j > 1.…”

Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

“…Hence, the leading order analysis is unchanged, which means that periodic orbits bifurcate/persist as for b = 0, although their symmetry properties may be broken. In particular, this applies to the Lorenz models with offsets from [27,17] for which one can also show that the transport is maximized in an equilibrium [16].…”

“…)K) = 0 as the remaining equation. Moving the term involving σ to the right hand side, using the definition λ = (σ + 1)/(β + 2), and after some arithmetic one obtains (27) 2λ…”

“…Together with E(1) = 0 and K → ∞ as k 1 → 1 the right-hand side of equation ( 24) is monotonically decreasing from ∞ to 1/3 when k 1 increases from k * to 1. This range corresponds to the values λ > 2/3 on the left-hand side of (27). Also, every k 1 ∈ (k * , 1) gives a unique positive value for B via formula (26).…”

“…for coefficients ĉ0 , ĉ1 , ĉ2 , ĉ3 , ĉ4 depending only on λ, β. Finally, applying (27) to eliminate λ yields…”