Lagrangian Statistics for Dispersion in
Magnetohydrodynamic Turbulence
J. Pratt1, A. Busse2, and W.-C. Müller3
1Department of Physics and Astronomy, Georgia State University, Atlanta, GA, USA, 2James Watt School of
Engineering, University of Glasgow, Glasgow, UK, 3Center for Astronomy and Astrophysics, Technische Universität
Berlin, Berlin, Germany
Abstract Measurements in the heliosphere and high-resolution fluid simulations give clear
indications for the anisotropy of plasma turbulence in the presence of magnetic fields. How this anisotropy
affects transport processes like diffusion and dispersion remains an open question. The first efforts to
characterize Lagrangian single-particle diffusion and two-particle dispersion in incompressible
magnetohydrodynamic (MHD) turbulence were performed a decade ago. We revisit those pioneering
results through updated simulations performed at higher Reynolds number. We present new investigations
that use the dispersion of many Lagrangian tracer particles to examine the extremes of dispersion and the
anisotropy in direct numerical simulations. We then point out directions in which Lagrangian statistics
need to be developed to address the fundamental problem of anisotropic MHD turbulence and transport in
solar and stellar winds.
1. Introduction
To understand solar or stellar winds, we first study the fundamental behavior of a turbulent plasma. Under-
standing the anisotropic dynamics of a turbulent plasma is key to producing predictions and models for the
scattering of energetic particles in a solar wind. In his famous book on turbulence (Lesieur, 1987), Marcel
Lesieur wrote, “Turbulence is a dangerous topic which is often at the origin of serious fights in the scientific
meetings devoted to it since it represents extremely different points of view, all of which have in common
their complexity, as well as an inability to solve the problem.” One point of view follows from adopting the
Lagrangian frame of reference, the natural point of view for studying diffusive processes such as turbulence
in the solar wind. This work develops Lagrangian statistics to quantify anisotropic MHD turbulence.
Anisotropy is introduced to a turbulent plasma by a macroscopic magnetic field (in fundamental studies the
macroscopic magnetic field is more often called a guide field or a mean field), typically designated B0, and
measured with respect to the root-mean-square (RMS) fluctuations of the turbulent magnetic field BRMS.
In the solar wind, ion foreshock, and magnetosheath, ranges have been reported such that the macroscopic
magnetic field is between 1 and 2.5 times the RMS fluctuations (Zimbardo et al., 2010). In Earth's plasma
sheet the macroscopic magnetic field is on the order of two times the RMS fluctuations of the turbulent mag-
netic field (B0≈2BRMS) (Borovsky, 2005). In the magnetotail, observational data indicate that the magnetic
field is stronger, between three and five times the RMS fluctuations (see Table 1 of Zimbardo et al., 2010).
We examine a system with a moderately strong anisotropy caused by a macroscopic magnetic field of mag-
nitude three times the average RMS magnetic field fluctuations. The observations from our simulations may
reveal characteristics of the anisotropy due to the strongest macroscopic magnetic fields of the solar wind
or the weakest magnetic fields in the magnetotail.
Motivated by new experimental techniques developed over the last two decades (e.g., Biferale et al., 2008;
Bourgoin et al., 2014; Bourgoin & Xu, 2014; La Porta et al., 2000; Lawson et al., 2018, 2019; Liot, Gay,
et al., 2016; Liot, Seychelles, et al., 2016; Mordant, Lévêque, et al., 2004; Polanco et al., 2018; Xu et al., 2006),
Lagrangian statistics of turbulent flows are attracting increasing attention. In oceanography and atmo-
spheric science, Lagrangian measurements have a rich history (e.g., as discussed by Aksamit et al., 2020;
Businger et al., 2006; Fossette et al., 2012; LaCasce, 2008a, 2008b). In space physics, the Cluster mis-
sion (Escoubet et al., 1997) and the CubeSat project (Poghosyan & Golkar, 2017) have demonstrated that
measurements made from multiple spacecraft will be feasible in the future.
RESEARCH ARTICLE
10.1029/2020JA028245
Special Section:
Solar and Heliospheric Plasma
Structures: Waves, Turbulence,
and Dissipation
Key Points:
• We perform direct numerical
simulations of isotropic and
anisotropic magnetohydrodynamic
turbulence
• We examine and compare
single-particle diffusion,
two-particle dispersion, and many
particle dispersion
• Our Lagrangian statistics reveal
the dependence of anisotropy on
separation time scales from the
Lagrangian point of view
Supporting Information:
• Supporting Information S1
• Supporting Information S2
Correspondence to:
J. Pratt,
jprat[email protected]
Citation:
Pratt, J., Busse, A., & Müller, W.-C.
(2020). Lagrangian statistics for
dispersion in magnetohydrodynamic
turbulence. Journal of Geophysical
Research: Space Physics,125,
e2020JA028245. https://doi.org/10.
1029/2020JA028245
Received 18 MAY 2020
Accepted 28 SEP 2020
Accepted article online 10 OCT 2020
©2020. American Geophysical Union.
All Rights Reserved.
PRATT ET AL. 1of12
Journal of Geophysical Research: Space Physics 10.1029/2020JA028245
Responding to the availability of high-quality measurements, high-resolution numerical simulations
are enabling increasingly detailed studies of the dynamics of Lagrangian tracer particles (e.g., Bianchi
et al., 2016; Biferale et al., 2005; Buaria et al., 2016; Sawford & Yeung, 2015; Schneide et al., 2018; Yeung &
Borgas, 2004). This explosion of work using Lagrangian tracer particles to explore the fundamental statis-
tics of turbulence in neutral fluids has been summarized in several reviews, including Meneveau (2011),
Pope (1994), Toschi and Bodenschatz (2009), Wilson and Sawford (1996), and Yeung (2002), and a compre-
hensive review dedicated to two-particle dispersion (Salazar & Collins, 2009).
The first program to investigate MHD turbulence from the Lagrangian point of view began over a decade
ago as a collaboration between a group at the Max Planck Institute for Plasma Physics and a group at the
Ruhr-Universität Bochum (Busse & Müller, 2008; Busse et al., 2007, 2010; Homann, Grauer, et al., 2007;
Homann et al., 2009; Müller & Busse, 2007a, 2007b). That work was based on direct numerical simulations
of three-dimensional incompressible homogeneous MHD turbulence. Here we revisit some of the funda-
mental results of those earlier studies, using new simulations at higher Reynolds number. We also expand
on progress that has been made more recently to use Lagrangian statistics for anisotropic turbulence driven
by convection and magnetoconvection (Pratt et al., 2017). We discuss further directions for future work.
2. Direct Numerical Simulations for Lagrangian Single-Particle Diffusion
and Two-Particle Dispersion
To study diffusion and dispersion, we produce simulations of statistically stationary, forced, homogeneous
incompressible MHD turbulence. In each direct numerical simulation presented in this work, we solve the
nondimensional equations for incompressible magnetohydrodynamics:
𝜕𝝎
𝜕t−∇×(
v×𝝎+j×B)=𝜈∇2𝝎+f𝜔,(1)
𝜕B
𝜕t−∇×(
v×B)=𝜂∇2B+fb,(2)
using a pseudospectral method in a three-dimensional rectangular simulation volume with periodic bound-
ary conditions. These equations include the solenoidal velocity field v, vorticity 𝝎=∇×v, magnetic field
B, and current density j=∇×B. Each of the quantities in Equations 1 and 2 has been nondimensionalized
using relevant time and length scales, commonly referred to as Alfvénic units. Two dimensionless parame-
ters, 𝜈 and 𝜂, appear in the equations. They derive from the kinematic viscosity 𝜈and the magnetic diffusivity
𝜂. A fixed time step and a low-storage third-order Runge-Kutta method (Williamson, 1980) are used for the
time integration. A static macroscopic magnetic field B0pointing in the positive zdirection may be imposed.
To maintain a statistically stationary turbulent steady state, both the vorticity and magnetic fields are forced
on the largest scales of the simulation volume using a method that allows the largest-scale motions of the
system to evolve. This deterministic homogeneous method of forcing establishes a constant injection of
energy at large scales (see Busse, 2009, for a detailed discussion of this forcing method). In Equations 1 and
2 forcing terms f𝜔and fbare introduced. For the simulations in Table 1, these forcing terms are nonzero
only for the wave-vector shell 1 ≤|k|≤2.5. The same amount of energy is injected into each forced mode
according to
f𝜔(k,t)=𝛾𝑓,𝜔
𝝎(k,t)
|
𝝎(k,t)|2,(3)
fb(k,t)=𝛾𝑓,B
B(k,t)
|
B(k,t)|2.(4)
The constants 𝛾f,𝜔and 𝛾f,Bregulate the energy injection rate and are equal for the simulations in Table 1.
Using homogeneous forcing, the cross helicity of the forced modes is maintained as a small nonzero value
to prevent the emergence of states dominated by Elsässer positive (z+) or negative (z−) interactions, that
is, states where the MHD turbulent system becomes maximally imbalanced. Such a state can lead to a
breakdown of the nonlinear energy cascade, for example, as discussed in Biskamp (2003). In a system in
quasi-stationary state, homogeneous forcing is expected to disturb the natural turbulent flow only mildly.
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Table 1
Parameters for Simulations of Lagrangian Single-Particle Diffusion and Two-Particle Dispersion
B0BRMS 𝜏𝜂(10−2)LE𝜂kol (10−3) Re Elongation
HNOVIS — — 4.47 3.56 3.46 7,571 1
ANOVIS0 0. 1.10 5.15 2.81 3.59 7,230 1
ANOVIS3 3. 1.09 5.74 5.56 3.79 7,570 2
Note. Magnitude of the static macroscopic magnetic field B0, the root-mean-square of magnetic fluctu-
ations BRMS averaged over the full simulation time, Kolmogorov time scale 𝜏𝜂, the large-eddy length
scale LE, and the Kolmogorov microscale 𝜂kol. Each simulation is performed on a grid of N3=1,0243
and uses 8.3 million Lagrangian tracer particles in a homogeneous random distribution.
This forcing method is distinct from those used by Busse and Müller (2008); Busse et al. (2010, 2007); and
Homann, Grauer, et al. (2007). Large-scale Alfvén waves are permitted by this forcing method and are
observed in the simulations examined in this work.
Table 1 provides a summary of the fundamental parameters of the simulations we consider. Each simulation
is performed on a grid of 1,0243collocation points. In the table, we record the strength of the macroscopic
magnetic field B0imposed in the zdirection, as well as the root mean square of the magnetic fluctua-
tions BRMS, averaged over the simulation time. We measure length in units of the Kolmogorov microscale
𝜂kol =(𝜈3∕𝜖v)1∕4and time in units of the Kolmogorov time scale 𝜏𝜂=(𝜈𝜖v)1∕2, where 𝜖v=𝜈⟨∑kk2
v2⟩is the
time-averaged rate of kinetic energy dissipation; the Kolmogorov microscales are the smallest length and
time scales that characterize turbulent flows.
The Kolmogorov microscales define the resolution requirements for a direct numerical simulation (DNS).
All of our simulations fulfill the classic criterion of Pope (2000) and Yeung and Pope (1989) for a DNS, that is,
kmax𝜂kol > 1.5. This criterion has been widely used to evaluate whether homogeneous isotropic turbulence is
sufficiently resolved (see also Yeung et al., 2018, for a recent study of the effect of resolution on homogeneous
isotropic turbulence). For reference, the Eulerian kinetic and magnetic energy spectra for the simulations
described in Table 1 are provided in Figure 1.
For simulations of MHD turbulence that are anisotropic because of the effect of a mean magnetic field, a
box that is elongated in the zdirection has been used in many earlier works, including, for example, Mason
et al. (2006). To determine the necessary elongation of the simulation volume in the zdirection, we consider
the correlation length of the velocity field in each direction. We measure a correlation length of the velocity
field in the zdirection, Lc,||, that is larger than in the xand ydirections, in agreement with previous studies,
(e.g., Boldyrev, 2005; Chandran, 2008; Cho et al., 2002). To accommodate this larger Lc,|| within our simula-
tion volume, we elongate the simulation volume in the zdirection, so that the condition on the box length
in the zdirection Lz≫Lc,|| is satisfied. The elongation of the simulation box, as measured by the ratio of
Figure 1. Energy spectra for the simulations in Table 1: (a) kinetic energy spectra and (b) magnetic energy spectra.
These spectra are compensated by k3/2.
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the box length in the zdirection to the box length in the xdirection, Lz/Lx, is listed in the table for each
simulation. The simulation volume has sides of length Lx=L𝑦=2𝜋in the perpendicular directions.
2.1. Lagrangian Tracer Particles
For the simulations in Table 1, the positions of Lagrangian tracer particles are initialized in a homogeneous
random distribution at a time when the turbulent flow is in a statistically stationary steady state. The total
number of particles in each simulation is np≈8.3 million. This is a high density of tracer particles, compa-
rable to earlier works (Busse & Müller, 2008; Busse et al., 2007, 2010; Homann, Grauer, et al., 2007; Homann
et al., 2009; Müller & Busse, 2007a, 2007b). The number of Lagrangian tracer particles that we use produces
statistics that are well resolved in space.
At each time step the particle velocities are interpolated from the instantaneous Eulerian velocity field using
a tricubic polynomial interpolation scheme (for a clear analysis of the impact of the interpolation scheme
on Lagrangian statistics, see Homann, Dreher, et al., 2007). Particle positions are calculated by numerical
integration of the equations of motion using a low-storage third-order Runge-Kutta method identical to
that used for the Eulerian fields. Each simulation is run for a sufficient time that Lagrangian particle pair
separations exhibit a diffusive trend. This length of time is approximately 400 𝜏𝜂, an amount of time that
allows the particles in our hydrodynamic simulations (HNOVIS and HPUFF) to cross the simulation volume
approximately once, on average. Because the Lagrangian tracer particles do not cross the simulation volume
multiple times, the periodic boundary conditions have no observed effect on the statistics produced.
2.2. Reynolds Number for Anisotropic MHD Turbulence
For an isotropic system with B0=0, we define the Reynolds number in the standard way as
Re =⟨E1∕2
vLE⟩∕𝜈, (5)
from the kinetic energy Ev, a characteristic length scale LE, and the viscosity 𝜈. For statistically homogeneous
isotropic turbulent flows, LEis commonly defined as a dimensional estimate of the size of the largest eddies,
LE=E3∕2
v∕𝜖v, using 𝜖v, the rate of kinetic energy dissipation. The calculated value of this length scale LEis
included in Table 1 for each of our simulations.
To compare isotropic and anisotropic turbulent flows, we use a more basic definition of the Reynolds number
(see chapter 6.1.2 of Pope, 2000)
Re =c(𝜂kol∕LF)−4∕3.(6)
This definition requires knowledge of the forcing length scale LFand a constant c. Our method of forcing
affects a minimum length scale
LF=2𝜋∕k𝑓,max =2𝜋∕3.(7)
We determine the constant cby comparison with the definition of the Reynolds number in the isotropic case
given in Equation 5. The Reynolds number is calculated in this way for simulation ANOVIS3 in Table 1. The
magnetic Reynolds number is defined from the Reynolds number and the magnetic Prandtl number, that is,
Rem=PrmRe. In all simulations in this work, the magnetic Prandtl number Prm=1 so that the magnetic
Reynolds number is equal to the Reynolds number.
3. Single-Particle Diffusion and Two-Particle Dispersion in MHD Turbulence
Perhaps the most fundamental result from studies of Lagrangian statistics in MHD turbulence is a com-
parison of single-particle diffusion and two-particle dispersion. While single-particle diffusion exhibits the
same essential behavior in hydrodynamic turbulence and MHD turbulence, two-particle relative dispersion
in MHD turbulence differs significantly from the hydrodynamic behavior (Busse et al., 2007). This is a sig-
nificant observation, because single-particle statistics are heavily impacted by the largest scales; two-particle
statistics effectively limit contributions from the largest scales of the flow. A single-particle diffusion curve
follows the evolution of the average square distance a particle has moved from its initial position, represented
by ⟨𝜉2⟩. We compare single-particle diffusion curves for two isotropic turbulence flows: in hydrodynamic
turbulence (simulation HNOVIS) and in MHD turbulence (simulation ANOVIS0), see Figure 2. These two
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Figure 2. Evolution of mean square distance from the initial position for
the three simulations described in Table 1. The mean square distance for
the anisotropic MHD simulation ANOVIS3 is separated into distances
parallel and perpendicular to the mean magnetic field. Each curve is
produced from the average of at least four independent initial times.
diffusion curves exhibit nearly identical behavior. This figure also
includes the curves for an anisotropic MHD turbulence simulation,
ANOVIS3, which is significantly different from either isotropic case. The
results from ANOVIS3 have been split into diffusion in the direction of
the mean magnetic field and diffusion in the direction perpendicular to
the mean magnetic field. Diffusion in both these distinct directions takes
a characteristically similar shape. Each of the curves in Figure 2 exhibits
a clear ballistic scaling as t2at early times and has entered a diffusive
regime with scaling close to tat late times. At early intermediate times,
both the diffusion parallel and perpendicular to the mean magnetic field
grow more slowly in anisotropic MHD than in either isotropic simulation.
This slowdown is larger in the direction parallel to the mean magnetic
field. During the later part of the intermediate time period, the separa-
tion process accelerates; diffusion then slows as the diffusive regime is
reached. These observations agree with results first reported in Busse and
Müller (2008) and Busse (2009).
The velocity autocorrelation function has a differential relation to diffu-
sion; these are Green-Kubo relations, for example, as discussed in Alder
et al. (1970) and Dubbeldam et al. (2009). Because of this, the velocity
autocorrelation function is typically used to shed new light on diffu-
sion by providing information about the relaxation of fluctuations over
long times and distances. For Brownian motion, the velocity autocor-
relation function is fit well by a single decaying exponential. A single
decaying exponential has also been shown to be a good fit for hydrodynamic turbulence in both exper-
imental and numerical studies (Sato & Yamamoto, 1987; Yeung & Pope, 1989). A single decaying expo-
nential is an excellent fit for the velocity autocorrelation function of simulation HNOVIS (see Figure 3).
However, for the MHD simulations ANOVIS0 and ANOVIS3, it is not clear whether a single decay-
ing exponential is a reasonable model. Isotropic MHD turbulence leads to a swifter overall decay of the
velocity autocorrelation function than hydrodynamic turbulence. In anisotropic MHD turbulence, this
swift decay is further exaggerated, fundamentally changing the initial shape of the decay of the velocity
autocorrelation function so that a decaying exponential produces a poor fit. In the anisotropic case,
Figure 3. Velocity autocorrelation function for the three simulations
described in Table 1. The velocity autocorrelation function for the
anisotropic MHD simulation ANOVIS3 is separated into velocities in the
direction parallel and perpendicular to the mean magnetic field. Each curve
is produced from the average of at least four independent initial times.
large-scale Alfvénic fluctuations are clearly visible in the velocity auto-
correlation function. The characteristic time of decay for the velocity
autocorrelation function is smaller in the anisotropic case than in the
isotropic case. In the direction aligned with the mean magnetic field
this decay time is slightly shorter than in the direction perpendicular.
Small-scale fluctuations in the velocity are therefore more probable in
the direction perpendicular to the magnetic field. Over long times, the
absolute diffusion parallel to the mean magnetic field is therefore smaller
because of the different prevalence in velocity fluctuations.
For two-particle dispersion, Busse et al. (2007) find considerable differ-
ences between isotropic hydrodynamic turbulence and isotropic MHD
turbulence. We examine pairs of particles that are initially separated
by 2𝜂kol, the smallest initial separation that is resolved by our grid
(see Figure 4). Each of these curves exhibits a clear ballistic scaling as
t2at early times and has entered a diffusive regime at late times. In
the diffusive regime of these dispersion curves, a mildly superdiffusive
slope is evident. The anisotropic MHD simulation ANOVIS3 is the most
superdiffusive with a slope near three at these late times. The dispersion
curves of simulation ANOVIS3 also show an oscillation that we attribute
to large-scale Alfvénic fluctuations. We find that the rate of dispersion
is slower for MHD turbulence than for hydrodynamic turbulence. This
rate of dispersion first slows down at early intermediate times, as the
dispersion curves depart from ballistic scaling. The slowdown is more
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Figure 4. Evolution of mean square separation of pairs of Lagrangian tracer particles in homogeneous isotropic
hydrodynamic turbulence (simulation HNOVIS), homogeneous isotropic MHD turbulence (simulation ANOVIS0), and
homogeneous anisotropic MHD turbulence (simulation ANOVIS3). The behavior for pairs with an initial separation
of 2ηkol is shown (a) for particles initially separated in the direction perpendicular to the mean magnetic field and
separation distance measured perpendicular to the field and (b) for particles initially separated in the direction parallel
to the mean magnetic field and separation distance measured aligned with the mean magnetic field.
significant for the anisotropic MHD simulation ANOVIS3 than for the isotropic MHD simulation ANOVIS0.
This slowdown feature that is common between isotropic and anisotropic MHD simulations may be
explained as an effect of the local, fluctuating magnetic field. This field appears to be sufficient to produce
a degree of anisotropy in the relative dispersion process, even in a globally isotropic simulation.
Figure 5. Diffusion of a single droplet with initial diameter 14ηkol, composed of approximately 16,000 Lagrangian
tracer particles, in simulation HPUFF. The series of six snapshots documents the dispersion of the particles over
approximately 40 τη. Particles are colored by the kinetic energy of the flow.
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Table 2
Parameters for Simulations of Lagrangian Many-Particle Dispersion
B0BRMS 𝜏𝜂(10−2)LE𝜂kol (10−3) Re Elongation
HPUFF — — 4.20 3.70 3.55 10,570 1
APUFF0 0. 1.78 4.89 2.49 3.83 5,640 1
APUFF3 3. 1.17 4.82 2.57 3.80 5,940 2
Note. Magnitude of the static macroscopic magnetic field B0, the root mean square of magnetic fluctu-
ations BRMS averaged over the full simulation time, Kolmogorov time scale 𝜏𝜂, the large-eddy length
scale LE, and the Kolmogorov microscale 𝜂kol. Each simulation is performed on a grid of N3=1,0243
and uses 4.5 million particles initialized into droplets.
4. Direct Numerical Simulations for Lagrangian Many-Particle Dispersion
To examine many-particle dispersion, we conduct a separate series of simulations that use a significantly
different initial set-up of Lagrangian tracer particles. The positions of the Lagrangian tracer particles are ini-
tialized into spherical volumes of a specified size and density of tracer particles, henceforth called droplets.
The droplets are homogeneously and randomly distributed throughout the simulation volume at a time
when the turbulent flow is in a statistically stationary steady state. A visualization of such a droplet dispers-
ing is shown in Figure 5. Simulations using this initial droplet setup for Lagrangian particles were published
in Pratt et al. (2017) for a study of convection, and a similar initial setup was used to study Navier-Stokes
turbulence in Bianchi et al. (2016). Each simulation described in Table 2 has this initial setup and a total
number of tracer particles of np≈4.5 million.
For the simulations in Table 2, a stochastic forcing method (as described by Busse, 2009; Eswaran &
Pope, 1988b, 1988a) is used. For simulations HPUFF and APUFF0, the forcing terms are nonzero only
for the wave-vector shell 1 ≤|k|≤2.5. For simulation APUFF3, this forcing wave-vector shell is shifted to
2.5 ≤|k|≤3.5; this adjustment was made because forcing wave vectors in the lower kshell in combination
with the mean magnetic field was found to lead to a buildup of energy at large scales, significantly chang-
ing the energy spectra. For reference, the Eulerian kinetic and magnetic energy spectra of the simulations
described in Table 2 are provided in Figure 6. As with the previous series of simulations, large-scale Alfvénic
fluctuations are permitted by the stochastic forcing method and are observed in our simulations. Aside
from the different initial setup of Lagrangian tracer particles and the use of a stochastic forcing method, the
simulations in Table 2 follow a similar setup to the simulations in Table 1.
5. Many-Particle Dispersion in MHD Turbulence
Lagrangian statistics are known to be sensitive to extreme events in the fluctuating turbulent fields, for
example, Boffetta and Sokolov (2002) and Yeung and Borgas (2004). In Pratt et al. (2017), we developed an
Figure 6. Energy spectra for the simulations in Table 2: (a) kinetic energy spectra and (b) magnetic energy spectra.
These spectra are compensated by k3/2.
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Figure 7. Evolution of mean square maximal ray rof the droplets for the simulations described in Table 2. Droplets
have identical initial particle density and an initial diameter of (a) 4ηkol and (b) 14ηkol.
analysis using the convex hull to calculate the extremes of dispersion of a group of many Lagrangian tracer
particles. The simplest diagnostic resulting from this approach is the maximal ray internal to a convex hull.
For a group of particles G, the maximal ray can be calculated:
r=max
i,𝑗∈G√(xi−x𝑗)2+(𝑦i−𝑦𝑗)2+(zi−z𝑗)2.(8)
This measure of separation is based on pairs of particles within the droplet that are furthest apart at a
given point in time, so that it always is defined as the largest extent of dispersion of the group. Thus, dif-
ferent particles may be used to determine the maximal ray as the convex hull evolves in time; in contrast,
two-particle dispersion always considers a fixed pair of particles. The maximal ray evolves with similar
ballistic, intermediate, and diffusive phases to two-particle dispersion (see Figure 7). However, because it
measures the extreme of dispersion, during the initial short ballistic regime the maximal ray dispersion
curve grows slightly more quickly than t2. In the intermediate regime, we observe that even the extremes
of dispersion are slowed by the presence of magnetic fields, with dispersion in the anisotropic MHD simu-
lation APUFF3 growing more slowly than in the isotropic case APUFF0. This has interesting overlap with
the earlier observation (Busse, 2009) that intermittency in particle accelerations is lower in isotropic MHD
turbulence than in isotropic hydrodynamic turbulence, and lower still in anisotropic MHD turbulence. The
extremes of dispersion, not just the averages, are suppressed by the anisotropy of the magnetic fluctuations.
In Pratt et al. (2017), we used the surface area sand volume vof the convex hull surrounding the Lagrangian
tracer particles to quantify the anisotropy in each simulation. For a perfect sphere, the nondimensional
ratio s/v2/3 takes a value of (36𝜋)1/3 ≈4.8; for an anisotropic shape like a pancake or needle, this ratio will be
larger. The magnitude of this ratio, compared with 4.8, indicates how anisotropic the convex hull around a
group of Lagrangian tracer particles is. During intermediate times corresponding to the early intermediate
range of time scales, the anisotropy ratio grows dramatically for all simulations. For the hydrodynamic sim-
ulation HPUFF, this period lasts between approximately 𝜏𝜂and 8𝜏𝜂; for the MHD simulations this period
lasts more than twice as long, between approximately 𝜏𝜂and 20𝜏𝜂. The time of the peak correlates with the
acceleration of dispersion in the pair dispersion curves or the maximal ray curves for each of these sim-
ulations. After this rise in the anisotropy ratio, it falls to lower levels (see Figure 8), eventually indicating
isotropic growth of the droplets in the early diffusive regime, with a shape near spherical. Many works have
argued that anisotropy in MHD turbulence is length scale dependent, for example, as discussed by Cho and
Vishniac (2000), Schekochihin et al. (2008), and Verdini et al. (2015). Figure 8 demonstrates that anisotropy
is time scale dependent, from the Lagrangian point of view. In addition, the evolution of the anisotropy ratio
reveals a new structure to that scale dependence.
The anisotropy ratio becomes larger in both isotropic and anisotropic MHD than in our hydrodynamic sim-
ulation. It is also slightly larger in isotropic MHD turbulence than in anisotropic MHD turbulence. However
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Figure 8. Average anisotropy ratio for the simulations described in Table 2, for droplets with (a) an initial diameter of
4ηkol and (b) an initial diameter of 14ηkol. The density of particles in both sizes of droplet is the same. A shaded area
indicating one standard deviation above and below each line is provided.
the strong overlap of the shaded regions in Figure 8, which indicate one standard deviation above and below
the average line, indicates that this difference between the average lines for these simulations is not sta-
tistically significant. The difference in the anisotropy ratio between hydrodynamic turbulence and MHD
turbulence is clearly statistically significant. A comparison between panels (a) and (b) of this figure demon-
strates that, for initially smaller groups of particles, a larger anisotropy ratio is achieved. This also gives an
indication that intermittency is different on different length scales of a turbulent flow.
6. Summary and Discussion
Lagrangian tracer particles provide a powerful tool for quantifying diffusion and dispersion in MHD turbu-
lence. In the simulations presented here, we have confirmed that single-particle diffusion curves are similar
for isotropic hydrodynamic and magnetohydrodynamic turbulence. However, for anisotropic MHD turbu-
lence the single-particle diffusion curves exhibit characteristic differences; they show slower diffusion at
intermediate separation times. Two-particle dispersion curves for isotropic MHD turbulence have clear dif-
ferences from isotropic hydrodynamic turbulence. Anisotropic MHD turbulence makes those differences
larger. We obtain these results from three simulations that have identical resolution and closely comparable
Reynolds numbers and general setup. These simulations have higher Reynolds numbers, and a weaker mean
magnetic field than earlier works (Busse, 2009; Busse & Müller, 2008; Busse et al., 2007), but our findings
agree in a broad sense.
Single-particle diffusion and two-particle dispersion curves provide critical information; however, they do
not provide a complete picture of transport processes for anisotropic MHD turbulence. We therefore extend
our examination of dispersion using the novel many-particle methods developed in Pratt et al. (2017). This
is a new application of statistical methods originally developed to examine anisotropy in convection sim-
ulations. These methods use a convex hull algorithm to describe the outer surface of a group of many
particles that are initially tightly packed into a spherical configuration, which we call a droplet. The maxi-
mal ray, which represents the extremes of dispersion, is found to follow a pattern similar to the two-particle
dispersion: The speed of dispersion can be ordered with hydrodynamic turbulence producing the fastest
dispersion, then isotropic MHD turbulence, and then anisotropic MHD turbulence. Thus, MHD turbu-
lence can be shown to suppress even the extremes of particle dispersion. When we examine the anisotropy
ratio for groups of many Lagrangian tracer particles, both isotropic and anisotropic MHD turbulence are
more anisotropic than hydrodynamic turbulence. The anisotropy ratio clearly reveals the dependence of
anisotropy on separation time scales, a concept that translates to length-scale dependent anisotropy, which
has been observed in the Eulerian frame of reference.
In order to fully understand how magnetic fields affect MHD turbulence and anisotropy, further develop-
ment of Lagrangian statistics is needed. One aspect in particular that would be helpful to clarify transport
processes is the analysis of particle trajectories in MHD turbulence; studies of particle trajectories in neutral
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Journal of Geophysical Research: Space Physics 10.1029/2020JA028245
fluid turbulence have produced interesting results in recent years (Bos et al., 2015; Choi et al., 2010; Ouellette
& Gollub, 2007; Siu & Taylor, 2011; Xu et al., 2007). Quantifying the trajectories involves calculating the cur-
vature and torsion that a particle experiences along its path. To better understand these trajectories, studies
of the extremes of acceleration (Homann, Grauer, et al., 2007) need to be further developed for anisotropic
MHD turbulence. Intriguing experimental results for the acceleration PDF of neutral-fluid turbulence
(e.g., Liot, Gay, et al., 2016; Mordant, Crawford, et al., 2004) make it likely that there is a great deal more to
understand about MHD turbulence through the peculiar behavior of the Lagrangian acceleration. In addi-
tion, studies of anisotropic MHD turbulence are often based on relatively few simulations and therefore do
not reveal the role of the mean magnetic field in generating anisotropy and changing transport properties.
A larger range of anisotropic simulations needs to be studied in conjunction so that the influence of a mean
magnetic field can be established. These areas of study are included in our ongoing and planned work.
Data Availability Statement
In accordance with the Enabling FAIR data Project guidelines, simulation data associated with this work are
made permanently available on the author's FigShare repository https://figshare.com/authors/Jane_Pratt/
2136082 (Pratt et al., 2020a, 2020b, 2020c, 2020d, 2020e).
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Acknowledgments
This material is based upon work
supported by the National Science
Foundation under Grant No. 1907876.
Simulations were performed on the
Konrad and Gottfried computer
systems of the Norddeutsche Verbund
zur Förderung des Hoch- und
Höchstleistungsrechnens (HLRN) by
the project bep00051 “Lagrangian
studies of incompressible turbulence
in plasmas.”
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