Bound of dissipation on a plane Couette dynamo

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.Comment: In this new version, amistake (in equation 23 of the first version) is correcte

Upper bound of heat flux in an anelastic model for Rayleigh-B\'enard convection

Bounds on heat transfer have been the subject of previous studies concerning convection in the Boussinesq approximation: in the Rayleigh-B\'enard configuration, the first result obtained by \cite{howard63} states that $Nu < (3/64 \ Ra)^{1/2}$ for large values of the Rayleigh number $Ra$, independently of the Prandtl number $Pr$. This is still the best known upper bound, only with the prefactor improved to $Nu < 1/6 \ Ra^{1/2}$ by \cite{DoeringConstantin96}. In the present paper, this result is extended to compressible convection. An upper bound is obtained for the anelastic liquid approximation, which is similar to the anelastic model used in astrophysics based on a turbulent diffusivity for entropy. The anelastic bound is still scaling as $Ra^{1/2}$, independently of $Pr$, but depends on the dissipation number $\mathcal{D}$ and on the equation of state. For monatomic gases and large Rayleigh numbers, the bound is $Nu < 146\, Ra^{\frac{1}{2}} / (2-\mathcal{D} )^{\frac{5}{2}}$.Comment: 12 pages, 1 figur

On the stability of the Hartmann layer

In this paper we are concerned with the theoretical stability of the laminar Hartmann layer, which forms at the boundary of any electrically conducting fluid flow under a steady magnetic field at high Hartmann number. We perform both linear and energetic stability analyses to investigate the stability of the Hartmann layer to both infinitesimal and finite perturbations. We find that there is more than three orders of magnitude between the critical Reynolds numbers from these two analyses. Our interest is motivated by experimental results on the laminar–turbulent transition of ducted magnetohydrodynamics flows. Importantly, all existing experiments have considered the laminarization of a turbulent flow, rather than transition to turbulence. The fact that experiments have considered laminarization, rather than transition, implies that the threshold value of the Reynolds number for stability of the Hartmann layer to finite-amplitude, rather than infinitesimal, disturbances is in better agreement with the experimental threshold values. In fact, the critical Reynolds number for linear instability of the Hartmann layer is more than two orders of magnitude larger than experimentally observed threshold values. It seems that this large discrepancy has led to the belief that stability or instability of the Hartmann layer has no bearing on whether the flow is laminar or turbulent. In this paper, we give support to Lock’s hypothesis [Proc. R. Soc. London, Ser. A 233, 105 (1955)] that “transition” is due to the stability characteristics of the Hartmann layer with respect to large-amplitude disturbances

A model for the turbulent Hartmann layer

Here we study the Hartmann layer, which forms at the boundary of any electrically-conducting fluid flow under a steady magnetic field at high Hartmann number provided the magnetic field is not parallel to the wall. The Hartmann layer has a well-known form when laminar. In this paper we develop a model for the turbulent Hartmann layer based on Prandtl’s mixing-length model without adding arbitrary parameters, other than those already included in the log-law. We find an exact expression for the displacement thickness of the turbulent Hartmann layer @also given by Tennekes, Phys. Fluids 9, 1876 ~1966!#, which supports our assertion that a fully-developed turbulent Hartmann layer of finite extent exists. Leading from this expression, we show that the interaction parameter is small compared with unity and that therefore the Lorentz force is negligible compared with inertia. Hence, we suggest that the turbulence present in the Hartmann layer is of classical type and not affected by the imposed magnetic field, so justifying use of a Prandtl model. A major result is a simple implicit relationship between the Reynolds number and the friction coefficient for the turbulent Hartmann layer in the limit of large Reynolds number. By considering the distance over which the stress decays, we find a condition for the two opposite Hartmann layers in duct flows to be isolated (nonoverlapping)

Experimental study of super-rotation in a magnetostrophic spherical Couette flow

We report measurements of electric potentials at the surface of a spherical container of liquid sodium in which a magnetized inner core is differentially rotating. The azimuthal angular velocities inferred from these potentials reveal a strong super-rotation of the liquid sodium in the equatorial region, for small differential rotation. Super-rotation was observed in numerical simulations by Dormy et al. [1]. We find that the latitudinal variation of the electric potentials in our experiments differs markedly from the predictions of a similar numerical model, suggesting that some of the assumptions used in the model - steadiness, equatorial symmetry, and linear treatment for the evolution of both the magnetic and velocity fields - are violated in the experiments. In addition, radial velocity measurements, using ultrasonic Doppler velocimetry, provide evidence of oscillatory motion near the outer sphere at low latitude: it is viewed as the signature of an instability of the super-rotating region

Zonal shear and super-rotation in a magnetized spherical Couette flow experiment

We present measurements performed in a spherical shell filled with liquid sodium, where a 74 mm-radius inner sphere is rotated while a 210 mm-radius outer sphere is at rest. The inner sphere holds a dipolar magnetic field and acts as a magnetic propeller when rotated. In this experimental set-up called DTS, direct measurements of the velocity are performed by ultrasonic Doppler velocimetry. Differences in electric potential and the induced magnetic field are also measured to characterize the magnetohydrodynamic flow. Rotation frequencies of the inner sphere are varied between -30 Hz and +30 Hz, the magnetic Reynolds number based on measured sodium velocities and on the shell radius reaching to about 33. We have investigated the mean axisymmetric part of the flow, which consists of differential rotation. Strong super-rotation of the fluid with respect to the rotating inner sphere is directly measured. It is found that the organization of the mean flow does not change much throughout the entire range of parameters covered by our experiment. The direct measurements of zonal velocity give a nice illustration of Ferraro's law of isorotation in the vicinity of the inner sphere where magnetic forces dominate inertial ones. The transition from a Ferraro regime in the interior to a geostrophic regime, where inertial forces predominate, in the outer regions has been well documented. It takes place where the local Elsasser number is about 1. A quantitative agreement with non-linear numerical simulations is obtained when keeping the same Elsasser number. The experiments also reveal a region that violates Ferraro's law just above the inner sphere.Comment: Phys Rev E, in pres

On the existence and structure of a mush at the inner core boundary of the Earth

It has been suggested about 20 years ago that the liquid close to the inner core boundary (ICB) is supercooled and that a sizable mushy layer has developed during the growth of the inner core. The morphological instability of the liquid-solid interface which usually results in the formation of a mushy zone has been intensively studied in metallurgy, but the freezing of the inner core occurs in very unusual conditions: the growth rate is very small, and the pressure gradient has a key role, the newly formed solid being hotter than the adjacent liquid. We investigate the linear stability of a solidification front under such conditions, pointing out the destabilizing role of the thermal and solutal fields, and the stabilizing role of the pressure gradient. The main consequence of the very small solidification rate is the importance of advective transport of solute in liquid, which tends to remove light solute from the vicinity of the ICB and to suppress supercooling, thus acting against the destabilization of the solidification front. For plausible phase diagrams of the core mixture, we nevertheless found that the ICB is likely to be morphologically unstable, and that a mushy zone might have developed at the ICB. The thermodynamic thickness of the resulting mushy zone can be significant, from $\sim100$ km to the entire inner core radius, depending on the phase diagram of the core mixture. However, such a thick mushy zone is predicted to collapse under its own weight, on a much smaller length scale ($\lesssim 1$ km). We estimate that the interdendritic spacing is probably smaller than a few tens of meter, and possibly only a few meters

Erratum: Reflections on dissipation associated with thermal convection (Journal of Fluid Mechanics (2014) 751 (749-751) DOI: 10.1017/jfm.2013.241))

cited By 2International audienc

Reflections on dissipation associated with thermal convection

cited By 9International audienceBuoyancy-driven convection is modelled using the Navier-Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, cp, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of -DP/Dt and not that of -αT(DP/Dt). Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, γ - 1 ≪ 1, independent of the dissipation number. © 2013 Cambridge University Press

Experimental study of the instability of the Hartmann layer

Hartmann layers are a common feature in magnetohydrodynamics, where they organize the electric current distribution in the flow and hence the characteristics of the velocity field. In spite of their importance their stability properties are not well understood, mainly because of the scarcity of experimental data. In this work we investigated experimentally the transition to turbulence in the Hartmann layers that arise in magnetohydrodynamic flows in ducts. From measurements of the friction factor a well-marked transition to turbulence was found at a critical Reynolds number, based on the laminar Hartmann layer thickness, of approximately 380, valid also for laminarization and for a wide range of intensities of the magnetic field. The sensitivity of this result to the roughness characteristics of the walls along which the Hartmann layers develop confirms that these layers are related to the transition observed and provides more information on its stability properties