Weak turbulence of anisotropic shear-Alfvén waves
S. Galtier , S.V. Nazarenko†, A.C. Newell and A. Pouquet‡
I.A.S., Université Paris-Sud, bâtiment 121, 91405 Orsay Cedex, France
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Department of Mathematics, University of Arizona, P.O. Box 210089, 617 N. S. Rita, Tucson, AZ 85721, USA
‡
ASP/NCAR, P.O. Box 3000, Boulder, CO 80307-3000, USA
†
Abstract. We review recent results obtained in [14] on weak turbulence of shear-Alfvén waves in the limit of strongly
anisotropic pulsations that are elongated along a strong external magnetic field. The kinetic equation for energy is presented
at the level of three-wave interactions. It is in agreement with the Galtier et al. [13] formulation of the full three–dimensional
helical case when taking the proper limit. The simplified formalism induced by the new approach renders the applicability
conditions for weak turbulence more transparent. It provides an attractive theoretical framework for describing anisotropic
MHD turbulence in astrophysical contexts where a strong magnetic field is present and for which shear-Alfvén waves are
important.
INTRODUCTION
has to be seen as a first attempt to describe such turbulent plasmas. It is a hard task to include compressibility
effects in a rigorous theory like weak wave turbulence.
Note that a first attempt has been realized by Kuznetsov
[25] in the low β limit. Weak MHD turbulence has to be
seen as a useful theoretical framework to understand media like the solar wind or the interstellar medium. It cannot be seen stricto sensu as a model for such media, since
we observe only a moderate anisotropy (moderate background magnetic field). However this asymptotic model
(limit of strong background magnetic field), by revealing
properties which are quite different to the isotropic case
(strong turbulence), tells us towards what the dynamics must tend when the degree of anisotropy increases
and thus it gives important physical information about
the medium itself. For example several observations reveal spectra with a power-law index between 1 9 and
2 [9, 10, 48], which are definitively steeper than the
(Kolmogorov or IK) phenomenological predictions for
isotropic turbulence. As indicated in Bhattacharjee and
Ng [5] such a steepening of the spectrum can be due to
shocks and discontinuities but it could also be due to turbulence.
IK proposed independently a phenomenology for incompressible, homogeneous, isotropic MHD turbulence
based on three-wave processes of interactions of counterpropagating Alfvén waves embedded in a local mean
magnetic field. The main differences with the neutral hydrodynamic turbulence case is a slowing down of the energy transfer to small scales and a k 3 2 scaling for the
energy spectrum when correlations between the velocity and magnetic field are negligible. In hydrodynamic
For many turbulent plasmas encountered in astrophysics
[45, 20, 30, 43, 49] as well as in laboratory devices such
as tokamaks [51, 17], magnetohydrodynamics (MHD)
offers a first satisfactory description. MHD turbulence
differs significantly from hydrodynamic turbulence in
the fact that a strong magnetic field has a non-trivial effect on the dynamics. It was Iroshnikov [21] and Kraichnan [24] (hereafter IK) who first recognised that the presence of Alfvén waves travelling in opposite directions
along a local large magnetic field leads to the weakening
of energy transfer to small scales and therefore to a modification of the scaling of the (Kolmogorov) energy spectrum from k 5 3 , for neutral fluids, to k 3 2 . In the IK
phenomenology MHD turbulence is assumed to be 3D
isotropic. However in many realistic situations the presence of strong magnetic fields is observed which makes
MHD turbulence strongly anisotropic. Similarly to rotating flows, anisotropy is manifested in a two dimensionalisation of the turbulence spectrum in a plane transverse
to the locally dominant magnetic field and in inhibiting
spectral energy transfer along the direction parallel to
the field [34, 35, 32, 22]. Replacing the 3D isotropy assumption by a 2D one, and retaining the rest of the IK
dimensional analysis, gives a k 2 spectrum (b0 b0 ê ,
the applied magnetic field, k k ê , k k k ê ,
k k ) [40].
Density variations are observed in the solar wind and
the interstellar medium [1, 3, 19, 28, 29] which means in
particular that any theory based on incompressible MHD
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
518
turbulence, the Kolmogorov k 5 3 energy spectrum prediction is well supported by many experimental and numerical observations (see for a review Frisch [12]). On
the other hand, in magnetized flows there is still a debate about the predicted scaling in the absence of a uniform magnetic field b 0 [7, 44, 8] or in the anisotropic
case [11, 27]. Indeed, the interpretation of the Alfvén
wave effect on the dynamics is still a subject of discussion [6], including in the case of the Solar Wind
and the interstellar medium [2, 18]. However it is well
known that the presence of a large scale applied magnetic field changes drastically the physics by imposing
a bi-dimensionalisation of the turbulence with a slower
spectral energy transfer along b 0 . This can be seen for
example from numerical simulations of incompressible
[34, 47, 41, 39, 33], reduced [22] and compressible MHD
[42, 26]. It is interesting to note that in the context of
the same regime of weak turbulence a tendency towards
bidimensionalisation is being analytically determined for
other type of waves like whistler waves described by
electron MHD equations [15] or inertial waves for incompressible fluid under rapid rotation [16].
In the present paper a weak wave turbulence formalism is reviewed [14] where only shear-Alfvén waves are
taken into account. To simplify further the derivation,
the problem will be considered in the limit of strongly
anisotropic pulsations elongated along a strong external
magnetic field. The result is a much simpler formalism
than the one derived in Galtier et al. [13] where the complexity in the algebra of deriving the kinetic equations
for the eight correlators involved in the full case of weak
MHD turbulence may hinder our understanding of the
physical mechanisms at play in weak MHD turbulence.
Part of the complexity stems from the inclusion of helicity Such a complexity of the weak turbulence in the general case makes it harder to use, to analyze and to establish the conditions of its applicability. On the other hand,
the kinetic equations become much easier to analyze in
the limit of nearly bi-dimensional turbulence, k k
[13]. In other words, we are able to show that making
an assumption about strong anisotropy (k k ) in the
original MHD equations allows, in particular, for a reliable control of the assumptions made during the derivation and results in transparency of the applicability conditions. The latter point is especially important because
of the active debate around the role of three-wave vs.
four-wave processes [50, 39, 13, 37]. The resulting kinetic equation is simple and it provides an attractive theoretical framework for applying anisotropic MHD turbulence theory to astrophysics.
DERIVATION OF THE KINETIC
EQUATION FOR SHEAR-ALFVÉN
WAVES
We introduce the “perturbed” Elsässer variables ε z s
v sb b0 where b0 b0 ê is a strong external magnetic field (ê is a constant unit vector), s 1 is a polarization index indicating the wave propagation direction
and ε is a small parameter measuring the intensity of the
wave turbulence. Here, we choose units such that the velocity v and the magnetic field b have the same physical
dimension. The inviscid 3D incompressible MHD equations written in terms of the “perturbed” Elsässer variables are
∂t
sb0 ∂ zsj
ε∂xm zm s zsj ∂x j P (1)
where P is the total pressure and ∂ is the derivative
along ê . After Fourier transforming equation (1) and
separating the fast and slow time dependencies, we have
∂t asj k
iε km Pjn
am s κ asn L ei
(2)
sωk sωκ sωL t
δkκ L dκ L
where dκ L d κ dL, δkκ L δ k κ L, zsj x t s
a j k t eikxsωk t dk and Pjn k δ jn k j kn k2 is the
projection operator which ensures the incompressibility
condition. The small parameter ε measures the strength
of the nonlinear coupling, and the Alfvén waves frequency is ω k b0 k b0 k .
The first simplification consists to assume in equation
(2) that the turbulence is strongly anisotropic, i.e. k k . Here we use explicitly the property of anisotropy
already observed in the presence of a strong b 0 [34].
The second assumption concerns the type of waves that
is going to be considered. Only shear-Alfvén waves are
taken into account; they are described by the transverse
part of as , namely as as1 as2 . The divergence free
condition allows one to express a s2 in terms of as1 as
as2 k1 k2 as1 k k2 as k1 k2 as1 . Introducing
the notation a s as1 and neglecting terms that contain
factors k on the right hand side (RHS), we finally obtain
∂t as k
iε
k2
k L kκ a
2
κ2 L2 k
(3)
s
κ as L e
2isb0 κ t
δkκ L dκ L where the solution of the resonance conditions has been
used. Note that the integration in this equation is still
over the three-dimensional vectors κκ and L and that a s
519
k where the condition is violated; this corresponds to the
non-uniform validity of the kinetic equation. Such behavior any three and four wave interactions is discussed
in Newell, Nazarenko and Biven [38].
Another applicability condition is that the spectrum
must change slowly when crossing the wavenumber cone
(7), that is that the spectrum q s k stays approximately
constant as a function of k in the range ε 2 k k depends on all three of the wavenumber components. We
note that the fundamental equation (3) is much simpler
than the one obtained in the most general case [13].
The next step consists in introducing the wave turbulence spectra qs k as follows,
as k as k qs k δ k k δ s s (4)
where the averaging is taken over an initial ensemble.
Spatial homogeneity means space averaging is equivalent. Further, because of the linear dynamics, the initial ensemble evolves to a state for which the random
phase approximation holds. Here, as usual, δ k k appears due to the turbulence homogeneity and δ s s is
due to the fast decorrelation of the oppositely propagating waves. Following standard weak turbulence approach
[4, 53] it is possible to derive the following kinetic equation (for more details, see Galtier et al. [14]),
∂t qs k
πε 2
b0
(5)
k2 k L 2 kκ 2
2 L κ2
k
2 2
ε 2 k at fixed k , which means that long spatial correlations along the external magnetic field are assumed to be
absent.
One of the well-known consequences of the threewave process for turbulence of Alfvén waves in the case
k k is the k2 energy spectrum. Recently, it was
shown to be an exact constant-flux solution of equation
(6) [13]. Perhaps, a less-known fact is that a stationary
constant-flux solution of equation (6) does not have to
be necessarily k2 : it may have its exponents anywhere
in the range from 1 to 3 depending on the degree of
asymmetry of the forcing of the s 1 and s 1 waves
[13]. Such asymmetric wave pumping is very common in
astrophysics (e.g. , there are more Alfvén waves traveling
away from the Sun than toward it in the Solar Wind (see
e.g. Matthaeus, Goldstein and Roberts [31]).
s
q
κ k2 s
L
q L 22 qs k δ κ δkκ L dκ L
2
L2 k
k2 L
The energy spectrum e s k of the shear-Alfvén waves
is the sum of the perpendicular components of the energy
tensor or, using the divergence free condition, e s k
k2 k22 qsk; therefore it obeys to the following kinetic
equation,
∂t es k
(6)
πε 2
b0
es L
k
L 2 kκ 2
2 L2 κ 2
k
e
s
DISCUSSION
The weak turbulence kinetic equation can be derived
in a much shorter way if the limit k k is taken
before the statistical averaging. The simplicity of our
derivation also allows for an easier understanding of the
two applicability conditions. The first one can always be
checked based on the solution of the kinetic equation. On
the other hand, the second condition cannot be checked
based on the weak turbulence theory itself because this
theory is invalid near k 0 and, therefore, cannot be
used to see if any spikes are present in this region or
not. However, at present it seems unlikely that any strong
turbulence mechanism could lead to long parallel correlations and to a strong condensation of turbulence near
k 0 that would preclude the validity of the weak turbulence theory in incompressible MHD.
It is shown in Galtier et al. [13] that the k 2 energy
spectrum is an exact finite flux Kolmogorov solution
of the weak wave turbulence kinetic equation at the
level of three–wave interactions. It is interesting to note
that in the context of kinetic Alfvén waves the same
scaling can also be predicted [52]. Note also that when
the four–wave interactions are dominant, the predicted
energy spectrum is k 7 3 [50]. For strong anisotropic
MHD turbulence, there is still a debate in the absence
of a rigorous theory; either k 5 3 [50] or k 3 2 [36]
κ es k δ κ δkκ L dκ L
Equation (6) is our main result and it describes a threewave process which dominates the turbulence dynamics at low turbulence intensities. This equation coincides with equation (46) of Galtier et al. [13] which
was obtained from the general kinetic equations of weak
Alfvénic turbulence in the limit k k . This shows in
particular that the taking of the two limits involved does
commute (anisotropic limit before or after having apply
the weak turbulence formalism to the MHD equations).
The conditions of validity of the kinetic equation are
already given in Galtier et al. [14]. The first condition is
k
k
ε2 (7)
which can be satisfied at any finite wavenumber for sufficiently weak turbulence. Note however that, for any turbulence intensity ε , there always exists a region of small
520
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magnetic loop) for which further applications have also
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Jovian magnetosphere [23] may be providing a new observational support for the present theory.
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ACKNOWLEDGMENTS
Grants from CNRS (PNST and PCMI) and from EC
(FMRX-CT98-0175) are gratefully acknowledged.
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