Do magnetic fields enhance turbulence at low magnetic Reynolds number ?
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Do magnetic fields enhance turbulence at low magnetic Reynolds number ?
Alban Poth´erat , and Rico Klein Applied Mathematics Research Centre, Coventry University,Priory Street Coventry CV1 5FB, United Kingdom Laboratoire National des Champs Magn´etiques Intenses- Grenoble,Universit´e Grenoble-Alpes/CNRS, 25 Rue des Martyrs, B.P. 166 38042 Grenoble Cedex, France ∗ Imposing a magnetic field on a turbulent flow of electrically conducting fluid incurs the Jouleeffect. A current paradigm is that the corresponding dissipation increases with the intensity of themagnetic field, and as a result turbulent fluctuations are all the more damped as the magnetic fieldis strong. While this idea finds apparent support in the phenomenology of decaying turbulence,measurements of turbulence in duct flows and other, more complex configurations have producedseemingly contradicting results. The root of the controversy is that magnetic fields promote sufficientscale-dependent anisotropy to profoundly reorganise the structure of turbulence, so their net effectcannot be understood in terms of the additional dissipation only. Here we show that when turbulenceis forced in a magnetic field that acts on turbulence itself rather than on the mechanisms thatgenerate it, the field promotes large, nearly 2D structures capturing sufficient energy to offset theloss due to Joule dissipation, with the net effect of increasing the intensity of turbulent fluctuations.This change of paradigm potentially carries important consequences for systems as diverse as theliquid cores of planets, accretion disks and a wide range of metallurgical and nuclear engineeringapplications.
I. INTRODUCTION
Turbulent flows are often exposed to magnetic fields,either externally applied, or self-generated. In strongmean fields, the induced Lorentz force alters the wayflows transport heat or mass, and dissipate energy [32].Among the vast array of processes that are concernedare the dynamics of liquid planetary cores [30], thesolidification of metallic alloys [18, 27], and the coolingof nuclear reactors [33]. The current paradigm is that theaction of the field on the flow is a damping one, becauseof the extra dissipation incurred by the Lorentz force [9].It finds support in the phenomenology of freely decayingmagnetohydrodynamic (MHD) turbulence, which decaysfaster under a higher externally imposed magnetic field[21, 25]. Nevertheless, the presence of an externalmagnetic field does not, in general, damp instabilitiesor turbulent fluctuations. Quite the opposite happensin axisymmetric flows subject to axial and toroidalmagnetic fields, where the magnetorotational and Taylerinstabilities occur (MRI and TI, respectively). Bothinstabilities have been theoretically predicted from linearstability analysis [19, 39] but only recently observed inliquid metal experiments [31, 37]. MRI might explainwhy accretion disks, which should remain laminar in theabsence of a magnetic field are in fact turbulent. TIis a good candidate to explain the mechanisms behindthe Sun’s dynamo [4]. In both cases, the instabilitymechanisms involve a two-way coupling between themagnetic field and the flow. Yet, enhancement ofturbulent fluctuations has been reported in a numberof examples where the magnetic field was imposed, and ∗ [email protected] not influenced by the flow too: in numerical simulationsand laboratory models for the continuous steel castingprocess, the complex DC magnetic field applied tocontrol the flow was found to drive high amplitudeturbulent fluctuations [8, 20, 40]. The picture is morecomplex in duct flows subject to an imposed magneticfield: when the flow is driven by pressure alone, theintensity of turbulent fluctuations decreases in thecentral part of the duct at higher magnetic fields [17].By contrast, earlier experiments clearly show an increasein turbulent intensity with an external magnetic fieldin duct flows where turbulence is generated by a grid[11, 16, 38], or in numerically forced flows in periodicdomains [5]. In pressure-driven duct flows, turbulence isproduced by shear layer instabilities, which are severelydamped by the magnetic field, so [17] arguably confirmsthat the magnetic field damps this particular mechanismof turbulence production [6]. However, the question ofthe effect of magnetic fields on turbulence itself remainsopen and needs to be addressed in conditions where itis not overshadowed by the magnetic field’s influence onthe mechanism forcing turbulence.We tackle this problem by considering a generic planechannel geometry pervaded by a uniform magnetic fieldperpendicular to it. We assume that (i) that the flow isdriven by an external force that is not affected by themagnetic field and (ii) the magnetic field induced bythe flow is sufficiently small for the externally appliedmagnetic field B to be considered constant, i.e. thatthe magnetic Reynolds number is small [29]. Thisapproximation applies to most laboratory experimentswith flows of moderate intensity. The key mechanism atplay in these conditions was first characterised by theauthors of Ref. [36], who showed that in inducing eddycurrents to oppose velocity gradients along the magneticfield, the Lorentz force diffuses momentum along B over a timescale τ = τ J ( l z /l ⊥ ) . Here, τ J = ρ/ ( σB )is the Joule dissipation time, l ⊥ and l z are lengthscalesacross and along the field, σ and ρ are the fluid’selectrical conductivity and density. In turbulent flows,diffusion is disrupted by inertial transfer acting over astructure turnover time τ U ∼ l ⊥ /U ( l ⊥ ), where U ( l ⊥ )is a typical velocity at scale l ⊥ . The competitionbetween both processes determines the scale-dependentanisotropy as l z ( l ⊥ ) ∼ l ⊥ N ( l ⊥ ) / , N ( l ⊥ ) = τ U /τ J being the interaction parameter at scale l ⊥ (see thetheory from Ref. [36] and experimental evidence in Ref.[24]). As a result of this process, velocity gradientsare reduced under the action of the magnetic field as ∂ z ∼ l − z ∝ B − , and so is the current density. In achannel of width h , perpendicular to the magnetic field,the flow dimensionality is therefore determined by thescale-dependent ratio l z ( l ⊥ ) /h [24]: if l z ( l ⊥ ) /h < l ⊥ is 3D. In the limit l z ( l ⊥ ) /h → ∞ ,it is quasi-2D. In the quasi-2D limit, no eddy currentsremain in the bulk, because diffusion then takes placeacross the entire channel, except in Hartmann boundarylayers along the channel walls [22]. Current that wouldbe directly induced in the bulk by an external force(through Lenz’s law) returns through the Hartmannlayers too. All three cases are illustrated in Fig. 1.To understand how this effect determines the intensity ofturbulence, we shall first translate this phenomenologyinto a scaling linking the relative turbulent intensitywith the forcing and magnetic field intensities (sectionII). Second, we shall experimentally drive turbulence ina liquid metal experiment where the forcing intensitycan be precisely set, and measure both relative andabsolute turbulent intensities (section III). II. THEORYA. Scaling for the local, event-averaged Reynoldsnumber.
We consider the generic configuration of an electri-cally conducting fluid (density ρ , electric conductivity σ , and viscosity ν ) confined between two parallel wallsdistant by h and pervaded by a uniform magnetic field B = B e z normal to them. The flow is driven by an ex-ternal force density field F non-dimensionally measuredby the Grashof number G = F l ⊥ / ( ρν ), where l ⊥ is alengthscale perpendicular to e z (Fig. 1). For simplic-ity, F is chosen normal to e z . The first step is to seeka scaling for the average flow intensity for a given exter-nal force and magnetic field. Away from the wall, F isbalanced by inertia and the Lorentz force. At low mag-netic Reynolds numbers, magnetic field fluctuations arenegligible compared to the externally imposed field [29];the governing equations can be expressed in the formof the incompressible Navier-Stokes equations with anadded term representing the Lorentz force density J × B and the constraints that the velocity field u and currentdensity field J must be solenoidal to ensure mass and charge conservation.( ∂ t + u · ∇ ) u + 1 ρ ∇ p = ν ∆ u + 1 ρ J × B + 1 ρ F , (1) ∇ · u = 0 , ∇ · J = 0 . (2)The system is closed with the addition of Ohm’s law1 σ J = −∇ φ + u × B , (3)where φ is the electric potential. Away from the walls,the curl of the Navier-Stokes equations and charge con-servation yield the gradient of current density J z along z [24]: − ∂ z J z = 1 B [ − ρ ∇ × ( u · ∇ ) u + ∇ × F ] · e z . (4)Equation (4) expresses that horizontally divergent eddycurrents ( lhs. ) are induced by the rotational part of theforcing (second term, rhs. ). Part of them returns throughthe bulk where the Lorentz force they induce balancesinertia (first term, rhs. ). Consider a structure of size l ⊥ and velocity U extending over height h V along the mag-netic field. From Eq. (4) the total current generated bythe forcing inside the structure and following this pathscales as I B ∼ ( πρl ⊥ h V /B )( U /l ⊥ ), and the current in-duced by the forcing scales as I F ∼ ( πρl ⊥ h V /B )( F/ρl ⊥ ).The remaining current I H ( lhs. of (4)) returns throughthe thin Hartmann boundary layers that develop alongthe walls [22]. The Lorentz force they generate thereis opposed by viscous friction only (see Fig. 1). Thisbalance determines the thickness δ H = B − ( ρν/σ ) / ofthese layers and the current density flowing through them J H ∼ σBU [22]. Consequently, the total current throughthe Hartmann layer scales as I H ∼ πl ⊥ J H δ H . From thisphenomenology, Eq. (4) reduces to the conservation ofthe global current induced by the forcing I F = I H + I B ,and from the scalings for each term, Eq.(4) is expressednon-dimensionally as G − Re ∼ hh V (cid:18) l ⊥ h (cid:19) HaRe, (5)where Re = U l ⊥ /nu is the Reynolds number, andthe Hartmann number Ha = h/δ H provides a non-dimensional measure of B .The height h V of the structure in the bulk is determinedby the anisotropic action of the Lorentz force as follows:In the limit Re/Ha ≫
1, the bulk of the flow is 3D, and h V is set by the momentum diffusion length defined inthe introduction h V ∼ l z ( l ⊥ ). In this case, inertia in thebulk consumes the larger part of the current induced byexternal forcing, so it remains from (5) that Re ∼ G / . (6)Since the contribution of the Hartmann layers in (5)is small in this limit, Eq. (6) remains valid whether l z ( l ⊥ ) ≪ h , h V = l z ( l ⊥ ) l z ( l ⊥ ) ∼ h , h V = h l z ( l ⊥ ) ≫ h , h V = h (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Viscously driven residual flow region B (a) PSfrag replacements I c I c ≃ I b I t I t = I b I F I F / UU b U t U t = U b δ H h l z l ⊥ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (b) PSfrag replacements
U I F (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (c) PSfrag replacements
UU I F / I F / l ⊥ driven by an external force F (in the green area) in an external magnetic field B . The Lorentz force diffuses momentum along the field over l z ( l ⊥ ), and induces electric currents (in red). (a) l z ( l ⊥ ) ≪ h :the current driven by the external forcing returns in the bulk. Only a residual flow exists near the walls. (b) l z ( l ⊥ ) ∼ h : thecurrent returns in the bulk and the Hartmann layers (in blue): the flow is still 3D but influenced by the walls. (c) l z ( l ⊥ ) ≫ h :the current returns equally in the top and bottom Hartmann layers, and not in the bulk: the flow is quasi-2D. or not the flow interacts with the walls (in this lattercase the current induced by the forcing returns entirelyover height l z , determined by the balance between theLorentz force and inertia ( i.e. I H = 0, as in Fig. 1-(a)).Experiments on electrically driven turbulence (See Fig.4-a in Ref. [24]) showed that this relation holds toa great precision for turbulent flows, if the Reynoldsnumber Re = h| u |i L f /ν is expressed in terms of theaverage velocity in the forced region and the forcinglengthscale L f ( i.e by setting l ⊥ = L f and U = h| u |i inthe derivation above). Even for Ha up to 2 × , theleft hand side correction in (5) due to current circulatingin the Hartmann layers remained small, which confirmsthe validity of (6). B. Scaling relation for the relative turbulentintensity.
We shall now seek an estimate for the turbulent in-tensity, for given control parameters G and Ha , i.e forgiven forcing and magnetic fields. For this, we firstnote that unlike in hydrodynamic turbulence, dissipa-tion in MHD turbulence is Ohmic and roughly scales as ǫ J ∼ ( ν/h ) Ha k u k ( l ⊥ /l z ) ( a more precise scaling willbe obtained in this section). Consequently, it preferen-tially affects large scales exhibiting three-dimensionality( i.e for which l z remains finite). Hence, the tendency to-ward two-dimensionality of MHD turbulence, which is aby-product of the dissipative process, is independent ofits spectral structure [13] and the global dissipation canbe captured by means of a power budget over a volume V of height h along B and size l ⊥ greater than the forcingscale. Decomposing the velocity field u = h u i + h u ′ i / into average and fluctuations, we shall write this bud-get separating the current density field J into its average and fluctuating parts returning through the Hartmannlayers h J D i + J ′ D and its average and fluctuating partsreturning through the bulk h J D i + J ′ D : h Z V F · h u i + ( h J D i + J ′ D + h J D i + J ′ D ) × B · ( h u i + u ′ )+ ρν ( h u i + u ′ ) · ∆ ( h u i + u ′ ) dV i = 0 , (7)This way, the budget reflects the structure of the eddycurrents identified previously: The power P F generatedby the forcing is consumed in three ways: Joule dissi-pation due to current returning through the bulk ǫ D ,Joule dissipation due to currents returning through theHartmann layers ǫ D , and viscous dissipation ǫ ν almostentirely produced in the Hartmann layers: P F + ǫ DJ + ǫ DJ + ǫ ν = 0 . (8)The contribution from inertial and pressure terms is ne-glected on the grounds that it is N ( l ⊥ ) times smallerthan Joule dissipation and that for strong enough mag-netic fields, N ( l ⊥ ) ≫
1. This condition sets a minimumlengthscale l ⊥ for Eq. (8) to be valid over V ( l ⊥ ). Theforcing power is evaluated by noticing that eddy currentsinduced by the forcing diffuse over the lengthscale l z ( l ⊥ )built on the average velocity: P F ∼ l ⊥ l z ( l ⊥ ) |h u i|| F | ∼ ρνh h u i G HaRe − / . (9)Joule dissipation in the bulk can be related to the velocityfield by virtue of the expression of the rotational part ofLorentz force put forward by [36]:[ J × B ] ∇× = − ρτ J ∆ − ∂ zz u . (10)Since bulk velocity gradients along the magnetic fieldsare controlled by the diffusion of momentum by theLorentz force, ∆ − ∂ zz ∼ ( l ⊥ /l z ) . Importantly, how-ever, mean flow and fluctuations may significantly dif-fer in intensity and may therefore diffuse over differentlengths under the action of the Lorentz force, respectively l z = l ⊥ N ( |h u i| ) / and l ′ z = l ⊥ N ( h u ′ i / ) / . Hence,separating average and fluctuating parts yields: ǫ DJ = − ρτ J Z V (cid:2) h u i · ∆ − ∂ zz h u i + h u ′ · ∆ − ∂ zz u ′ i (cid:3) dV, ∼ ρνh h u i Re (1 + α ) , (11)where α = h u ′ i / / |h u i| (12)is the relative turbulent intensity.Dissipation in the Hartmann layers is equally Ohmic andviscous ǫ DJ = R V ( h J D i + J ′ D ) × B · u dV ≃ ǫ ν = R V ρν u · ∆ u dV ≃ R V ρν u · ∂ zz u dV and is estimated fromthe thickness of the Hartmann layer δ H = h/Ha [22]: ǫ DJ ≃ ǫ ν ∼ − ρνh h u i (cid:18) l ⊥ h (cid:19) Ha (1 + α ) . (13)By virtue of (9,11,13), and (7) becomes G HaRe − / (cid:18) l ⊥ h (cid:19) ∼ Re (1+ α )+2 HaRe (cid:18) l ⊥ h (cid:19) (1+ α ) . (14)Lastly, Re , which is not known a priori , is expressed interms of control parameter G through scaling (6). Thisyields a simplified relation linking the relative turbulentintensity α to control parameters Ha and G : Ha G − / (cid:18) l ⊥ h (cid:19) ∼ (1 + α ) + 2 Ha G − / (cid:18) l ⊥ h (cid:19) (1 + α ) . (15)These estimates reflect that increasing B (or Ha )extends the thickness of the forced region by l z ∝ B .Consequently, the forcing power increases linearly with B . On the other hand, the power dissipated in thebulk does not vary with B because the intensity of theeddy currents there is governed by velocity gradientsalong z that decrease as l − z ∝ B − , thus exactlycompensating the increase in B . Plots of (15) in Fig. 3show that the relative intensity of turbulence increaseswith the externally imposed magnetic field (or Ha ), with α ∝ Ha / for α >
1. The underlying phenomenologyreflects anisotropic mechanisms that are the hallmark ofMHD turbulence: in 3D flows, high velocity gradientsalong the magnetic field incur Joule dissipation thatdamps velocity fluctuations. As the magnetic field in-creases, turbulent structures elongate as l ′ z ( l ⊥ ) increases,velocity gradients weaken, so does the Joule dissipationthey incur, and turbulent fluctuations retain moreenergy. Through this process, the relative intensity ofturbulent fluctuations is higher for higher magnetic fields. III. EXPERIMENTSA. Experimental approach.
Our second step is to test scaling relation (15) on theFLOWCUBE facility, which reproduces the generic con-figuration from Fig. 1 in a 10 cm-cubic vessel filled withliquid metal pervaded by the near-homogeneous mag-netic field generated in the bore of a superconductingsolenoidal magnet. The metal is Gallinstan, an eutec-tic alloy of gallium, indium and tin that is liquid aroom temperature. Full details and validation of theexperimental setup are provided in [3, 24]. An electri-cally generated force of lengthscale L f keeps turbulencein a statistically steady state as follows: Electric cur-rent I locally injected through one of the channel walls(arbitrarily the bottom wall) forces horizontally diver-gent currents through the bulk and the Hartmann layers.Their interaction with the externally imposed magneticfield exerts a driving Lorentz force on the flow. Thismechanism ensures that the forcing is directly controlledby the injected current and does not vary with the ex-ternal magnetic field. This somewhat counter-intuitiveresult comes as a consequence of the diffusion by theLorentz force over l z ∼ L f N / = L f Ha/Re / ( L f /h )of the current I injected through the wall. Becauseof this effect, the volume of forced fluid increases lin-early with the magnetic field. Hence, the horizontalcurrent density driving fluid motion scales as J ⊥ ∼ I/ (2 πL f l z ). It follows that the force density drivingturbulence is controlled by the injected current I only,as F ∼ IB/ (2 πL f l z ) = I ( ρν/σ ) / / (2 πL f ) Re / , anddoes not depend on the magnetic field. By virtue of (6), G = [ I/ (2 π ( ν σρ ) / )] / is varied solely by adjustingthe injected current I . This method can be understoodas an extension to 3D turbulence of the method of [35]to force quasi-2D MHD flows, where the forcing was alsocontrolled by the current measured non-dimensionally bythe parameter Re = I π ( σρν ) / = G / . (16)Based on this principle, turbulence is driven inFLOWCUBE by injecting current through electrodes em-bedded flush in the bottom wall, arranged in a 10 × L f = 1 cm and alternately connectedto the positive and negative poles of a DC current supply.The corresponding average flow is a crystal of columnarvortices attached to the bottom wall extending by l z ( L f )into the bulk through diffusion by the Lorentz force (see[14, 24] for a detailed analysis of this flow, and [2, 23, 26]for a theory explaining how its componentality is con-trolled by the magnetic field). The region just outsidethe bottom Hartmann layer is always in the forcing re-gion and is representative of the bulk of forced turbu-lence. On the other hand, the region just outside of theHartmann layer near the top wall may lay within the forc-ing region if l z ( L f ) /h = N ( L f ) / L f /h > FIG. 2. Simplified sketch of the FLOWCUBE platform for thestudy of electrically driven liquid metal flows representing themain vessel filled with liquid metal with Hartmann walls (inyellow) and walls parallel to the magnetic field (transparent).The flow is forced by injecting DC electric current through asquare array of electrodes embedded flush in the bottom wall.This creates a crystal of columnar vortices, which in destabil-ising, gives rise to turbulence. An external magnetic field isapplied by means of a large solenoidal cryomagnet (not rep-resented). Depending on the intensities of the magnetic fieldand turbulence, turbulence extends to part of the bulk (if l ′ z ( L f ) < h ) or up to the top wall (if l ′ z ( L f ) & h ). Velocitiesare reconstructed just outside the boundary layers near thetop and bottom walls, from measurements of electric potentialat 384 contact probes embedded flush in the top and bottomwalls (black dots) and connected to a 768-channel, high pre-cision acquisition system through printed circuit built-in theHartmann walls. if l z ( L f ) /h <
1. Since L f is the scale of the average flow, Re , N and G are evaluated taking l ⊥ = L f . As in SectionII, we shall distinguish between N and N ′ , built on theaverage velocity and the fluctuations, respectively.Bulk velocities are measured locally just outside the bot-tom and top Hartmann layers by electric potential ve-locimetry [15] through 192 electric potential probes fit-ted flush in each of the Hartmann walls [24]. Averageand RMS fluctuations near bottom and top walls (de-noted by indices b and t respectively) are obtained fromspatial and time averages of time-dependent signals of u b,t ( x, y, t ), recorded in a statistically state. B. Relative turbulent intensity.
Relative turbulent intensities near the bottom andtop walls α b,t = h u ′ b,t i / / |h u b,t i| are shown in Fig. 3.Within the forced region of the flow ( i.e. near the bottom Ha=1822.2 Ha=3644.3 Ha=10933 Ha=14577 Ha=18222 α b Re α =0.37Re α b α b H a − / α =0.014Re Ha α t α =35Re Ha −1 Re α t H a (d)(c)(b)(a)(e) PSfrag replacements Re = G / FIG. 3. Relative turbulent intensity near the bottom wall(forced region) (from (a) Eq. (15), (b,c): experiment) and (d)top wall, α b,t = h u ′ b,t i / / |h u b,t i| . The red rectangle indicatesthe regime of strongest three-dimensionality, where turbulentfluctuations increase with the external magnetic field and de-crease as G − / with the forcing. The slope Re / ( ∝ G / ) isindicative only, as the solution of (15) for α ( G ) is not a powerlaw. Ha=1822.2 Ha=3644.3 Ha=10933 Ha=14577 Ha=18222 λ b /L i =0.034Re ± λ b / L i λ t /L i =0.21Re ± λ t /L i =0.15Re ± λ t / L i R λ ,b =0.9Re ± Ha R λ ,b =1.4Re ± Ha R λ , b H a − / R λ ,t =0.85Re ± Ha R λ ,b =0.5Re ± Ha R λ ,b =0.4Re ± Ha Re R λ ,t H a − / (a)(b)(c)(d) PSfrag replacements Re = G / FIG. 4. Taylor microscale near (a) the bottom wall ( λ b ,forced region), and (b) the top wall ( λ t , outside the forcedregion if l z ( L f ) > h ), and Reynolds number R λ based on (c) λ b and (d) λ t , showing increasing absolute turbulent intensitywith the external magnetic field both within and outside theforced region. wall, see Fig. 3-b), α b is found to always increase with Ha and its variations are very well reproduced by thescaling relation (15): in both theory and experiments, α b ∝ Ha / for α b >
1, a regime where both the topand the bottom walls lie within the forced region ( i.e l z ( L f ) > h ). When α b .
1, the dependence on Ha iseven stronger than Ha / . In this regime, l z ( L f ) < h so the forcing region progressively extends across thechannel as Ha is increased. The increase in actualvortex length along B causes a drastic reduction ofvelocity gradients along z and a drop in bulk dissipationthat translates into the sharper increase of turbulentfluctuations observed.Equation (15) and its underlying phenomenology arefurther validated in the most 3D regimes ( Ha = 1822,high G ), where α b decreases with G (Region inside the redrectangle in Fig. 3-b). This behaviour is captured in the limit where Joule dissipation mostly occurs in the bulk,in which case (15) reduces to α b ∝ G − / ∝ Re − / .By contrast Eq. (15) does not apply to relative fluc-tuations α t near the top wall, which lays outside theforced region if l z ( L f ) . h . Indeed relative turbulentfluctuations there first sharply increase with Ha for Ha < Ha increases, to progressively reach the top wall. Fluctua-tions then decrease slightly with Ha : rather than a lossof intensity in turbulence, we shall see that this reflectsan increasing intensity of the mean flow near the top wall. C. Absolute turbulent intensity.
To isolate the influence of the magnetic field on tur-bulent fluctuations from that on the mean flow, absoluteturbulent intensity was measured through the microscaleReynolds number R λ,b,t = h u ′ b,t i / λ b,t /ν, (17)based on the Taylor microscale λ b,t = " h u ′ b,t ( x ) · u ′ b,t ( x + r ) i ∂ rr h u ′ b,t ( x ) · u ′ b,t ( x + r ) i / ( r = 0) , (18)which is representative of the inertial range of turbulence[12]. The variations of λ near the bottom ( λ b , Fig. 4-a)and top walls ( λ t , Fig. 4-b) turn out to be very weak.Hence, the variations of R λ are driven by the velocityfluctuations. Just like the relative turbulent intensity,the microscale Reynolds number (Figs. 4-c and 4-d)follows a scaling of R λ,b,t ∝ Ha / in the limit of high Ha . There are, however, two important differences.First, unlike relative fluctuations, absolute fluctuationsalways increase with Ha , both inside and outside theforcing region. Second, they always increase with theforcing too, confirming that the decrease in relativeturbulent intensity with either the forcing (for low Ha )or the magnetic field outside the forcing region reflectsan intensification of the average flow, and not weakeningturbulent fluctuations. IV. DISCUSSION.
These results provide robust evidence that the inten-sity of forced MHD turbulence increases with an exter-nally applied magnetic field, when the forcing densityis kept constant and when the damping of energy trans-fer from mean flow to turbulence by the magnetic fielddoes not mask its influence on turbulence itself. In theFLOWCUBE setup, constant forcing density is obtainedby driving turbulence with a constant electric current butthe same result can be achieved with other types of forc-ing mechanisms: If the flow had been locally forced bymeans of a propeller, since the forced volume increasesas l z ( L f ) ∝ B , the total torque on the propeller wouldhave to increase with B to keep the force density constantover the entire forced volume. Regardless of the forcingmechanism, in any given volume of forced turbulence, ve-locity gradients along the field are smoothed out by theLorentz force as the field increases. This mechanism re-duces Joule dissipation. It allows turbulent fluctuationsto retain more energy, and thereby leads turbulent inten-sity to increase as the magnetic field is increased. In gridturbulence experiments [1, 11, 16, 38], the average fluidvelocity relative to the grid was kept constant. This im-pled increasing the driving force with the magnetic fieldto overcome the obstacle resistance. Equation (6) im-plies that keeping the average velocity constant is indeedequivalent to keeping the force density constant, so thephenomenology identified in the FLOWCUBE applies tothese experiments on duct flows too. Indeed, [11] showsan increase of approximately α ∝ B . ± . for Ha & ,which is consistent with our own findings at low valuesof Ha . Hence, while magnetic fields may damp instabil-ities in duct flows, they enhance turbulent fluctuationsin general, in duct flows and other configurations. Thisphenomenon stems from the promotion of larger, moretwo-dimensional scales at higher magnetic fields. Thehigher the fields, the wider the range of quasi-2D scalesand the more intense the turbulent fluctuations. Thismechanism is further reinforced as large quasi-2D scalespromote an inverse energy cascade that channels the en-ergy injected by the forcing to them [10, 28, 34].In our experiment, the forcing, rather than turbulent pro-duction itself was kept constant and independent of thefield. Hence, the increase in turbulent intensity tookplace despite a possible loss in the energy transferred from the mean flow to turbulence. A more precise charac-terisation of turbulence enhancement would be obtainedby holding turbulent production constant, which can beenvisaged in numerical simulations, rather than experi-ments.An important point to notice is that the reduction of ve-locity gradients with increasing magnetic field is the coreeffect driving turbulence enhancement. As a direct con-sequence, this mechanism cannot act when both the aver-age flow and fluctuations cease to exhibit any gradientsalong the magnetic field, i.e. when all flow structuresare quasi-2D (as on Fig. 1-(c)). Such a regime is notreached in FLOWCUBE for the magnetic fields consid-ered here. However, [11] precisely reports a decrease inturbulent intensity with the fluctuations in the quasi-2Dregime, which provides further support of the turbulenceenhancement scenario we put forward.Finally, two-dimensionalisation and energy cascades to-ward large scales exist in other systems. For example,rotation promotes less dissipative quasi-2D turbulencetoo, as the Coriolis force plays a similar role to that ofthe Lorentz force in MHD turbulence [7]. Hence, it canbe expected that the intensity of turbulent fluctuationsshould increase with rotation too, as unlike the Lorentzforce, the Coriolis force generates no dissipation to op-pose this mechanism. ACKNOWLEDGMENTS
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