https://insu.hal.science/insu-00446579Alboussiere, ThierryThierryAlboussiereLGIT - Laboratoire de Géophysique Interne et Tectonophysique - OSUG - Observatoire des Sciences de l'Univers de Grenoble - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - INSU - CNRS - Institut national des sciences de l'Univers - IRSTEA - Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture - USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc - CNRS - Centre National de la Recherche Scientifique - LCPC - Laboratoire Central des Ponts et Chaussées - CNRS - Centre National de la Recherche ScientifiqueBound of dissipation on a plane Couette dynamoHAL CCSD2009[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 ChangesTalour, Pascale2022-05-06 13:51:362023-05-01 03:58:252022-05-06 13:51:47enJournal articleshttps://insu.hal.science/insu-00446579/document10.1103/PhysRevE.79.066304application/pdf1Variational 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.