arXiv:1206.4902v1 [physics.flu-dyn] 21 Jun 2012 yao nti yeo yao,a antcPrandtl magnetic at dynamos, Pm of with scale Pm type small numbers a this possi- is where In seems flow value therefore Couette dynamo. critical It spherical turbulent a [6]. that fields to ble unstable magnetic becomes up flow generate Re the not numbers does Couette Reynolds spherical contrary [5] for The the James dynamos. and on be Dudley flow to by known studied are flows which equiv- s2t1 topologically the is he- to and opposite alent hemispheres with two The the helical in is in licities sphere. It loops inner two circulation. and meridional two the rotation the differential the to of flow of back consists thus poles A fluid flow the the plane connecting outward. brings equatorial axis radially spheres the rotation centrifuged in the is along jet fluid in- a which the with by in driven together rotation sphere, differential ner a mainly is sphere Mary- in those like [4]. experiments nu- land The spherical directly flows. to are the paper applicable this of to in part presented assumed investigations averaged merical still time is the magnetic experiments from to rele- come these more contribution in is main generation action field the dynamo Nevertheless, on turbulence vant. that so of experiments, [2], effect first con- the Karlsruhe the less than flows shapes and specific Af- implement [1] to 4] fined Riga laboratory. [3, experiments the in recent in experiments more dynamo first a the reproduce ter built to been order or have stars in experiments in Several example for interiors. fluids, planetary conducting of flows in fields n yrdnmcRyod ubr o Pm For increas- number. with Reynolds increases hydrodynamic Rm ing critical number the and Reynolds scale resistive magnetic the at occurs field magnetic nraigR hnad ml otcswt magnetic with because vortices enough, small critical large the adds Re that then for so Re Re 8], of increasing [7, independent scale is scale viscous Rm some the at occurs than field larger magnetic of production largest iuain fteknmtcdnm ne fshrclCoue spherical of onset dynamo kinematic the of Simulations h peia oet o ihasainr outer stationary a with flow Couette spherical The magnetic of generation the describes theory Dynamo ASnmes 76.d 72.k 72.i 91.25.Cw 47.27.-i, 47.20.-k, 47.65.-d, spherica numbers: improve simu o PACS to forcing thickness the help boundary may of on the which layer thickness rate, that a layer rotation show necessary within boundary simulations The the sphere inten of inner regime. force effect the volume boun the a to no-slip investigate series next through second seri fluid sphere a first inner In the a smooth stationary. In a is spheres. sphere of concentric rotation two the between by gap the filling esuynmrclytednm rniino nincompress an of transition dynamo the numerically study We .INTRODUCTION I. ≫ ,telretpouto fthe of production largest the 1, rerc-udPaz1 77 ¨tign Germany G¨ottingen, 37077 1, Friedrich-Hund-Platz nttt fGohsc,Uiest fG¨ottingen, of University Geophysics, of Institute n og boundaries rough and .FneadA Tilgner A. and Finke K. Dtd aur 9 2018) 29, January (Dated: ≪ ,the 1, enlsnme e n h antcPadlnumber the Prandtl numbers, magnetic two the by and governed Re, number is Reynolds system induction The the incompressible by an equation. described in is fluid field conducting magnetic electrically the of evolution The h ue onayi trs.Tefli scharacterized is fluid The viscosity rest. kinematic at its by is boundary outer the bandb edn ldso h ne peei or- the in sphere. and inner fluid sphere the the inner between of coupling rotation the the on strengthen improvement to this blades possible der in welding the by obtained considers answer obtained IV section pessimistic section in the investigated section, of be will mag- Because self-generated question III. this a This mo- sustain whether field. to predict One netic able to be is [10]. will study metals experiment present liquid the for in tivation flow Couette spherical [7–9]. always Pm dynamo is and the Re it of on that dependence threshold so the understand Re, to larger important to extrapolate to results need numerical a always from is magnetic There enough the used high numbers. commonly above achieve Reynolds to liquid far order the values in is experiments) to (which in sodium adjusted liquid be of to Pm has Prandtl Re magnetic Pm number the number Reynolds limited, the is simulations could Since numerical flows in dynamos. Couette scale spherical small that be suspicion the further to which adds range, parameter investigated entire the for rcsee ihradii with speres tric fmgei ed e.6fudRm mixing found or 6 creation Ref. either field. to so magnetic contribute scales of not length do these they shows on that field dynamics magnetic diffusive The but 1. nothing than less numbers Reynolds R h a sdie ytertto fteinrcr,which core, inner Ω the frequency of angular rotation at the by rotates driven is gap the i neprmn sudrcntuto hc realizes which construction under is experiment An h ytmudrivsiaincnit ftoconcen- two of consists investigation under system The − R I H AHMTCLMODEL MATHEMATICAL THE II. o yaoexperiments. dynamo l e osmlt og ufc drives surface rough a simulate to ded = so iuain,tefli sdriven is fluid the simulations, of es yaotrsodi h turbulent the in threshold dynamo aigteruhsraelwr the lowers surface rough the lating d aycniin,weesteouter the whereas conditions, dary n setratio aspect and etnho h a it.We width. gap the of tenth ne beeetial odcigfluid conducting electrically ible R i t oswt smooth with flows tte and ν n t antcdiffusivity magnetic its and R R i o i bu the about /R omn a fwidth of gap a forming c o oices ihRe with increase to 1 = / .Tefli in fluid The 3. APS/123-QED z − xs while axis, λ . 2

Pm or alternatively the magnetic Rm This forcing qualitatively reproduces the flow driven defined by by an inner sphere with blades of height one tenth the 2 gap size mounted along meridians. The non-dimensional Ωid ν ′ Re = , Pm = , Rm = Re Pm. (1) rotation rate of the inner sphere, Ωi, has to be found a ν λ posteriori in these simulations by computing From now on, only adimensional variables will be used in this paper. Choosing as units of time and length the 2π π ′ 3 2 reciprocal of the inner core’s rotation rate and the gap Ωi = dϕ dθ sin θhvϕ(r = ri,θ,ϕ,t)i (8) width, the induction equation reads for non-dimensional 8πri Z0 Z0 B v magnetic and velocity fields and : with which a Reynolds number Re′, analogous to Re for the no slip boundaries, is defined as 1 2 ′ ′ ∂tB + ∇× (B × v)= ∇ B , ∇ · B =0. (2) Re Pm Re =Re Ωi (9) The velocity field itself is a solenoidal vector field deter- In the following Re is going to be increased up to mined by the Navier-Stokes equation and the continuity 1.67 · 104 for the no-slip boundary conditions and up to equation: 2.5 · 103 in the simulations utilizing the volume force. The magnetic is of order unity for the 1 simulations just at the onset of magnetic amplification. ∂ v + (v · ∇)v = −∇Φ+ ∇2v + F , ∇ · v =0. (3) t Re The numerical method is a spectral method, in which the fields are expanded in spherical harmonics and Cheby- Eq. (3) contains a volume force F and Φ stands for chev polynomials [11]. the pressure variable. Eqs. (2) and (3) describe the kine- matic dynamo problem, in which one assumes the system to be near the onset of magnetic field generation, so that III. NO-SLIP the magnetic field strength is small and the Lorentz force is negligible in equation (3). A. Hydrodynamic Characteristics The parameters Re and Rm depend on the inner core’s rotation rate. More revealing parameters are the The hydrodynamic properties of the spherical Cou- Reynolds numbers Re and Rm based on the rms velocity ette flow with stationary outer sphere have already been vrms defined as investigated in detail in [6, 12, 13]. At low Reynolds 1 2 numbers the basic spherical Couette flow is axisymmet- vrms = 2Ekin/V , Ekin = h v dV i (4) 2 ric and we find a critical Reynolds number of Reh = 1500 p Z (Reh = 105) beyond which small non-axisymmetric per- and turbations increase and an instability develops as a prop- agating wave on the equatorial jet with a dominant az- Re = Re vrms , Rm = Re Pm (5) imuthal wavenumber m = 2, which is in agreement with where V is the volume of the shell, brackets denote time [6, 13]. At Res = 2800 (Res = 178) a second transi- average and Ekin the kinetic energy. tion occurs. Beyond this value, the amplitudes of odd In the following the effect of different surface proper- wavenumbers m of the kinetic energy develop as well ties on the dynamo threshold are compared. The inner and the power spectrum begins to flatten and approaches −5/3 −5/3 and outer spheres have radii ri = 1/2 and ro = 3/2, power laws in m and l , indicative of Kolmogorov respectively. The inner sphere and the space surround- turbulence (see Figure 1). Large scales of the velocity ing the outer sphere are assumed to be insulating. Two field, however, have comparable spectra for all Re. The different types of boundary forcing are implemented in bottom panel shows the power spectra of an estimate of order to simulate both smooth and rough surfaces. For the turbulent rate of strain of spherical harmonic order 2 the smooth surface no-slip boundary conditions l, which is simply the kinetic energy multiplied by l and has its maximum at the viscous scale. At Re = 970 the v z r v = ˆ × at r = ri , =0 at r = ro (6) inertial range reaches up to l ≈ 40. The wavenumber of the most unstable mode depends are chosen together with F = 0 in eq. (3). In the second on the aspect ratio of the shell and switches from m =2 case, the inner boundary is assumed free slip and the to m = 3 at an aspect ratio close to the one chosen here fluid is driven in one tenth of the gap width by a toroidal [13]. It is possible to generate flows with m = 3 as domi- force field F of spherical harmonic degree l = 1 and order nating mode by starting from an equilibrated solution at m = 0, given in spherical polar coordinates (r, θ, ϕ) by: Re > Res and lowering Re to some value in between Reh and Res. 1 60 d Figure 2 (top) shows the thickness of the boundary F = 1 − tanh (r − ri − ) sin θ eˆϕ. (7) 2   d 10  layer of uϕ near the inner core. In order to produce this 3

0.04

d −1/2 B Re 0.02

0 0.01 0.02 0.03 Re−1/2 0.04

E kin 0.02

0 −4 −3 10 10 Re−1

FIG. 1: (Color online) Power spectrum of kinetic energy plot- −2 m 10 ted against spherical harmonic degree (top) and the turbu- Re−0.62 lent rate of strain versus spherical harmonic order l (bottom) ε for Re = 300 (black continuous line) and 970 (thick red dashed − line), together with the Kolmogorov power laws m 5/3 and l1/3 (blue dashed line).

−3 10 3 4 10 10 figure, uϕ has been averaged over spherical surfaces and Re the boundary layer thickness defined as the distance from the inner sphere at which this averaged velocity drops to FIG. 2: (Color online) Boundary layer thickness dB plotted − − the arithmetic mean of its value at the boundary and its against Re 1/2 (top), kinetic energy plotted against Re 1 value averaged over radius. At high Re the evolution of (middle), and energy dissipation rate plotted against Re (bot- −1 2 the boundary layer thickness is approaching Re / , as tom). The vertical dashed lines mark Reh and Res. one would expect from theory. Figure 2 (middle) shows the dependence of the dimen- sionless kinetic energy on Re, which converges to a con- related to the energy dissipation ǫ by ǫ = hτi/Re. As −1 −0 62 stant in the limit of Re → 0. The viscosity becomes shown in fig. 2, the dissipation scales as Re . . The irrelevant for high Re and the only remaining control pa- bracket in the integrand in eq. (11) is dominated by the rameters entering the dimensional kinetic energy are the derivative ∂rvϕ which can be estimated as vrel/dB, where 2 rel density and Ωi, so that the kinetic energy scales with Ωi dB is the boundary layer thickness and v the velocity of and the dimensionless kinetic energy becomes constant. the inner boundary relative to the bulk fluid. Since dB ∝ −1/2 −1/2 A quantity of direct relevance to experiments is the Re , we have ǫ ∝ Re vrel. If the motion of the energy dissipated in the flow. The energy budget, ob- inner core was completely decoupled from the rotation tained by taking the scalar product of eq. (3) with v and of the inner boundary, vrel would be independent of Re −1 2 integrating over space, reads and ǫ ∝ Re / . There is of course some entrainment of the fluid by the rotation rate of the inner sphere and one finds an exponent of -0.62 instead of -0.5. 1 2 1 1 2 ∂t v dV = τ − (∂ivj ) dV + F ·vdV (10) Z 2 Re Re Z Z with B. Dynamo transition 2π π 3 2 vϕ τ = − dϕ dθr sin θ ∂rvϕ − , (11) The dynamo transition for the spherical Couette flow Z0 Z0 r   with stationary outer sphere has already been computed evaluated on the inner boundary r = ri, being the torque in [6] with different magnetic boundary conditions and on the inner boundary. F = 0 for the simulations in this an aspect ratio of ri/ro = 0.35. These results will be section so that the time averaged torque hτi is directly compared with ours in the last section. Figure 3 shows 4 the dynamo simulations in the (Re, Rm)−plane. Aster- of Re, but neither the frequency of the oscillation of the isks mark working dynamos and dots are failed dynamos. magnetic energy nor the growth rate agree exactly. How- Linear interpolation between the growth rates computed ever, the amplitude of the m = 4 contributions to the at those points allows us to find the locus of zero growth kinetic energy are only one order of magnitude smaller rate. The onset of magnetic field amplification obtained than the dominant mode and they introduce a slight time in this way is indicated by the dashed line. The mag- dependence in the codrifting frame of reference, so that netic fields are dominated by modes with an azimuthal the simulations of the drifting snapshot can only catch wavenumber of m = 2. As mentioned above, suitable the qualitative features of the full simulation. initial conditions lead to magnetic fields dominated by 0 modes with m = 3. The onset for these dynamos is in- 10 Em dicated by the thick solid line. The two lines coincide kin Em for Re > Res. The straight line indicates Pm = 1. Ex- B cept for the highest Re the is always larger than one. In agreement with [6] there −5 are no working dynamos for axisymmetric flows. For 10 Reh < Re < Res the critical magnetic Reynolds num- ber Rmc for m = 2 first increases, reaches a maximum and finally decreases, until at the second transition odd −10 modes of the kinetic energy become unstable and the dy- 10 0 1 namo threshold again increases. 10 10 m 0 10 Em kin Em B

−5 10

−10 FIG. 3: (Color online) Dynamo transition for magnetic fields 10 0 1 dominated by modes with m = 2 (black dashed line) and 10 10 m m = 3 (blue bold line). Failed dynamos are indicated by red dots, working dynamos by blue asterisks. The vertical dashed lines seperate the different hydrodynamic regimes: Axissym- FIG. 4: (Color online) Power spectrum of the kinetic energy metric flow (left), first nonaxisymmetric instability (middle) (black continuous line) and the magnetic energy (red dashed and turbulent regime (right). line) for Re = 130 (top) and Re = 160 (bottom)

The non-monotonous dependence of Rm on Re in the 0 interval Reh < Re < Res appears to be related to two 10 m E other features of the dynamo: The drift frequency of the kin Em unstable mode and the dominating wavenumber in the B magnetic field. The velocity field for Reh < Re < Res is dominated by −5 one azimuthal wavenumber and its harmonics. Time se- 10 ries of the radial velocity field at a fixed point show that the phase velocity of the propagting wave diminishes for increasing Re. Similarly to previous studies of kinematic dynamos [14, 15], it seemed useful to run simulations in −10 10 which the velocity field is a snapshot taken from the full 0 1 10 10 simulation, and which is set to drift at arbitrary phase m velocities in order to investigate the dependence of the growth rate on the phase velocity of the instability. It FIG. 5: (Color online) Power spetrum of the kinetic energy turned out that the critical magnetic Reynolds number (black solid line) and the magnetic energy (red dashed line) increases with the phase velocity, as does Rmc in Figure for Re = 130 in a simulation in which modes with m = 3 3. The time evolution of the magnetic energy using the dominate. artificially drifting velocity field shows the same charac- teristic superposition of an exponential growth and an In Figure 4 power spectra of the kinetic and magnetic oscillation as in the dynamic simulations in that region energy of the simulations at Re = 130 and Re = 160 are 5 shown. In the first case only even wavenumbers of the doubts because the spectrum of magnetic energy reaches magnetic energy are amplified, whereas odd wavenum- a second local maximum at l = 11 in this case, which bers decreases in time. The opposite is seen in the sec- nevertheless is still within the inertial range (see Fig. 1) ond case, where odd wavenumbers of the magnetic en- and at any rate below the global maximum at l =1. Pm ergy are amplified and even wavenumbers decrease. The is already slightly below 1 in this computation so that a different symmetry of the magnetic field seems to be re- further increase in Re should only add a high wavenum- sponsible for the decrease of the dynamo threshold, since ber tail to the kinetic energy spectrum corresponding to this decrease correlates with the appearence of the odd eddies with too small a magnetic Reynolds number to wavenumbers in the magnetic field. In addition, Figure contribute to the dynamo effect. In order to further test 5 shows the spectra of the simulations with m = 3 as the whether the small scales are responsible for magnetic field dominant wavenumber, which have for Reh < Re < Res generation, eqs. (2,3) were solved simultaneously, but the the same symmetry and accordingly the dynamo thresh- axisymmetric components of the velocity were removed old is monotonously increasing. before the induction term in eq. (2) was computed. The For Re > Res, the critical Rm first increases but then magnetic field decayed in this simulation, which shows reaches a plateau at Rm = 600. Pm is less than 3 that the turbulent eddies in this velocity field are unable throughout the plateau region. It will now be argued to support the magnetic field by themselves. that even more turbulent flows will not alter the critical Rm significantly. Since the magnetic field is generated at large scales even for Pm ≈ 1, an increase in Re and concomittant decrease in Pm adds scales which contribute neither to the creation nor the destruction of the magnetic field. It is concluded that Rm = 600 is the critical value of Rm in more turbulent flows as well. In an experiment with Pm = 10−5 this corresponds to Re = 6 × 107. For Reynolds numbers this large, Ekin is nearly inde- pendent of the Reynolds number according to fig. 2 and equal to 0.023, from which one deduces vrms =0.058 and Re ≈ 109. For the sodium experiment in Maryland with a gapwidth of d = 1m and a viscosity ν ≈ 10−6m2/s, this corresponds to a rotation period of about only 6ms.

To conclude this section, we compare the stability di- agram in fig. 3 with the results of similar studies. All numerical simulations report a general increase of the critical Rm with the Reynolds number for Pm larger than 1, an effect which is also found in the analytical model of ref. [16]. The dependence of the dynamo threshold on the Reynolds number is not monotonous, however, and other published results have in common with fig. 3 a FIG. 6: (Color online) Power spectrum of magnetic energy of maximum in the critical Rm for Pm around 1. This fea- spherical harmonic degree m (top) and order l (bottom) for ture is seen in refs. [8, 9]. Explanations for this effect Re = 300 (black continuous line) and 970 (red dashed line). have been attempted in refs. [17, 18] who attributed it to either the bottleneck effect in turbulent spectra or the helicity at the viscous scale in the velocity field. In the plateau region, the Reynolds number is high enough for the kinetic energy spectrum to decay as a power law as a function of both m and l with an exponent Despite the similarities, it is doubtful whether there is of -5/3. The phenomenolgy developed for small scale dy- a universal mechanism for the dependence of the dynamo namos in Kolmogorov turbulence [7, 8] therefore applies. threshold. For example, the plateau in Rm is reached for According to this phenomenology, Rm can increase as a values of Pm ranging from 0.1 to 1 across the different function of Re if Pm is larger than 1 and the eddies near studies, so that there are variations by an order of mag- the viscous length scale generate the magnetic field. If nitude in the Pm at which the critical Rm is maximum. either Pm is small or the field generation occurs whithin There is also no indication of a bottleneck in the spec- the inertial range or at the integral scale, Rm is indepen- tra presented here which shows that the bottleneck is dent of Re. In spherical Couette flow, the large scales are not necessary for the observed variation of the dynamo already capable of dynamo action. Not surprisingly, the threshold. It remains an open question why there is a spectra of magnetic energy peak at small m and l (see maximum in the critical magnetic Reynolds number as Fig. 6). Only the computation at the largest Re raises shown in fig. 3 in the spherical Couette system. 6

IV. ROUGH SURFACE

A. Hydrodynamic characteristics

The critical Re found in the previous section is chal- lenging to realize in experiments. The problem arises from the fact that the inner sphere needs to rotate faster in order to increase Re. But a faster rotation rate also thins the boundary layer at the inner sphere and reduces the volume of fluid directly coupled to the boundary mo- tion. An obvious remedy is to mount blades on the inner surface. The boundary layer cannot be smaller than the surface roughness which ensures a better coupling of the fluid to the inner sphere at high rotation rates. In the following, the blades are modelled by a volume force as described in section II. Due to the increased momentum transport, turbulence develops at lower rotation rates. The first nonaxisym- metric instability arises at Reh = 425 (Reh = 95) with a dominant wavenumber of m = 2, and beyond Res = 465 (Res = 108) odd wavenumbers increase, too. The evolu- tion of the boundary layer thickness is shown in Figure 7 (top). As expected, the thickness of the boundary layer does not drop below one tenth of the gap width and ap- proaches the thickness of the forced layer at large Re. Since the thickness of the layer in which momentum is injected into the fluid remains nearly constant, indepen- dend from the rotation rate, the kinetic energy increases 1/2 with Re. It does so approximately in Re , which is shown in Figure 7 (middle). The torque on the inner sphere is zero for free slip FIG. 7: (Color online) Boundary layer thickness dB plotted boundary conditions so that one deduces from the energy − −1 against Re 1/2 (top), kinetic energy plotted against Re budget (10) that the dissipation rate is given by ǫ = ′ F · vdV . As can be seen in the bottom panel of fig. 7, (middle), and energy dissipation rate plotted against Re (bot- ǫ behaves differently depending on whether the boundary tom). The vertical dashed lines show transitions to different R regimes. Results obtained for smooth and rough surfaces are layer is thicker than the forced layer or not. The interval indicated by the symbols × and +, respectively. of Reynolds number in each regime is too small to deduce ′ a law relating ǫ with Re .

′ B. Dynamo transition Ωi =9.4.

The diagram of the dynamo threshold plotted in the (Re, Rm)−plane in Figure 8 is similar to Figure 3. There is no dynamo action for axisymmetric flows. Similar to the simulations with no-slip conditions, even wavenum- bers (m = 2 and harmonics) increase in the magnetic energy spectra, whereas the total magnetic energy is in- creasing exponentially with superposed oscillations. In contrast to the simulations of the previous section, it was not possible to find a case in which the odd wavenumbers dominate the magnetic spectrum. Like in the simulations with no-slip conditions the dynamo threshold shows a plateau at Rm = 600. In experiments with Pm = 10−5 FIG. 8: (Color online) Dynamo transition (black dashed line) this corresponds to Re = 6 × 107. By extrapolating the with the same symbols as in figure 3. kinetic energy in Figure 7 (bottom) up to this Re, one ′ 7 gets Ekin ≈ 300 and vrms ≈ 6.6, or Re ≈ 9 × 10 with 7

plotted in terms of Reynolds numbers, magnetic and hy-

4 drodynamic, based on the rms velocity rather than the 10 boundary velocity. The data are more difficult to inter- pret and extrapolate if they are given as functions of the Rm Reynolds numbers computed with the inner boundary velocity as seen in Figure 9. This plot also includes the data from ref. [6], which should be compared to our sim- smooth rough ulations with smooth boundaries. The curves are broadly 3 10 C.Guervilly similar apart from a shift by roughly a factor 1.5 along P. Cardin the Re−axis, which must be attributed to the different 3 4 10 10 aspectio ratio and the different magnetic boundary con- Re ditions.

FIG. 9: (Color online) Dynamo threshold of both surface types and results of [6]. If the flow is driven by the volume force which simu- lates blades of height one tenth of the gap size mounted on the inner sphere, the dynamo threshold expressed as V. CONCLUSION critical magnetic Reynolds number based on the rms ve- locity is reduced by one sixth. Such a change is plau- Laminar spherical Couette flow with a stationary outer sible because the topology of the flow is the same as sphere is axisymmetric and consists of a differential ro- for smooth boundaries, but the kinetic energy is more tation superimposed on a meridional circulation, which evenly distributed in space. Both the boundary layer flows from the inner to the outer sphere in the equato- and the equatorial jet are thicker. From an experimen- rial plane, continues towards the poles of the outer sphere tal point of view, the much more important improve- along the outer boundary in each hemisphere, and finally ment brought about by the blades is the reduction of the returns to the inner sphere. Dudley and James [5] showed rotation rate of the inner sphere at the dynamo onset. that flows of this type are capable of dynamo action. At The critical Reynolds number based on the rotation fre- the magnetic Reynolds numbers accessible by numerical quency of the inner sphere is reduced by a factor 10 by computation, laminar spherical Couette flow nonetheless the blades. Using the results obtained with the simu- is not a dynamo due to the unfavourable velocity pro- lated blades and the numbers for the Maryland experi- file. For instance, the outward flow is concentrated in ment with a gapwidth of one meter and the viscosity of a narrow jet in the equatorial plane. Unstable spherical liquid sodium ν ≈ 10−6m2/s, a Reynolds number num- ′ Couette flow on the other hand readily generates mag- ber of Re = 9 × 107 is reached for a rotation period of netic field. However, the field generation process still the inner sphere of 0.07 s. A spherical Couette dynamo occurs at large scales, the small turbulent scales on their is more readily attainable with blades mounted on the own do not support the magnetic field in these dynamos. inner sphere because they improve the coupling between The numerical data allow us to extrapolate to more the fluid and the motion of the inner sphere and at the turbulent flows than the simulated flows if the data are same time modify little the structure of the flow.

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