23P138-B
Dissipative Structures in Isotropic Hall MHD Turbulence
一様Hall MHD乱流における散逸構造
Hideaki Miura1), Keisuke Araki2)
三浦 英昭1), 荒木 圭典2)
1)National Institute for Fusion Science, 322-6 Orsohi, Toki, Gifu 509-5292, Japan1)核融合科学研究所 〒509-5292 岐阜県土岐市下石町322-6
2)Okayama University of Science, 1-1 Ridaicho, Okayama 700-0005, Japan2)岡山理科大学 〒700-0005 岡山県岡山市理大町1-1

Dissipative structures in homogeneous, isotropic and incompressible Hall MHD turbulence are studied bymeans of direct numerical simulations. A data analysis by the use of the low-pass filter shows that theenstrophy density, which is the dissipative structure in the velocity field, is significantly modified at themoderate scales by the Hall effect. The large scale structures can be different between Hall and single-fluid
MHD turbulence, while the small scale structures are similar to each other.

1. Introduction
Recent Magnetohydrodynamic (MHD) simulations resolve very fine scales numerically. However,
the applicability of the single-fluid MHD equations
is intrinsically restricted to some limited length andtime scales in the sense of so-called the MHDordering. For example, the length scales of the shortwave number ballooning modes resolved in full-3D
MHD simulations of the Large Helical Device canbe shorter than the ion skin depth[1], and thedescription of such short wave number modes
should be incomplete. Furthermore, the plasma
motions in the small scales are somewhat turbulent
in nonlinear time evolutions, and a physical modelis required to describe small-scale turbulence more
precisely. For such studies, we can consider
studying by the use of either extended MHD or
two-fluid equations.
The Hall-MHD model is among the simplest
fluid models. The difference from the single-fluid
MHD model appears only in the induction equation
but it can bring about significant modifications ofdynamics. For example, unstable Rayleigh-Taylor
modes can be stabilized over some wave numbers[2]. The introduction of the Hall term also brings
about the whistler-Alfven waves[3], which limit thetime step width significantly in simulations. For thepurpose of avoiding the time step width limitation,a hyper viscosity or a sort of numerical smoothingis often adopted. However, such an artificialtruncation of small scales often changes plasma
motions to those far from a well-resolved
simulation. To avoid such a violent truncation, weneed to clarify how the Hall term influences
turbulent field and develop a smart numericalmodel. In this presentation, we study small scale
structures in Hall MHD turbulence through directcomparison with single-fluid MHD turbulence.
2. Structure Analysis
Outlines of our numerical simulations are seen in
Ref.[5]. The point of the analysis here is to makeuse of a low-pass filter and see spatial structures ateach resolution. While a similar analysis has been
carried out in Ref.[4] only for one cut-off wave
number, we carry out here the analysis for various
cut-off wave numbers and see how the hierarchy of
turbulent structures appears or disappears. Here wefocus on the dissipative structures, the enstrophy
density Q =| ω |2 / 2 and the current density J =| j |2 / 2 as the representatives of local structures
where ω = ∇ × u and j = ∇ × B are the vorticity andthe current density vector, respectively.
In Fig.1, we see isosurfaces of Q and J. A regionof 256 2 × 128 grid points is shown out of the512 3 simulation box. In Fig.1(a), the low-pass filter
is not operated. The spatial structures are sheet-like,as we often expect in isotropic MHD turbulence. InFig.1(b), however, some of vortex structures(isosurfaces in dark color) are tubular, while thecurrent structures (isosurfaces in bright gray)remain sheet-like. The point here is that neither thevortex structures nor the current structures insingle-fluid MHD turbulence are modified
significantly by operating the low-pass filter. Only
the vortex structures in Hall MHD turbulence ismodified by the low-pass filtering. (More detailed
observations with some cut-off wave numbers arereported in Ref.[6].) It is noteworthy here that, thecurrent-sheets are often observed at almost the same
locations independent to the cut-off wave number,while tubular vortices do not necessarily appear in
the same locations.
(a)single-fluid MHD respectively) by an appropriate
normalization [7]. Thus the observation here can beapplicable to all the decaying process after atransient stage before t=1.0 in Fig.2.
3. Summary
Numerical data analyses by the use of thelow-pass filters are carried out. The scale-hierarchyof the vortex structures appears to have beenchanged by the Hall term, while that of the current
structures does not appear being changed. Thoughfurther numerical studies on the scale-hierarchy
should be required in order to construct appropriate
numerical models to simulate Hall MHD events
quickly, the observations presented here provide
some insights for the purpose.

(b)
Hall MHD vs (single-fluid) MHD
0.1
JxB term in dEK(k)/dt

0-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
Hall MHD, t=1.0
MHD, t=1.0

-0.7
-0.8
0.01

Fig.1: Isosurfaces of the enstrophy density Q and thecurrent density J in Hall MHD turbulence, (a) withoutand (b) with low-pass filtering of k c = 32 .

In Fig.2, the kinetic energy transfer function ofHall MHD and single-fluid MHD turbulence areshown. While we usually normalize the wave
number and the transfer function by theKolmogorov length and the energy dissipation ratein order to compare two different computations, thecomparison without the normalization in Fig.2 canmake sense in this case because the initial conditionof the two simulations is completely the same, andthe difference comes only from the Hall term. It isclear that the energy transfer in Hall MHD
turbulence is smaller than that in single-fluid MHD
turbulence. Furthermore, it appear that the profile ofthe transfer function is shifted in the lower wave
number side due to the Hall effect, because thezero-crossing wave number of the function in HallMHD turbulence is lower than that of single-fluid
MHD turbulence. Though the transfer function does
not depict what kind of structures to be formed in
the real space, the change of the profile should beclosely related with the change of the spatial
structures observed in Fig.1. Note here that theenergy transfer functions at different time snap
shots collapse to one profile (Hall MHD and0.1

1kFig.2. The kinetic energy transfer function of Hall MHD
and the single-fluid MHD turbulence.

Acknowledgments
This work is performed under the auspices of the NIFSCollaboration Research Program (NIFS10KNSS011,NIFS11KNTS005, NIFS11KNSS016) and partially
supported by KAKENHI (Grant-in-Aid for Scientific
Research(C)) 22540509, 23340182 and 23540583.
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2305.
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898.
[4] H. Miura and D. Hori: Plasma Fusion Res. SERIES
8 (2009) 73.
[5] D. Hori and H. Miura: Plasma Fusion Res. 3 (2008)
S1053.
[6] H. Miura and K. Araki: Proc. 13th European
Turbulence Conference (Warsaw, Poland, Sep.
12-15, 2011), to appear in J. Physics: ConferenceSeries.
[7] K. Araki and H. Miura: to appear in Plasma Fusion
Res. (2011).