J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Orthonormal Divergence-free Wavelet Analysis of Energy Transfer
in Magnetohydrodynamic Turbulence
Keisuke ARAKI and Hideaki MIURA1)
Okayama University of Science, 1-1 Ridai-cho, Okayama, OKAYAMA 700-0005 JAPAN1)
National Institute for Fusion Science, Oroshi-cho Toki, GIFU 509-5292 JAPAN(Received: 1 September 2008 / Accepted: 11 January 2009)

For the purpose of clarifying mechanism of local structure formations in magnetohydrodynamic (MHD)
turbulence, energy transfers among various scales and positions of magnetic/kinetic energies in the course ofroll-up processes of vortices are studied by direct numerical simulations (DNS) and orthonormal wavelet analysis.
Initial shear layers are perturbed by three-dimensional random perturbations, and roll up to form tubular vortices
which are connected to each other by blade structures. The energy transfers in this process is considered to be
modiﬁed from that in the neutral ﬂuid turbulence in the presence of large-scale magnetic ﬁelds. Information onscales provided by the wavelet analysis are used to study scale-to-scale energy transfers in the roll-up processes.
The scale-to-scale analysis suggests that large scale ﬂow structures directly excite magnetic ﬁelds with variousscales. Energy exchange between the kinetic and magnetic ﬁeld energies will also be examined from the pointsof views of local analysis of scale-to-scale energy transfers.
Keywords: Wavelet analysis, turbulence, coherent structures, scale-to-scale energy transfer, nonlocal interaction∂B= ∇ × [u × B] + η∇2 B∂t∇·u =0

1. Introduction
Turbulent motions of plasmas are considered to play
key roles in magnetic conﬁnement system. For example,the plasma and energy transports across the magnetic ﬁeld
lines are closely related to turbulence[1]. For a neutral
ﬂuid, turbulent transport of kinetic energy is well known as“turbulent energy cascade” and its dynamical mechanism
has been intensively investigated (cf.Refs.[2, 3]). In termsof wavelet analysis, Kishida et al. have found that local
interaction dominates the transfer process[5]. For MHD
ﬂuids, Alexakis et al. have carefully conducted DNS studies and investigated turbulent energy transfer processes between the velocity and magnetic ﬁelds and between larger
and smaller scales [6] and similar analysis is carried out
for Hall-MHD case[7]. They found that nonlocal interaction is important for the energy supplying process from thekinetic energy to the magnetic one.
In the present study, we will focus the relation between motions of some isolated coherent structures andmagnetic induction process under the existence of a uniform background magnetic ﬁeld, which is not treated inRefs.[6, 7]. For this purpose, wavelet analysis, which captures the information of spatial scale and position simultaneously, is adopted. We use here orthonormal divergencefree wavelet, which demonstrates clearly that coherent
structures are relevant to the energy transfer process[8].
In the present study we use only the scale information ofwavelets and analyze scale-to-scale energy transfer.

(3)

where B is the magnetic ﬁeld (normalized by a representative value B0 ), j = ∇ × B is the current (normalized by
B0 /L0 ; L0 is the characteristic length), u is the velocity√(normalized by the Alfvén speed VA = B0 / μ0 ni Mi ; μ0is the permeability of vacuum, Mi is the ion mass and ni is
the ion number density, which is assumed to be constant forsimplicity), ν is the viscosity and η is the resistivity (normalized by VA L0 ), and p is the pressure (normalized byB20 /μ0 ). The pressure p is given as the solution of the Poisson equation which comes from the divergence of eq.(1).

3. Numerical method
A DNS is carried out using pseudospectral methodwith 5123 number of grids in physical space, whose system
length is 2π in each direction, and 2/3-dealiasing method
for the mode interaction calculation and using RungeKutta-Gill method for time stepping [4]. The kinematic
viscosity and normalized resistivity are set to ν = η =0.001. The initial condition is a uniform magnetic ﬁeld
given by B0 = (0, 0, 0.1) and a pair of shear layers givenby��� 32π �(y − )u = 0.33 arctanπ4���� 323π(y − ) − 1.54 , 0, 0− arctanπ4

2. Basic equations

with small isotropic disturbances
Fourier spectrum⎡ �whose�2 ⎤� �2⎢⎢⎢⎥⎥⎥kk⎥⎥. No forcing is inis given by E(k) ∝exp ⎢⎢⎣−k0k0 ⎦troduced during the calculation.
While the DNS is carried out in terms of Fourier

Incompressible MHD equations are described as∂u= −(u · ∇)u − ∇p + j × B + ν∇2 u,
∂t

(2)

(1)

author’s e-mail: araki@are.ous.ac.jp

96©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

K. Araki and H. Miura, Orthonormal Divergence-Free Wavelet Analysis of Energy Transfer in Magnetohydrodynamic Turbulence

0.1

101

wavelet U
Fourier U
wavelet B
Fourier B

0.01

0.1
0.001

0.001

0.0001energy

w2,2j

0.01
1e-04

1e-0051e-05
1e-006

1e-06
1e-07
1e-08

enstrophy
current
051015t20

1e-0072530

1e-008110100wavenumber

Fig. 2 Fourier and wavelet scale spectra of kinetic and magnetic energy. Open and solid symbols denote the kinetic
and magnetic energy spectra, respectively. The abscissaof each wavelet spectrum
is determined by the mean����wavenumber k :=| f |2 d3 �x where f|∇ × f |2 d3 �xstands for u j or B j .

Fig. 1 Time development of enstrophy and current. In thepresent work the amplitudes of quantities are measured indimensionless units that are normalized by the constants
given in §2.

modes, the mode interaction analysis is carried out in termsof wavelet modes. Orthonormal divergence-free wavelets,
which we call “helical wavelets” and denote by ψ̃ j��lσ (�x, t)
hereafter, are obtained by a unitary transform of complex
helical waves[5]. The velocity and magnetic ﬁelds are expanded in helical wavelet modes as follows:�(4)f˜j��lσ (�x, t).
f (�x, t) =larger wavenumbers the magnetic energy become larger
than the kinetic one. Several strong vortices are developedaround the initial shear layers and current is developedaround the vortices and the shear layers between them. It
is expected that the ﬁelds have some characteristic featuresof interactions between the velocity and magnetic ﬁelds
drived by coherent structures.

where f stands for u or B, f˜j��lσ (�x, t) is a helical wavelet
mode which is given by

4.1

j,�,�l,σf˜j��lσ (�x, t):=f˜j��lσ (t):=f˜j��lσ (t) ψ j��lσ (�x),�f (�x, t) · ψ j��lσ (�x) d3 �x,(5)Energy budget in wavelet scale spectra
representation

Taking inner product of Eqs.(1) and (2) with helicalwavelet modes and summation with respect to all the indices but j, which implies spatial scale in physical space,
we obtain energy budget equations with respect to thewavelet scale spectra(6)

and j, �, �l, σ are the indices of helical wavelet. The implication of indices are summarized in Ref.[8].

d (u)Edt jd (B)E
dt j

4. Numerical result
Time development of the enstrophy and current are
shown in Fig.1. Disturbances are grown gradually andformation of rolling-up vortices become prominent aroundt = 15. About t = 18 and the later time the net current
exceeds the enstrophy. We analyzed mode interactions at
the time t = 16, 18, 20, 22 and show the result of t = 22 in
the following.
Fourier and wavelet scale spectra of the velocity andmagnetic ﬁelds at t = 22 is shown in Fig.2 and the enstrophy and current density ﬁelds are shown in Fig.3. In Fig.2
Fourier kinetic and magnetic energy spectra are drawn simultaneously with wavelet ones in order to compare theamplitude of wavelet spectra and spatial scale of wavelets
with Fourier ones which may be more familiar to the readers.
At this time, the lowest wavenumber Fourier modes,
i.e. large scale ﬂow structures have most of energy and for=

N j + L j + L(0)j + D j,(7)=

I j + I (0)j + Rj(8)(0)
where the terms N j , L j , L(0)j , D j , I j , I j , and R j are the inner product with −(u · ∇)u, (∇ × B) × B, (B0 · ∇)B, ν∇2 u,∇ × (u × B), (B0 · ∇)u, and η∇2 B, respectively. The pressure term is vanished because helical wavelet modes aredivergence-free.
The contribution of each term to the energy budgetof kinetic and magnetic ﬁelds is shown in Fig.4. Velocity
ﬁeld losts its energy at the scale with wavelet scale index
j = 1, at which the kinetic energy spectrum has its peak,
and the other scales it acquires energy. On the other hand,magnetic ﬁeld acquires energy at all the scales. That is,
the ﬁelds are in such a process that the energy is redistributed from the j = 1 mode of the velocity ﬁeld to theother wavelet modes of the velocity and magnetic ﬁelds asa whole.

97

K. Araki and H. Miura, Orthonormal Divergence-Free Wavelet Analysis of Energy Transfer in Magnetohydrodynamic Turbulence

0.01((U,grad)U,U)
losingacquiring((B,grad)B,U)
losingacquiring((B0,grad)B,U)
losingacquiring
net transfer
losingacquiringinteraction modulus

0.001

0.0001

1e-0051e-0061e-007

01

2

3

4

5wavelet scale

6

interaction modulus

0.001

78(rot(UxB),B)
acquiringlosing(rot(UxB0),B)
acquiringlosingnet transfer
acquiring

0.0001

1e-0051e-0061e-007

0

1

2

3

4

5

wavelet scale

678Fig. 4 Wavelet spectra of kinetic and magnetic energy budget
equations at the time t = 22; (top) the kinetic energy budget, (bottom) magnetic one. Ordinate of each plot is determined by the modulus of each term and the sign of itis shown by open (acquiring) or solid (losing) symbols.
Fig. 3 Isosurfaces of enstrophy density (top) and current density(bottom)

For this purpose the mode interaction terms that appear in
the rhs of the energy budget equations (7) and (8) are de�composed into subcomponents as follows: N j = k N jk ,����(0)= k L(0)= k I (0)L j = k L jk , L(0)k I jk , I jjjk , I j =jk
where the subcomponents are deﬁned by�(9)
N jk = − u j · (u · ∇)uk d3 �x,
�u j · (∇ × Bk ) × Bd 3 �x,(10)
L jk =
�=u j · (B0 · ∇)Bk d3 �x,
(11)
L(0)
jk
�
B j · ∇ × (uk × B)d 3 �x,(12)I jk =
�=B j · (B0 · ∇)uk d3 �x.
(13)
I (0)
jk

The ﬂuid motion is most aﬀected by the nonlinear interaction for larger scales ( j ≤ 5) and Alfvén wave forsmaller scales ( j ≥ 6). As for the contribution of Lorentz
force term to the kinetic energy budget L j , Lorentz force
reduces the kinetic energy at 0 ≤ j ≤ 4 and j = 7. Thisimplies that the motion of ﬂuid induces magnetic ﬁeld asa whole though some small backscatter from magnetic tokinetic energy is seen at j = 5, 6 and 8.
The energy budget due to the Alfvén waves L(0)j andI (0)j tend to transport the energy from velocity ﬁeld to magnetic one for larger spatial scales cases ( j ≤ 2). For smallerscales ( j ≥ 3) the tendency become opposite. This implies
that Alfvén wave suppresses the growth of smaller scalesof the magnetic ﬁeld. This feature is also seen at the earlier
times t = 16, 18, 20.

These subcomponents are carefully deﬁned to satisfy thebalance conditions

4.2 Inter-scale mode interaction between u jand b j

N jk=L jk

=L(0)
jk

Next we will see the debit and credit relation between(B)the kinetic and magnetic scale energy spectra E (u)j and E k .

=−Nk j ,−Ik j ,−Ik(0)j .

(14)
(15)
(16)

The ﬁrst balance equation (14) implies the energy debit andcredit between the budget of the kinetic energies E (u)j and98

K. Araki and H. Miura, Orthonormal Divergence-Free Wavelet Analysis of Energy Transfer in Magnetohydrodynamic TurbulenceFig. 5 Wavelet spectra of nonlinear kinetic energy transfer N jk .
Solid circles: N jk < 0, i.e. energy is transferred from u j touk ; open circles: N jk > 0. Contours are drawn at −0.95,−0.85, ..., 0.85, 0.95 of max{|N jk |} to grasp the amplitudeof N jk visually.

Fig. 6 Wavelet scale spectra of energy transfer from velocity tomagnetic ﬁeld L jk (= −Ik j ). Solid circles: L jk < 0, i.e.
energy is transferred from u j to Bk ; open circles: L jk > 0.
Contours are drawn by the same way as Fig.5.

netic ﬁeld under the existence of uniform background maginetic ﬁeld. Wavelet analysis of energy redistribution process via the nonlinear and induction/Lorentz force termshows that the motion of ﬂuid is dominated by local interactions while the magnetic ﬁeld is induced mainly by
the large scale ﬂuid motions.
The small scale ﬂuid motions are enhanced by all theterms, i.e. nonlinearity, Lorentz force and Alfvén waves.
This implies that MHD dynamics may promote the development of turbulent ﬂuid motion, which is characterized
by the highly excited small scale ﬂuid motions.
In the present study we used only the informationon the spatial scale of wavelets. Wavelet scale-to-scale
analysis of the energy transfers gives consistent results toFourier ones. Analysis using the spatial location information of wavelets, which is expected to clarify the relation
between coherent structures and energy transfers in detail,is now underway.

Ek(u) is exactly balanced by the modulus of N jk .

The rest
two equations (15) and (16), on the other hand, imply theenergy balance relation between the budget of the kinetic(B)
and magnetic energies E (u)j and E k .
Distribution of scale-to-scale wavelet nonlinear energy transfer N jk is shown in Fig.5. Since N jk satisﬁes
the balance condition (14), the graph is odd with respect
to the line j = k. If N jk has positive value at ( j, k), it im(u)plies that energy is transferred from E (u)j to E k by N jk .
Since positive peaks are aligned at k = j + 1, the following
two characteristics are concluded: (1) energy is transferredfrom larger scales to smaller ones, and (2) the dominantnonlinear transfers are local. Although the ﬂow is not fully
developed turbulent state but has only several large rollingup vortices, this tendency is quite similar to that of fully
developed three dimensional turbulent ﬂow [5].
Energy transfer between the velocity and magnetic
ﬁelds by magnetic induction L jk is shown in Fig.6. It isremarkable that, irrespective of the scale of magnetic ﬁeld,
transfer to the magnetic energy is induced by the large
scale wavelet modes of the velocity ﬁeld, i.e. u1 , u2 andu3 . This result directly shows that the induction of magnetic ﬁeld is dominated by large scale ﬂow structures. Especially for the smaller scales of magnetic ﬁeld, this result
implies that the inductive energy transfer is dominated by
non-local interactions. It should be remarked that u2 , themode that most intensively excites the magnetic ﬁeld does
not agree with the peak of the kinetic energy spectrum E1(u) .
It seems interesting that weak backscatters from magnetic
to kinetic energy are seen at ( j, k) = (2, 2), (5, 5). Similarbackscatter is observed for t = 16, 18, 20.

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[3] M. Lesieur, Turbulence in Fluids 4th Ed., (Springer,Dortrecht, 2008).
[4] C. Canuto et al., Spectral Methods in Fluid Dynamics,(Springer, New-York, 1988).
[5] K. Kishida, K. Araki, K. Suzuki, and S. Kishiba, Phys. Rev.
Lett., 83, 5487 (1999).
[6] A. Alexakis, P. D. Mininni, and A. Pouquet, Phys. Rev. E,
72, 046301 (2005).
[7] P. D. Mininni, A. Alexakis, and A. Pouquet, J. Plasma Phys.,73, 377 (2007).
[8] K. Araki and H. Miura, “Orthonormal Divergence-free
Wavelet Analysis of Nonlinear Energy Transfer in RollingUp Vortices”, Y. Kaneda (ed.), IUTAM symposium on Computational Physics and New Perspectives in Turbulence, 149(2008).

5. Discussion
In the present study, we investigated the eﬀect of coherent isolated vortices on the excitation process of mag-99