On the peculiar nature of turbulence in planetary dynamos
aa r X i v : . [ phy s i c s . g e o - ph ] A ug Physics or Astrophysics/Header
On the peculiar nature of turbulence inplanetary dynamos
Henri-Claude Nataf, Nad`ege Gagni`ere
University of Grenoble - Centre National de la Recherche Scientifique, LGIT, BP53, Maison des Geosciences, 38041 Grenoble cedex 9, France
Received *****; accepted after revision +++++
Abstract
Under the combined constraints of rapid rotation, sphericity, and magnetic field,motions in planetary cores get organized in a peculiar way. Classical hydrodynamicturbulence is not present, but turbulent motions can take place under the actionof the buoyancy and Lorentz forces. Laboratory experiments, such as the rotatingspherical magnetic Couette
DTS experiment in Grenoble, help us understand whatmotions take place in planetary core conditions.
To cite this article: H.-C. Natafand N. Gagni`ere, C. R. Physique XXX (2008).
R´esum´eSur la nature particuli`ere de la turbulence dans les noyaux plan´etaires.
Sous les contraintes combin´ees de la rotation rapide, de la sph´ericit´e et du champmagn´etique, les ´ecoulements dans les noyaux plan´etaires s’organisent d’une mani`ereparticuli`ere. La turbulence hydrodynamique classique n’est pas pr´esente mais desmouvements turbulents peuvent se mettre en place sous l’action des forces d’Ar-chim`ede et de Lorentz. Des exp´eriences de laboratoire comme l’exp´erience
DTS deCouette sph´erique sous champ magn´etique `a Grenoble, nous aide `a comprendre les´ecoulements qui peuvent exister dans les conditions des noyaux plan´etaires.
Pourciter cet article : H.-C. Nataf and N. Gagni`ere, C. R. Physique XXX (2008).Key words:
Dynamo ; Planetary core ; DTS ; spherical Couette
Mot-cl´es :
Dynamo ; Noyau plan´etaire ; DTS ; Couette sph´erique
Email addresses:
[email protected] (Henri-ClaudeNataf),
[email protected] (Nad`ege Gagni`ere).
Preprint submitted to Elsevier Science October 24, 2018
Introduction
The recent success of the
VKS dynamo experiment [1,2], after the harvest ofthe pioneer experiments in Riga [3] and Karlsruhe [4], brings hope to betterunderstand the mechanisms at work in the dynamo process. It is believed thatthe self–sustained dynamo process generates the magnetic fields of most as-trophysical objects, including our planet, the Earth. In this process, a givenvelocity field ~u can produce a magnetic field controlled by the magnetic in-duction equation (1), which shows that if the velocity is large enough for theinduction term ~ ∇ × (cid:16) ~u × ~B (cid:17) to dominate over the diffusion term η ∆ ~B , themagnetic field ~B can grow. ∂ ~B∂t = ~ ∇ × (cid:16) ~u × ~B (cid:17) + η ∆ ~B (1)The magnetic field of the Earth originates in its core, made of liquid iron.As the Earth cools down, on a geological time–scale, heat is carried from thecore to the mantle by convective motions. These motions are governed bythe Navier–Stokes equation, which, in the Boussinesq–approximation may bewritten as: ρ ∂~u∂t + ~u · ∇ ~u ! = − ~ ∇ P − ρ~ Ω × ~u + µ ∆ ~u + ~j × ~B + ~f a (2)where the various symbols have their usual meaning. On the right hand side,one recognizes the Coriolis force − ~ Ω × ~u , the viscous force µ ∆ ~u , the Lorentzforce ~j × ~B , and the buoyancy force ~f a .When plugging in typical values of the properties of liquid iron and of the ve-locity ~u and magnetic ~B fields, one finds that, because of the fast rotation Ω,the dominant forces are the Coriolis and Lorentz forces. The resulting regimeis called “magnetostrophic”, and it has received the attention of theoreticiansfor decades. Taylor [5] was the first to point out that, in this regime, if oneconsiders tubes of liquid co–axial with the rotation axis, there is no torqueto resist the torque applied by the Lorentz forces. He concluded that, in thesteady–state, the torque applied by the Lorentz forces must vanish, yieldingwhat is now called a “Taylor–state”. The magnetostrophic regime and theexpected Taylor–state remain very difficult to reach in numerical simulations(because it is technically difficult to neglect viscous terms) and their explo-ration in laboratory experiments is recent [6,7,8].In this article, we will highlight some striking properties of the “magne-tostrophic” regime, as derived from measurements performed in the DTS ex-periment. In section 2, we describe the
DTS setup and present the relevantparameters. The time–averaged flow is analyzed in section 3, and the (weak)2 igure 1. Sketch of the central part of the
DT S experiment. The 7 . − ◦ , 10 ◦ and ± ◦ receive interchangeable assemblies, which canbe equipped with ultrasonic transducers. fluctuations in section 4. Implications for turbulence in planetary cores is dis-cussed in section 5. DTS experiment: exploring the magnetostrophic regime
The
Derviche Tourneur Sodium (DTS) experiment has been designed for theexploration of the magnetostrophic regime, in which the Coriolis and Lorentzforces are dominant [9]. It consists in a spherical Couette flow where boththe inner and outer spheres can rotate rapidly at separate angular velocitiesaround a common vertical axis. Forty liters of liquid sodium fill the shellbetween the two spheres and the inner sphere contains a strong magnet. Theset–up is depicted in figure 1. The fluid flow is thus strongly influenced by boththe Coriolis force and the Lorentz force. Dimensions and typical dimensionlessnumbers are given in Table 1.The radius of the outer sphere is a = 21cm. Because of this moderate di-mension, the Joule dissipation time τ J of magnetic fluctuations by diffusion issmall (less than one tenth of a second). The Alfv´en wave velocity U a depends3 ymbol expression units value a outer radius cm 21 b inner radius cm 7 . τ J a /π η s 0 . B = B i B = B o Ha aB/ √ µ ρνη aB/η √ µ ρ
12 0.56 U a B/ √ µ ρ m s − f = 5HzE ν/ Ω a . − B = B i B = B o Λ σB /ρ Ω 9.2 0.02 λ U a /a Ω 0.77 0.04∆ f = 5Hz U b ∆Ω m s − U a/η
U a/ν B = B i B = B o N σaB /ρU
26 0.006Table 1Typical values of the relevant parameters and dimensionless numbers for given im-posed rotation frequencies f = Ω / π of the outer sphere and differential rotationof the inner sphere ∆ f = ∆Ω / π with respect to the outer sphere. For the num-bers that depend on the magnetic field strength, two values are given, the first onewith B = B i = 0 .
175 T at the equator of the inner sphere, the second one with B = B o = 0 .
008 T at the equator of the outer sphere. on the strength of the magnetic field, and is thus much stronger near theinner sphere than at the outer sphere. Comparing the Alfv´en time to the dif-fusion time yields the Lundquist number S. It reaches 12 on the inner sphere,suggesting that Alfv´en waves can survive there, while they must be severelydamped near the outer boundary, even though the Hartmann number Ha islarge everywhere in the shell.The rotation frequency f of the outer sphere has been varied between 0 and15Hz. The small value of the Ekman number for a typical rotation frequencyof 5Hz illustrates that the Coriolis force dominates over viscous forces in the4ulk of the fluid. A very thin Ekman layer (one tenth of a millimeter) mustform beneath the outer shell. The value of the Elsasser number Λ, whichcompares the Lorentz force to the Coriolis force, shows that rotation effectsdominate in that region, while the opposite holds near the inner sphere. Thedomain of magnetostrophic equilibrium Λ ≃ λ , called Lehnert number in [10], compares thefrequencies of Alfv´en modes to that of rotation (inertial) modes. It is smallerthan 1 everywhere.Other effects depend upon the strength of the forcing, measured by ∆ f , thedifferential rotation frequency of the inner sphere with respect to the outersphere. It can be varied between about − U is of the order of 2m/s. The Reynoldsnumber Re is thus very large, while the magnetic Reynolds number Rm islarger than 5, indicating that the magnetic field is modified by the flow. Thelarge value of the interaction parameter N near the inner sphere demonstratesthat, in turn, the magnetic field deeply influences the flow.The DT S set–up is well suited for exploring the magnetostrophic regime. It isnot a dynamo experiment, but the imposed magnetic field is strong enough togive rise to Lorentz forces of the same order as the Coriolis force. Both forceshave a strong influence on the flow. The magnetic Reynolds number is largeenough for induction effects to be well developed. We now examine what flowresults from this complex combination, which is typical in planetary cores.
We use ultrasonic Doppler velocimetry to measure the mean axisymmetricflow. A transducer is placed in one of the interchangeable assemblies at latitude10 ◦ , and shoots a beam with declination and inclination angles of 60 ◦ and66 ◦ , respectively. After a 38cm–long linear trajectory, the beam hits the outersphere at a latitude of about − ◦ . Azimuthal velocities being more than 10times larger than radial velocities, the along–beam measured velocity directlyyields an angular velocity profile.Such a profile is shown in figure 2 for f ≃ f ≃ ◦ , we see that the angular velocity risesregularly from a value of about 0 . . et al. [8] explain that the velocity bump near theinner sphere is a signature of the magnetic wind: the Elsasser number is largeand the angular velocity isolines follow the magnetic field lines; the magnetic5oupling between the rotating inner sphere and the liquid sodium is efficient.Away from that bump, the flow is geostrophic as the Elsasser number getsbelow 1 (see Table 1). The geostrophic flow is driven by the magnetic torquethat results from the shearing of the magnetic field lines and slowed down byfriction in the Ekman layers beneath the outer sphere. This situation has beenwell studied by Kleeorin et al. [11], in the asymptotic limit of small Ekman,Rossby, Reynolds and Elsasser numbers. At small cylindrical radius s , theimposed magnetic field is large and friction in the Ekman layers of the outersphere is weak. As a consequence, the shear must be very small in order forthe magnetic torque to remain small enough to balance the friction torque:the fluid is in rigid body rotation. The regular decrease in angular velocity asone gets farther away from the inner sphere was predicted by Kleeorin et al. [11]: friction increases while the imposed magnetic field decreases, implyingan increasing shear. It is important to realize that the flow is geostrophic, asa consequence of strong rotation, but entirely driven by magnetic forces. Amodified Taylor state is achieved, where the magnetic torque on cylindricaltubes parallel to the axis of rotation only balances the (weak) friction inthe Ekman boundary layers. Note that, under this mechanism, the inducedtoroidal magnetic field is expected to be larger in the external region, whereshear is large, than in the region near the inner sphere, although the imposedmagnetic field is much larger there.We find similar profiles for all values of ∆ f as long as the Rossby numberRo = ∆ f /f remains smaller than a few units. A good quantitative fit betweenthe theory and the measurements is achieved, when one takes into account thatthe Ekman layers become turbulent in the experimental conditions (see [8]).Therefore, we think that the same dynamics prevails for the very small Rossbynumbers that characterize flow in the Earth’s core. The profile shown in figure 2 is, in fact, a histogram constructed from over 1000shots. Although the Reynolds number reaches 7 · for this run (see Table 1),the fluctuations around the average profile are amazingly small. Besides, thepower spectra of the fluctuations reveal peculiar bumps at several frequencies.This is best seen on the spectrogram of a whole run, where f ≃ f is varied in steps from 0 to − − ◦ and distant of 40 ◦ in longitude (see figure 1). Thespectrogram is shown in figure 3. Frequency is on the horizontal axis and timeon the vertical axis. The power spectral density of the signal is color–coded(log–scale). Thin lines reflect the two forcing frequencies f and − ∆ f and some6 ile42 PDF BRUT Velocity (m/s) D i s t a n ce f r o m t h e o u t er b o und a r y ( c m )
0 0.4 0.8 1.20 4 8 121620 s/ a Figure 2. Profile of the angular velocity (velocity along the beam) of the flow asa function of distance from the outer sphere, obtained by ultrasonic Doppler ve-locimetry. The angular velocity rises smoothly to a plateau before it increases againas the ultrasonic beam gets closest to the inner sphere. f ≃ f ≃ s . overtones. For − ∆ f < f , the spectrum is featureless, while bands of higherpower are clearly visible when ∆ f reaches − et al [7] reveal that they correspond to wavesthat propagate in the direction of the flow (in the frame of reference rotatingwith the outer sphere) but at a slower velocity. The lowest band correspondsto a wave or mode with an azimuthal wave–number m = 1, the followingone to m = 2, and so on. The origin of these waves is not clear yet. They aredifferent from the inertial modes identified by Kelley et al [12] in a similar set–up but with a weak magnetic field. Indeed, the derivation of their dispersionrelationship by Schmitt et al [7] reveal that their frequencies can be higherthan that of inertial modes (which are bounded by 2 f fluid , where f fluid is therotation frequency of the fluid in the laboratory frame of reference).Once again, useful information can be gained from ultrasonic Doppler velocitymeasurements. The bands that were first seen on power spectra of the inducedmagnetic field are also present in the power spectra of velocity, as illustratedin figure 4. They clearly extend deep into the sphere.We have seen that, in the modified Taylor state, the maximum induced toroidalfield is not near the inner sphere. However, it is clear that a local perturbationof the azimuthal velocity near the inner sphere will induce a strong azimuthalmagnetic field, which will in turn change the magnetic field and the torquethat drives the azimuthal flow. It is tempting to see in this effect the origin ofthe waves we observe in DT S , but further work is needed to corroborate thishypothesis. 7 igure 3. Spectrogram of electric potential measurement. The signal is the electricpotential measured between two electrodes situated at latitude − ◦ and distantof 40 ◦ in longitude. The colors give the log–amplitude of the short–time spectraldensity. The horizontal axis is frequency and the vertical axis is time. During thistime–frame, the outer sphere was rotating at a constant angular frequency f = 5Hz,while the inner sphere differential rotation rate ∆ f was varied in steps from 0 to − f and ∆ f , reveal these frequencies and some over-tones. The spectrum is almost void until ∆ f reaches about − | ∆ f | increases. requency (Hz) r / a Radial velocity spectrum
Figure 4. Radial profile of power spectral density of radial velocity versus frequency.Radial velocities are measured by ultrasonic Doppler velocimetry along a radialbeam shot from a transducer at − ◦ in latitude. We then derive this image ofcolor–coded log of power–spectral density as a function of adimensional radius r/a and frequency. The rotation frequency of the sphere is f = 4 . f = − l og o f s p ec t r a l p o w er Radial velocity spectrum f = 4.3 Hz f = −23 Hz ∆ Figure 5. Power spectral density of radial velocity versus frequency at mid–depth.For the same record as in figure 4, we plot the log of power spectrum measured atmid–depth. Peaks at several frequencies dominate the spectrum. Implications for turbulence in the Earth’s core
Our experiments illustrate that, due to the strong influence of the Coriolisforce, motions are geostrophic when the Elsasser number is less than 1, eventhough the magnetic torque on co–axial tubes is the only driving mechanism.Jault [10] argues that this property remains true even for large Elsasser num-bers if one considers time scales that are short compared to the Alfv´en time.Pais and Jault [13] apply this idea to retrieve geostrophic motions in the corethat explain the observed secular variation of the magnetic field of the Earthbetween years 2000 and 2005.The idea that motions are organized in cylinders around the axis of rotationdates back to Taylor [5]. He pointed out that, in the steady–state, the mag-netic torque acting on such cylinders must vanish since, in the limit of smallEkman and Rossby numbers, there is no torque to balance it. The
DT S ex-periment supports this idea, with the modification that the magnetic torqueis balanced by friction at the core–mantle boundary [11], and that this fric-tion takes into account that the Ekman boundary layer becomes turbulent[8]. Although the local Reynolds number of the Ekman layers remain small inthe core [14], enhanced friction at the core–mantle boundary can result froma rough boundary [15,16], or from electromagnetic coupling between the coreand the mantle [16]. In any case, nutation observations require a 10000–foldenhancement of core–mantle coupling, as compared to linear Ekman friction[17]. It would be worth it to take this effect into account in numerical simu-lations, in particular in quasi–geostrophic dynamo simulations [18], which areparticularly efficient for exploring Earth–like fluid properties, and in whichEkman friction is already parameterized.Our experiments also suggest that the only fluctuations left in the magne-tostrophic regime, apart from localized turbulence in the Ekman layers, arewaves that propagate azimuthally in a retrograde fashion. These global wavephenomena are usually excluded from the local turbulence analyses used tocalibrate sub–grid algorithms in 3D numerical simulations [19]. This shouldprobably be re–assessed once the origin of these waves is clearly established.In that respect, one should note one important limitation of our
DT S experi-ment: the time–scales that characterize the various kinds of waves are all verysimilar. For typical values of f and ∆ f , the period of inertial waves and Alfv´enwaves, as well as the dissipation time of Alfv´en waves are all of the order of0 . DT S experiment suggest to us that the importance of torsional oscillationsin the Earth’s core could have been overemphasized. However, we should noteagain that, because the Alfv´en time in our experiment is of the same order asthe Joule dissipation time, torsional oscillations must be severely damped. Itwould be of great value to conduct experiments where Joule dissipation is lessimportant.Finally, we emphasize that fluctuations are very minute when the flow is un-der the combined influence of a strong magnetic field and rapid rotation (seefigure 2 and 3). Basically, the flow is entirely constrained once the magneticfield and the rotation are given. When the Elsasser number is small, or whenthe timescale of the motions is short [10], the motions are constrained to beinvariant along the rotation axis (quasi–geostrophy) but are driven by theLorentz forces. As magnetic diffusion is small at short time–scales, only theflow can modify the magnetic field, and the same holds for the buoyancy field.One is thus led to conclude that the turbulence at work in planetary cores haslittle to do with classical turbulence. Our
DT S experiment is missing someof the key ingredients of a planetary dynamo, in fact it is not a dynamo. The
V KS experiment, which does behave as a dynamo [1], is also missing key in-gredients of planetary dynamos: turbulence is almost entirely hydrodynamicand the produced magnetic field has a negligible effect on the flow. The mech-anisms of the spectacular magnetic reversals observed in
V KS [2], as those ofthe pioneer experiments of Lowes and Wilkinson [25], probably have little todo with those at work in the Earth’s core.
Laboratory experiments such as our
DT S experiment help us better under-stand the peculiar organization of fluid flow and magnetic field at work inplanetary dynamos. When the Coriolis and Lorentz forces are dominant, andthe Elsasser number less than 1, a modified Taylor state is observed, where11he geometry is given by the rotation (invariance along the axis of rotation)and the driving by the Lorentz force. In a Taylor state, the magnetic torque onco–axial tubes self–adjusts to zero, since there is no balancing torque. In ourexperiments, as foreseen by Kleeorin et al [11], Ekman friction at the externalsurface balances the magnetic torque. In this modified Taylor state, the flowremains geostrophic (azimuthal) and its variation with cylindrical radius isentirely controlled by the balance between the magnetic torque the resultingshear produces and the friction at the external boundaries. Using ultrasonicDoppler velocimetry, we measure time–averaged azimuthal velocity profiles inperfect agreement with this analysis.Under the combined constraints of the imposed rotation and magnetic field,velocity fluctuations are very minute even though the Reynolds number is inthe range 10 . When present, the fluctuations reveal a very peculiar behavior:the power spectra are dominated by peaks at several frequencies. The detailedanalysis of the signals reveal that they correspond to waves that propagate in aretrograde fashion with respect to the fluid [7]. The spectral peaks correspondto various azimuthal wave numbers. It is not yet clear what are the drivingforces of these magneto–inertial waves.We infer that turbulence at work in planetary dynamos is of a very specialkind: the flow is a complete slave of rotation and magnetic forces, Ekmanfriction at the boundaries providing a balancing torque. But the magneticfield itself is advected by the flow, and the same holds for the buoyancy field.Two main limitations of the DT S experiment prevent it of approaching regimesmore relevant for planetary situations. The first one is that it is not a dynamoexperiment, hence the magnetic field is not free to be advected by the flow.However, in most experimental dynamos, the produced magnetic field is tooweak to substantially alter the flow, in contrast with what occurs in
DT S .The second limitation is that all relevant time–scales (Alfv´en, Joule dissipa-tion, rotation, Rossby times) are in the same range (of order 0 . Acknowledgements
We are thankful to all members of the geodynamo team in Grenoble for theircontribution to the results and ideas presented in this article. We acknowledgeuseful comments from an anonymous reviewer. The
DT S project is supported12y Fonds National de la Science, Institut National des Sciences de l’Univers,Centre National de la Recherche Scientifique, R´egion Rhˆone-Alpes and Uni-versit´e Joseph Fourier.
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