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Abstract

Eulerian and Lagrangian tools are used to detect coherent structures in the velocity and magnetic fields of a mean–field dynamo, produced by direct numerical simulations of the three–dimensional compressible magnetohydrodynamic equations with an isotropic helical forcing and moderate Reynolds number. Two distinct stages of the dynamo are studied, the kinematic stage, where a seed magnetic field undergoes exponential growth, and the saturated regime. It is shown that the Lagrangian analysis detects structures with greater detail, besides providing information on the chaotic mixing properties of the flow and the magnetic fields. The traditional way of detecting Lagrangian coherent structures using finite–time Lyapunov exponents is compared with a recently developed method called function M. The latter is shown to produce clearer pictures which readily permit the identification of hyperbolic regions in the magnetic field, where chaotic transport/dispersion of magnetic field lines is highly enhanced.

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eurm10 \checkfontmsam10 Coherent structures in a nonlinear dynamo]Coherent structures and the saturation of a nonlinear dynamo E. L. Rempel, A. C.-L. Chian, A. Brandenburg and P. R. Muñoz]E\lsR\lsI\lsC\lsO\nsL.\nsR\lsE\lsM\lsP\lsE\lsL1,\nsA\lsB\lsR\lsA\lsH\lsA\lsM\nsC.\ls-L.\nsC\lsH\lsI\lsA\lsNA\lsX\lsE\lsL\nsB\lsR\lsA\lsN\lsD\lsE\lsN\lsB\lsU\lsR\lsG and P\lsA\lsB\lsL\lsO\nsR.\nsM\lsU\lsÑ\lsO\lsZ 2010 \volume650 \pagerange119–126

agrangian coherent structures, nonlinear dynamo, magnetohydrodynamics, chaotic mixing

1 Introduction

The description of chaotic and turbulent flows by means of embedded coherent structures is a topic of great interest in the study of transport and mixing in fluids, since these structures act as organizing units in the flow, defining attracting and repelling directions, transport barriers and regions of high or low dispersion of passive scalars. There is no standard way of defining what a coherent structure is, but from the Eulerian point of view, they are often defined based on some measure related to vorticity. An example is the highly popular –criterion, first introduced by Hunt et al. (1988) to identify vortex cores based on the difference between the rate of strain and vorticity. Some other criteria define coherent structures or vortices based on local pressure minima (Jeong & Hussain, 1995) or on quantities involving the eigenvalues of the gradient tensor of the velocity field (Chong et al., 1990; Zhou et al., 1999; Chakraborty et al., 2005; Varun et al., 2008). From a Lagrangian point of view, coherent structures are seen as material surfaces that form the boundaries between regions of the flow with different behavior, such as vortex surfaces. They are found by following trajectories of fluid particles, while computing quantities such as the maximum rate of divergence of neighboring trajectories (Haller, 2001; Shadden et al., 2005) or the arc–length of the trajectory (Madrid & Mancho, 2009). Lagrangian tools are naturally suited for unsteady flows, since they take into account the temporal variations of the vector field, not just instantaneous snapshots. For a recent list of applications, see Peacock & Dabiri (2010).

Most works on Lagrangian coherent structures (LCSs) have focused on hydrodynamic turbulence, mainly in two–dimensions. A few papers have computed LCSs for three–dimensional magnetohydrodynamic (MHD) systems in the conservative (Leoncini et al., 2006) and dissipative (Rempel et al., 2011, 2012) regimes. In the aforementioned dissipative cases, only velocity field (kinetic) structures were explored. Here, we expand our previous results by computing the kinetic and magnetic coherent structures in a MHD model of mean–field dynamo. Dynamo action consists in the amplification of magnetic field by the motion of an electrically conducting fluid, being the mechanism responsible for the equipartition–strength magnetic fields observed in planets and stars (Brandenburg & Subramanian, 2005). Initially, a weak magnetic field undergoes an exponential growth in the kinematic dynamo phase until is strong enough to impact the fluid velocity , and eventually the magnetic energy saturates. The saturation process is closely related to the suppression of Lagrangian chaos in the velocity field; a comparison between the chaoticity of the velocity field during the growth and saturation phases of the dynamo has been performed in previous works (Brandenburg et al., 1995; Cattaneo et al., 1996; Zienicke et al., 1998). In this paper, the emphasis is on the detection of coherent structures and the transport of passive scalars and magnetic field lines in the transition from the kinematic to the saturated phase. Eulerian structures are detected using the –criterion and, for the detection of LCSs, the traditional technique of finite–time Lyapunov exponents (FTLEs) is compared with the recently proposed function M (Madrid & Mancho, 2009).

Section 2 of this paper describes the dynamo model adopted. The numerical results are presented in section 3, where the Eulerian and Lagrangian coherent structures are computed for the velocity and magnetic fields. Some conclusions are given in section 4.

2 The Model

The model is the prototype of dynamo used by Brandenburg (2001), where a compressible isothermal gas is considered, with constant sound speed , constant dynamical viscosity , constant magnetic diffusivity , and constant magnetic permeability . The following set of compressible MHD equations is solved

(1)
(2)
(3)

where is the density, is the fluid velocity, is the magnetic vector potential, is the current density, is the pressure, is an external forcing, and , where is assumed to be constant. Nondimensional units are adopted by setting , where is the spatial average of and is the smallest wavenumber in the box, which has sides and periodic boundary conditions. Thus, the time unit is , space is measured in units of , in units of , in units of , in units of and the unit of viscosity and magnetic diffusivity is . Equations (1)–(3) are solved with the PENCIL CODE 2, which employes an explicit sixth–order finite differences scheme in space and a third–order Runge–Kutta scheme for time integration.

The initial conditions are , and is a set of normally distributed, uncorrelated random numbers with zero mean and standard deviation equal to . The forcing function is given by

(4)

where is a time–dependent wavevector, is position, and with is a random phase. On dimensional grounds the normalization factor is chosen to be , where is a nondimensional factor, , and is the length of the integration timestep. We focus on the case where is around and randomly select, at each timestep, one of 350 possible vectors in . The operator is given by

(5)

where is an arbitrary unit vector needed in order to generate a vector that is perpendicular to . Note that and the helicity density satisfies , which is an important condition for the production of a mean–field dynamo (Moffatt, 1978). The forcing function is delta–correlated in time, i. e., all points of are correlated at any instant in time but are different at the next time step. Following Brandenburg (2001), the control parameters are set as , and the numerical resolution is .

3 Results

3.1 Mean–field dynamo

Figure 1 shows the time series of (light line) and (dark line), where and . During the first time units up to , the magnetic energy is too week to significantly impact the velocity field and , thus the Reynolds number is . During this kinematic phase, increases exponentially, with a growth rate obtained from the fitted line (dashed line). After , starts to decay due to the contribution of the Lorentz force (second term in the right side of Eq. (2)). Eventually, the r.m.s. quantities saturate due to nonlinear effects, with while the magnetic field reaches a super–equipartition value . The arrows indicate the times and , respectively, which will be used later to represent the kinematic and saturated phases. In turnover time units (), the referred times are and , respectively, and the growth rate is .

Figure 1: Time series of (light line) and (dark line) of MHD dynamo simulations for . The arrows indicate the kinematic phase at and the saturated nonlinear regime at , respectively. The growth rate during the kinematic phase is .

During the kinematic stage, the magnetic field displays low–amplitude stochastic fluctuations, as shown in the upper panels of Fig. 2. As grows, small–scale velocity and magnetic field fluctuations combine to produce a robust large–scale mean–field pattern (lower panels). The physics behind the rise of this mean–field is related to the so–called –effect (Moffatt, 1978) and has been explored in this model by Brandenburg (2001).

Figure 2: Intensity plot of magnetic field components at (upper panel) and (lower panel).

3.2 Eulerian coherent structures

Eulerian coherent structures can be extracted from the velocity field by decomposing the gradient tensor as

(6)

where and are the symmetric and antisymmetric parts of , respectively. The symmetric part is the rate–of–strain tensor and the antisymmetric part is the vorticity tensor. One way to define an Eulerian coherent structure is by finding regions of where vorticity dominates over strain, which can be measured by the –criterion (Hunt et al., 1988; Zhong et al., 1998; Haller, 2005; Lawson & Barakos, 2010)

(7)

Thus, an Eulerian coherent structure or vortex is defined as a region where .

Figure 3 shows the isosurfaces of the –criterion, using maximum (contour surfaces enclose high values). These plots are highly dependent on the threshold chosen for , but it is possible to see that the fluid is more intermittent at the kinematic dynamo phase () than after saturation (), since in the right panel the coherent structures fill the space in a more homogeneous way. There are fewer regions for where is much higher than the average, thus the presence of fewer vortices for this threshold in Fig. 3(a) than in 3(b), where local values of are closer to the average . Figure 4 shows the corresponding plots of for the magnetic field, where the coherent structures represent magnetic vortices or current structures Brandenburg et al. (1996).

Figure 3: Eulerian coherent structures in the velocity field, detected by instantaneous isosurfaces of the –criterion. The isosurfaces are defined using maximum .

Figure 4: Eulerian coherent structures in the magnetic field, detected by isosurfaces of the –criterion. The isosurfaces are defined using maximum .

In Fig. 5, intensity plots of the –criterion are shown for two–dimensional slices of the box at planes (upper panels) and (lower panels) at times (left panels) and (right panels), respectively. Coherent structures with strong vorticity are observed as bright spots, such as the one highlighted by a box in Fig. 5(a). Notice that at a large number of bright spots is seen in the –plane, but they are rare in the –plane, revealing a preferential alignment of coherent structures in the vertical direction in the saturated regime. A similar plot is shown for the magnetic field in Fig. 6, where some of the same coherent structures found in the velocity field can be observed, reflecting the strong coupling between and in Eqs. (2) and (3).

Figure 5: Eulerian coherent structures in the velocity field, detected by the –criterion at (left panel) and (right panel).

Figure 6: Eulerian coherent structures in the magnetic field, detected by the –criterion at (left panel) and (right panel).

Although some coherent structures are clearly detected by this Eulerian technique, the –criterion relies on a user defined threshold to determine their boundaries. In order to precisely identify the boundaries and the main transport barriers in the flow, the next section proceeds with a Lagrangian analysis.

3.3 Lagrangian coherent structures

This section describes two tools that can be employed to define/detect coherent structures in the Lagrangian frame, the finite–time Lyapunov exponents and the recently proposed function M.

Finite–time Lyapunov exponents

In the Lagrangian point of view, coherent structures are seen as material surfaces around which trajectory patterns are formed. In Haller & Yuan (2000), these surfaces are simply called Lagrangian coherent structures and are distinguished from other material surfaces in that a LCS exhibits locally the strongest attraction, repulsion or shearing in the flow. Repelling LCSs are responsible for generating stretching, attracting LCSs for folding, and shear LCSs for swirling and jet-type tracer patterns (Haller, 2011).

Attracting LCSs have commonly been associated with local maximizing curves (ridges) in the backward–time finite–time Lyapunov exponent (FTLE) field and repelling LCSs to ridges in the forward–time FTLE field (Shadden et al., 2005; Green et al., 2007; Beron–Vera & Olascoaga, 2010). There are limitations in such definition, as pointed out by Haller (2011) and Farazmand & Haller (2012), e. g., a ridge in the FTLE field may indicate the presence of a shear LCS or no LCS at all. But in general, ridges in the FTLE fields provide a good approximation to the true LCSs of the flow.

Let be the domain of the fluid to be studied, let denote the position of a passive particle at time and let be the velocity field defined on . The motion of the particle is given by the solution of the initial value problem

(8)

Let the flow map for be defined as . The deformation gradient is given by and the finite-time right Cauchy-Green deformation tensor is given by . Let be the eigenvalues of . Then, the finite-time Lyapunov exponents or direct Lyapunov exponents of the trajectory of the particle are defined as

(9)

A positive is the signature of chaotic streamlines in the velocity field, being a measure of the stretching of fluid elements (although it also incorporates shear (Haller, 2011)). In this work, the deformation gradient is computed with second order centered finite–differences.

Function M

Madrid & Mancho (2009) proposed a function to define “distinguished trajectories” (DTs), which are a generalization of the concept of fixed points for aperiodically time–dependent flows. In stationary flows, hyperbolic fixed points are responsible for particle dispersion and nonhyperbolic fixed points for particle confinement. Invariant stable and unstable manifolds of hyperbolic fixed points are barriers to transport and divide the phase space in regions with qualitatively different behaviors. The proposed function, named function M, can reveal both hyperbolic and nonhyperbolic flow regions of time–dependent flows. Moreover, M is also useful to detect the stable and unstable manifolds of distinguished hyperbolic trajectories (DHTs), defined as the set of points such that trajectories passing through these points at will approach the DHTs at an exponential rate as time goes to infinity or minus infinity, respectively (Branicki et al., 2011). The stable and unstable manifolds of DHTs correspond to the repelling and attracting Lagrangian coherent structures, respectively, as defined in the section 3.3.1.

Consider the system given by Eq. (8), where . For all initial conditions in at a given time , let us define the function as

(10)

Thus, the function M is a measure of the arc length of the curve traced by . Local minima of M represent trajectories that “move less”, being related either to hyperbolic or nonhyperbolic DTs. The manifolds of DHTs are also visible in the M field, since one expects a sharp distinction in the lengths of trajectory curves for particles in regions with different behaviors, separated by stable and unstable manifolds, as noted by Mendoza & Mancho (2010). The technique has been successfully applied to the detection of DTs and manifolds in oceanic (Mendoza & Mancho, 2010; Mendoza et al., 2010) and stratospheric (de la Cámara et al., 2012) flows.

Velocity field structures and chaotic mixing

The FTLEs are computed from a series of fully 3D snapshots of the velocity field taken at different times from to . Linear interpolation in time and third-order splines in space are used to obtain the continuous vector fields necessary to obtain the particle trajectories. Figure 7 depicts the probability distribution functions (PDFs) of the three FTLEs at (left) and (right) computed for particle trajectories from Eq. (9) with a value of corresponding to 9 turnover time units, where for the kinematic phase and for the saturated regime. Therefore, time units for the kinematic phase and time units for the saturated phase. One can see a clear reduction of Lagrangian chaos in the velocity field at , with the PDF of being shrunk and shifted to the left. There are also fewer regions with two or three positive exponents. Overall, chaotic mixing is diminished due to the growth of and the action of the Lorentz force. The asymmetry in the distributions is typical of heterogeneous mixing, where both regular and irregular trajectories coexist (in finite–time), which means that trajectories cannot uniformly sample the phase space (see, e. g., Beron–Vera & Olascoaga (2010)).

Figure 7: Probability density functions (PDFs) of the FTLEs of the velocity field at the kinematic (, left panel) and saturated (, right panel) regimes. The solid line represents , the dotted line, , and the dashed line, .

From Fig. 7 it is clear that most trajectories display two positive Lyapunov exponents. Zel’dovich et al. (1984) and Chertkov et al. (1999) state that in such a case, the total magnetic energy in a kinematic dynamo should behave as , therefore, one has for the growth rate

(11)

At , and , which from Eq. (11) provides (or in dimensional units), which agrees to within an order of magnitude with the fitted value , given in Fig. 1.

The remainder of this paper focuses on the backward–time maximum FTLE field, since they reveal the attracting LCSs, which correspond to structures seen using flow visualization in experiments (Green et al., 2007). Figure 8 shows the backward–time maximum FTLE field computed for corresponding to 9 turnover time units at (left) and (right) from a grid of initial conditions with particles. The bright lines represent the attracting LCSs. While the LCSs at reveal no preferred direction, consistent with an isotropic forcing, at there is a clear vertical alignment of LCSs in the –slice (Fig. 8(b)). This is due to the super–equipartition magnetic field at , which develops a large–scale vertical pattern in this plane (see Fig. 2), affecting the alignment of velocity field vectors.

A comparison between Figs. 5 and 8 shows that the FTLE field provides a clearer depiction of coherent structures, with finer details and more precise detection of structure boundaries. Moreover, some coherent structures are only apparent in the FTLE field, such as the large eddy indicated in Fig. 8(d).

Figure 8: Attracting Lagrangian coherent structures in the velocity field, given by the backward–time FTLE at (left panel) and (right panel).

From our experience, one of the problems with FTLE plots in turbulent flows is that pictures usually become increasingly complex for larger , with material lines “growing” and filling the entire phase space. In that sense, it is easier to use function M to detect the main coherent structures of the flow. Figure 9 is a plot of function M with in turnover time units, equivalent to Fig. 8. For the kinematic phase at (left panel) the eddies are clearly identified as closed regions with distinguished shades of gray. At (right panel) the borders between regions are not so sharp and there are wide smooth regions in the flow. Smoothness in the M field indicates that trajectories in those regions do not reach nearby hyperbolic regions during turnover time units, since hyperbolic trajectories are the ones responsible for dispersion and for producing sharp changes in M (Mendoza & Mancho, 2010). For larger , the boundaries become sharper and more foldings of manifolds are seen, but we keep in all our pictures to facilitate the comparison between both methods in different regimes. Overall, the function M seems to be less sensitive to the choice of than the FTLE. Figure 9 corroborates with Fig. 7 in revealing that there is less chaotic mixing after the nonlinear saturation of .

Figure 9: Lagrangian coherent structures in the velocity field, given by the function at (left panel) and (right panel).

Magnetic field structures and transport of field lines

Our simulations reveal that the magnetic field displays smooth and complex regions (see Fig. 6). If one applies the Lagrangian techniques discussed in the previous section to the magnetic field, the identification of magnetic LCSs provides the main barriers to the transport of field lines, a topic of great interest in magnetic reconnection studies (Evans et al., 2004; Grasso et al., 2010; Borgogno et al., 2011; Yeates & Hornig, 2011).

To obtain the magnetic LCSs, the magnetic field at a fixed dynamic time is used and the maximum FTLE field is computed by integrating

(12)

where the parameter (position) along the field line is seen as an effective time, or field–line–time (Borgogno et al., 2011). The flow map for is defined as . Equation (12) is integrated from to with fixed at . Lagrangian chaos in the magnetic field is responsible for the transport of magnetic field lines between different regions of the box. Here, the term “transport” is used to refer to motion of field lines in field–line–time, not in dynamic time. Therefore, the maximum FTLE provides a measure of the exponential separation between two neighboring field lines after a finite field–line–time , i. e., after a finite distance along the field line.

Figure 10 shows the backward–time maximum FTLE field for the kinematic (left panel) and saturated (right panel) regimes. The high–intensity lines represent attracting magnetic LCSs which act as barriers to field line transport. No transport of magnetic field lines occurs across invariant LCSs and large–scale transport is possible only through homoclinic and heteroclinic crossings of attracting and repelling LCSs, where a lobe dynamics mechanism takes place (Grasso et al., 2010; Borgogno et al., 2011; Yeates & Hornig, 2011; Rempel et al., 2012). Both FTLE fields are obtained by fixing the evolution (dynamic) time ( for the left panel and for the right panel) and setting , where for and for . In the kinematic regime () the LCSs display no preferred direction, and randomly fill the simulation box. Note that, at least for this value of , it is difficult to identify the coherent structure marked in the box in Fig. 10(a) due to the many foldings of attracting lines. After growth and saturation of (), the randomness of field line orientation is diminished and there is a preferential direction of alignment of field lines which, as mentioned before, directly affects the velocity field. The growth of is also reflected in the presence of many smooth regions in Figs. 10(c) and (d). Consequently, there is less chaotic mixing of field lines.

Figure 10: Attracting Lagrangian coherent structures in the magnetic field, given by the backward–time maximum FTLE at (left panel) and (right panel).

Once again, to obtain a clearer picture of magnetic coherent structures, we plot in Fig. 11 the function M for . It is easier to spot coherent structures from this field, such as the one in the box in Fig. 11(a). Function M seems to be better than the FTLE field in highlighting the main transport barriers, filtering out spurious lines that are not so important for mixing (Mendoza & Mancho, 2010).

Figure 11: Lagrangian coherent structures in the magnetic field, given by the function at (left panel) and (right panel).

As mentioned before, another feature of function M plots is that they provide both the stable and unstable manifolds of DHTs in the same picture. In order to illustrate this feature, three distinct regions are marked in Fig. 11(d). Regions A and B are located in smooth parts of the M field and region C in a region where manifolds are crossing. Smoothness of M in regions A and B indicates that initial conditions in these regions do not perceive nearby hyperbolic regions for (Mendoza & Mancho, 2010). An enlargement of region C is shown in Fig. 12, where the presence of manifolds indicates that field lines in this region either were dispersed in or will disperse in . We define three sets of initial conditions inside the small white squares A, B and C in Fig. 11(d), with each square containing 25 initial conditions. The result of integrating Eqs. (12) with each set of initial conditions for field–line–time units is shown in Fig. 13. Figures 13(a) and (b) show the trajectories of initial conditions in regions A and B, respectively, where it can be seen that all magnetic field lines stay close to each other, forming a magnetic flux tube that is not dispersed in this field–line–time interval. The apparent discontinuities in field lines are due to the periodic boundary conditions. In Fig. 13(c) the trajectories of initial conditions at region C are shown and one can see that there is great chaotic dispersion of field lines due to the crossings of manifolds in this region.

We conclude that function M can efficiently detect transport barriers and dispersion regions in a magnetic field.

Figure 12: Enlargement of the upper rectangle in Fig. 11(d).

Figure 13: Magnetic field lines produced by advecting a small blob of initial conditions in the magnetic field at . (a) The initial blob is located at , the point inside a magnetic vortex in Fig. 11(d); (b) the initial blob is located at , the point in a smooth region of Fig. 11(d); (c) the initial blob is located at , the point at a crossing of manifolds in Fig. 11(d).

4 Conclusions

Magnetohydrodynamic coherent structures have been identified in direct numerical simulations of a nonlinear dynamo. It was shown that both Eulerian and Lagrangian tools are able to extract vortices from velocity and magnetic field data. Although the Eulerian tool adopted is less computationally expensive, Lagrangian plots show finer details and can better locate the boundaries of vortices. In addition, the Lagrangian analysis provides important information about the mixing properties of the flow. Regarding the numerical tools employed to detect Lagrangian coherent structures (LCSs), the function M seems to be less sensitive to the choice of the integration time in comparison to the maximum finite–time Lyapunov exponent (FTLE). Thus, pictures obtained with the FTLE can become increasingly “noisy” with increasing due to the complex folding of material lines. Although function M provides “cleaner” pictures, the manifolds (transport barriers) are often not as clearly traced as in a FTLE field. Both tools reveal the strong impact of the magnetic field on the mixing properties of the velocity field when the system moves from the kinematic to the saturated dynamo phases. After the appearance of a strong mean–field, the kinetic and magnetic coherent structures are shown to align in a preferred direction, revealing the anisotropy developed in the vector fields.

Function M is also shown to be useful to detect manifolds of hyperbolic trajectories in the magnetic field, where intense transport of magnetic field lines takes place, a feature that can be further explored to study magnetic reconnection phenomena in plasmas. In relation to this, Lagrangian coherent structures in photospheric velocity fields have been shown to be associated with quasi–separatrix layers in the magnetic field (Yeates et al., 2012), which are regions of strong gradients in stretching and squashing of magnetic flux tubes, being identified as the preferential regions for magnetic reconnection (Démoulin, 2006; Santos et al., 2008). Magnetic reconnection is an important phenomenon in nonlinear dynamos, since it is believed that it can reduce the backreaction of the Lorentz force on the velocity field (Blackman, 1996). Essentially, turbulent motions can cause the stretching, twisting and folding of weak magnetic field lines in such a way as to produce the growth of magnetic flux. After the magnetic field reaches equipartition with the velocity field, the field lines can restrict fluid motions and transport of material is significantly reduced. This suppression of motions may also inhibit the dynamo. However, if there is rapid reconnection between magnetic flux tubes, this could prevent the tube from backreacting. For other works on the role of magnetic reconnection in dynamo models, see Archontis et al. (2003) and Baggaley et al. (2009).

We acknowledge Prof. R. A. Miranda, of the University of Brasilia, for his support with numerical codes. E. L. R. acknowledges the support of FAPESP (Brazil), CNPq (Brazil) and NORDITA (Sweden). A. C.-L. C. acknowledges support from CNPq (Brazil), the award of a Marie Curie International Incoming Fellowship and the hospitality of Paris Observatory. P. R. M. acknowledges the support of FAPESP (Brazil).

Footnotes

  1. thanks: Email address for correspondence: rempel@ita.br
  2. http://pencil-code.googlecode.com

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