Coherent Vorticity and Current Density Simulation ofMagnetohydrodynamic Turbulence
K. Yoshimatsu1 , N. Okamoto2,3 , Y. Kawahara1 , K. Schneider4 , and M. Farge5
121

Department of Computational Science and Engineering, Nagoya University, Nagoya, 464-8603, Japan
Center for Computational Science, Graduate School of Engineering, Nagoya University, Nagoya, 464-8603, Japan
3JST, CREST, 5, Sanbancho, Chiyoda, Tokyo 102-0075, Japan4
M2P2–CNRS & CMI, Aix-Marseille Université, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France
5
LMD–IPSL–CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

Introduction

Turbulence, which is ubiquitous in our daily life, has alarge number of degrees of freedom, a wide range of dynamically active scales and strong nonlinearity. A key featureof turbulence is strong intermittency, which is attributed
to coherent structures, e.g. vortex tubes and sheets. Observations show its self-organization into structures, evenat large Reynolds number superimposed to a background
ﬂow. This motivates us to split turbulent ﬂows into twocontributions: i.e., a coherent ﬂow and a random background ﬂow. Both contributions are multiscale and exhibit no scale separation. The wavelet representation isan eﬃcient tool to perform such a multiscale decomposition, since wavelets are well localized functions in space,scale and direction. The wavelet representation has beenutilized to extract coherent vorticity from turbulence witha reduced set of degrees of freedom. To simulate hydrodynamic (HD) turbulent ﬂow using this reduced set, thecoherent vorticity simulation (CVS) method has been proposed [1, 2, 3].
CVS is a deterministic computation of HD turbulent
ﬂows based on the wavelet ﬁltered Navier-Stokes equations, using an adaptive wavelet basis while either modeling or neglecting the inﬂuence of the incoherent background ﬂow. At each time step the vorticity ﬁeld is decomposed into two orthogonal components using an orthogonal
wavelet basis: the coherent vorticity, corresponding to thecoeﬃcients whose modulus is larger than a threshold, andthe remaining incoherent vorticity. The threshold value
depends on the total enstrophy, which evolves in time, andon the maximal resolution, which remains constant. Thevalue is directly related to Donoho’s criterion [4] which
supposes the incoherent ﬂow to be Gaussian and decorrelated. To track the translation of the coherent vorticity
and the generation of smaller scales, a safety zone is required in wavelet space [5, 6].
The CVS approach is diﬀerent from the ﬁltering approach of large eddy simulation, where only the evolutionof the large-scale ﬂow is computed while modeling the inﬂuence of small-scale motion onto the large-scale motion.
The aim of the present work is a generalization of CVS
in order to compute the time evolution of coherent ﬂow inmagnetohydrodynamic (MHD) turbulence which exhibits
other types of intermittent dynamics. We call this simulation method coherent vorticity and current density simulation (CVCS). CVCS is carried out for three-dimensional(3D) forced homogeneous incompressible MHD turbulence

without any imposed uniform magnetic ﬁeld. Homogeneous turbulence is chosen here in order to demonstrate
the eﬃciency of CVCS in the worst possible case wherestructures are spread all over physical space in contrast toinhomogeneous turbulence. To assess CVCS, the results
are compared with direct numerical simulation (DNS) using the same maximal resolution.

2Methodology of CVCSIn CVCS, the vorticity and current density ﬁelds are respectively decomposed at each time step into two orthogonal components, the coherent and incoherent ﬁelds, using
an orthogonal wavelet representation. Each of the coherent
ﬁelds is reconstructed from the wavelet coeﬃcients whosemodulus is larger than a threshold, while their incoherent
counterparts are obtained from the remaining coeﬃcients.
The two threshold values depend on the instantaneous kinetic and magnetic enstrophies as well as the maximal resolution [7]. The induced coherent velocity and magnetic
ﬁelds are computed from the coherent vorticity and current
density using the Biot-Savart kernel. In order to computethe ﬂow evolution, one should retain not only the coherent wavelet coeﬃcients but also their neighbors in wavelet
space, i.e., a safety zone.
CVCS is performed for 3D forced incompressible homogeneous MHD turbulence without mean magnetic ﬁeld in
a 2π periodic box for a magnetic Prandtl number equal tounity and with N = 2563 grid points. We also carry out
DNS of the turbulence to assess the CVCS. These computations are integrated over the time period tf = 3.31τ ,where τ is a large-eddy turnover time. The numerical codeuses a Fourier pseudo-spectral method with a fourth-order
Runge-Kutta scheme for time marching. The aliasing errors are removed by a phase-shift method. Solenoidal random forces are imposed only at large scale, i.e. in thewavenumber range 1 ≤ k < 2.5. They have the same time
history in all presented computations. We use the same
statistically quasi-stationary initial ﬂow ﬁeld which wasobtained by a preceding DNS computation of MHD turbulence. Compactly supported Coiﬂet wavelets with ﬁlter
width 12 are used. We here perform CVCS using pseudoadaptive computations, as done for CVS in [5] and [6],because the motivation is to get insight into the feasibilityof fully adaptive CVCS computations.

PSfrag repla ements

E u (k )

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uE b (k )PSfrag repla ements

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Assessment of CVCSCVCS retains only 13[%] of the total number of degrees of freedom after a transient decay at early times, fort/τ ≳ 0.1 (ﬁgure omitted). However, the time evolution ofthe kinetic and magnetic energies in CVCS is in excellent
agreement with that in DNS, during the time evolution
period studied here. In Fig. 1, we ﬁnd that for CVCS
both of the kinetic and magnetic energy spectra, E u (k)and E b (k), agree well with those for DNS, respectively.
Figure 2 shows ﬂow visualizations of intense vorticity
regions and current density regions. Vorticity sheets andcurrent sheets are observed. We can see that CVCS wellpreserves the positions of these regions of DNS at the ﬁnal
time tf . This observation is in contrast to what is foundin CVS of 3D homogeneous incompressible HD turbulence[6]. The CVS computations show a statistically similar
picture of entangled vortex tubes as in DNS. However,the position of these intense vorticity regions in CVS iscompletely diﬀerent from those in DNS because of the ﬂow
sensitivity. It was conﬁrmed that impressions obtained
from the visualization are the same as those at diﬀerent
time instants t ≤ tf , say, t = tf /2.
Further details on the assessment of CVCS are presented
in [8]. It was also found that the wavelet representation
better predicts the turbulence statistics, such as the vorticity and current density probability density functions, thanthe Fourier representation.
4(b)10-2Fig. 1. (a) Kinetic and (b) magnetic energy spectra E u (k)and E b (k) at t = tf . The wavenumber k is normalized bythe Iroshnikov and Kraichnan microscale ηIK of the DNS
at t = tf .

3

CVCS

10-1(a)E (k )

DNS

10-3
10-4(b)CVCS(a)10-2(a)E b (k )DNSConclusion

We developed the CVCS method to track the time evolution of coherent vorticity and current density for 3D incompressible MHD turbulence, and examined the feasibility of CVCS for 3D forced homogeneous incompressible
MHD turbulence in the absence of imposed uniform mag-Fig. 2. Visualization of (a) the intense vorticity and (b)current density regions for DNS and CVCS at t = tf .
Isosurfaces of vorticity ω and current density j are shownfor |ω| = ⟨|ω|⟩ + 3σω and |j| = ⟨|j|⟩ + 3σj , where ⟨·⟩expresses the mean value of ·. Here σω and σj denote thestandard deviations of |ω| and |j|, respectively.

netic ﬁeld. We found that the statistics of the reference
DNS are well preserved by CVCS with safety zone, while
the number of the degrees of freedom retained by CVCS isreduced by a factor eight in comparison to DNS. It was veriﬁed that information which travels further than the safetyzone is not crucial to track the evolution of the nonlineardynamics.
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Fluid Dyn., DOI:10.1080/03091929.2012.654790