insu-00446579
https://insu.hal.science/insu-00446579
https://insu.hal.science/insu-00446579/document
https://insu.hal.science/insu-00446579/file/10.1103%40PhysRevE.79.066304.pdf
arxiv:0801.1623
doi:10.1103/PhysRevE.79.066304
[SDE] Sciences De l'Environnement
[INSU] INSU - Institut National des Sciences de l'Univers
[UNIV-SAVOIE] Université Savoie Mont Blanc
[UGA] HAL Grenoble Alpes
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-GRENOBLE1] Université Joseph Fourier - Grenoble I
[INPG] Institut polytechnique de Grenoble
[IRSTEA] IRSTEA - Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (<b>anciennement Cemagref</b>)
[GIP-BE] GIP Bretagne Environnement
[IFSTTAR] Ifsttar
[AGREENIUM] Archive ouverte en agrobiosciences
[INRAE] Institut National de Recherche en Agriculture, Alimentation et Environnement
[USMB-COMUE] Université Savoie Mont Blanc - UGA [* - 2019]
[UNIV-EIFFEL] Université Gustave Eiffel
[IFSTTAR-UNIVEIFFEL] Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux
Bound of dissipation on a plane Couette dynamo
Alboussiere, Thierry
[SDU.STU.GP] Sciences of the Universe [physics]/Earth Sciences/Geophysics [physics.geo-ph]
[PHYS.PHYS.PHYS-GEO-PH] Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph]
[SDE.MCG] Environmental Sciences/Global Changes
ART
Variational turbulence is among the few approaches providing rigorous results in turbulence. In addition, it addresses a question of direct practical interest, namely, the rate of energy dissipation. Unfortunately, only an upper bound is obtained as a larger functional space than the space of solutions to the Navier-Stokes equations is searched. Yet, in some cases, this upper bound is in good agreement with experimental results in terms of order of magnitude and power law of the imposed Reynolds number. In this paper, the variational approach to turbulence is extended to the case of dynamo action and an upper bound is obtained for the global dissipation rate (viscous and Ohmic). A simple plane Couette flow is investigated. For low magnetic Prandtl number Pm fluids, the upper bound of energy dissipation is that of classical turbulence (i.e., proportional to the cubic power of the shear velocity) for magnetic Reynolds numbers below Pm−1 and follows a steeper evolution for magnetic Reynolds numbers above Pm−1 (i.e., proportional to the shear velocity to the power of 4) in the case of electrically insulating walls. However, the effect of wall conductance is crucial: for a given value of wall conductance, there is a value for the magnetic Reynolds number above which energy dissipation cannot be bounded. This limiting magnetic Reynolds number is inversely proportional to the square root of the conductance of the wall. Implications in terms of energy dissipation in experimental and natural dynamos are discussed.
2009-06-05
2022-05-06
en
Physical Review E : Statistical, Nonlinear, and Soft Matter Physics
American Physical Society