Maximal transport in the Lorenz equations

Souza, Andre N.; Doering, Charles R.

doi:10.1016/j.physleta.2014.10.050

Cited by 19 publications

(46 citation statements)

References 24 publications

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“…In any case the flows constructed there indicate that the background bounds computed here are essentially sharp in the sense that there exist "admissible" flows that transport heat very close to the estimated rate. This is in accord with recent observations that the background method produces bounds that correspond precisely to optimally transporting flows in truncated versions of Rayleigh's model, specifically the Lorenz equations [39] and some distinguished higher order truncations that respect energy and enstrophy conservation in the inviscid limit [40]. We would emphasize that this correspondence seems to be special to Rayleigh's model with stress-free boundaries and, for the background method, in 2D explicitly exploiting the enstrophy balance.…”

Section: Discussionsupporting

confidence: 88%

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

et al. 2015

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A new computational procedure for numerically solving a class of variational problems arising from rigorous upper bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number Nu, as a function of the Rayleigh number Ra, Prandtl number Pr, and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu ∼ Pr α Ra β scaling relation. The computations clearly show that for Ra ≤ 10 10 at fixed L = 2 √ 2, Nu ≤ 0.106Pr 0 Ra 5/12 , which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra

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Section: Discussionsupporting

confidence: 88%

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

et al. 2015

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“…In analogy with Rayleigh-Bénard 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 Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude.…”

mentioning

confidence: 56%

“…Thus, we study the influence of noise in the Lorenz system [9], which is an archetype of deterministic nonlinear dynamics. Moreover, Souza and Doering [1] have recently determined the maximal (upper bounds) transport in the Lorenz equations, thereby providing us with a rigorous test bed for stochastic extensions. In §II we describe the stochastic Lorenz model, followed by the derivation of the stochastic upper bounds in §III.…”

Section: Introductionmentioning

confidence: 99%

Maximal stochastic transport in the Lorenz equations

Agarwal

Wettlaufer

2016

Physics Letters A

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We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-Bénard 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 Souza and Doering [1], 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.

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“…For A = 0 these reduce to the upper bounds of Souza and Doering [14]. However, unlike the deterministic case, the fixed point solutions do not exist so that the optimum is never truly attained.…”

Section: A Upper Bounds Of the Stochastic Lorenz Systemmentioning

confidence: 93%

“…Averaging time derivatives of 1 2 x 2 , 1 2 (y 2 + z 2 ), and −z (see [14] for details) we get the balances…”

Section: Appendix A: Offset Upper Boundsmentioning

confidence: 99%

Circuit bounds on stochastic transport in the Lorenz equations

et al. 2018

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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.

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Maximal transport in the Lorenz equations

Cited by 19 publications

References 24 publications

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

Maximal stochastic transport in the Lorenz equations

Circuit bounds on stochastic transport in the Lorenz equations

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