Bifurcations and dynamo action in a Taylor–Green flow

We report successive bifurcations in direct numerical simulations (DNSs) of a Taylor–Green flow, in both a hydro- and a magneto-hydrodynamic case. Hydrodynamic bifurcations occur in between different metastable states with different dynamo action, and are triggered by the numerical noise. The various states encountered range from stationary to chaotic or turbulent through possible oscillatory states. The corresponding sequence of bifurcations is reminiscent of the sequence obtained in the von Karman (VK) flow, at aspect ratio Γ=2 (Nore et al 2003 J. Fluid Mech. 477 51). We then use kinematic simulations to compute the dynamo thresholds of the different metastable states. A more detailed study of the turbulent state reveals the existence of two windows of dynamo action. Stochastic numerical simulations are then used to mimic the influence of turbulence on the dynamo threshold of the turbulent state. We show that the dynamo threshold is increased (respectively decreased) by the presence of large scale (resp. small scale) turbulent velocity fluctuations. Finally, DNSs of the magneto-hydrodynamic equations are used to explore the linear and nonlinear stage of the dynamo instability. In the linear stage, we show that the magnetic field favours the bifurcation from the basic state directly towards the turbulent or chaotic stable state. The magnetic field can also temporarily stabilize a metastable state, resulting in cycles of dynamo action, with different Lyapunov exponents. The critical magnetic Reynolds number for dynamo action is found to increase strongly with the Reynolds number. Finally, we provide a preliminary study of the saturation regime above the dynamo threshold. At large magnetic Prandtl number, we have observed two main types of saturations, in agreement with an analytical prediction of Leprovost and Dubrulle (2005 Eur. Phys. J. B 44 395): (i) intermittent dynamo, with vanishing most probable value of the magnetic energy; (ii) dynamo with non vanishing mean value of the magnetic energy. We describe a sequence of bifurcation of the dynamo at low Reynolds number and at increasing magnetic Prandtl number, where the dynamo switches from stationary mode to chaotic modes in a complex manner, involving intermittency in a way reminiscent to what is observed in dynamical systems with low number of degrees of freedom.


Introduction
The Taylor-Green (TG) flow results from a forcing with the two-dimensional (2D) space periodic force proportional to the TG vortex v TG = (sinx cosy cosz, −cosx siny cosz, 0). Despite its simple structure, only few analytical results are presently available regarding its stability properties, or its capability of dynamo action. However, its somewhat special symmetry allows for substantial reduction of the computation work [1], allowing hydro or magnetohydrodynamic (MHD) simulations at large kinetic or magnetic Reynolds numbers. In the past, it has therefore often been used to explore challenging issues such as singularities in Euler equations [1] or dynamo action at small magnetic Prandtl number [2]- [5].
Another interesting property of the TG flow is its similarity with the von Karman (VK) flow, obtained between two-counter rotating disks. This experimental configuration is used 3 within the von Karman Sodium (VKS) experiment [6]- [10] to study the dynamo onset and magneto hydrodynamic turbulence in a high Reynolds number unconstrained flow. We therefore used the TG flow as a prototype to understand the mechanism of dynamo onset and dynamo saturation, for use in the analysis of the VKS experimental results. In this process, we obtained several results in the hydrodynamic or magneto-hydrodynamic regime, as side consequences of our exploration of issues related to the dynamo onset. We also report an aspect of the TG flow, we discovered while running MHD simulations: a complex parameter space at low Reynolds number, with coexistence of several metastable states, resulting in possibly non-trivial transition to turbulence or dynamo action. This feature explains several intriguing aspects of dynamo action that had been observed in previous simulations of TG vortex, but never published [11].
The plan of the paper is as follows: in the first part, we set the magnetic field to zero and explore the parameter space of the TG flow as a function of the Reynolds number in the hydrodynamical case. We identify several branches of solution and the bifurcations between them. We sketch the similarity between the sequence of bifurcations we obtain and the successive bifurcations observed in the VK flow. We finally study the convergence from the turbulent state towards the mean flow. In the second part, we also integrate the magnetic field equations and study various aspects connected with dynamo action. After recalling the main equations and definitions, we recall a few analytical results regarding the stability of the TG flow in the MHD case. We then compute the kinematic dynamo thresholds for the different metastable states observed in the hydrodynamic limit. We study the influence of the noise on the dynamo threshold of the turbulent state, in a kinematic limit where there is no back reaction of the magnetic field on the flow. In the last part, we turn to full dynamic simulations of the MHD-TG flow. In the linear stage, we identify interesting dynamo cycles in between metastable states, and show that the magnetic field can favour direct transition from the basic state to the turbulent state. We compute the dynamic dynamo threshold as a function of the Reynolds number. Finally, we explore the possible saturation states of the dynamo and describe an interesting sequence of bifurcations of dynamo states when increasing the magnetic Prandtl number. A discussion of our results in connection to the VKS experiment is finally presented.

Earlier analytic and numerical results about stability
The TG flow is characterized by the presence of both hyperbolic and elliptic stagnation points. The shortwave asymptotic developed by Lifschitz [12,13] shows that both types of stagnation points are unstable with respect to shortwave instabilities, no matter which type of flow surrounds them. These instabilities have been studied in the planar TG vortex (with velocity (sinx cosy, −cosx siny, 0)), by Sipp and Jacquin for the elliptical case [14] and Leblanc and Godeferd [15] for the hyperbolic case. They showed that the elliptic instability deforms the core of the vortices, while the hyperbolic instability favours the formation of longitudinal vortices (the so-called rib). In an earlier numerical study of the TG cell, Lundgren and Mansour [16] derived neutral stability curves and explored the transition to turbulence. They obtained a critical Reynolds number of the order of 10 (respectively 30) for odd (resp. even) modes at low wavenumber (between 2/3 and 1). The transition to turbulence occurs via the hyperbolic 4 instability (with counter-rotating swirls perpendicular to the axis), developing into secondary instabilities leading to a turbulent state made of thin elongated vortex tubes and weaker vortex layers which spiral around them.

Description of the simulations and their parameter
The Navier-Stokes (NS) equations for incompressible fluids are: Here, u is the velocity, P the pressure, ν the viscosity and v TG = (sinx cosy cosz, −cosx siny cosz, 0) is the TG vortex and f (t) is set by the condition that the modes that correspond to v TG are kept with constant amplitude. The equations are integrated on a triply periodic cubic domain using a pseudo-spectral method. The aliasing is removed by setting the solution of the 1/3 largest modes to zero. The time marching is done using a second-order finite difference scheme. An Adams-Bashforth scheme is used for the non-linear terms while the dissipative terms are integrated exactly [17]. The control parameter is the Reynolds number Re = V rms L int /ν, where V rms = √ 2E/3 is the rms velocity based on the total kinetic energy E = E(k) dk and L int = (3π/4) E(k)/k dk/E is the integral scale of the turbulent flow. Both V rms and L int fluctuates with time. Thus, viscosity is dynamically monitored so as to keep Re constant. We have checked that Re is a simple linear function of a geometric Reynolds number (usually used in experiments) based on maximum velocity and half the simulation box Re = 7.41V max π/ν. Detailed comparison with the previous results obtained by Nore or Ponty can be obtained by multiplying our Reynolds number by a factor 8/ √ 3, due to our slightly different definition of V rms and L int .
The instabilities and bifurcations we report are obtained in the range 6 < Re < 100. Below this range, we detected no hydrodynamic instability. Above this range, the computation needs of the direct numerical simulation (DNS) exceeds our computational resources. For DNS at Re < 46, the equations were integrated with 64 3 Fourier modes before desaliasing. This resolution was increased up to 96 3 for 46 Re < 75 and up to 128 3 for 75 Re 100. The spectral model [18] was used for large-eddy simulations (LES) at Re = 500 on 128 3 Fourier modes.

Successive temporal transitions
During our simulations, we partially explored the phase space of the TG flow by monitoring the time dependence of V rms . When initializing the flow with the TG vortex (in which V rms = 0.28 and k = √ 3 ), the typical transition scenario we observed is the following (figure 1): first, a very short (lasting typically one or two eddy turnover (ETO) times) relaxation towards a first state, with a low value of V rms , between 0.33 and 0.4. For Re < 50, this basic state (state B) is 2D (with zero vertical velocity, see figure 2(a)), stationary and metastable. Its lifetime depends on the Reynolds number (see figure 3): varying from over 250 ETO times at Re = 8-33 ETO for Re = 25, and then 22 for Re = 46. At Re = 50, this stationary state is not observed anymore, and is replaced by a quasi-oscillatory metastable state (O1) (see figure 4(a)), with lifetime independent of the Reynolds number, of the order of 25 ETO time. Due to the numerical noise, these two metastable states bifurcates towards a second state, with a slightly larger V rms , but very different spatial structure. For Re < 25, it is another quasi-bidimensional stationary state (state M), with negligible vertical velocity (figure 2(b)), while it becomes chaotic or turbulent (state T1)   The lifetime is expressed in ETO time. The vertical arrow, at Re = 8 indicates that the indicated lifetime is a lower limit. The basic state B becomes unstable between Re = 40 and 50, where it is not observed anymore.
for Re 25, with an average spatial structure, see figure 5(a), but no energy in the gravest mode k = 1. The M state observed at low Reynolds number is also metastable, with a lifetime of the order of 50 ETO times. At Re = 8 in the presence of additional external noise, we observed a bifurcation towards an oscillatory state, with a period of the order of 50 turnover time. At Re = 15, with only the numerical noise, we observed a bifurcation towards a third metastable, stationary state (state P), with nonzero vertical velocity, see figure 2(c). However, when restarting the simulation in the M case, and continuing it on another computer (see figure 1), we obtained instead two successive transitions towards oscillatory states O2 and O3 (see figures 4(b) and (c)). This is an indication that the numerical noise is the engine of the exploration of the phase space and prompted us to introduce external noise to try and induce a bifurcation towards other states (figure 6). We then obtained a bifurcation towards an additional state T2, with a large value of V rms (of the order of 0.4 at Re = 15 to 0.51 at Re = 100). This state is stable, and oscillatory or quasi periodic until Re = 8, and becomes turbulent for a Reynolds number larger than Re = 8. An example is provided in figure 6. Looking at the spectra of velocity with and without noise, we noticed that the introduction of noise favours the building of a mode k = 1, absent in our initial conditions with k = √ 3 (see figure 7). This mode apparently favours the apparition of a stable, turbulent or chaotic state. It is then tempting to attribute this transition to a short wave instability, identical to the one observed in simulations of the 2D TG vortex (cf previous section), and which should occur above Re = 10.
A schematic bifurcation diagram summarizing the various states we encountered in our simulations is drawn in figure 8, using the V rms as order parameter and Re as control parameter.

Comparison with sequences of bifurcations in VK flow
There is a geometric similarity between the TG flow forced at k = √ 3 and the VK flow with aspect ratio = 2 (so as to obtain square boxes for velocities in a plane) [19]. In both cases, it is a shear layer between two-counter rotating eddies. Boundary conditions, however, are different: periodic with free-slip (normal component equals zero at the surface) for TG, while the VK flow is enclosed in between two counter rotating disks. Because of the periodicity, and symmetries, the mean helicity in TG is zero, in contrast with nonzero helicity of VK. However, the mean helicity in the box defined by the three planes of mirror symmetry (x = 0, π, y = 0, π, z = 0, π) of TG flow is nonzero and its distribution resembles the helicity in VK flow [19]. Similarities between the two flows suggest that the sequence of bifurcations we observe in TG flow could have a counter-part in the VK flow. These sequences of bifurcations have been numerically studied in [20], and experimentally in VK flow driven by smooth disks in [21]. In these publications, the Reynolds number is based on the cylinder radius, and the propeller azimuthal velocity. Taking into account the fact that maximal typical poloidal velocities are of the order of 0.1 the propeller azimuthal velocity, we infer that a conversion factor (division by) of 7.41/0.1 ≈ 70 must be applied to transform their Reynolds number into ours. With this convention, their result can be summarized as follows. Three different steady states with different symmetries are observed: the basic state for Re < 5; a mixed mode M with both odd and even harmonics for 5.87 < Re < 6.1 and a pure mode P (m = 2) with only even harmonics for Re > 6.5. Travelling waves TW are also observed while increasing the Reynolds number. They originate from a drift pitchfork bifurcation from the mixed mode, occurring at Re = 6.1, and give rise to nearly periodic oscillations, with a period of 190 ETO times. When the Reynolds number is increased above Re = 6.5, near heteroclinic cycles appear.
This bifurcation diagram bears strong similarity with our diagram. In fact, the portion of our figure between Re = 6 and Re = 15 looks very similar to the figure 9 of Nore et al [20]. Clearly, it would be very interesting to study this analogy in more detail.

Convergence towards the mean flow
In the course of our work, we came across the problem of computing time-averaged flows, and the question of how long one needs to average a given velocity field to get reliable (time-average independent) results. We thus tried to quantify this effect by defining a global convergence , where t < T . As t → T , one expects δ(t, T ) to converge towards 1 in an exponential fashion, with a typical time of the order of the correlation time of the slowest structures. Besides, the typical order of magnitude level reached before δ(t, T ) − 1 exactly reaches zero is a decreasing function of T . An illustration of this effect is provided in figure 9, for three different Reynolds numbers, Re = 25, 46 and 100, and with T chosen as the longest available time (the simulation duration). In the case Re = 25, where a very long integration time has been used (typically 240 ETO times), one sees that the convergence coefficient δ(t, T ) − 1 reaches values of the order of 10 −4 , while, for the larger Reynolds numbers only 125 ETO times were used, the convergence only reaches values of the order of 10 −3 : the field is ten times less converged. We tried to detect how this convergence occurs, by looking at spectra (figure 10) and velocity field structure (figures 11 and 12). One sees that the effect of averaging is to maintain the constant level of the modes k = 2 and k = 3, while decreasing the mode k = 1 and the modes k > 3. In the physical space, the effect is less obvious. In the case Re = 46, we had the feeling that averaging leads to a more and more symmetrical

Equations and definitions
In the MHD case, the equations of motion are now: (2) In addition to variables already defined in previous sections, the new symbols are now B the Alfven velocity, η the magnetic diffusivity and j = ∇ × B the current density. The forcing is done like in the hydrodynamical case. Our simulations therefore differ with simulations reported in [3,4] that are at constant force. The two control parameters are now the Reynolds number Re and the magnetic Reynolds number Rm = V rms L int /η. Both viscosity and diffusivity are dynamically monitored so as to keep Re and Rm constant. The ratio Rm/Re = Pm is the magnetic Prandtl number.

Analytical results
There is no systematic analytical study of the TG flow in the MHD case. Partial results regarding the stability of the TG vortex with respect to large scale perturbation have been obtained through multi-scale techniques by Lanotte et al [22]). In this mean field type of approach, one finds that the major effect of a given incompressible flow u on the mean (large scale) magnetic field B is to generate a mean electromotive force E in the mean induction equation, so that: where α i j and β i jk are functions of u. where A and B are free parameters. When A = B = 0, the velocity field is just the TG vortex, equal to the TG flow in the limit of infinite viscosity. When A ∝ Re and B ∝ Re 2 , this velocity field can be seen as the solution of the expansion of the TG vortex to second-order in Re.
Due to the parity invariance of the flow, the α coefficient is zero. The β coefficient however depends on the velocity field: it is always positive for the TG vortex (A = B = 0), but can take negative values when A and B are different from zero, for large enough magnetic Reynolds number. This induces a large-scale instability for the magnetic field, for Rm > 4.4 for A = 1, B = 0 or Rm > 3 for A = B = 1. The most unstable mode lies in the x y-plane, with a corresponding eigen-component of the form B = (0, 0, B z exp ik · x). This instability corresponds to a dynamo action through negative magnetic eddy diffusivity, and occurs at wavenumbers smaller than the scale of the basic flow.
where T is less than the lifetime of the considered state. Kinematic dynamos have been studied for different metastable states observed in our simulations. For this, we proceed in two steps. First, compute a time-averaged velocity field from a time series of a DNS in a given metastable state. Then, plug this (time-independent) velocity field in the induction equation (5) and study the evolution of a given seed magnetic field of given amplitude (with an energy of the order of 10 −4 of the kinetic energy of the averaged field). For this, we fix Re, vary Rm and monitor the time behaviour of the magnetic energy B 2 and the finite-time Lyapunov  magnetic Reynolds number between Rm c = 3.3 and 7. This means that it is less sensitive to the spatial structure of the flow, than in a non-periodic flow, with boundary layers [23,24]. This could be traced to another mechanism of dynamo action, via stagnation or hyperbolic points (fast dynamo mechanisms), rather than through boundary layers (slow dynamo mechanism).

Dynamo windows.
Because of its possible relevance to the analysis of the dynamical dynamos, we have studied the turbulent mean field T2 in greater detail, by computing Lyapunov exponents over a large range of magnetic Reynolds numbers. The result is provided in figure 15, for T2 fields obtained at Re = 25, 46 and 100. For comparison, the Lyapunov exponent for the basic state B at Re = 6 is also provided. One sees that the Lyapunov exponents do actually cross the value = 0 at three locations Rm (1) c < Rm (2) c < Rm (3) c , indicating the presence of at least two dynamo windows, illustrated in figure 16 and detailed in table 2. The first dynamo window, occurring around Rm = 10 for all Reynolds numbers, corresponds to a stationary mode. The second dynamo window, occurs for decreasing values Rm as Re is increased beyond Re = 25. This situation is therefore very similar to the ABC flow with A = B = C = 1, where two windows of dynamo action have been reported: one around Rm = 6 [25], and one starting at Rm > 14 [26]. The latter corresponds to an oscillatory mode until Rm = 100, where it is replaced by a stationary mode [27]. Such a mode crossing could occur in the TG flow, a possibility that could be worth exploring. Note that when looking at the Lyapunov exponent as a function of averaging, we indeed detected oscillatory dynamos near the border of the second dynamo window, with oscillations of the magnetic energy over a very long period (between 200 and 500 ETO) ( figure 17). If such oscillatory modes are present in the second dynamo window, they may have a very large oscillation period and may prove very difficult to detect. For the time being, neither our group, nor the Boulder group [28] were able to detect any such oscillations in the second dynamo window.

Convergence of Lyapunov exponent
To understand the influence of convergence on the dynamo threshold, we have also studied the influence of the averaging on the Lyapunov exponent by performing kinematic simulations with velocity fields averaged over different timescales. In order to allow the removing of many scales through the averaging, we have conducted this study at a rather large Reynolds number Re = 46, where the flow is already fully turbulent and quite different from the mean velocity field (see figures 11 and 12). The result is provided in figure 18. Interestingly enough, one sees that the averaging mostly affects the large Rm part of the Lyapunov exponent.
For the instantaneous field (no averaging), the decrease of the Lyapunov exponent between Rm = 10 and 18 (responsible for the apparition of a second dynamo of dynamo action) is either removed or shifted towards larger Rm (out of reach to simulations). As the averaging process is operating, the Lyapunov exponent slowly bends downwards, ensuring the apparition of the second dynamo window. Note however that the location of the first dynamo window is not affected by the length of the averaging, so that the kinematic simulations with the frozen velocity field gives a lower critical magnetic Reynolds number than the dynamical velocity field. This seems to contradict a belief in the dynamo community, but we have no obvious explanation for this fact.

Stochastic-kinematic dynamos
3.5.1. Motivation. Many recent dynamo experiments are based on liquid sodium, characterized by a very low magnetic Prandtl number. At the dynamo threshold, the flow is then necessarily highly turbulent, with a large Reynolds number and large fluctuation level (as high as 50%). From a theoretical point of view, the corresponding dynamo can be regarded as an instability (driven by the mean flow) in the presence of a multiplicative noise (turbulent fluctuations). Based on this remark, Leprovost and Dubrulle [29] derived an analytical prediction regarding the turbulent dynamo using tools borrowed from the study of a noisy Duffing oscillator [30]. They found that noise could favor or impede the magnetic field growth depending on the fluctuation intensity or correlation time. They also found that the transition to a turbulent dynamo (at Rm = Rm ct with a nonzero mean field value does not occur directly from the trivial, zero-field state, but is preceded by an intermittent regime of the magnetic field, occurring at Rm = Rm ci . There is then a whole range of magnetic Reynolds numbers for which the intermittent behaviour is observed: Rm ci < Rm < Rm ct . In the isotropic case, Rm ci is a non-monotonic function of the noise intensity: Rm ci is larger than the critical magnetic Reynolds number with no-noise Rm c0 at small noise amplitude, while it becomes Rm ci < Rm c0 at large noise amplitude. Furthermore, in a numerical study of the noisy Bullard dynamo (a 2D solid dynamo) [31], Rm ct is found to be a monotonic increasing function of noise intensity, so that noise always delays transition towards a turbulent dynamo state. Some of these findings have recently been reproduced experimentally in a Bullard-VK dynamo, where the intermittent regime (often referred to on-off regime) has been observed [32].
However, their simple model ignores the molecular magnetic diffusivity. This motivates new types of simulations-stochastic-kinematic dynamo (KS)-to investigate the validity and limitation of their results, when molecular diffusivity is taken into account. Stochastic-kinematic simulations will also prove useful to interpret the outcome of DNSs within the Kraichnan-Kazantsev model of MHD flows, where turbulence is replaced by a suitable noise.

Notations and definitions.
In the kinematic-stochastic case, the velocity field u = u (Re) + v (k I , τ c ) is the sum of a time averaged velocity field at a given Re and of an external Markovian Gaussian noise, with fixed amplitude v , correlation time τ c and typical scale k I . The control parameters are therefore Rm, v , τ c and k I . To make an easier connection with the DNSs, where Re is an essential control parameter, we introduce a noise intensity parameter, δ ≡ δ(0, T ) = u 2 / u 2 . This parameter depends on the noise amplitude, as well as its correlation time and characteristic scale, and needs to be computed for each stochastic simulation. It can also be computed in the direct simulations, and is found to depend on the Reynolds number, increasing from a value of 1 at low Reynolds number, until about 3 at the largest Reynolds number ( figure 19(a)). Note that δ − 1 is also connected to the difference between the total kinetic energy and the kinetic energy of the mean flow, a quantity which increases with the Reynolds number, see figure 19(b). Another interesting non-dimensional parameter is given by the variation of the instantaneous total kinetic energy (see figure 20). This variation can be quantified by a second parameter δ 2 = u 2 2 − u 2 2 / u 2 . In the sequel, we show that all our results can be interpreted using δ and δ 2 as a control parameter.

The parameter space.
We have considered noise at large (k I = 1) and small scale (k I = 16), with correlation time ranging from 0 to 50 s, i.e. from 0 to 17 ETO time. An example of our noise power-spectrum is provided in figure 21 for different correlation times. It is essentially a white noise (flat spectrum) until f 0 ∼ 1/6τ c , with rapid decrease after f 0 . From now on, we express all values of τ c in units of the ETO time. The spatial resolutions of stochastic-kinetic simulations are equivalent to the resolutions of the kinetic simulations without noise at the same Reynolds numbers.
Since the kinematic dynamo threshold is essentially constant for all values of the Reynolds number (see Rm c,1 on table 2), we first focus on the study of the case where the time-averaged field is fixed as u(Re = 6) and vary the noise amplitude, characteristic scale or correlation time, to explore their influence on the dynamo threshold. Our exploration of the parameter space is provided in figure 22, for different kinds of noise and u(Re = 6). Note that by using our external  noise, we are able to produce noise intensities of the order of the noise intensities measured in experiments (δ ∼ 10 at Re ∼ 10 6 for the VK flow), and that are out of reach of DNS.

Dynamo threshold.
For low correlation time or injection scale, we are actually able to follow the deformation of the two windows of dynamo action. One sees that a noise does not destroy them, but rather distorts them. In the case where the noise is at small scale (k I = 16, i.e. at a scale smaller than the dissipative length scale for any Rm 40), the two windows are slightly tilted downwards, while they are lifted upwards in the case of large scale noise (k I = 1). The influence of the noise on the first dynamo bifurcation (the dynamo threshold) can be summarized by plotting the critical magnetic Reynolds numbers as a function of the noise intensity ( figure 23(a)). Large scale (resp. small-scale) noise tends to increase  (resp. decrease) the dynamo threshold. Furthermore, one sees that for small-scale noise, the decrease in the dynamo threshold is almost independent of the noise correlation time τ c , while for the large scale noise, the increase is proportional to τ c at small τ c . At τ c 0.3, all curves Rm c (δ) collapse onto the same curve. We have further investigated this behaviour to understand its origin. Increasing δ firstly increases the flow turbulent diffusivity η t = V rms L int with respect to its mean flow value, η MF , computed using the energy and the integral length of the mean flow. Since the magnetic Reynolds number is essentially the ratio of the (mean or turbulent) flow diffusivity over the true diffusivity, this naturally produces an increase in the critical value. This effect can be corrected by considering the equivalent mean-field magnetic Reynolds number Rm * c = Rm c η MF /η t . Secondly, an increase of δ produces an increase of the fluctuations of kinetic energy (see figure 20), quantified by δ 2 = u 2 2 − u 2 2 / u 2 . This last effect is more pronounced at k I = 1 than at k I = 16. It is amplified through increasing noise correlation time.
We thus re-analysed our data by plotting Rm * c as a function of δ 2 ( figure 23(b)). All results tend to collapse onto a single curve, independently of the noise injection scale and correlation time. This curve tends to a constant equal to Rm MF c at low δ 2 . This means that the magnetic diffusivity needed to achieve dynamo action in the mean flow is not affected by spatial velocity fluctuations. This is achieved for small-scale noise, or large scale noise with small correlation time scale. In contrast, the curve diverges for δ 2 of the order of 0.2, meaning that timefluctuations of the kinetic energy superseding 20% of the total energy annihilate the dynamo. In a real flow, or in the DNS, such fluctuations are more likely reached via non-stationary large scale, since they are the most energetic.

Comparison with theory.
At small values of noise, δ → 1, one may compare our numerical results to an asymptotic theory by Fauve and Petrelis [33], linking the dynamo threshold to the noise amplitude (δ − 1) 1/2 .
Their prediction is that the correction to the dynamo threshold is zero at first-order O((δ − 1) 1/2 ), and depends on the noise helicity at the next-order O(δ − 1). The correction is positive (resp. negative) if the noise helicity u · (∇ × u) has same sign (resp. opposite sign) than the  helicity of the basic flow. As a first check of this prediction, we therefore plotted (δ − 1) n/2 as a function of Rm c − Rm c (δ = 1) for n = 1 (first-order), n = 2 (second-order) and n = 3 (thirdorder), and looked for linear behaviour. Examples for large scale and small-scale noise are provided in figure 24). In all cases, we obtain the same kind of behaviour. One sees that indeed, the best linear behaviour until δ − 1 = 2 is obtained for n = 2, indicating that the correction to threshold is at second-order in the noise amplitude. We have also tried to see if the sign of the correction depends on the helicity of the noise. This quantity strongly fluctuates in time. It is plotted in figure 25  Fauve-Petrelis theory, which requires a non-zero helicity of the mean flow. It therefore seems more likely that the corrections are due to the effects we mention above (mean energy and variance of the energy).

Lyapunov exponent.
Another way to look at the influence of noise on the dynamo threshold is through the noise-induced deformation of the Lyapunov exponent. Figure 26 shows this deformation of the curve (Rm) for various values of δ, and different noise time correlation scales or injection scales. We have actually checked on our data (not shown) that the deformation of the Lyapunov exponent observed in the KS simulations is comparable to the deformation observed in DNS, at comparable noise intensity (i.e. δ) and for τ c = 0.3; k I = 1. This reinforces the similitude between the DNS and a KS simulation at τ c = 0.3, k I = 1.

Influence of mean flow.
To check that our results are not affected by the choice of u, we ran additional KS simulations with u computed at Re = 25, 46 and 100. Since the computational cost in these cases is much larger than in the case Re = 6, we focused on the case where the noise has a correlation time τ c = 0.3 or 2.7 and injection scale k I = 1 for a few characteristic values of δ. Figure 27 shows that the dynamo threshold at various Re coincides with the dynamo thresholds of KS simulations at Re = 6, indicating that the influence of the mean flow on the dynamo threshold derived using the KS simulations is probably negligible.

Influence of spatial correlation: NS-I noise.
The Markovian noise we used has no spatial correlation, at variance with real turbulence. Following [31], we therefore ran a few additional stochastic simulations using NS-noise computed as v = A( u(Re) − u(Re)), where u(Re) is a velocity field computed in a DNS simulation at Reynolds Re, and A is an amplitude, tuned so as to get a desirable value for δ. The noise characteristic scale can be furthermore selected by appropriate filtering of v . Figure 28 gives Rm c = 9 (resp. Rm c = 11). We have compared these values with the thresholds obtained for Markovian correlated noise in figure 28(b). The value for the large-scale NS-noise is larger than the asymptotic value obtained at the same δ in the Markovian large-scale noise when τ c > 0.3. The small-scale NS-noise has a lower threshold, that is above the value of the small-scale Markovian noise, but is comparable with the delta-correlated largescale noise. In any case, the large-scale filtering applied to the NS-noise does indeed lead to thresholds that are substantially smaller than in the DNS. This reinforces the idea that it may be a good idea in real experiments to filter out the large-scale noise through an appropriate device.

Full coupled problem
Non-linear dynamos have also been studied, using simultaneous integrations of equation (2). Several interesting features have been found, at both linear and non-linear stages of the dynamo process. A summary of the DNS and LES we performed is provided in figure 29.
3.6.1. Linear stage: cycle between two states and oscillatory states for Re < 10. At a low kinetic Reynolds number, the basic flow is, as previously seen, characterized by several metastable states. The introduction of a seed magnetic field (of energy of the order of 10 −3 of the kinetic energy) has a strong influence on the stability of the various states of the TG flow, even in the no dynamo regime. At low kinetic and magnetic Reynolds numbers (Re, Rm < 10), the magnetic field can temporarily stabilize the basic metastable state B, resulting in cycles in between the state B and the state M. The velocity field then oscillates in between the two states, and the magnetic field growth is made of portions of exponential growth or decays, corresponding to growth rates of the two different states (figure 30). Increasing further the magnetic Reynolds number Rm > 20, one observes the appearance of complex Lyapunov exponents, with magnetic energy made of a combination of exponential growth and oscillation (figure 31) in the magnetic energy, accompanied by oscillation in the kinetic energy. The location of apparition of these oscillatory modes roughly corresponds to the end of the first dynamo window in the kinematic case.

Linear stage: noise versus magnetic field.
Around Re = 15, the magnetic field allows exploration of all the various metastable states, resulting in short periods of exponential growth, followed by random wandering as the system searches for the largest eigen-component of the instantaneous state visited. The total Lyapunov exponent is then less than any Lyapunov exponent of any metastable state present in the system. An example is provided in figure 32. This weakening of the Lyapunov exponent due to the noisy nature of the velocity field is reminiscent of the disorientation effect predicted by Leprovost and Dubrulle [29] in their stochastic model of dynamo.
For larger values of Re > 15, the magnetic field plays the role of a noise, allowing direct transition of the velocity field towards the turbulent state, even in the absence of the dynamo effect. Looking at the spectra, we observed that even a small magnetic field favors the building of a k = 1 component in the kinetic energy spectra (figure 33), in a way similar to an external noise. Note that a similar population of the low frequency spectrum has been observed in the time spectra of a VK flow, during the transition from chaos to turbulence [34].
At the end of a no-dynamo MHD simulation with an asymptotically vanishing magnetic field, one then obtains a velocity field different than the velocity field obtained by integration of the hydrodynamic equation (1) with the same initial condition but no noise, as if the magnetic field had vaccinated the flow. This special feature has been recognized a few years ago by [11], who gave it this name.  table 3. One can see that they are very much larger than the values obtained in the kinematic case. An explanation of this effect could be that the turbulence is akin to a noise, with a specific correlation time and characteristic scale [35]. Indeed, it can be shown in figure 27 that the dynamo threshold obtained in the DNS coincides with the dynamo thresholds of KS simulations at Re = 6 obtained for large-scale noise with correlation time τ c = 0.3. Note that another interpretation is proposed by [3] in the present issue.
3.6.4. Link with the negative eddy-diffusivity. One may also notice that below Re = 3, no dynamo is observed. This is reminiscent of the critical Reynolds number for presence of a large-scale negative eddy-diffusivity instability computed by Lanotte et al [22], indicating that this mechanism could be at work, at least at a low Reynolds number. A partial confirmation of this suspicion can be obtained by looking at the structure of the neutral mode, and see whether it has the planar structure characteristic of the negative-eddy-diffusivity. This has been done by Ponty et al [4] who indeed observe a planar structure for the dynamo mode in the first window of instability, suggestive of the negative-eddy-viscosity instability. Note that this instability seems to be forbidden in a symmetric TG code since such spatial structure is forbidden because of overall rotational invariance. One is then potentially left only with the second dynamo mechanism (corresponding to the second window of dynamo action), resulting in larger dynamo threshold. This therefore explains previous results of Nore et al [19], who noticed that the critical magnetic Reynolds number for symmetric simulations are larger than in a non-symmetric case, but can be substantially decreased by using forcing at larger wavenumber: from Rm c = 144 at k 0 = √ 3, up to Rm c = 11 at k 0 = 12. They noticed that the increase of the forcing wavenumber could favor the apparition of large-scale instabilities, presumably the negative-eddy-viscosity instability.
3.6.5. Non-linear stage: intermittent versus mean field dynamo. After the initial exponential growth, the magnetic field becomes large enough to react back on the velocity field through the Lorentz force, initializing a nonlinear saturation. We have observed two main types of saturation: intermittent dynamo and mean field dynamo (see figure 35). In the first case, the magnetic field behavior is very intermittent, made of bursts of magnetic activity followed by (relatively) long lasting periods of inactivity (see figure 36). In this case, the probability  distribution of magnetic energy is sharply peaked towards the origin: the most probable value is zero (see figure 36). This type of dynamo is reminiscent of the intermittent dynamo proposed by Leprovost and Dubrulle [29]. In our case, the intermittent dynamo was clearly identified only in the low Re, large Rm regime, in the parameter range corresponding to the transition between the two dynamo windows for the mean field. More precisely, the parameter range for an intermittent dynamo seems to coincide with the no-dynamo window of the mean field, distorted by noise (see figure 35), reinforcing the analogy between DNS-MHD simulations and KS simulations with Markovian noise. We do not exclude the possibility of an intermittent dynamo near the onset of the dynamo action at larger Re (typically for Re 30, corresponding to our undecided state), but their study and confirmation would involve computation time beyond our capacity.
Note that the intermittent dynamos we observe are seemingly quite different from the previously reported intermittent dynamo (see e.g. [36]). In these cases, the dynamo was observed to be spatially intermittent, in the sense that the magnetic field structures were irregularly disposed over the space. The magnetic energy, however, fluctuates around a mean value. In our case,  the dynamics of the magnetic fields proceeds through bursts in time, meaning that there is actually a large probability that there is no magnetic field at all anywhere in the box. This strange behaviour is related to on-off dynamics (typical of systems with multiplicative noise [29]) and has been reported before in a truncated simulation of an ABC flow [37], near the dynamo threshold, or, more recently, in an experimental Bullar-VK dynamo [32], or in the VKS2 experiment [6].
Dynamos with a non vanishing mean field were observed most of the time. They are characterized by a saturation occurring around a well-defined mean value. At low Reynolds numbers (typically Re and Rm < 15), the saturation is stationary and evolves towards a turbulent state for large Reynolds numbers (see figure 37). In that case, the probability distribution is distributed around the mean value, and decreases towards zero or infinity (see figure 37).
3.6.6. Nonlinear stage: a bifurcation sequence at low Re. We have followed a bifurcation sequence at Re = 6, and 4 < Rm < 100 and found that the transition to turbulence occurs via successive bifurcations (figure 38) reminiscent to what is observed in systems with low degrees of freedom: the stationary dynamo excited nonlinearly at Rm = 5.78 bifurcates into an oscillatory state at Rm ∼ 7 until Rm = 12 where cycles between the unstable ground state and an unstable upper dynamo state begin to appear. These cycles are observed until the beginning of the second dynamo window Rm ∼ 25 where a second sequence of bifurcations begins. The resulting dynamos are oscillatory, switching towards irregular (intermittent) dynamos around Rm ∼ 40. At Rm = 100, the dynamo is turbulent, with a well defined mean value. This sequence of bifurcations is reminiscent of what has been previously observed in periodic simulations of the ABC flow [38]. Oscillations in low Reynolds number TG flow have also been reported by Mininni et al in [39]. They noted that as the Reynolds number is increased, the oscillations disappear and the energies become more stationary. However, they were able to detect some vestige of the periodic behaviour in the magnetic energy. It is therefore likely that the sequence of bifurcations we observe at a low Reynolds number disappears at a higher Reynolds number due to fluctuations.

Discussion
The present study of the TG flow reveals several distinguished features that are worth summarizing: (i) a complex parameter space, with coexistence of several metastable states and possible transitions in between them, that may be mediated by a magnetic field; (ii) the existence of two windows of dynamo instability; (iii) the increase of the dynamo threshold with respect to the kinematic dynamo threshold for increasing the Reynolds number that bears strong similarities with the increase observed in the presence of large-scale noise with increasing kinetic energy; (iv) the existence of different regimes of dynamo saturation, and a sequence of bifurcations in between these regimes that is reminiscent of systems with a few degrees of freedom.
An interesting open question is therefore whether these features are of any relevance to the VKS experiments, that are operated in a range of parameters quite different from our numerical simulations, with a kinematic Reynolds number about 10 4 larger than ours. Regarding the complex parameter space, there is indeed indication that such high Reynolds number VK flow without a magnetic field is characterized by different metastable states, with different symmetry properties [34]. Whether a magnetic field can influence the transition in between these states is currently a question explored in the VKS experiment. The influence of the large-scale noise on the dynamo action is also a question that can be addressed in future VKS experiments, thanks to devices monitoring large-scale vortices that are presently tested and implemented. For example, with stainless propellers, no dynamo action has yet been achieved, but it has been noted that the Figure 38. Sequence of bifurcations observed at Re = 6 for increasing Rm. Note the transient unstable high-dynamo mode occurring after the linear growth at low Rm before saturation of the dynamo towards a lower amplitude. This second dynamo mode is stationary at Rm = 7, oscillatory at Rm = 11, chaotic at Rm = 20. Starting around Rm = 25, new dynamo modes are seen, after a sequence of oscillatory growth towards a second unstable high dynamo mode. These modes are oscillatory for Rm = 25, intermittent for Rm = 40 and turbulent for Rm = 100. addition of an annulus at the mid-plane (so as to stabilize large-scale vortices) strongly increases the induction [40]. In the configuration with iron propellers and the annulus at the mid-plane, dynamo action has been observed [8] at magnetic Reynolds numbers slightly below critical magnetic Reynolds numbers computed in kinematic codes for stainless propellers [24,41,42]. An interesting open question is whether the dynamo threshold will change if the annulus is removed, and large-scale motions of the mixing layer are permitted. Finally, different regimes of dynamo saturation have indeed been observed in the VKS experiment, with dynamics similar to systems with few degrees of freedom [6]. These regimes are mostly observed in situations where a mean rotation is superimposed to the flow. Since one of the influences of the rotation is to damp velocity fluctuations, one may wonder whether the complex dynamos with the rotation of VKS are related to the sequence of bifurcations we observe in TG flow at low Reynolds number, with few fluctuations.
This confirms the interest of the TG flow as a prototype to understand, at least qualitatively, some results of VKS experiments as first suggested in [5].