Structures and Lagrangian statistics of the Taylor–Green dynamo

The evolution of a Taylor–Green forced magnetohydrodynamic system showing dynamo activity is analyzed via direct numerical simulations. The statistical properties of the velocity and magnetic fields in Eulerian and Lagrangian coordinates are found to change between the kinematic, nonlinear and saturated regime. Fluid element (tracer) trajectories change from chaotic quasi-isotropic (kinematic phase) to mean magnetic field aligned (saturated phase). The probability density functions (PDFs) of the magnetic field change from strongly non-Gaussian in the kinematic to quasi-Gaussian PDFs in the saturated regime so that their flatness give a precise handle on the definition of the limiting points of the three regimes. Also the statistics of the kinetic and magnetic fluctuations along fluid trajectories changes. All this goes along with a dramatic increase of the correlation time of the velocity and magnetic fields experienced by tracers, significantly exceeding one turbulent large-eddy turn-over time. A remarkable consequence is an intermittent scaling regime of the Lagrangian magnetic field structure functions at unusually long time scales.


introduction
The magnetic field of stars and telluric planets is explained by the dynamo instability, produced by a turbulent conducting fluid where the induction due to the motion takes over the magnetic diffusion.A system with dynamo action passes through different stages from the linear (kinematic) to the non-linear and finally to the saturated phase.During the kinematic phase the magnetic energy grows exponentially but has no influence on the flow.During the non-linear phase the Lorentz force changes the flow and at the saturation state the diffusion and the electromotive force reach an equilibrium.The system reaches the fully non-linear magnetohydrodynamic (MHD) regime, driven by an energy exchange between the velocity and magnetic field.
In the last decade, several experimental groups have investigated dynamo action in laboratory experiments using liquid sodium [1,2,3,4,5,6].The instability threshold and a rich non-linear behavior along the saturation regime have been largely observed [7,8,9,10].Investigations of the fast kinematic dynamo [11] in the 90's, showed the important role of the chaotic properties of the fluid trajectories for the dynamo threshold.The effect of turbulent fluctuations on the dynamo onset has also been studied in Navier-Stokes flows [12,13] and some noisy models [14], distinguishing the important role between the mean flow dynamo and fluctuation dynamo modes [15].
The transition between the linear and the saturation regime has been studied measuring the finite-time Lyapunov exponent of the flow trajectories [16,17].These results showed a strong reduction of the chaotic properties in the saturation phase, due to the action of the Lorentz force.Fluid trajectories thus change along the different dynamo phases.This naturally motivates the use of a Lagrangian description where the properties of the flow are investigated by using tracers [18].
Lagrangian statistics describe the dynamical evolution of physical quantities along tracer trajectories in contrast to the Eulerian perspective in which such quantities are analyzed on fixed spatial points.During the last decades Lagrangian studies revealed new aspects of homogeneous and isotropic turbulence [19].The trajectory point of view has shown to be especially useful for the study of coherent structures and intermittency [20,21,22] as well as for diffusion and dispersion properties of hydrodynamic and homogeneous and isotropic MHD turbulence [23,24,25].This approach has produced many experimental results by tracking solid inertial particles or bubbles [26,27,28,29], which can have a different dynamics compared to simple tracers.Lagrangian statistics has also been used in MHD simulations [30,31] to compare the anomalous exponents of the structure function to their hydrodynamic counterparts and to understand the relation between Eulerian and Lagrangian statistics.
In this study we use a Lagrangian approach in the context of turbulent dynamo action.In the Kraichnan-Kazantsev framework it is used to determine the dynamo onset as a function of the roughness of the flow [32].In this work the Lagrangian perspective allows us to precisely define the limiting points of each regime, to highlight the correlation of the magnetic field and the fluid trajectories and to discover a to our knowledge unknown scaling regime at time scales well beyond the large-eddy turn-over time of the turbulent flow.
The dynamo systems under consideration in this work is induced mainly by the so called Taylor-Green forcing [33].For comparison, a second forcing where the large scales of the velocity field are frozen in time is also considered.The Taylor-Green flow, which can produce dynamo action, is a very well documented study case involving rich dynamo behaviors in both the linear and the non-linear regime [34,35,36,37,12,14,38,39]. The Taylor-Green flow contained in a periodic box has several properties that mimic the von Kàrmàn flow, driven by two counter-rotating impellers.The von Kàrmàn flow has a strong experimental history inside the turbulent and the dynamo scientific communities.Indeed, several teams set up such experiments with different designs in incompressible [40,41] and compressible flows [42,43,44,45,46,47].This type of experiment was also one of the first setup used to study Lagrangian statistics of turbulent flows [26,27,28,29].After the intensive water campaign experiments of the Saclay group [48,49,50,51,52], this setup has lead to the von Kàrmàn Sodium (VKS) dynamo campaigns also cited above.
This paper is organized as follows: In the next section the dynamo system under consideration and the numerical method are explained.In section 3 the evolution of the flow structure and trajectories within the three different regimes are discussed.Section 4 investigates the changes of the velocity and magnetic field fluctuations from one regime to another.A magnetic field scaling regime at very large temporal scales is reported in section 5. Conclusions are drawn in section 6.

Model and methods
We perform direct numerical simulations of turbulent MHD flows with large-scale forcing in a periodic box.We integrate the three-dimensional incompressible MHD equations that, expressed in Alfvénic units, read where ν and η are the kinematic viscosity and the magnetic diffusivity respectively.The density of the fluid is set to unity and F is the constant volume force which generates and maintains the turbulent flow.For most simulations (see table 1) we consider the so called Taylor-Green (TG) flow that is generated by forcing with the TG vortex with F 0 = 3 and k 0 = 1 or k 0 = 2.When k 0 = 1, this forcing leads to a subdivision of the total domain in eight fundamental boxes that, when symmetries are preserved, can be related to each other by mirror-symmetric transformations.Each fundamental box contains a swirling flow composed of a shear layer between two counter-rotating eddies.The TG flow mimics some aspects of the von Kàrmàn flow, largely used in hydrodynamics turbulence and dynamo action experiments.As we will see in the following sections, the TG dynamo action presents some particular properties due to its anisotropy.In order to distinguish universal and non-universal properties of the TG dynamo action we also consider a mechanical force F (see run ffH and ff in table 1) obtained by keeping constant all Fourier modes of the velocity field in the two lowest shells (henceforth called frozen force simulation (FF)).This force is much less structured than the TG forcing and the corresponding flow is found to be nearly isotropic.
We use classical global quantities to characterize the different dynamo phases.The kinetic energy E kin , the magnetic energy E mag , the enstrophy Ω, the cross helicity H C , and the magnetic helicity H M are defined as where stands for spatial average and B = ∇ × A with A the magnetic potential.In the ideal case (ν = η = 0) and without forcing (F = 0), E tot = E kin + E mag , H C and H M are conserved by the MHD equations.
The TG forcing (4) is purely horizontal, leading to anisotropy in both the hydrodynamic and the MHD regime.In order to quantify this anisotropy we compute the root mean square (rms) values of the perpendicular (xy-plane) and parallel (z) components of the velocity fields and define the global isotropy coefficient ρ iso u as With these definitions the average rms velocity is u rms = [(2 (u rms ⊥ ) 2 + (u rms ) 2 )/3] 1/2 .We use analog definitions for the magnetic field.
There are also three important dimensionless numbers, namely the kinematic Reynolds number Re, the magnetic Reynolds number Rm and the magnetic Prandtl u rms : large-eddy turnover time.For all runs: size of domain = 2π.For run ffH and ff only the mean rms value of the three components u rms ⊥ = u rms = u rms is listed which differ by less than 10%.
number P m defined as A useful quantity that measures the alignment of velocity and magnetic fields is the normalized cross helicity defined as Numerical integration of the MHD equations (1-3) is carried out by using a fully MPI-parallel pseudo-spectral code (LaTu [53]) with a strongly stable third order Runge-Kutta temporal scheme.De-aliasing is performed using the standard 2/3 rule.
In this work we are concerned with the Lagrangian aspects of dynamo action.Lagrangian statistics are obtained by tracers obeying the equations where X(x, t) and v(t) are the position and velocity of a tracer which started x for t = 0.The magnetic field along a tracer path is denoted by b(t) = B(X(x, t)).The values of the velocity fields and other physical quantities at the particle positions are evaluated and stored using a tricubic interpolation which is numerically efficient and accurate [54].The equation ( 6) is integrated with the same third order Runge-Kutta scheme as the equations (1) for the velocity and magnetic fields.Statistical data is obtained from a large number of tracer trajectories varying from 2 • 10 4 to 2 • 10 6 .A detailed list of the physical parameters for the different runs is presented in table 2.

Dynamo action: structures and trajectories
During dynamo action, three phases are clearly distinguished.The linear phase, where a seed of magnetic field grows exponentially by the dynamo instability without backreaction on the velocity field.The second stage, called the non-linear phase, corresponds to the one where the Lorentz force starts to act on the flow and the kinetic energy slightly drops by the transfer of kinetic energy to the magnetic field.Finally the full MHD system reaches a statistically stationary state, refereed to as the saturation regime.In Fig. 1 (left) the kinetic and magnetic energy is displayed for the TG flow.The three phases of the dynamo action are separated by dashed vertical lines.In Fig. 1 (right) the enstrophy is shown for the same simulation.When the magnetic energy attains a magnitude of the order of one percent of the kinetic energy the enstrophy drops.This drop is larger the smaller is P r (or Rm) that is the closer one is to the dynamo onset.This rapid change of Ω marks the transition between the linear phase and the non-linear phase.The saturated regime is characterized by strong fluctuations in both energies and enstrophy.Note in Fig. 1 (right) that the normalized cross helicity h C starts to fluctuate in the non-linear phase, showing a tendency towards an alignment of the u and B fields.We find h 2 C 1/2 = 0.03 during the kinematic phase and h 2 C 1/2 = 0.23 during the saturated phase.This alignment leads to a less efficient global electromotive force u × b.In the sequel we will present another quantity whose statistics strongly changes from one regime to another and which allows for a more precise definition of the different phases of dynamo action.This quantity is in fact used to plot the vertical dashed lines in Fig. 1.
As mentioned before, the TG flow presents a manifest anisotropy coming from the very definition of the forcing.In the hydrodynamic regime we find a global isotropy coefficient ρ iso u = 0.74 (see table 2).Anisotropy is even enhanced to ρ iso u ≈ 0.36 in the saturation regime, where large scale magnetic coherent structures appear in the two shear vortex plans and large tubes of kinetic energy are along the diagonal direction (see Fig. 2).We note that for the frozen force dynamo ρ iso u is close to unity during all phases of the simulation.
We now turn to the geometry of tracer trajectories.The linear phase is characterized by chaotic trajectories showing spiral motions around vorticity filaments In the saturated phase the flow structure and the geometry of the trajectories change (see top right panel of Fig. 2).Vorticity filaments of the kinematic phase are quenched to vortex sheets in the saturated phase (see inset).The magnetic field has an ordering and smoothing effect (see multimedia supplementary material).Trajectories become highly aligned with the structures of the mean (time-averaged) kinetic and magnetic energy.These large-scale structures, well known in the literature, interconnect the fundamental boxes of the Taylor-Green flow.In the vicinity of the diagonally oriented high kinetic energy tubes, we observe streams of tracers with high velocities.They are separated by regions of chaotic motions (see Fig. 2 (bottom left)) which coincides with regions of low mean magnetic energy.From the bottom right panel of Fig. 2 one sees that an approximately isotropic tracers motion of the fluid is preserved perpendicular to the diagonal structures.For the frozen force (run ff), there is no large scale structure traversing the fundamental box (not shown).Blobs of increased magnetic energy are randomly distributed all over the cube.Tracer trajectories are in this case statistically homogeneous and isotropic.

Eulerian statistics
A turbulent flow naturally leads to magnetic field fluctuations which originate from stretching, twisting and diffusion of magnetic field lines.The statistical properties of these fluctuations change significantly from one regime to another.Especially those of the magnetic field whose probability distribution functions (PDFs) are shown in Fig. 3.The data of the saturated Taylor-Green regime is from run tg3.
In the linear phase, the PDFs of the perpendicular magnetic field components present fat tails, far away from the Gaussian distribution (Fig. 3 left).Even at moderate Reynolds numbers we observe fluctuations of the magnetic field which are 15 times larger than its root mean square value.We emphasize that this non-Gaussian character of the magnetic field during the linear phase is also observed for the frozen large scale forcing (run ffH), as apparent in Fig. 3 (left).It has also been observed by a model using the Recent Fluid Deformation closure [55].This strongly suggests that the non-Gaussianity of the growing magnetic field is a generic property of the linear phase.In contrast, the velocity field has nearly Gaussian statistics (see Fig. 3 right) as usually observed in hydrodynamic turbulence.
When the flow enters the saturated phase the normalized PDFs of the velocity and magnetic field components become similar.The tails of the perpendicular component b ⊥ of the magnetic field PDF shrink to quasi-Gaussian tails (Fig. 3 left).The perpendicular component of the velocity field v ⊥ remains Gaussian.Such magnetic-field PDFs along the saturated regime have also been observed in the VKS experiment [7].The parallel component of the velocity field seems to follow the magnetic field: The PDF of b remains non-Gaussian and that of v develops non-Gaussian tails.A reason for this matching of velocity and magnetic field PDFs might be the increased alignment of v and b in the saturated regime which couples kinetic and magnetic fluctuations.Scale dependent alignment has been recently evoked in a model of the scaling properties of MHD turbulence [56].
In order to estimate the non-Gaussianity of a field f , the temporal evolution of its flatness f 4 / f 2 2 is measured.The flatness of the magnetic field starts from a large value (see Fig. 4) originating from their stretched tails.During the non-linear phase its flatness strongly reduces and reaches a slightly sub-Gaussian value for the perpendicular component b ⊥ while it is super-Gaussian for the parallel component b .A significant change can also be observed for the flatness of the velocity field PDFs (see also Fig. 4).The perpendicular component v ⊥ starts slightly sub-Gaussian and fluctuates around the Gaussian value 3, while the flatness of the component v is clearly larger and comparable to that of b during the saturated phase.These changes in the behavior of the flatness temporal evolution allow to clearly distinguish the three phases of the dynamo action indicated by the vertical lines drawn in Fig. 1 and 4).We now turn to the statistics of a small scale quantity of the flow, namely the fluid acceleration, whose PDFs are displayed in Fig. 5 (left).Contrarily to the PDFs of the velocity, a large scale quantity, there are no significant differences among perpendicular and parallel components.This is agreement with the general observation that anisotropy of large scales decreases towards the smaller scales in turbulent flows.In order to improve statistics, the three acceleration components have been averaged.The acceleration has strongly non-Gaussian tails as usually observed in turbulent flows [57] and decrease from the kinematic to the saturated phase (see Fig. 5 (left)).This is an agreement with the known fact that the saturation regime smoothes out the small scales of the velocity.Indeed, the rms acceleration reduces from a rms = 80 for the hydrodynamic phase to a rms = 16 and a rms = 10 for the saturated phase for P m = 1 and P m = 1/4, respectively.Smaller magnetic Prandtl numbers imply thus smaller accelerations.However, when comparing the normalized PDFs (normalized to standard deviation, Fig. 5 (right)), more extreme accelerations events are observed for the saturated regime than the pure hydrodynamic case.The tails of the PDFs become significantly fatter.The flatness of the PDFs reaches 25 in the saturated regime, while it is 14 in the hydro case.Here, the Prandtl number has a negligible effect.The normalized PDFs distinguish therefore clearly the saturated from the kinematic dynamics.

Lagrangian statistics
We now turn to study the properties of the magnetic field along fluid element trajectories, i.e. the magnetic field seen by a tracer.We will first focus on changes in the perpendicular component of the magnetic field, the dominant component of the mean field.For this, we consider the temporal increment δb ⊥ (τ, t, x) = B ⊥ (X(x, t + τ ), t + τ ) − B ⊥ (X(x, t), t) where B ⊥ denotes a magnetic field component from the perpendicular plane (B x or B y ) and X(x, t) a fluid trajectory starting at x at a time t.To improve statistics we average over x and t which leads to the study of the magnetic field increment The normalized PDFs of the increment δb ⊥ during the saturated phase are shown in Fig. 6 (left).Stretched tails were also observed for small scale magnetic fluctuations in the solar wind [58].With increasing time-lag τ the PDFs show a transition to Gaussian statistics for large time lags.This is simply due to time decorrelation of b ⊥ (τ ) and b ⊥ (0) so that one recovers for large τ the one-point Eulerian PDF shown in Fig. 3

(left).
It is important to stress that the time scale τ at which this decorrelation happens is unusually long as it exceeds significantly one large-eddy turn-over time.The associated long time regime will be analyzed in detail in the subsequent section.
Another type of Lagrangian increment has recently attracted interest in the hope of a better understanding of the energy cascade and energy flux in turbulent flows [59], namely the kinetic (and magnetic) energy increment In Fig. 6 (right) the corresponding PDFs are shown for a short time lag τ .The mean of all PDFs but that of the magnetic field energy during the linear phase is zero.Only the mean of the latter is positive because of the exponential increase of magnetic energy.More interesting is their skewness S f = f 3 / f 2 3/2 .For the hydro case the PDF of the kinetic energy has a negative skewness of S v ≈ −0.4.We remark that the Lagrangian derivative d|u| 2 /dt can be written in terms of Eulerian quantities d|u| 2 /dt = 2(|u| 2 ∇ u u + u∂ t u), where ∇ u u is the longitudinal velocity gradient ∇ r u ≡ r • ( r • ∇u) evaluated in the direction of the local velocity û. ∇ r u averaged over r is known to have negative skewness close to the one measured for d|u| 2 /dt.A negative skewness means that violent decelerations are more probable than violent accelerations.The skewness of the velocity PDFs decreases towards the saturated regime (S v ≈ −1.0 for P m = 1/4, S v ≈ −1.2 for P m = 1/2 and S v ≈ −1.2 for P m = 1).In contrast, the magnetic field PDF has a positive skewness S b ≈ 2.5 during the kinematic phase, which decreases in the saturated phase (S b ≈ 0.43 for P m = 1/4, S b ≈ 0.32 for P m = 1/2 and S b ≈ 0.34 for P m = 1).The evolution of the energy increment skewness along the different dynamo regimes, is related to the energy exchange between the kinetic and the magnetic energies in the non-linear and saturation regimes.Note that energy transfers are non-linear exchanges, with local and non local transfer in a large range of scale [60,61].

Long time scaling regime
In the previous section we observed that the transition to Gaussian statistics of the perpendicular magnetic field takes very long times during the saturated regime while this happens on time scales of the order of T L in the kinematic regime.The reason are long lasting correlations.The autocorrelation of the Lagrangian velocity increments , is displayed in Fig. 7 (left).The Lagrangian correlation time is indeed of the order T L for the hydrodynamic regime.During the saturated regime the perpendicular component of the velocity field remains correlated for a longer time (see Fig. 7) (left)) of the order of several T L .Note that the parallel component becomes anti-correlated at times of the order of π/v rms z (see table 2).This corresponds to the typical time of a fluid particle crossing a TG fundamental box in the z direction and starting to feel the mirror symmetries of the TG flow.The long correlation time scale is also clearly observed for the magnetic increments in Fig. 7 (right) and corresponds to the time-lag for which the PDF of the magnetic field increments becomes Gaussian.We note that this time scale is a specific property of the TG flow which is not observed with the frozen force (run ff, Fig. 6 (right)) and it does only weakly depend on P m.
In order to better understand the long time correlations and the non-Gaussianity of the magnetic field increments we study Lagrangian structure functions (LSF) of the form during the saturated phase.Here again, B ⊥ denotes a magnetic field component from the perpendicular plane (B x or B y ).In turbulent flows structure functions are usually measured to identify and to analyze the scales of the inertial range (for u i instead of B i ) in which they behave as a power law S p (τ ) ∼ τ ζ p .The corresponding scaling exponents ζ p provide a handle on the phenomenon of intermittency which characterizes the occurrence of extreme events in the dynamical evolution of the system [62].In hydrodynamic turbulence, even for very large Reynolds numbers, the range of scales where a power law is observed and where thus the local slope is constant is very narrow if visible at all [63].We therefore do not expect to observe a power-law behavior on time-scales shorter than 1 T L .Surprisingly, for larger times scales we observe a clear plateau for the Taylor-Green runs.Its width is similar (from approximately 1 In order to better understand the origin of the observed long time correlations we computed the auto-correlations of the terms on the right hand side of (1) multiplied by u.They corresponds to the correlation of the energy sources and sinks of the velocity field.We find that the correlation time of the mean external energy injection rate u•F becomes longer during the non-linear phase and extends well beyond one T L during the saturated phase.It is then also on this time scale that the total energy of the system fluctuates.This is in agreement with the observed changes of the correlation time of u ⊥ in Fig. 7 (left) as the Taylor-Green force is constant.The energy exchange time scale u • (∇ × B) × B connected to the Lorentz force is shorter.We note that the mean external energy injection rate is a large-scale quantity.Apparently, the Lorentz force changes the global flow structure in such a way that the resulting large-scale velocity field and the (constant) Taylor Green force remain correlated for times much longer than without Lorentz force.The observed scaling regime is probably not a classical turbulent scaling regime.
We complement the discussion of the long time scaling regime by analyzing the power density spectrum (PDS) of the perpendicular magnetic field b ⊥ along tracer trajectories.For the PDS we find a low frequency band showing a power-law decay (see Fig. 9 (left)).The slope ζ 2 of a second order structure function and that of the corresponding PDS (α) are connected via the formula α = −ζ 2 − 1 [62].The measured slopes of the PDS are consistent with the scaling exponents of the structure functions.
Finally, we would like to note that we measured slopes of the spectrum of the temporal fluctuations of the total magnetic energy E mag (t) (see Fig. 9 (right)) which are compatible with the previously presented data.Keeping in mind that the variable of measured PDS has a dimension of B 2 and not of B like that of the LSF we find an agreement with ζ L 2 by simple dimensional arguments.

Conclusions
By means of direct numerical simulations of the magnetohydrodynamic equations forced by the Taylor-Green vortex we analyzed Lagrangian statistics during the three stages of a dynamo system: the kinematic phase with negligible magnetic field energy, the nonlinear phase with awakening Lorentz force, and the saturated phase when the non-linear dynamics lead to a statistically stationary equilibrium.Lagrangian data allows for a clear identification of these three regimes.The statistics of the magnetic and the velocity field changes drastically from one phase to another.The magnetic field PDFs are strongly non-Gaussian in the kinematic phase and quasi-Gaussian in the saturated regime.PDFs of magnetic and velocity energy increments along tracer trajectories are skewed.We find a positive skewness for the magnetic energy which reduces from the kinematic to the saturated phase.The skewness of the kinetic energy is negative during the kinematic phase and is even smaller during the saturated phase.The evolution of the skewness of PDF energy increment is clearly due the energy transfer between the kinetic and magnetic energies.
Temporal correlations extend much longer in the saturated regime than in the kinematic one and exceed significantly one large-eddy turn-over time.This is in agreement with increased temporal correlations of the external kinetic energy injection rate due to the action of the Lorentz force.Remarkably, we find a clear scaling regime of magnetic field increments along particle trajectories at time scales approximately ranging from one to ten large-eddy turn-over times.High-order scaling exponents of the Lagrangian structure functions show that these long-time magnetic field fluctuations are intermittent and that intermittency is increasing with increasing magnetic Prandtl number.The second-order scaling is consistent with a power-law observed for the corresponding frequency spectrum and that of fluctuations of the total magnetic energy of the system.
Finally, we would like to briefly mention that we do not observe a long time scaling regime for simulations with an ABC force.If it is specific to the Taylor-Green force or if also other forces develop it has to be investigated in future work.

Figure 1 :
Figure 1: Temporal evolution of kinetic and magnetic energy (left).In the right panel the enstrophy and normalized cross helicity are shown.(all data for run tg3) The vertical dashed lines, representing the transition between the different phases have been determined by using the flatness of the magnetic field (see section 4 and Fig 4).

Figure 2 :
Figure 2: Volume rendering of the time-averaged kinetic energy (green) and magnetic energy (blue) for run tg3 together with tracer trajectories.Their speed is given in colors from black (low) to yellow (high).Left top : Linear phase.The inset shows a snapshot of the enstrophy, where vortex filaments can be observed.Right top : Saturated phase.The inset shows a snapshot of the en-trophy, where MHD vortex filaments and sheets are visible.Left bottom : Top view (along z-axis) of the saturated phase.Right bottom : Transverse view (along x,y-diagonal) of the saturated phase.

Figure 3 :
Figure 3: Probability distribution function of the magnetic field (left) and the velocity field (right) where the magnetic field component b , sat (TG) is added for comparison.The data of the saturated Taylor-Green regime is from run tg3.

Figure 4 :
Figure 4: Temporal evolution of the flatness of the magnetic and velocity field PDFs for run tg3.

Figure 5 :
Figure 5: Probability distribution function of acceleration.Non-normalized (left) and normalized to standard deviation (right) for a hydrodynamic simulation (run tgH) and saturated regimes (run tg1, tg2, and tg3).

Figure 6 :
Figure 6: Left: PDFs of the magnetic field increments in the perpendicular direction along fluid element trajectories for several time lags in the saturated phase.For comparison the PDF of the b ⊥ is shown.(data from run tg3) Right: PDFs of the magnetic field energy increments for τ of the order of 5 • 10 −3 T L ∼ τ K .(data from run tg3)

Figure 7 :
Figure 7: Left: Autocorrelation functions C v (τ ) of the velocity along trajectories in the hydrodynamic phase (run tgH) and saturated phase (run tg3) Right: Autocorrelation functions C b (τ ) of the magnetic field measured along trajectories for the saturated phase for run tg3, and ff.

Figure 8 :
Figure 8: Local slope of the second order Lagrangian structure functions of the magnetic field S B 2 (τ ) for all runs in the saturated regime.The inset shows the measured scaling exponents ζ B p different orders p. Error bars give the maximal variation around the mean in the interval where S B 2 is flat.(data from run tg1, tg2, and tg3)

Figure 9 :
Figure 9: Left: Power density spectrum (PDS) of b ⊥ for run tg1, tg2, and tg3.The frequency f is given in terms of the large scale frequency 1/T L .Right: PDS of the total magnetic energy E mag (t) for run tg1, tg2, and tg3.All curves are shifted for clarity.

Table 1 :
List of the numerical simulations.P m = ν/η: magnetic Prandtl number, Re = u rms L/ν: Reynolds number, u rms root mean square velocity (defined in table 2), N number of collocation points, N p number of tracer particles.

Table 2 :
Parameters of the numerical simulations.