Bound of dissipation on a plane Couette dynamo

TL;DR

This paper extends the variational turbulence approach to dynamo action, deriving upper bounds for energy dissipation in a plane Couette flow, highlighting the influence of wall conductance and magnetic Reynolds number.

Contribution

It introduces a novel variational bound for energy dissipation in dynamo flows, considering wall conductance effects and magnetic Reynolds number regimes.

Findings

Upper bound matches classical turbulence for low magnetic Reynolds numbers.

Dissipation scales with shear velocity to the fourth power above a critical Reynolds number.

Wall conductance critically influences the maximum sustainable magnetic Reynolds number.

Abstract

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 general, 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 $P_m$ 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 $P_m^{-1}$ and follows a steeper evolution for magnetic Reynolds numbers above $P_m^{-1}$ (i.e. proportional to the shear velocity to the power four) 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.