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Collisionless magnetohydrodynamic turbulence in two dimensions
Annals of Physics 317 (2005) 1–23 www.elsevier.com/locate/aop
Collisionless magnetohydrodynamic turbulence in two dimensions Bhimsen K. Shivamoggi* University of Central Florida, Orlando, FL 32816-1364, USA Received 4 June 2004; accepted 20 December 2004
This paper is dedicated to my mother for her 75th birthday.
Abstract In this paper we give a formulation of two-dimensional (2D) collisionless magnetohydrodynamic (MHD) turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. We place particular emphasis on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We investigate the fractal/multi-fractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k1 spectrum in the solar wind. Multi-fractal scaling behaviors in the inertial range are first deduced, and are then extrapolated down to the dissipative microscales. We then consider a parabolic-profile model for the singularity spectrum f (a), as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence. 2004 Elsevier Inc. All rights reserved. PACS: 47.27.Ak; 47.27.Gs Keywords: Magnetohydrodynamics; Turbulence
*
Fax: +1 4078236253. E-mail address: [email protected].
0003-4916/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.12.006
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1. Introduction Many magnetohydrodynamic (MHD) flows that occur naturally (like in astrophysical systems, notably the solar wind) and in modern technological systems (like fusion reactors) show turbulence.1 However, laboratory experiments on MHD turbulence are scarce because of the difficulty of achieving sufficiently high magnetic Reynolds numbers. Nuclear fusion devices present some hope for the latter aspect but detailed measurements in them are extremely difficult. Advective stretching of magnetic field lines leads to amplification of magnetic field energy at small scales—dynamo action [3–8] and magnetic field lines that have been highly stretched experience typically stronger Ohmic dissipation [9]. If the magnetic field is weak, the advection-diffusion mechanism controls the statistical properties of the small-scale components of the magnetic field (somewhat similar to the case with passive-scalar transport). The influence of spatial dimension on the nature of MHD turbulence is less stringent than on that of hydrodynamic turbulence thanks to the Lorentz force which breaks the vorticity conservation in two dimensions. This leads to an energy cascade to small scales in both two dimensions and three dimensions and the behavior of twodimensional (2D) and three-dimensional (3D) MHD turbulence is phenomenologically similar.2 2D MHD turbulence cases have therefore been considered extensively in direct numerical simulations (DNS) thanks also to the ease of attainment of high Reynolds numbers in 2D [10–16]. Most of the theoretical investigations of 3D MHD turbulence have considered the isotropic3 case which is appropriate for the case where the average magnetic field is zero. In this case, Iroshnikov [18] and Kraichnan [19] made arguments similar to those involved in KolmogorovÕs [20] theory of isotropic hydrodynamic turbulence to propose that the statistical properties of the small-scale components of the velocity and magnetic fields are controlled by shear Alfve´n-wave dynamics, in the limit of large viscous and magnetic Reynolds numbers, have some universality in the inertial range, and are influenced only weakly by the large-scale features of the fields. The Iroshnikov–Kraichnan (IK) theory uses dimensional arguments to derive a k3/2 power law for the spectrum of the kinetic and magnetic energy densities in the stationary state. Pouquet et al. [21] used an eddy-damped quasi-normal Markovian (EDQNM) closure theory to show that, in the absence of the cross helicity, an inertial range occurs with a cascade of energy to small scales with a 3/2 power law. However, Goldreich and Sridhar [22] argued that the shear Alfve´n-wave dynamics does not control 3D MHD turbulence and Muller and Biskamp [23] argued that
1
MHD turbulence plays a major role in the current models on the dynamo action for magnetic field generation in planets and stars [1,2]. 2 Some essential differences persist, however, like the absence of the dynamo effect in the 2D case. 3 The isotropic assumption for the 3D MHD case is not a very sound one because of a preferred direction imposed, at least in a local sense, by the magnetic field of the large-scale eddies or by the mean magnetic field. An axisymmetric model for MHD turbulence was considered by Shivamoggi [17] via a statistical approach.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
3
the latter is instead dominated by vortical motions in the plane perpendicular to the local large-scale magnetic field B0ˆiz.4 But, in the absence of a rigorous theory, this issue is still open. The IK theory does not take into account the spatial intermittency in the flow that arises due to the spatial scarcity of intense small-scale dissipative structures. This was dealt with by Shivamoggi [29] who used fractal [30,31] and multi-fractal [32] models for the dissipative structures. The DNS of 2D MHD turbulence [12] showed strongly intermittent situations with large magnetic islands (where magnetic field is strong) surrounded by a sea of small-scale turbulence (where the magnetic field is weak but the magnetic field gradients are large). Strong intermittency effects were also shown in several other DNS [11,15,16]. The high-temperature plasmas in both space and fusion systems have been found to be collisionless, so considerable work has gone into collisionless plasma dynamics [33–45], among others. In a collisionless plasma, the electron inertia: leads to the decoupling of the plasma motion from that of the magnetic field lines, limits the election current and prevents it from becoming unbounded as the resistivity g ) 0. In the electron-inertia regime, the conservation of the magnetic flux is replaced by the conservation of the generalized electron-fluid momentum. Physically, this allows for the localized violation of the topological constraint on the magnetic flux and hence magnetic field reconnection to occur in the electron-inertia regime and an exchange between magnetic and kinetic energies. Collisionless plasma dynamics cannot therefore be described adequately by the single-fluid formulation of resistive MHD. In a low-b (b being the ratio of the plasma pressure to the magnetic pressure) plasma (such as the strongly magnetized plasmas in fusion systems), the plasma dynamics is also governed by the electron-pressure gradient term in the electron momentum-balance equation [37,46–48]. In this paper, we give a formulation of 2D collisionless MHD turbulence that includes the effects of both electron inertia and electron pressure and is applicable to strongly magnetized collisionless plasmas. We adopt a fluid treatment for both electrons and ions [37,48]. We take the statistical properties of the small-scale components of the velocity and magnetic fields to be controlled by shear (and kinetic) Alfve´n-wave dynamics because the IK phenomenology based on the Alfve´n-effect mechanism is believed to be more valid for the 2D case rather than the 3D case [12]. We consider fractal and multi-fractal models to describe the effects of spatial intermittency in 2D fully developed collisionless MHD turbulence. We will then extrapolate multi-fractal scaling in the inertial range down to the dissipative 4
Sridhar and Goldreich [24] sought to justify their argument on the grounds that the three Alfve´n-wave resonant interactions underlying the IK model are non-existent. But, this was refuted by Montgomery and Matthaeus [25], Ng and Bhattacharjee [26], and Nazarenko et al. [27]. Goldreich and Sridhar [28] subsequently conceded the flaw in their argument. So, the details of the shear Alfve´n-wave dynamics in the IK model are still controversial.
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microscale. We then consider a parabolic-profile model for the singularity spectrum f (a) [49], as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence.
2. Governing equations of collisionless plasma dynamics We adopt a fluid treatment for both electrons and ions5 [37,48]. We assume the dynamics to preserve quasi-neutrality, so ni ne = n, n being the number density of the particles, the subscripts i and e being associated with ions and electrons, respectively. The governing equations are: (i) Mass conservation of ions: on þ r ðnvÞ ¼ 0 ot
ð2:1Þ
v being the ion-flow velocity. (ii) FaradayÕs law: oB ¼ cr E ot
ð2:2Þ
E and B being the electric and magnetic fields, respectively. (iii) AmpereÕs law: 1 rB¼ J c J being the electric current density. (iv) Momentum-balance equation of ions: ov 1 þ ðv rÞv ¼ e E þ v B mi n ot c
ð2:3Þ
ð2:4Þ
m being the particle mass. We take the ions to be cold. (v) Momentum-balance equation of electrons (generalized OhmÕs law): 1 1 oJ Te E ¼ ve B þ 2 þ ðve rÞJ rn ð2:5Þ c xpe ot ne pffiffiffiffiffiffiffiffiffiffiffiffiffiffi xpe being the plasma frequency, xpe ne2 =me , and T is the temperature. We take the electrons to be isothermal (Te = constant), and we neglect the diamagnetic effects 5 Theoretical analysis of collisionless plasma processes usually require a kinetic model. This is true a fortiori if the particle distribution functions deviate strongly from Maxwellian as in a long-tail development due to acceleration of particles to highly superthermal energies. On the other hand, kinetic effects become important even for nearly isotropic distributions if the Larmor radii exceed the gradient length scales in question. However, if superthermal particles are negligible and the Larmor radii are sufficiently small, the fluid model is adequate in describing the collisionless plasma dynamics.
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associated with the electron-pressure gradient. Eq. (2.5) shows that the electron-pressure gradient parallel to the guide magnetic field B0ˆiz causes compression of the electron population in the z-direction—parallel electron compressibility. Let us introduce the magnetic flux w and the flow stream function / according to: ) B ¼ B0^iz þ rw ^iz ; ð2:6Þ v ¼ r/ ^iz ; where B0ˆiz is the constant guide field.6 We assume B0 is large so that z is the ignorable coordinate. Eqs. (2.1)–(2.5) then lead to [37,48] o ðw d 2e r2 wÞ þ ½w d 2e r2 w; / q2s ½w; r2 / ¼ 0; ot
ð2:7Þ
o ðr2 /Þ þ ½r2 /; / þ ½w; r2 w ¼ 0: ot
ð2:8Þ
Here, ½f ; g ^iz rf rg: de is pthe ffiffiffiffiffiffiffiffiffielectron skin depth d e c=xpe , and qs is the ion sound gyro-radius T e =mi qs xc , xci being the ion gyro-frequency, xci eB0 =mi c. Eqs. (2.7) and (2.8) i are adequate for describing several dynamical aspects of collisionless plasmas in space (e.g., the magnetosphere) and fusion systems [37,48].
3. Elsa¨sser variables Eqs. (2.7) and (2.8) have the following Hamiltonian representation [48] Z Z 1 2 2 2 2 H¼ ½ðr/Þ þ ðrwÞ þ q2s ðr2 /Þ þ d 2e ðr2 wÞ dx; 2
ð3:1Þ
v
where V R2 is the appropriate 2D domain on which the plasma flow and the magnetic field evolve. In order to be able to rewrite 3.1 in terms of Elsa¨sser [51] variables (which is a crucial step prior to formulation of a theory of collisionless MHD turbulence), let us assume qs = de.7 The present results can be expected to serve as a qualitative guide even for those situations (see Footnote 7) where this condition does not hold. We then introduce Elsa¨sser variables W w / 6
ð3:2Þ
An embedded magnetic field is also a frequent feature in astrophysical turbulent plasmas [50]. This condition may be expected to occur in strongly magnetized plasma situations like those in some astrophysical contexts. Cafaro et al. [48] in fact also considered cases qs < de. For the magnetospheric [38] and current fusion-reactor [41] plasmas qs/de > 1. 7
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so 3.1 may be written as Z Z n o 1 n o 1 1 2 2 2 2 2 2 2 ðrWþ Þ þ ðrW Þ þ d e ðr Wþ Þ þ ðr W Þ H¼ dx: 2 2 2 ð3:3Þ Eq. (3.3) shows that there are no self-interaction terms in the 2D collisionless MHD case (under the assumption qs = de), as in classical MHD. Hence, Alfve´n waves (and their variants—kinetic Alfve´n waves in the presence of parallel electron compressibility) do not undergo non-linear wave steepening and only those waves propagating in opposite directions along the background magnetic field interact8—the so-called Alfve´n effect. Let us introduce further ðrWÞ2 12½ðrWþ Þ2 þ ðrW Þ2
ð3:4aÞ
2
~ via and a Laplacian-like operator r 2
2
2
2
~ WÞ 1½ðr2 Wþ Þ þ ðr2 W Þ : ðr 2
ð3:4bÞ
2
~ turns out not to be relevant for the ensuing considerations in The detailed form of r this paper. Eq. (3.3) may then be rewritten further as Z Z 1 2 ~ 2 WÞ2 dx: ½ðrWÞ þ d 2e ðr H¼ ð3:5Þ 2
4. The MHD eddy turn-over time: Iroshnikov–Kraichnan hypothesis The IK hypothesis can be shown to follow also from the formal analogy between the hydrodynamic and MHD spectral energy density expressions. To see this, let us write spectral energy density E (k) as EðkÞ
V ; k2s
ð4:1Þ
V being the characteristic velocity of the spectral element k. The hydrodynamic eddy turn-over time s given by s
1 kV
ð4:2Þ
then becomes9 s
8
1 : k E1=2 3=2
ð4:3Þ
Such interactions are non-local. Eq. (4.2) implies that the energy transfer in the hydrodynamic case is local in the spectral space which also reflects the fact that a large-scale velocity field can be transformed away via Galilean invariance. 9
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
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If we use the relation, s
Ek
ð4:4Þ
being the mean energy dissipation rate, Eq. (4.3) leads to the Kolmogorov spectrum: EðkÞ 2=3 k 5=3 :
ð4:5Þ
If, on the other hand, we write for the MHD case, in analogy with (4.1) [29], CA k 2^s
ð4:6Þ
1 : kC A
ð4:8Þ
EðkÞ
~ 0 =pffiffiffi ^s being the MHD eddy turn-over time and C A B q being the velocity of Alfve´n ~ waves in the magnetic field B0 of the large-scale eddies, we immediately obtain s ^s s ; ð4:7Þ sA where sA
Eq. (4.7) implies that the energy transfer in the MHD case is non-local in the spectral space and represents the Alfve´n effect10 (see Section 3) postulated in the IK theory! If we again use the relation (4.4), (4.7) leads to the IK spectrum: 1=2
EðkÞ 1=2 C A k 3=2 :
ð4:9Þ
5. Inertial-range scaling laws The influence of spatial dimension is less stringent in MHD turbulence than in hydrodynamic turbulence because the Lorentz force breaks the vorticity conservation in 2D, allowing an energy cascade to small scales in 2D (as well as 3D). One may then consider the possibility of an inertial range of the Kolmogorov type which is in a state of statistical equilibrium and the energy is assumed to cascade smoothly through non-linear processes in a stationary state. Consider a discrete sequence of scales ‘n ‘0 2n ;
n ¼ 0; 1; 2; . . .
ð5:1Þ
Let us assume that we have a statistically stationary turbulence, where energy is introduced into the plasma at scales ‘0, and is then transferred successively to scales
10 Alfve´n effect can therefore be viewed as a non-local effect of the large-scale magnetic field and is a kind of sweeping action of the large eddies on the small eddies [52].
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‘1, ‘2, . . .until some scale ‘d is reached where dissipative effects are able to compete with non-linear transfer. The energy per unit mass in the nth scale is given by ! W2n d 2e ð5:2Þ En 2 1 þ 2 : ‘n ‘n The rate of transfer of energy per unit mass from the nth scale to the (n + 1)th scale is given by En en
ð5:3Þ ^sn and using (4.7), (5.3) becomes en
E n sA : s2n
ð5:4Þ
Noting, sA
‘n CA
and
sn
‘2n ; Wn
ð5:5Þ
and using (5.2), (5.4) becomes ! W4n d 2e en 5 1þ 2 : ‘n C A ‘n
ð5:6Þ
In the inertial range, we assume a stationary process in which the energy transfer rate is constant: en ¼ constant ¼ e;
‘d 6 ‘n 6 ‘0 :
ð5:7Þ
Using (5.6), (5.7) leads to Wn
1=4 e1=4 ‘1=4 n CA
d2 1 þ 2e ‘n
!1=4 :
ð5:8Þ
Using (5.8), (5.2) gives En e
1=2
1=2 C A ‘n1=2
d2 1 þ 2e ‘n
from which, we have ( 1=2 e1=2 C A ‘n1=2 ; En
1=2 e1=2 C A ‘n1=2 ;
!1=2
d e ‘n ; d e ‘n :
Eq. (5.10) leads to the following energy spectra: ( 1=2 e1=2 C A k 3=2 ; kd e 1; EðkÞ
1=2 e1=2 C A k 1=2 ; kd e 1:
ð5:9Þ
ð5:10Þ
ð5:11Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
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The energy spectrum for the classical MHD regime (kde 1) corresponds to the IK spectrum in 2D MHD turbulence. The DNS of 2D classical MHD turbulence [10,53,12–14,54] and closure calculations [55] confirm this.11,12 There seems to be no available observational data confirming the scaling law for the collisionless MHD regime (kde 1) yet. However, the intermittency-corrected version of this scaling law appears to have some observational record (see Section 6.1). The shallower energy spectrum in the collisionless MHD regime (kde 1) signifies the appearance of energy generation (say, via electron inertia) in the system for length scales below de. Further, it can lead to energy pile-up at scales ‘n de and, hence, can cause creation of an ordered vortex/current filament phase similar to the one numerically shown in the hydrodynamic case by Kukharin et al. [56]. 6. Spatial intermittency 6.1. Homogeneous fractal model The theory given in Section 5 does not take into account the spatial intermittency in the flow that would cause systematic departures from the scaling laws (5.11) which use mean transfer rates. One may follow [30] and argue that the spatial intermittency effects in the present system are related to the fractal aspects of the geometry of turbulence. This reflects the fact that the dissipative structures are strongly convoluted. Indeed, since small-scale structures in MHD turbulence are primarily produced by flows perpendicular to the large-scale magnetic field, the most intense regions of dissipation may be expected to occur as large sheets, like the current sheets13 which are also vorticity sheets [58] and also [12]. This may be simulated in a first approximation by representing the dissipative structures via a homogeneous fractal with non-integer Hausdorff dimension D0. This amounts to assuming the energy flux to be transferred to only a fixed fraction b of the eddies downstream in the cascade—the b-model [31]. Consider a discrete sequence of scales as in (5.1), but now assume that at the nth step, only a fraction bn of the total space has an appreciable excitation with a fractal dimension D0. The energy per unit mass in the nth scale is given by ! bn W2n d 2e En 2 1þ 2 ; ð6:1Þ ‘n ‘n 11
The Alfve´n effect seems to be the dominant mechanism underlying 2D MHD turbulence [12] unlike the situation with 3D MHD turbulence (see Section 1). 12 The DNS [10] further showed that the IK spectrum prevails only for small values of the correlation between the velocity and the magnetic fields. Large values of this correlation (and concomitantly, the dynamic alignment of the velocity and the magnetic fields) introduce certain phase relations between the velocity and magnetic fields and further inhibit both exchanges of kinetic and magnetic energies and the energy transfer to small scales. 13 The advective stretching of magnetic field lines intensify the current sheets until magnetic reconnection processes set in consequent to tearing-mode instabilities [57].
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where bn
2D0 ‘n : ‘0
ð6:2Þ
The rate of transfer of energy per unit mass from the nth scale to the (n + 1)th scale is ! given by 4 E n sA d 2e n Wn ð6:3Þ 1þ 2 : en 2 b 5 sn ‘n C A ‘n In the inertial range, the energy transfer rate is constant for a stationary process, so on using (6.2) and (6.3), and assuming the scaling behavior, Wn ‘a we have, from (5.7), 4a þ 2 D0 5 ¼ 0; 4a þ 2 D0 7 ¼ 0; from which ( 3þD
0
a¼
4 5þD0 4
ð6:4Þ d e ‘n ; d e ‘n
; d e ‘n ; ; d e ‘n :
Using (6.4) and (6.6), we have from (6.1), 8 1 1 < ‘2þ2ð2D0 Þ ; d e ‘n ; n En
: 12þ12ð2D0 Þ ‘n ; d e ‘n : Eq. (6.7) leads to the following energy spectra: 8 < k 3212ð2D0 Þ ; kd 1; e EðkÞ
: 1212ð2D0 Þ k ; kd e 1:
ð6:5Þ
ð6:6Þ
ð6:7Þ
ð6:8Þ
Observe that the intermittency corrections (D0 < 2) make the spectra steeper, as expected. The result in (6.8) corresponding to the classical MHD regime (kde 1) agrees with that given by Biskamp [14]. If the dissipative structures correspond to current sheets, i.e., D0 = 1, then (6.8) yields ( k 2 ; kd e 1; EðkÞ
ð6:9Þ k 1 ; kd e 1: The classical MHD (kde 1) result is well known, while the collisionless MHD (kde 1) result appears to have been observed in the solar-wind plasma [59].14 14 There have been several controversial attempts in the past (see Goldstein et al. [60]) to explain this k1 spectrum. But these have been hampered by the objection that the plasma will have much less than one eddy turn-over time to evolve and the structures involved are too big to fit in the ‘‘box’’ between the sun and the point of observation.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
On the other hand, the structure function of order p, Sp (‘), behaves as p oW S p ð‘Þ d ð‘Þ ‘fp : ox
11
ð6:10Þ
Using (6.4), and noting that the probability to belong to this fractal at scale ‘ goes like ‘2D0 , (6.10) leads to ( ‘ða1Þpþ2D0 ; d e ‘; S p ð‘Þ
ð6:11Þ ‘ða2Þpþ2D0 ; d e ‘: So, the characteristic exponent fp is given by
ða 1Þp þ 2 D0 ; d e ‘; fp ¼ ða 2Þp þ 2 D0 ; d e ‘: Using (6.6), (6.12) becomes 8 p p > < þ 1 ð2 D0 Þ; fp ¼ 4 p 4 p > : þ 1 ð2 D0 Þ; 4 4
ð6:12Þ
d e ‘; ð6:13Þ d e ‘:
Observe that the b-model cannot describe any non-linear behavior (as is usually the case) of the characteristic exponents fp of higher-order structure functions. It is, therefore, pertinent to consider the multi-fractal model [32] to address this. 6.2. Multi-fractal model Let us follow [32] and assume that the energy dissipation field is a multi-fractal object. A singular measure defined on a multi-fractal is represented in terms of interwoven fractal subsets corresponding to different measure levels having a continuous spectrum of scaling exponents. Let S (a) be the support of interwoven sets of singularities in the measure e of strength a and fractal dimension f (a). Noting the scaling behavior of the Elsa¨sser-variable increment dW over a distance ‘ given by (6.4), the singularity spectrum f (a) is obtained via a Legendre transformation of the scaling exponent fp of the pth-order structure function: (R p oW dlðaÞ‘ða1Þpþ2f ðaÞ ; d e ‘; fp S p ð‘Þ d ð‘Þ ‘ R ð6:14Þ ox dlðaÞ‘ða2Þpþ2f ðaÞ ; d e ‘; where ‘2f (a) dl(a) represents the probability of encountering the set S (a) within a 2D circle of radius ‘. One may use the method of steepest descent to extract the dominant contribution to the integral in (6.14), in the limit of small ‘. This gives 8 < inf ½ða 1Þp þ 2 f ðaÞ; d e ‘; a fp
: inf ½ða 2Þp þ 2 f ðaÞ; d e ‘ a
12
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
or
fp ¼
ða 1Þp þ 2 f ða Þ;
ða 2Þp þ 2 f ða Þ;
d e ‘; d e ‘;
ð6:15aÞ
where f 0 ða Þ ¼ p:
ð6:15bÞ
To relate the singularity spectrum f (a) to the generalized fractal dimension of the energy dissipation field, consider a coarse-grained probability measure given by the total energy dissipation occurring in a square of size ‘: ( ‘4a3 ; d e ‘; 2 U ð‘Þ eð‘Þ‘
ð6:16Þ ‘4a5 ; d e ‘: We then cover the support of the measure e with squares of size ‘ and sum [U (‘)]q over all squares. If the energy dissipation field is a multi-fractal, such a sum exhibits the asymptotic scaling behavior [61] of moments: (R X dlðaÞ‘ð4a3Þqf ðaÞ ; d e ‘; q ðq1ÞDq
R ½U ð‘Þ ‘ ð6:17Þ dlðaÞ‘ð4a5Þqf ðaÞ ; d e ‘; where we have assumed that the number of iso-a squares for which a takes on values between a and a + da is proportional to dl (a)‘f (a). Dq is the generalized fractal dimension of the e-field. Implicit in (6.16) and (6.17) is a refined similarity hypothesis, analogous to that introduced by Kolmogorov [62] for hydrodynamic turbulence, connecting the Elsa¨sser variable increment with that of the locally averaged energy dissipation in 2D MHD turbulence. The dominant terms in the integral in (6.17), in the limit of very small ‘, may again be extracted using the method of steepest descent to yield the Legendre transformation:
ð4a 3Þq f ða Þ; d e ‘; ðq 1ÞDq ¼ ð6:18aÞ ð4a 5Þq f ða Þ; d e ‘; where f 0 ða Þ ¼ 4q:
ð6:18bÞ
Eliminating f (a) from (6.18a) and (6.18b), and putting q = p/4, we obtain 8 p p > < þ 1 ð2 Dp=4 Þ; d e ‘; fp ¼ 4 p 4 p > : þ 1 ð2 Dp=4 Þ; d e ‘: 4 4
ð6:19Þ
For a fractally homogeneous turbulence, Dp=4 ¼ D0
8p:
ð6:20Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
Eq. (6.19) then reduces to 8 p p > < þ 1 ð2 D0 Þ; fp ¼ 4 p 4 p > : þ 1 ð2 D0 Þ; 4 4
13
d e ‘; ð6:21Þ de ‘
in agreement with the result (6.13) for the b-model. 6.3. Multi-fractal scaling at the dissipative microscale We will now consider extrapolation of multi-fractal scaling in the inertial range discussed in Section 6.2 down to the dissipative microscale. The following development is predicated on the assumption that an inertial behavior persists at scales smaller than de—this assumption may not be too restrictive for tenuous plasmas like those in space (de 10 km for the magnetospheric plasma). On taking into account the spatial intermittent character of the energy dissipation field into account, the MHD dissipative microscale 2 1=3 m CA g
ð6:22Þ eðgÞ implies (along the lines of the development of Nelkin [63] for hydrodynamic turbulence) 8 1 g < hRi 2a1 ; d e g;
ð6:23Þ 1 L : hRi 2a2 ; d e g; where ÆRæ is the Reynolds number, 1=3
ðheiL4 Þ ; ð6:24Þ m mis the kinematic viscosity of the plasma and we have assumed, for the sake of simplicity, the magnetic Prandtl number to be unity. Observe the reduction in the dissipative microscale g produced by spatial intermittency (D0 < 2), as expected. hRi
6.3.1. Degrees of freedom Following Landau and Lifshitz [64], one may give an estimate for the number of degrees of freedom necessary for describing 2D fully developed MHD turbulence 2 L N ðRÞ
R4=3 S 2=3 ; ð6:25Þ g where S is the Lundquist number, S
CA ðeLÞ
1=3
:
In the presence of spatial intermittency this estimate is modified as follows.
ð6:26Þ
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Let us assume that the energy dissipation field is a multi-fractal object for which one has a continuous spectrum of scaling exponents a, with each a distributed on a set S (a) of Hausdorff dimension f (a) embedded in 2D space. Let us assume further that multi-fractal scaling in the inertial range, discussed in Section 6.2, can be extrapolated down to the dissipative microscale g. Eq. (6.25) then becomes f ðaÞ Z L N dlðaÞ : ð6:27Þ g On using (6.23), (6.27) becomes 8R f ðaÞ < dlðaÞhRi2a1 ; d g; e N
f ðaÞ R : dlðaÞhRi2a2 ; d e g: In the limit of large ÆRæ, the dominant exponents in (6.28) correspond to ð2a 1Þf 0 ðaÞ 2f ðaÞ ¼ 0; d e g; ð2a 2Þf 0 ðaÞ 2f ðaÞ ¼ 0; d e g:
ð6:28Þ
ð6:29Þ
Eqs. (6.29), in conjunction with (6.18a) and (6.18b), imply N hRi2Q ;
ð6:30aÞ
where Q¼
DQ DQ þ 1
8g=d e :
ð6:30bÞ
Eqs. (6.30a) and (6.30b) show that the effect of intermittency is to reduce the number of degrees of freedom, as to be expected! Observe the robust nature of (6.30a) and (6.30b)—it is identical for both 2D classical MHD (de g) and collisionless MHD (de g) regimes!15 In the absence of intermittency (DQ ) 2) (6.30a) and (6.30b) reduce to (6.25). 6.3.2. Moments of the Elsa¨sser-variable-gradient distribution Consider moments of the Elsa¨sser-variable-gradient distribution. Assuming again that the multi-fractal scaling of the inertial range can be extrapolated down to the dissipative microscale g, we have 8 * 1 +
15
A similar result occurs also in geostrophic turbulence [65].
ð6:32Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
Eqs. (6.32), in conjunction with (6.18a) and (6.18b), imply 8 ðDQ 2Þðp4Þ3p > 2DQ þ3p4 > < hRi 2DQ þ2 ; where Q ¼ 2D ; d e g; Q þ2 Ap
ðD 2Þðp4Þp Q > > : hRi 2DQ þ2 ; where Q ¼ 2DQ þ2p4 ; d e g: 2DQ þ2 For p = 2, (6.33) implies 8 1 d e g; < hRi ; where Q ¼ 1; DQ 1 A2
: hRiDQ þ1 ; where Q ¼ DQ ; d g: e DQ þ1 Thus, the mean energy dissipation has the following scaling behavior: 8 < hRi0 ; where Q ¼ 1; d e g; 2 mA2
: hRiDQ þ1 ; where Q ¼ DQ ; d g: e DQ þ1
15
ð6:33Þ
ð6:34Þ
ð6:35Þ
Observe that the mean energy dissipation rate, in the limit m ) 0, remains finite in the classical MHD regime but vanishes in the collisionless MHD regime! The former result is confirmed by the DNS [12,13] while the latter result is further consistent with the fact (indicated in Section 5) that the collisionless MHD flows are less dissipative. 6.3.3. Probability distribution function of the Elsa¨sser-variable gradient Spatial intermittency in the energy dissipation field accentuates non-Gaussian statistics at small scales. To derive the probability distribution function of the Elsa¨sservariable gradients, note first that the scaling behaviors (6.23) of the dissipative microscale g can be reexpressed in the form 8 1 > m 2a1 > > > ; d e g; < W0 ð6:36Þ g
1 > 2a2 > > m > : ; d e g; W0 where (W0/L) is the Elsa¨sser-variable increment on a macroscopic length L which we have set equal to unity, for the sake of simplicity. The scaling behavior of Elsa¨sser-variable gradient is then given by 8 aþ1 a2 W < W02a1 m2a1 ; d e g; s 2
ð6:37Þ a a2 : 2a2 g W0 m2a2 ; d e g: The probability distribution function (PDF) of the Elsa¨sser-variable gradient may be determined in terms of that for the characteristic Elsa¨sser-variable increment W0 for large scales as follows:
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B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
P ðs; aÞ ¼ P ðW0 Þ
dW0 : ds
ð6:38Þ
Taking P (W0) to be Gaussian, ! W20 P ðW0 Þ exp 2hW20 i and using (6.37), (6.38) becomes 0 2 42a 4a2 31 8 2a > aþ1 m m aþ1 jsj aþ1 5A > > > exp @4 ; > > jsj 2hW20 i < 0 2 42a 4a4 31 P ðs; aÞ
2a > > a > m m a jsj a 5A > > exp @4 ; > : jsj 2hW20 i
ð6:39Þ
d e g; ð6:40Þ d e g:
For the zero intermittency regime (D0 = 2), where the energy dissipation field is space-filling, we have, from (6.6), ( 5 ; d e g; a ¼ 47 ð6:41Þ ; d e g: 4 Eq. (6.40) then becomes 8 " #! 1=3 > m m2=3 jsj4=3 > > > exp ; d e g; > < jsj 2hW20 i " #! P ðs; aÞ 1=7 12=7 > > m m2=7 jsj > > exp ; d e g: > : jsj 2hW20 i
ð6:42Þ
Observe that the PDF for the collisionless MHD (de g) regime is less non-Gaussian than the PDF for the classical MHD (de g) regime which is consistent with the fact (indicated in Sections 5 and 6.3.2) that the collisionless MHD flows are less dissipative. For a fractally homogeneous MHD turbulence, on the other hand, using (6.6) for the scaling exponent a, (6.40) now becomes 8 0 2 102D0 4D0 þ4 31 5D0 > > 7þD0 m m 7þD0 jsj D0 þ7 5A > > > exp @4 ; d e g; > > 2hW20 i < jsj ð6:43Þ P ðs; aÞ
0 2 62D0 4D0 þ4 31 > 3D0 > 5þD0 jsj D0 þ5 > m 5þD0 m > > 5A; d e g; exp @4 > > : jsj 2hW20 i where D0 is the Hausdorff dimension of the fractal set on which the energy dissipation is concentrated. Observe that the spatial intermittency effects (D0 < 2) accentuate non-Gaussian statistics at small scales.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
17
Next, suppose instead that the energy dissipation is a multi-fractal set. Then, the PDF of the Elsa¨sser-variable gradient may be determined as follows. Let us first note, using (6.36) and (6.37), that the dissipative microscale g can be reexpressed as 8 1 > m aþ1 > > > ; d e g; < jsj ð6:44Þ g
1 > > a > m > : ; d e g: jsj Now, for a multi-fractal set, we have a continuous spectrum of scaling exponents a, with each a distributed on a set S (a) of Hausdorff dimension f (a). Assuming again that multi-fractal scaling in the inertial range, discussed in Section 6.2, can be extrapolated down to the dissipative microscale g, and using (6.40) and (6.44), we have for the PDF of the Elsa¨sser-variable gradient: 8 0 2 42a 4a2 31 Z 4af ðaÞ > > aþ1 m m aþ1 jsj aþ1 5A > > > exp @4 dlðaÞ; d e g; > > jsj 2hW20 i < P ðs; aÞ
0 2 42a 4a4 31 > Z 4af ðaÞ > > a m m a jsj a 5A > > exp @4 dlðaÞ; d e g: > > : jsj 2hW20 i
ð6:45Þ
Now, according to (6.37), 8 a2 < m2a1 ; d e g; jsj
a2 : 2a2 m ; de g so that the exponents in (6.45) are O (1), even for strong singularities, namely, a ) 0. This rules out, therefore, the use of the method of steepest descent to evaluate the integrals in (6.45) analytically. One has to resort to numerical computations, as in the case of hydrodynamic turbulence [66]. 6.3.4. Parabolic-profile model for f (a) Based on the general theoretical knowledge of the f (s) vs. a curve, one may consider the following parabolic profile for f (a) [49]:16 2 f ðaÞ ¼ aða a0 Þ2 ;
a and a0 > 0:
ð6:46Þ
Let us now consider the implications of (6.46) for the inertial and the dissipative microscale regimes.
16
Eq. (6.46) has been confirmed experimentally for hydrodynamic turbulence [49].
18
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
6.3.4.1. Inertial regime. Using (6.46), (6.15b) yields p a ðpÞ ¼ a0 : 2a Using (6.47), (6.15a) becomes 8 p2 > > < ða0 1Þp ; d e ‘; 4a fp ¼ 2 > > : ða0 2Þp p ; d e ‘: 4a Now, (5.8) implies that
1; d e ‘; f4 ¼ 1; d e ‘: Using (6.49), (6.48) leads to 8 5 1 > < þ ; d e ‘; a0 ¼ 4 a > : 7 þ 1 ; d ‘: e 4 a Using (6.50), (6.47) gives 8 5 1 > < þ ð2 pÞ; a ðpÞ ¼ 4 2a > : 7 þ 1 ð2 pÞ; 4 2a while (6.48) becomes 8 p ð4 pÞp > > ; < þ 4a fp ¼ 4 > p ð4 pÞp > : þ ; 4 4a
ð6:47Þ
ð6:48Þ
ð6:49Þ
ð6:50Þ
d e ‘; ð6:51Þ de ‘
d e ‘; ð6:52Þ d e ‘:
On comparing (6.52) with the multi-fractal result (6.19) for the inertial regime, we obtain p Dp=4 ¼ 2 8d e =‘ ð6:53Þ a which shows that the parabolic-profile model (6.46) mimics the log-normal model. On the other hand, (6.51) implies that 8 5 > < ; de ‘
a ðpÞ < 4 8p P 2 ð6:54Þ > : 7; d ‘ e 4 which shows that the velocity/magnetic field singularities are weaker in the collisionless MHD case than in the classical MHD case—this was confirmed in Section 6.3.2.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
19
Eqs. (6.50)–(6.53) consistently show that the zero-intermittency limit corresponds to the limit a ) 1. On the other hand, using (6.50) and 6.51, (6.46) yields 2 f ða Þ ¼
p2 ; 4a
ð6:55aÞ
so lim 2 f ða Þ ¼ 0
a)1
ð6:55bÞ
as to be expected! On the other hand, (6.53) implies D0 ¼ 2 which is also confirmed by (6.55a) that yields f ða ð0ÞÞ ¼ 2: f (a* (0)) being the fractal dimension of the support of the measure, namely, D0. Thus, in the parabolic-profile model given by (6.46) the multi-fractality manifests itself via the way the measure is distributed rather than the geometrical properties like the support of the set. 6.3.4.2. Dissipative microscale regime. Here, we sketch the development for only a collisional (de g) plasma. The results of Section 6.3.2 show that we do not have sufficient number of conditions to accomplish closure in the parabolic-profile model (6.46) for a collisionless (de g) plasma. Using (6.46), (6.32) now yields 2aa 2 2aa þ ð3p þ 2aa0 2aa20 Þ ¼ 0
ð6:56aÞ
from which
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 6p 2 a ðpÞ ¼ 1 ð2a0 1Þ : 2 a
ð6:56bÞ
Now, noting that A0 1
ð6:57Þ
we have from (6.31) and (6.46) that að0Þ ¼ a0 :
ð6:58Þ
Eq. (6.58) dictates immediately that we discard the negative root in (6.56b), and hence " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 6p 2 1 þ ð2a0 1Þ a ðpÞ ¼ : ð6:59Þ 2 a If we write Ap hRicp
ð6:60Þ
20
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
then we have from (6.31) and (6.32), cp ¼ 12½p f 0 ða Þ
ð6:61Þ
and using (6.46) cp ¼ 12½p þ 2afa ðpÞ a0 g:
ð6:62Þ
Now, Section 6.3.2 showed that the mean energy dissipation rate remains finite in the ideal limit m ) 0, for a collisional plasma (de g), i.e., c 1
mA2 hRi 2 constant which implies that c2 1 ¼ 0: Using (6.62) and (6.59), (6.64) yields 5 1 a0 ¼ þ 4 a
ð6:63Þ ð6:64Þ
ð6:65Þ
which is identical to the one, namely (6.50), that ensued consequent to the condition (6.49). This appears to indicate that the ideal dissipation of energy (6.63) has been incorporated into the exact-like result (6.49) in MHD in a manner similar to that for classical 3D hydrodynamic result [67]. Further, on using (6.65), (6.59) and (6.46) lead to: 5 9 1 a ðpÞ ¼ þ ð1 pÞ þ O 2 ; ð6:66Þ 4 a a 2 ð8 9pÞ 1 þO 2 : 2 f ða Þ ¼ a a
ð6:67Þ
Eqs. (6.65)–(6.67) consistently show that the zero-intermittency limit corresponds again to the limit a ) 1. 6.3.4.3. Probability distribution function. The multi-fractal model was seen in Section 6.3 not to afford an analytic calculation of PDF of the Elsa¨sser-variable gradient. The parabolic-profile model turns out to be fruitful on this aspect. Noting the scaling behavior of the Elsa¨sser-variable gradient given by (6.37), and assuming W0 to be Gaussian distributed, as per (6.39), we observe 4a 2
W20 W a þ1 : So, a* (p) corresponds to a ð~pÞ where ~ p is the solution of 4a ð~ pÞ 2 ~ : p¼ a ð~ pÞ þ 1
ð6:68Þ
ð6:69Þ
Using (6.69), and assuming the intermittency to be weak (i.e., a is large) to simplify the calculations, we have from (6.66), 5 3 1
a ð~ pÞ ¼ þ O 2 : ð6:70Þ 4 a a
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
21
Using (6.70), the PDF of the Elsa¨sser-variable gradient given by (6.40) becomes
0 2 2 32 4 32 31 1þ16 3 9a m m 3þ9a jsj 39a 5A P ðs; a ð~ pÞÞ
exp @4 : ð6:71Þ jsj 2hW20 i Incidentally, using (6.70), (6.69) gives 4 32 1 ~ p ¼ þO 2 3 9a a
ð6:72Þ
which is of course the exponent of jsj in the argument of the exponential in (6.71). Note the accentuation of the non-Gaussianity of the PDF due to intermittency (a „ 1). It should be mentioned that the physical principle (namely, the ideal dissipation of energy) underlying the calculation of the intermittency correction to the PDF of the Elsa¨sser-variable gradient given above is also the same as that used previously in the homogeneous-fractal based result (6.43). Since the parabolic-profile model given by (6.46) incorporates the multi-fractal aspects only via the measure distribution arrangement the PDF (6.71) is complementary to the PDF (6.43) which incorporates multi-fractal aspects only via the geometrical properties like the support of the set.
7. Discussion In this paper, we have given a formulation of 2D collisionless MHD turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. Particular emphasis has been placed on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We have shown that the energy spectrum is shallower in the collisionless MHD regime which implies that the collisionless MHD flows are less dissipative than the classical MHD flows and leads to an energy pile-up at scales ‘n de in the energy cascade. This can cause creation of an ordered vortex/current filament phase similar to the one numerically shown in the hydrodynamic phase [56]. We have then investigated the fractal/multifractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k1 spectrum in the solar wind. Multi-fractal scaling behavior in the inertial range is first deduced, and is then extrapolated down to the dissipative microscale. Intermittency is found to reduce the degrees of freedom, and this result is a robust one in that it is identical even quantitatively for both the 2D classical MHD and collisionless MHD cases. The mean energy dissipation rate, in the ideal limit, remains finite in the classical MHD regime but vanishes in the collisionless MHD regime. The latter result is consistent with the fact that the collisionless MHD flows are less dissipative than the classical MHD flows, as mentioned above. Finally, spatial intermittency has been shown to accentuate non-Gaussian statistics at small scales in MHD turbulence even in the collisionless regime. Further, the PDF of the Elsa¨sser-variable gradient for the collision-
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B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
less MHD (de g) regime has been found to be less non-Gaussian than the PDF for the classical MHD (de g) regime which is again consistent with the fact that the collisionless MHD flows are less dissipative. A parabolic-profile model for the singularity spectrum f (a) is considered as a special case of the multi-fractal scenario. This model affords not only considerable insight into qualitative aspects of the intermittency problem but also an analytical calculation of PDF of the Elsa¨sser-variable gradient.
Note added in proof For the linearized version of the system (3.1), for which the dispersion relation is x2 ð1 þ k 2 d 2e Þ ¼ k 2z C 2A ð1 þ k 2 q2s Þ the condition qs = de corresponds to a degenerate case wherein the electron-inertia and electron-pressure effects cancel one another. This appears to be related to the feasibility of Elsa¨sser-variable formulation for the nonlinear case when qs = de.
Acknowledgments Some of the work contained in this paper was done while I was in residence at the International Centre for Theoretical Physics, Trieste, Italy. I am very thankful to Professor K.R. Sreenivasan for the hospitality and helpful information. My thanks are due to the referee for his helpful remarks that led to an improvement in the presentation in this paper.
References [1] Ya.B. Zeldovich, A.A. Ruzmaikin, D.D. Sokoloff, Magnetic Fields in Astrophysics, Gordon and Breach, 1983. [2] S. Childress, A.D. Gilbert, Stretch, Twist and Fold: the Fast Dynamo, Springer Verlag, Berlin, 1995. [3] G.K. Batchelor, Proc. Roy. Soc. (London) A 201 (1950) 405. [4] L. Biermann, A. Schlu¨ter, Phys. Rev. 82 (1951) 863. [5] H.P. Pao, Phys. Fluids 6 (1963) 632. [6] P.G. Saffman, J. Fluid Mech. 18 (1964) 449. [7] A. Tsinober, G. Galanti, Phys. Fluids 15 (2003) 3514. [8] K. Ohkitani, Phys. Rev. E 65 (2002) 046304. [9] S. Kida, S. Yanase, J. Mizushima, Phys. Fluids A 3 (1991) 457. [10] H. Politano, A. Pouquet, P.L. Sulem, Phys. Fluids B 1 (1989) 2330. [11] T. Gomez, H. Politano, A. Pouquet, Phys. Fluids 11 (1999) 2298. [12] D. Biskamp, H. Welter, Phys. Fluids B 1 (1989) 1964. [13] D. Biskamp, H. Welter, Phys. Fluids B 2 (1990) 1787. [14] D. Biskamp, H. Welter, M. Walter, Phys. Fluids B 2 (1990) 3024. [15] R. Kinney, J.C. McWilliams, T. Tajima, Phys. Plasmas 2 (1995) 3623. [16] R. Grauer, C. Marliani, Phys. Plasmas 2 (1995) 41. [17] B.K. Shivamoggi, Ann. Phys. 272 (1999) 167. [18] I.S. Iroshnikov, Sov. Astron. 7 (1964) 568.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23 [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67]
23
R.H. Kraichnan, Phys. Fluids 8 (1965) 1385. A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 30 (1941) 301. A. Pouquet, U. Frisch, J. Leorat, J. Fluid Mech. 77 (1976) 321. P. Goldreich, R. Sridhar, Astrophys. J. 438 (1995) 763. W. Muller, D. Biskamp, Phys. Rev. Lett. 84 (2000) 475. R. Sridhar, P. Goldreich, Astrophys. J. 432 (1994) 612. D. Montgomery, W.H. Matthaeus, Astrophys. J. 447 (1995) 706. C.S. Ng, A. Bhattacharjee, Astrophys. J. 465 (1996) 845. S.V. Nazarenko, A.C. Newell, S. Galtier, Physica D 152–153 (2001) 646. P. Goldreich, R. Sridhar, Astrophys. J. 485 (1997) 680. B.K. Shivamoggi, Ann. Phys. 253 (1997) 239; Erratum in, Ann. Phys. 312 (2004) 270. B. Mandelbrot, in: R. Temam (Ed.), Turbulence and Navier–Stokes Equations, Lecture Notes in Mathematics, vol. 565, Springer-Verlag, Berlin, 1975. U. Frisch, P.L. Sulem, M. Nelkin, J. Fluid Mech. 87 (1978) 719. U. Frisch, G. Parisi, in: M. Ghil, R. Benzi, G. Parisi (Eds.), Turbulence and Predictability in Geophysical Fluid Dynamics and Climatic Dynamics, North-Holland, Amsterdam, 1985. B. Coppi, Phys. Lett. A 11 (1964) 226. K. Schindler, J. Geophys. Res. 70 (1974) 2803. J.F. Drake, Y.C. Lee, Phys. Fluids 20 (1977) 134. J.F. Drake, R.G. Kleva, Phys. Rev. Lett. 66 (1991) 1458. R.G. Kleva, J.F. Drake, F.L. Waelbroeck, Phys. Plasmas 2 (1995) 23. J.F. Drake, in: P. Song, B.U.O. Sonnerup, M.F. Thomsen (Eds.), Physics of the Magnetopause, Geophys. Monogr. Ser., Vol. 90, Amer. Geophys. Union, Washington, DC, 1995. A.A. Galeev, Phys. Fluids 21 (1978) 1353. J. Wesson, Nucl. Fusion 30 (1990) 2545. F. Porcelli, Phys. Rev. Lett. 66 (1991) 425. M. Ottaviani, F. Porcelli, Phys. Rev. Lett. 71 (1993) 3802. B.K. Shivamoggi, J. Plasma Phys. 58 (1997) 329. B.K. Shivamoggi, Phys. Lett. A 292 (2001) 134. N. Al-Salti, B.K. Shivamoggi, Phys. Plasmas 10 (2003) 4271. A.Y. Aydemir, Phys. Fluids B 4 (1992) 3469. L.E. Zakharov, B. Rogers, S. Migliulo, Phys. Fluids B 5 (1993) 2498. E. Cafaro, D. Grasso, F. Pegoraro, F. Porcelli, A. Saluzzi, Phys. Rev. Lett. 80 (1998) 4430. C. Meneveau, K.R. Sreenivasan, Nucl. Phys. B 2 (1987) 49. J.W. Armstrong, B.J. Rickett, S.R. Spangler, Astrophys. J. 443 (1995) 209. W.M. Elsa¨sser, Phys. Rev. 79 (1950) 183. W.H. Matthaeus, P. Dmitruk, L.J. Milano, Phys. Plasmas 9 (2002) 2440. H. Politano, A. Pouquet, V. Carbone, Europhys. Lett. 43 (1998) 516. D. Biskamp, E. Schwarz, Phys. Plasmas 8 (2001) 3282. K. Nakayama, Astrophys. J. 556 (2001) 1027. N. Kukharin, S.A. Orszag, V. Yakhot, Phys. Rev. Lett. 75 (1995) 2486. H.P. Furth, J. Killeen, M.N. Rosenbluth, Phys. Fluids 6 (1963) 459. H. Politano, A. Pouquet, P.L. Sulem, Phys. Plasmas 2 (1995) 2931. E. Marsch, C.Y. Tu, J. Geophys. Res. 95 (1990) 8211. M.L. Goldstein, D.A. Roberts, W.H. Matthaeus, Annu. Rev. Astron. Astrophys. 33 (1995) 283. T.C. Halsey, M.H. Jensen, L. Kadanoff, I. Proccacia, B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82. M. Nelkin, Phys. Rev. A 42 (1990) 7226. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959. B.K. Shivamoggi, Ann. Phys. 270 (1998) 263. R. Benzi, L. Biferale, G. Paladin, A. Vulpiani, M. Vergassola, Phys. Rev. Lett. 67 (1991) 2299. A.N. Kolmogorov, Dokl. Akad. Nauk SSR 32 (1941) 16.
Collisionless magnetohydrodynamic turbulence in two dimensions Bhimsen K. Shivamoggi* University of Central Florida, Orlando, FL 32816-1364, USA Received 4 June 2004; accepted 20 December 2004
This paper is dedicated to my mother for her 75th birthday.
Abstract In this paper we give a formulation of two-dimensional (2D) collisionless magnetohydrodynamic (MHD) turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. We place particular emphasis on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We investigate the fractal/multi-fractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k1 spectrum in the solar wind. Multi-fractal scaling behaviors in the inertial range are first deduced, and are then extrapolated down to the dissipative microscales. We then consider a parabolic-profile model for the singularity spectrum f (a), as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence. 2004 Elsevier Inc. All rights reserved. PACS: 47.27.Ak; 47.27.Gs Keywords: Magnetohydrodynamics; Turbulence
*
Fax: +1 4078236253. E-mail address: [email protected].
0003-4916/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.12.006
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1. Introduction Many magnetohydrodynamic (MHD) flows that occur naturally (like in astrophysical systems, notably the solar wind) and in modern technological systems (like fusion reactors) show turbulence.1 However, laboratory experiments on MHD turbulence are scarce because of the difficulty of achieving sufficiently high magnetic Reynolds numbers. Nuclear fusion devices present some hope for the latter aspect but detailed measurements in them are extremely difficult. Advective stretching of magnetic field lines leads to amplification of magnetic field energy at small scales—dynamo action [3–8] and magnetic field lines that have been highly stretched experience typically stronger Ohmic dissipation [9]. If the magnetic field is weak, the advection-diffusion mechanism controls the statistical properties of the small-scale components of the magnetic field (somewhat similar to the case with passive-scalar transport). The influence of spatial dimension on the nature of MHD turbulence is less stringent than on that of hydrodynamic turbulence thanks to the Lorentz force which breaks the vorticity conservation in two dimensions. This leads to an energy cascade to small scales in both two dimensions and three dimensions and the behavior of twodimensional (2D) and three-dimensional (3D) MHD turbulence is phenomenologically similar.2 2D MHD turbulence cases have therefore been considered extensively in direct numerical simulations (DNS) thanks also to the ease of attainment of high Reynolds numbers in 2D [10–16]. Most of the theoretical investigations of 3D MHD turbulence have considered the isotropic3 case which is appropriate for the case where the average magnetic field is zero. In this case, Iroshnikov [18] and Kraichnan [19] made arguments similar to those involved in KolmogorovÕs [20] theory of isotropic hydrodynamic turbulence to propose that the statistical properties of the small-scale components of the velocity and magnetic fields are controlled by shear Alfve´n-wave dynamics, in the limit of large viscous and magnetic Reynolds numbers, have some universality in the inertial range, and are influenced only weakly by the large-scale features of the fields. The Iroshnikov–Kraichnan (IK) theory uses dimensional arguments to derive a k3/2 power law for the spectrum of the kinetic and magnetic energy densities in the stationary state. Pouquet et al. [21] used an eddy-damped quasi-normal Markovian (EDQNM) closure theory to show that, in the absence of the cross helicity, an inertial range occurs with a cascade of energy to small scales with a 3/2 power law. However, Goldreich and Sridhar [22] argued that the shear Alfve´n-wave dynamics does not control 3D MHD turbulence and Muller and Biskamp [23] argued that
1
MHD turbulence plays a major role in the current models on the dynamo action for magnetic field generation in planets and stars [1,2]. 2 Some essential differences persist, however, like the absence of the dynamo effect in the 2D case. 3 The isotropic assumption for the 3D MHD case is not a very sound one because of a preferred direction imposed, at least in a local sense, by the magnetic field of the large-scale eddies or by the mean magnetic field. An axisymmetric model for MHD turbulence was considered by Shivamoggi [17] via a statistical approach.
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3
the latter is instead dominated by vortical motions in the plane perpendicular to the local large-scale magnetic field B0ˆiz.4 But, in the absence of a rigorous theory, this issue is still open. The IK theory does not take into account the spatial intermittency in the flow that arises due to the spatial scarcity of intense small-scale dissipative structures. This was dealt with by Shivamoggi [29] who used fractal [30,31] and multi-fractal [32] models for the dissipative structures. The DNS of 2D MHD turbulence [12] showed strongly intermittent situations with large magnetic islands (where magnetic field is strong) surrounded by a sea of small-scale turbulence (where the magnetic field is weak but the magnetic field gradients are large). Strong intermittency effects were also shown in several other DNS [11,15,16]. The high-temperature plasmas in both space and fusion systems have been found to be collisionless, so considerable work has gone into collisionless plasma dynamics [33–45], among others. In a collisionless plasma, the electron inertia: leads to the decoupling of the plasma motion from that of the magnetic field lines, limits the election current and prevents it from becoming unbounded as the resistivity g ) 0. In the electron-inertia regime, the conservation of the magnetic flux is replaced by the conservation of the generalized electron-fluid momentum. Physically, this allows for the localized violation of the topological constraint on the magnetic flux and hence magnetic field reconnection to occur in the electron-inertia regime and an exchange between magnetic and kinetic energies. Collisionless plasma dynamics cannot therefore be described adequately by the single-fluid formulation of resistive MHD. In a low-b (b being the ratio of the plasma pressure to the magnetic pressure) plasma (such as the strongly magnetized plasmas in fusion systems), the plasma dynamics is also governed by the electron-pressure gradient term in the electron momentum-balance equation [37,46–48]. In this paper, we give a formulation of 2D collisionless MHD turbulence that includes the effects of both electron inertia and electron pressure and is applicable to strongly magnetized collisionless plasmas. We adopt a fluid treatment for both electrons and ions [37,48]. We take the statistical properties of the small-scale components of the velocity and magnetic fields to be controlled by shear (and kinetic) Alfve´n-wave dynamics because the IK phenomenology based on the Alfve´n-effect mechanism is believed to be more valid for the 2D case rather than the 3D case [12]. We consider fractal and multi-fractal models to describe the effects of spatial intermittency in 2D fully developed collisionless MHD turbulence. We will then extrapolate multi-fractal scaling in the inertial range down to the dissipative 4
Sridhar and Goldreich [24] sought to justify their argument on the grounds that the three Alfve´n-wave resonant interactions underlying the IK model are non-existent. But, this was refuted by Montgomery and Matthaeus [25], Ng and Bhattacharjee [26], and Nazarenko et al. [27]. Goldreich and Sridhar [28] subsequently conceded the flaw in their argument. So, the details of the shear Alfve´n-wave dynamics in the IK model are still controversial.
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microscale. We then consider a parabolic-profile model for the singularity spectrum f (a) [49], as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence.
2. Governing equations of collisionless plasma dynamics We adopt a fluid treatment for both electrons and ions5 [37,48]. We assume the dynamics to preserve quasi-neutrality, so ni ne = n, n being the number density of the particles, the subscripts i and e being associated with ions and electrons, respectively. The governing equations are: (i) Mass conservation of ions: on þ r ðnvÞ ¼ 0 ot
ð2:1Þ
v being the ion-flow velocity. (ii) FaradayÕs law: oB ¼ cr E ot
ð2:2Þ
E and B being the electric and magnetic fields, respectively. (iii) AmpereÕs law: 1 rB¼ J c J being the electric current density. (iv) Momentum-balance equation of ions: ov 1 þ ðv rÞv ¼ e E þ v B mi n ot c
ð2:3Þ
ð2:4Þ
m being the particle mass. We take the ions to be cold. (v) Momentum-balance equation of electrons (generalized OhmÕs law): 1 1 oJ Te E ¼ ve B þ 2 þ ðve rÞJ rn ð2:5Þ c xpe ot ne pffiffiffiffiffiffiffiffiffiffiffiffiffiffi xpe being the plasma frequency, xpe ne2 =me , and T is the temperature. We take the electrons to be isothermal (Te = constant), and we neglect the diamagnetic effects 5 Theoretical analysis of collisionless plasma processes usually require a kinetic model. This is true a fortiori if the particle distribution functions deviate strongly from Maxwellian as in a long-tail development due to acceleration of particles to highly superthermal energies. On the other hand, kinetic effects become important even for nearly isotropic distributions if the Larmor radii exceed the gradient length scales in question. However, if superthermal particles are negligible and the Larmor radii are sufficiently small, the fluid model is adequate in describing the collisionless plasma dynamics.
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associated with the electron-pressure gradient. Eq. (2.5) shows that the electron-pressure gradient parallel to the guide magnetic field B0ˆiz causes compression of the electron population in the z-direction—parallel electron compressibility. Let us introduce the magnetic flux w and the flow stream function / according to: ) B ¼ B0^iz þ rw ^iz ; ð2:6Þ v ¼ r/ ^iz ; where B0ˆiz is the constant guide field.6 We assume B0 is large so that z is the ignorable coordinate. Eqs. (2.1)–(2.5) then lead to [37,48] o ðw d 2e r2 wÞ þ ½w d 2e r2 w; / q2s ½w; r2 / ¼ 0; ot
ð2:7Þ
o ðr2 /Þ þ ½r2 /; / þ ½w; r2 w ¼ 0: ot
ð2:8Þ
Here, ½f ; g ^iz rf rg: de is pthe ffiffiffiffiffiffiffiffiffielectron skin depth d e c=xpe , and qs is the ion sound gyro-radius T e =mi qs xc , xci being the ion gyro-frequency, xci eB0 =mi c. Eqs. (2.7) and (2.8) i are adequate for describing several dynamical aspects of collisionless plasmas in space (e.g., the magnetosphere) and fusion systems [37,48].
3. Elsa¨sser variables Eqs. (2.7) and (2.8) have the following Hamiltonian representation [48] Z Z 1 2 2 2 2 H¼ ½ðr/Þ þ ðrwÞ þ q2s ðr2 /Þ þ d 2e ðr2 wÞ dx; 2
ð3:1Þ
v
where V R2 is the appropriate 2D domain on which the plasma flow and the magnetic field evolve. In order to be able to rewrite 3.1 in terms of Elsa¨sser [51] variables (which is a crucial step prior to formulation of a theory of collisionless MHD turbulence), let us assume qs = de.7 The present results can be expected to serve as a qualitative guide even for those situations (see Footnote 7) where this condition does not hold. We then introduce Elsa¨sser variables W w / 6
ð3:2Þ
An embedded magnetic field is also a frequent feature in astrophysical turbulent plasmas [50]. This condition may be expected to occur in strongly magnetized plasma situations like those in some astrophysical contexts. Cafaro et al. [48] in fact also considered cases qs < de. For the magnetospheric [38] and current fusion-reactor [41] plasmas qs/de > 1. 7
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so 3.1 may be written as Z Z n o 1 n o 1 1 2 2 2 2 2 2 2 ðrWþ Þ þ ðrW Þ þ d e ðr Wþ Þ þ ðr W Þ H¼ dx: 2 2 2 ð3:3Þ Eq. (3.3) shows that there are no self-interaction terms in the 2D collisionless MHD case (under the assumption qs = de), as in classical MHD. Hence, Alfve´n waves (and their variants—kinetic Alfve´n waves in the presence of parallel electron compressibility) do not undergo non-linear wave steepening and only those waves propagating in opposite directions along the background magnetic field interact8—the so-called Alfve´n effect. Let us introduce further ðrWÞ2 12½ðrWþ Þ2 þ ðrW Þ2
ð3:4aÞ
2
~ via and a Laplacian-like operator r 2
2
2
2
~ WÞ 1½ðr2 Wþ Þ þ ðr2 W Þ : ðr 2
ð3:4bÞ
2
~ turns out not to be relevant for the ensuing considerations in The detailed form of r this paper. Eq. (3.3) may then be rewritten further as Z Z 1 2 ~ 2 WÞ2 dx: ½ðrWÞ þ d 2e ðr H¼ ð3:5Þ 2
4. The MHD eddy turn-over time: Iroshnikov–Kraichnan hypothesis The IK hypothesis can be shown to follow also from the formal analogy between the hydrodynamic and MHD spectral energy density expressions. To see this, let us write spectral energy density E (k) as EðkÞ
V ; k2s
ð4:1Þ
V being the characteristic velocity of the spectral element k. The hydrodynamic eddy turn-over time s given by s
1 kV
ð4:2Þ
then becomes9 s
8
1 : k E1=2 3=2
ð4:3Þ
Such interactions are non-local. Eq. (4.2) implies that the energy transfer in the hydrodynamic case is local in the spectral space which also reflects the fact that a large-scale velocity field can be transformed away via Galilean invariance. 9
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
7
If we use the relation, s
Ek
ð4:4Þ
being the mean energy dissipation rate, Eq. (4.3) leads to the Kolmogorov spectrum: EðkÞ 2=3 k 5=3 :
ð4:5Þ
If, on the other hand, we write for the MHD case, in analogy with (4.1) [29], CA k 2^s
ð4:6Þ
1 : kC A
ð4:8Þ
EðkÞ
~ 0 =pffiffiffi ^s being the MHD eddy turn-over time and C A B q being the velocity of Alfve´n ~ waves in the magnetic field B0 of the large-scale eddies, we immediately obtain s ^s s ; ð4:7Þ sA where sA
Eq. (4.7) implies that the energy transfer in the MHD case is non-local in the spectral space and represents the Alfve´n effect10 (see Section 3) postulated in the IK theory! If we again use the relation (4.4), (4.7) leads to the IK spectrum: 1=2
EðkÞ 1=2 C A k 3=2 :
ð4:9Þ
5. Inertial-range scaling laws The influence of spatial dimension is less stringent in MHD turbulence than in hydrodynamic turbulence because the Lorentz force breaks the vorticity conservation in 2D, allowing an energy cascade to small scales in 2D (as well as 3D). One may then consider the possibility of an inertial range of the Kolmogorov type which is in a state of statistical equilibrium and the energy is assumed to cascade smoothly through non-linear processes in a stationary state. Consider a discrete sequence of scales ‘n ‘0 2n ;
n ¼ 0; 1; 2; . . .
ð5:1Þ
Let us assume that we have a statistically stationary turbulence, where energy is introduced into the plasma at scales ‘0, and is then transferred successively to scales
10 Alfve´n effect can therefore be viewed as a non-local effect of the large-scale magnetic field and is a kind of sweeping action of the large eddies on the small eddies [52].
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‘1, ‘2, . . .until some scale ‘d is reached where dissipative effects are able to compete with non-linear transfer. The energy per unit mass in the nth scale is given by ! W2n d 2e ð5:2Þ En 2 1 þ 2 : ‘n ‘n The rate of transfer of energy per unit mass from the nth scale to the (n + 1)th scale is given by En en
ð5:3Þ ^sn and using (4.7), (5.3) becomes en
E n sA : s2n
ð5:4Þ
Noting, sA
‘n CA
and
sn
‘2n ; Wn
ð5:5Þ
and using (5.2), (5.4) becomes ! W4n d 2e en 5 1þ 2 : ‘n C A ‘n
ð5:6Þ
In the inertial range, we assume a stationary process in which the energy transfer rate is constant: en ¼ constant ¼ e;
‘d 6 ‘n 6 ‘0 :
ð5:7Þ
Using (5.6), (5.7) leads to Wn
1=4 e1=4 ‘1=4 n CA
d2 1 þ 2e ‘n
!1=4 :
ð5:8Þ
Using (5.8), (5.2) gives En e
1=2
1=2 C A ‘n1=2
d2 1 þ 2e ‘n
from which, we have ( 1=2 e1=2 C A ‘n1=2 ; En
1=2 e1=2 C A ‘n1=2 ;
!1=2
d e ‘n ; d e ‘n :
Eq. (5.10) leads to the following energy spectra: ( 1=2 e1=2 C A k 3=2 ; kd e 1; EðkÞ
1=2 e1=2 C A k 1=2 ; kd e 1:
ð5:9Þ
ð5:10Þ
ð5:11Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
9
The energy spectrum for the classical MHD regime (kde 1) corresponds to the IK spectrum in 2D MHD turbulence. The DNS of 2D classical MHD turbulence [10,53,12–14,54] and closure calculations [55] confirm this.11,12 There seems to be no available observational data confirming the scaling law for the collisionless MHD regime (kde 1) yet. However, the intermittency-corrected version of this scaling law appears to have some observational record (see Section 6.1). The shallower energy spectrum in the collisionless MHD regime (kde 1) signifies the appearance of energy generation (say, via electron inertia) in the system for length scales below de. Further, it can lead to energy pile-up at scales ‘n de and, hence, can cause creation of an ordered vortex/current filament phase similar to the one numerically shown in the hydrodynamic case by Kukharin et al. [56]. 6. Spatial intermittency 6.1. Homogeneous fractal model The theory given in Section 5 does not take into account the spatial intermittency in the flow that would cause systematic departures from the scaling laws (5.11) which use mean transfer rates. One may follow [30] and argue that the spatial intermittency effects in the present system are related to the fractal aspects of the geometry of turbulence. This reflects the fact that the dissipative structures are strongly convoluted. Indeed, since small-scale structures in MHD turbulence are primarily produced by flows perpendicular to the large-scale magnetic field, the most intense regions of dissipation may be expected to occur as large sheets, like the current sheets13 which are also vorticity sheets [58] and also [12]. This may be simulated in a first approximation by representing the dissipative structures via a homogeneous fractal with non-integer Hausdorff dimension D0. This amounts to assuming the energy flux to be transferred to only a fixed fraction b of the eddies downstream in the cascade—the b-model [31]. Consider a discrete sequence of scales as in (5.1), but now assume that at the nth step, only a fraction bn of the total space has an appreciable excitation with a fractal dimension D0. The energy per unit mass in the nth scale is given by ! bn W2n d 2e En 2 1þ 2 ; ð6:1Þ ‘n ‘n 11
The Alfve´n effect seems to be the dominant mechanism underlying 2D MHD turbulence [12] unlike the situation with 3D MHD turbulence (see Section 1). 12 The DNS [10] further showed that the IK spectrum prevails only for small values of the correlation between the velocity and the magnetic fields. Large values of this correlation (and concomitantly, the dynamic alignment of the velocity and the magnetic fields) introduce certain phase relations between the velocity and magnetic fields and further inhibit both exchanges of kinetic and magnetic energies and the energy transfer to small scales. 13 The advective stretching of magnetic field lines intensify the current sheets until magnetic reconnection processes set in consequent to tearing-mode instabilities [57].
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where bn
2D0 ‘n : ‘0
ð6:2Þ
The rate of transfer of energy per unit mass from the nth scale to the (n + 1)th scale is ! given by 4 E n sA d 2e n Wn ð6:3Þ 1þ 2 : en 2 b 5 sn ‘n C A ‘n In the inertial range, the energy transfer rate is constant for a stationary process, so on using (6.2) and (6.3), and assuming the scaling behavior, Wn ‘a we have, from (5.7), 4a þ 2 D0 5 ¼ 0; 4a þ 2 D0 7 ¼ 0; from which ( 3þD
0
a¼
4 5þD0 4
ð6:4Þ d e ‘n ; d e ‘n
; d e ‘n ; ; d e ‘n :
Using (6.4) and (6.6), we have from (6.1), 8 1 1 < ‘2þ2ð2D0 Þ ; d e ‘n ; n En
: 12þ12ð2D0 Þ ‘n ; d e ‘n : Eq. (6.7) leads to the following energy spectra: 8 < k 3212ð2D0 Þ ; kd 1; e EðkÞ
: 1212ð2D0 Þ k ; kd e 1:
ð6:5Þ
ð6:6Þ
ð6:7Þ
ð6:8Þ
Observe that the intermittency corrections (D0 < 2) make the spectra steeper, as expected. The result in (6.8) corresponding to the classical MHD regime (kde 1) agrees with that given by Biskamp [14]. If the dissipative structures correspond to current sheets, i.e., D0 = 1, then (6.8) yields ( k 2 ; kd e 1; EðkÞ
ð6:9Þ k 1 ; kd e 1: The classical MHD (kde 1) result is well known, while the collisionless MHD (kde 1) result appears to have been observed in the solar-wind plasma [59].14 14 There have been several controversial attempts in the past (see Goldstein et al. [60]) to explain this k1 spectrum. But these have been hampered by the objection that the plasma will have much less than one eddy turn-over time to evolve and the structures involved are too big to fit in the ‘‘box’’ between the sun and the point of observation.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
On the other hand, the structure function of order p, Sp (‘), behaves as p oW S p ð‘Þ d ð‘Þ ‘fp : ox
11
ð6:10Þ
Using (6.4), and noting that the probability to belong to this fractal at scale ‘ goes like ‘2D0 , (6.10) leads to ( ‘ða1Þpþ2D0 ; d e ‘; S p ð‘Þ
ð6:11Þ ‘ða2Þpþ2D0 ; d e ‘: So, the characteristic exponent fp is given by
ða 1Þp þ 2 D0 ; d e ‘; fp ¼ ða 2Þp þ 2 D0 ; d e ‘: Using (6.6), (6.12) becomes 8 p p > < þ 1 ð2 D0 Þ; fp ¼ 4 p 4 p > : þ 1 ð2 D0 Þ; 4 4
ð6:12Þ
d e ‘; ð6:13Þ d e ‘:
Observe that the b-model cannot describe any non-linear behavior (as is usually the case) of the characteristic exponents fp of higher-order structure functions. It is, therefore, pertinent to consider the multi-fractal model [32] to address this. 6.2. Multi-fractal model Let us follow [32] and assume that the energy dissipation field is a multi-fractal object. A singular measure defined on a multi-fractal is represented in terms of interwoven fractal subsets corresponding to different measure levels having a continuous spectrum of scaling exponents. Let S (a) be the support of interwoven sets of singularities in the measure e of strength a and fractal dimension f (a). Noting the scaling behavior of the Elsa¨sser-variable increment dW over a distance ‘ given by (6.4), the singularity spectrum f (a) is obtained via a Legendre transformation of the scaling exponent fp of the pth-order structure function: (R p oW dlðaÞ‘ða1Þpþ2f ðaÞ ; d e ‘; fp S p ð‘Þ d ð‘Þ ‘ R ð6:14Þ ox dlðaÞ‘ða2Þpþ2f ðaÞ ; d e ‘; where ‘2f (a) dl(a) represents the probability of encountering the set S (a) within a 2D circle of radius ‘. One may use the method of steepest descent to extract the dominant contribution to the integral in (6.14), in the limit of small ‘. This gives 8 < inf ½ða 1Þp þ 2 f ðaÞ; d e ‘; a fp
: inf ½ða 2Þp þ 2 f ðaÞ; d e ‘ a
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or
fp ¼
ða 1Þp þ 2 f ða Þ;
ða 2Þp þ 2 f ða Þ;
d e ‘; d e ‘;
ð6:15aÞ
where f 0 ða Þ ¼ p:
ð6:15bÞ
To relate the singularity spectrum f (a) to the generalized fractal dimension of the energy dissipation field, consider a coarse-grained probability measure given by the total energy dissipation occurring in a square of size ‘: ( ‘4a3 ; d e ‘; 2 U ð‘Þ eð‘Þ‘
ð6:16Þ ‘4a5 ; d e ‘: We then cover the support of the measure e with squares of size ‘ and sum [U (‘)]q over all squares. If the energy dissipation field is a multi-fractal, such a sum exhibits the asymptotic scaling behavior [61] of moments: (R X dlðaÞ‘ð4a3Þqf ðaÞ ; d e ‘; q ðq1ÞDq
R ½U ð‘Þ ‘ ð6:17Þ dlðaÞ‘ð4a5Þqf ðaÞ ; d e ‘; where we have assumed that the number of iso-a squares for which a takes on values between a and a + da is proportional to dl (a)‘f (a). Dq is the generalized fractal dimension of the e-field. Implicit in (6.16) and (6.17) is a refined similarity hypothesis, analogous to that introduced by Kolmogorov [62] for hydrodynamic turbulence, connecting the Elsa¨sser variable increment with that of the locally averaged energy dissipation in 2D MHD turbulence. The dominant terms in the integral in (6.17), in the limit of very small ‘, may again be extracted using the method of steepest descent to yield the Legendre transformation:
ð4a 3Þq f ða Þ; d e ‘; ðq 1ÞDq ¼ ð6:18aÞ ð4a 5Þq f ða Þ; d e ‘; where f 0 ða Þ ¼ 4q:
ð6:18bÞ
Eliminating f (a) from (6.18a) and (6.18b), and putting q = p/4, we obtain 8 p p > < þ 1 ð2 Dp=4 Þ; d e ‘; fp ¼ 4 p 4 p > : þ 1 ð2 Dp=4 Þ; d e ‘: 4 4
ð6:19Þ
For a fractally homogeneous turbulence, Dp=4 ¼ D0
8p:
ð6:20Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
Eq. (6.19) then reduces to 8 p p > < þ 1 ð2 D0 Þ; fp ¼ 4 p 4 p > : þ 1 ð2 D0 Þ; 4 4
13
d e ‘; ð6:21Þ de ‘
in agreement with the result (6.13) for the b-model. 6.3. Multi-fractal scaling at the dissipative microscale We will now consider extrapolation of multi-fractal scaling in the inertial range discussed in Section 6.2 down to the dissipative microscale. The following development is predicated on the assumption that an inertial behavior persists at scales smaller than de—this assumption may not be too restrictive for tenuous plasmas like those in space (de 10 km for the magnetospheric plasma). On taking into account the spatial intermittent character of the energy dissipation field into account, the MHD dissipative microscale 2 1=3 m CA g
ð6:22Þ eðgÞ implies (along the lines of the development of Nelkin [63] for hydrodynamic turbulence) 8 1 g < hRi 2a1 ; d e g;
ð6:23Þ 1 L : hRi 2a2 ; d e g; where ÆRæ is the Reynolds number, 1=3
ðheiL4 Þ ; ð6:24Þ m mis the kinematic viscosity of the plasma and we have assumed, for the sake of simplicity, the magnetic Prandtl number to be unity. Observe the reduction in the dissipative microscale g produced by spatial intermittency (D0 < 2), as expected. hRi
6.3.1. Degrees of freedom Following Landau and Lifshitz [64], one may give an estimate for the number of degrees of freedom necessary for describing 2D fully developed MHD turbulence 2 L N ðRÞ
R4=3 S 2=3 ; ð6:25Þ g where S is the Lundquist number, S
CA ðeLÞ
1=3
:
In the presence of spatial intermittency this estimate is modified as follows.
ð6:26Þ
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Let us assume that the energy dissipation field is a multi-fractal object for which one has a continuous spectrum of scaling exponents a, with each a distributed on a set S (a) of Hausdorff dimension f (a) embedded in 2D space. Let us assume further that multi-fractal scaling in the inertial range, discussed in Section 6.2, can be extrapolated down to the dissipative microscale g. Eq. (6.25) then becomes f ðaÞ Z L N dlðaÞ : ð6:27Þ g On using (6.23), (6.27) becomes 8R f ðaÞ < dlðaÞhRi2a1 ; d g; e N
f ðaÞ R : dlðaÞhRi2a2 ; d e g: In the limit of large ÆRæ, the dominant exponents in (6.28) correspond to ð2a 1Þf 0 ðaÞ 2f ðaÞ ¼ 0; d e g; ð2a 2Þf 0 ðaÞ 2f ðaÞ ¼ 0; d e g:
ð6:28Þ
ð6:29Þ
Eqs. (6.29), in conjunction with (6.18a) and (6.18b), imply N hRi2Q ;
ð6:30aÞ
where Q¼
DQ DQ þ 1
8g=d e :
ð6:30bÞ
Eqs. (6.30a) and (6.30b) show that the effect of intermittency is to reduce the number of degrees of freedom, as to be expected! Observe the robust nature of (6.30a) and (6.30b)—it is identical for both 2D classical MHD (de g) and collisionless MHD (de g) regimes!15 In the absence of intermittency (DQ ) 2) (6.30a) and (6.30b) reduce to (6.25). 6.3.2. Moments of the Elsa¨sser-variable-gradient distribution Consider moments of the Elsa¨sser-variable-gradient distribution. Assuming again that the multi-fractal scaling of the inertial range can be extrapolated down to the dissipative microscale g, we have 8 * 1 +
15
A similar result occurs also in geostrophic turbulence [65].
ð6:32Þ
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
Eqs. (6.32), in conjunction with (6.18a) and (6.18b), imply 8 ðDQ 2Þðp4Þ3p > 2DQ þ3p4 > < hRi 2DQ þ2 ; where Q ¼ 2D ; d e g; Q þ2 Ap
ðD 2Þðp4Þp Q > > : hRi 2DQ þ2 ; where Q ¼ 2DQ þ2p4 ; d e g: 2DQ þ2 For p = 2, (6.33) implies 8 1 d e g; < hRi ; where Q ¼ 1; DQ 1 A2
: hRiDQ þ1 ; where Q ¼ DQ ; d g: e DQ þ1 Thus, the mean energy dissipation has the following scaling behavior: 8 < hRi0 ; where Q ¼ 1; d e g; 2 mA2
: hRiDQ þ1 ; where Q ¼ DQ ; d g: e DQ þ1
15
ð6:33Þ
ð6:34Þ
ð6:35Þ
Observe that the mean energy dissipation rate, in the limit m ) 0, remains finite in the classical MHD regime but vanishes in the collisionless MHD regime! The former result is confirmed by the DNS [12,13] while the latter result is further consistent with the fact (indicated in Section 5) that the collisionless MHD flows are less dissipative. 6.3.3. Probability distribution function of the Elsa¨sser-variable gradient Spatial intermittency in the energy dissipation field accentuates non-Gaussian statistics at small scales. To derive the probability distribution function of the Elsa¨sservariable gradients, note first that the scaling behaviors (6.23) of the dissipative microscale g can be reexpressed in the form 8 1 > m 2a1 > > > ; d e g; < W0 ð6:36Þ g
1 > 2a2 > > m > : ; d e g; W0 where (W0/L) is the Elsa¨sser-variable increment on a macroscopic length L which we have set equal to unity, for the sake of simplicity. The scaling behavior of Elsa¨sser-variable gradient is then given by 8 aþ1 a2 W < W02a1 m2a1 ; d e g; s 2
ð6:37Þ a a2 : 2a2 g W0 m2a2 ; d e g: The probability distribution function (PDF) of the Elsa¨sser-variable gradient may be determined in terms of that for the characteristic Elsa¨sser-variable increment W0 for large scales as follows:
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B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
P ðs; aÞ ¼ P ðW0 Þ
dW0 : ds
ð6:38Þ
Taking P (W0) to be Gaussian, ! W20 P ðW0 Þ exp 2hW20 i and using (6.37), (6.38) becomes 0 2 42a 4a2 31 8 2a > aþ1 m m aþ1 jsj aþ1 5A > > > exp @4 ; > > jsj 2hW20 i < 0 2 42a 4a4 31 P ðs; aÞ
2a > > a > m m a jsj a 5A > > exp @4 ; > : jsj 2hW20 i
ð6:39Þ
d e g; ð6:40Þ d e g:
For the zero intermittency regime (D0 = 2), where the energy dissipation field is space-filling, we have, from (6.6), ( 5 ; d e g; a ¼ 47 ð6:41Þ ; d e g: 4 Eq. (6.40) then becomes 8 " #! 1=3 > m m2=3 jsj4=3 > > > exp ; d e g; > < jsj 2hW20 i " #! P ðs; aÞ 1=7 12=7 > > m m2=7 jsj > > exp ; d e g: > : jsj 2hW20 i
ð6:42Þ
Observe that the PDF for the collisionless MHD (de g) regime is less non-Gaussian than the PDF for the classical MHD (de g) regime which is consistent with the fact (indicated in Sections 5 and 6.3.2) that the collisionless MHD flows are less dissipative. For a fractally homogeneous MHD turbulence, on the other hand, using (6.6) for the scaling exponent a, (6.40) now becomes 8 0 2 102D0 4D0 þ4 31 5D0 > > 7þD0 m m 7þD0 jsj D0 þ7 5A > > > exp @4 ; d e g; > > 2hW20 i < jsj ð6:43Þ P ðs; aÞ
0 2 62D0 4D0 þ4 31 > 3D0 > 5þD0 jsj D0 þ5 > m 5þD0 m > > 5A; d e g; exp @4 > > : jsj 2hW20 i where D0 is the Hausdorff dimension of the fractal set on which the energy dissipation is concentrated. Observe that the spatial intermittency effects (D0 < 2) accentuate non-Gaussian statistics at small scales.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
17
Next, suppose instead that the energy dissipation is a multi-fractal set. Then, the PDF of the Elsa¨sser-variable gradient may be determined as follows. Let us first note, using (6.36) and (6.37), that the dissipative microscale g can be reexpressed as 8 1 > m aþ1 > > > ; d e g; < jsj ð6:44Þ g
1 > > a > m > : ; d e g: jsj Now, for a multi-fractal set, we have a continuous spectrum of scaling exponents a, with each a distributed on a set S (a) of Hausdorff dimension f (a). Assuming again that multi-fractal scaling in the inertial range, discussed in Section 6.2, can be extrapolated down to the dissipative microscale g, and using (6.40) and (6.44), we have for the PDF of the Elsa¨sser-variable gradient: 8 0 2 42a 4a2 31 Z 4af ðaÞ > > aþ1 m m aþ1 jsj aþ1 5A > > > exp @4 dlðaÞ; d e g; > > jsj 2hW20 i < P ðs; aÞ
0 2 42a 4a4 31 > Z 4af ðaÞ > > a m m a jsj a 5A > > exp @4 dlðaÞ; d e g: > > : jsj 2hW20 i
ð6:45Þ
Now, according to (6.37), 8 a2 < m2a1 ; d e g; jsj
a2 : 2a2 m ; de g so that the exponents in (6.45) are O (1), even for strong singularities, namely, a ) 0. This rules out, therefore, the use of the method of steepest descent to evaluate the integrals in (6.45) analytically. One has to resort to numerical computations, as in the case of hydrodynamic turbulence [66]. 6.3.4. Parabolic-profile model for f (a) Based on the general theoretical knowledge of the f (s) vs. a curve, one may consider the following parabolic profile for f (a) [49]:16 2 f ðaÞ ¼ aða a0 Þ2 ;
a and a0 > 0:
ð6:46Þ
Let us now consider the implications of (6.46) for the inertial and the dissipative microscale regimes.
16
Eq. (6.46) has been confirmed experimentally for hydrodynamic turbulence [49].
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B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
6.3.4.1. Inertial regime. Using (6.46), (6.15b) yields p a ðpÞ ¼ a0 : 2a Using (6.47), (6.15a) becomes 8 p2 > > < ða0 1Þp ; d e ‘; 4a fp ¼ 2 > > : ða0 2Þp p ; d e ‘: 4a Now, (5.8) implies that
1; d e ‘; f4 ¼ 1; d e ‘: Using (6.49), (6.48) leads to 8 5 1 > < þ ; d e ‘; a0 ¼ 4 a > : 7 þ 1 ; d ‘: e 4 a Using (6.50), (6.47) gives 8 5 1 > < þ ð2 pÞ; a ðpÞ ¼ 4 2a > : 7 þ 1 ð2 pÞ; 4 2a while (6.48) becomes 8 p ð4 pÞp > > ; < þ 4a fp ¼ 4 > p ð4 pÞp > : þ ; 4 4a
ð6:47Þ
ð6:48Þ
ð6:49Þ
ð6:50Þ
d e ‘; ð6:51Þ de ‘
d e ‘; ð6:52Þ d e ‘:
On comparing (6.52) with the multi-fractal result (6.19) for the inertial regime, we obtain p Dp=4 ¼ 2 8d e =‘ ð6:53Þ a which shows that the parabolic-profile model (6.46) mimics the log-normal model. On the other hand, (6.51) implies that 8 5 > < ; de ‘
a ðpÞ < 4 8p P 2 ð6:54Þ > : 7; d ‘ e 4 which shows that the velocity/magnetic field singularities are weaker in the collisionless MHD case than in the classical MHD case—this was confirmed in Section 6.3.2.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
19
Eqs. (6.50)–(6.53) consistently show that the zero-intermittency limit corresponds to the limit a ) 1. On the other hand, using (6.50) and 6.51, (6.46) yields 2 f ða Þ ¼
p2 ; 4a
ð6:55aÞ
so lim 2 f ða Þ ¼ 0
a)1
ð6:55bÞ
as to be expected! On the other hand, (6.53) implies D0 ¼ 2 which is also confirmed by (6.55a) that yields f ða ð0ÞÞ ¼ 2: f (a* (0)) being the fractal dimension of the support of the measure, namely, D0. Thus, in the parabolic-profile model given by (6.46) the multi-fractality manifests itself via the way the measure is distributed rather than the geometrical properties like the support of the set. 6.3.4.2. Dissipative microscale regime. Here, we sketch the development for only a collisional (de g) plasma. The results of Section 6.3.2 show that we do not have sufficient number of conditions to accomplish closure in the parabolic-profile model (6.46) for a collisionless (de g) plasma. Using (6.46), (6.32) now yields 2aa 2 2aa þ ð3p þ 2aa0 2aa20 Þ ¼ 0
ð6:56aÞ
from which
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 6p 2 a ðpÞ ¼ 1 ð2a0 1Þ : 2 a
ð6:56bÞ
Now, noting that A0 1
ð6:57Þ
we have from (6.31) and (6.46) that að0Þ ¼ a0 :
ð6:58Þ
Eq. (6.58) dictates immediately that we discard the negative root in (6.56b), and hence " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 6p 2 1 þ ð2a0 1Þ a ðpÞ ¼ : ð6:59Þ 2 a If we write Ap hRicp
ð6:60Þ
20
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
then we have from (6.31) and (6.32), cp ¼ 12½p f 0 ða Þ
ð6:61Þ
and using (6.46) cp ¼ 12½p þ 2afa ðpÞ a0 g:
ð6:62Þ
Now, Section 6.3.2 showed that the mean energy dissipation rate remains finite in the ideal limit m ) 0, for a collisional plasma (de g), i.e., c 1
mA2 hRi 2 constant which implies that c2 1 ¼ 0: Using (6.62) and (6.59), (6.64) yields 5 1 a0 ¼ þ 4 a
ð6:63Þ ð6:64Þ
ð6:65Þ
which is identical to the one, namely (6.50), that ensued consequent to the condition (6.49). This appears to indicate that the ideal dissipation of energy (6.63) has been incorporated into the exact-like result (6.49) in MHD in a manner similar to that for classical 3D hydrodynamic result [67]. Further, on using (6.65), (6.59) and (6.46) lead to: 5 9 1 a ðpÞ ¼ þ ð1 pÞ þ O 2 ; ð6:66Þ 4 a a 2 ð8 9pÞ 1 þO 2 : 2 f ða Þ ¼ a a
ð6:67Þ
Eqs. (6.65)–(6.67) consistently show that the zero-intermittency limit corresponds again to the limit a ) 1. 6.3.4.3. Probability distribution function. The multi-fractal model was seen in Section 6.3 not to afford an analytic calculation of PDF of the Elsa¨sser-variable gradient. The parabolic-profile model turns out to be fruitful on this aspect. Noting the scaling behavior of the Elsa¨sser-variable gradient given by (6.37), and assuming W0 to be Gaussian distributed, as per (6.39), we observe 4a 2
W20 W a þ1 : So, a* (p) corresponds to a ð~pÞ where ~ p is the solution of 4a ð~ pÞ 2 ~ : p¼ a ð~ pÞ þ 1
ð6:68Þ
ð6:69Þ
Using (6.69), and assuming the intermittency to be weak (i.e., a is large) to simplify the calculations, we have from (6.66), 5 3 1
a ð~ pÞ ¼ þ O 2 : ð6:70Þ 4 a a
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23
21
Using (6.70), the PDF of the Elsa¨sser-variable gradient given by (6.40) becomes
0 2 2 32 4 32 31 1þ16 3 9a m m 3þ9a jsj 39a 5A P ðs; a ð~ pÞÞ
exp @4 : ð6:71Þ jsj 2hW20 i Incidentally, using (6.70), (6.69) gives 4 32 1 ~ p ¼ þO 2 3 9a a
ð6:72Þ
which is of course the exponent of jsj in the argument of the exponential in (6.71). Note the accentuation of the non-Gaussianity of the PDF due to intermittency (a „ 1). It should be mentioned that the physical principle (namely, the ideal dissipation of energy) underlying the calculation of the intermittency correction to the PDF of the Elsa¨sser-variable gradient given above is also the same as that used previously in the homogeneous-fractal based result (6.43). Since the parabolic-profile model given by (6.46) incorporates the multi-fractal aspects only via the measure distribution arrangement the PDF (6.71) is complementary to the PDF (6.43) which incorporates multi-fractal aspects only via the geometrical properties like the support of the set.
7. Discussion In this paper, we have given a formulation of 2D collisionless MHD turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. Particular emphasis has been placed on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We have shown that the energy spectrum is shallower in the collisionless MHD regime which implies that the collisionless MHD flows are less dissipative than the classical MHD flows and leads to an energy pile-up at scales ‘n de in the energy cascade. This can cause creation of an ordered vortex/current filament phase similar to the one numerically shown in the hydrodynamic phase [56]. We have then investigated the fractal/multifractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k1 spectrum in the solar wind. Multi-fractal scaling behavior in the inertial range is first deduced, and is then extrapolated down to the dissipative microscale. Intermittency is found to reduce the degrees of freedom, and this result is a robust one in that it is identical even quantitatively for both the 2D classical MHD and collisionless MHD cases. The mean energy dissipation rate, in the ideal limit, remains finite in the classical MHD regime but vanishes in the collisionless MHD regime. The latter result is consistent with the fact that the collisionless MHD flows are less dissipative than the classical MHD flows, as mentioned above. Finally, spatial intermittency has been shown to accentuate non-Gaussian statistics at small scales in MHD turbulence even in the collisionless regime. Further, the PDF of the Elsa¨sser-variable gradient for the collision-
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less MHD (de g) regime has been found to be less non-Gaussian than the PDF for the classical MHD (de g) regime which is again consistent with the fact that the collisionless MHD flows are less dissipative. A parabolic-profile model for the singularity spectrum f (a) is considered as a special case of the multi-fractal scenario. This model affords not only considerable insight into qualitative aspects of the intermittency problem but also an analytical calculation of PDF of the Elsa¨sser-variable gradient.
Note added in proof For the linearized version of the system (3.1), for which the dispersion relation is x2 ð1 þ k 2 d 2e Þ ¼ k 2z C 2A ð1 þ k 2 q2s Þ the condition qs = de corresponds to a degenerate case wherein the electron-inertia and electron-pressure effects cancel one another. This appears to be related to the feasibility of Elsa¨sser-variable formulation for the nonlinear case when qs = de.
Acknowledgments Some of the work contained in this paper was done while I was in residence at the International Centre for Theoretical Physics, Trieste, Italy. I am very thankful to Professor K.R. Sreenivasan for the hospitality and helpful information. My thanks are due to the referee for his helpful remarks that led to an improvement in the presentation in this paper.
References [1] Ya.B. Zeldovich, A.A. Ruzmaikin, D.D. Sokoloff, Magnetic Fields in Astrophysics, Gordon and Breach, 1983. [2] S. Childress, A.D. Gilbert, Stretch, Twist and Fold: the Fast Dynamo, Springer Verlag, Berlin, 1995. [3] G.K. Batchelor, Proc. Roy. Soc. (London) A 201 (1950) 405. [4] L. Biermann, A. Schlu¨ter, Phys. Rev. 82 (1951) 863. [5] H.P. Pao, Phys. Fluids 6 (1963) 632. [6] P.G. Saffman, J. Fluid Mech. 18 (1964) 449. [7] A. Tsinober, G. Galanti, Phys. Fluids 15 (2003) 3514. [8] K. Ohkitani, Phys. Rev. E 65 (2002) 046304. [9] S. Kida, S. Yanase, J. Mizushima, Phys. Fluids A 3 (1991) 457. [10] H. Politano, A. Pouquet, P.L. Sulem, Phys. Fluids B 1 (1989) 2330. [11] T. Gomez, H. Politano, A. Pouquet, Phys. Fluids 11 (1999) 2298. [12] D. Biskamp, H. Welter, Phys. Fluids B 1 (1989) 1964. [13] D. Biskamp, H. Welter, Phys. Fluids B 2 (1990) 1787. [14] D. Biskamp, H. Welter, M. Walter, Phys. Fluids B 2 (1990) 3024. [15] R. Kinney, J.C. McWilliams, T. Tajima, Phys. Plasmas 2 (1995) 3623. [16] R. Grauer, C. Marliani, Phys. Plasmas 2 (1995) 41. [17] B.K. Shivamoggi, Ann. Phys. 272 (1999) 167. [18] I.S. Iroshnikov, Sov. Astron. 7 (1964) 568.
B.K. Shivamoggi / Annals of Physics 317 (2005) 1–23 [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67]
23
R.H. Kraichnan, Phys. Fluids 8 (1965) 1385. A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 30 (1941) 301. A. Pouquet, U. Frisch, J. Leorat, J. Fluid Mech. 77 (1976) 321. P. Goldreich, R. Sridhar, Astrophys. J. 438 (1995) 763. W. Muller, D. Biskamp, Phys. Rev. Lett. 84 (2000) 475. R. Sridhar, P. Goldreich, Astrophys. J. 432 (1994) 612. D. Montgomery, W.H. Matthaeus, Astrophys. J. 447 (1995) 706. C.S. Ng, A. Bhattacharjee, Astrophys. J. 465 (1996) 845. S.V. Nazarenko, A.C. Newell, S. Galtier, Physica D 152–153 (2001) 646. P. Goldreich, R. Sridhar, Astrophys. J. 485 (1997) 680. B.K. Shivamoggi, Ann. Phys. 253 (1997) 239; Erratum in, Ann. Phys. 312 (2004) 270. B. Mandelbrot, in: R. Temam (Ed.), Turbulence and Navier–Stokes Equations, Lecture Notes in Mathematics, vol. 565, Springer-Verlag, Berlin, 1975. U. Frisch, P.L. Sulem, M. Nelkin, J. Fluid Mech. 87 (1978) 719. U. Frisch, G. Parisi, in: M. Ghil, R. Benzi, G. Parisi (Eds.), Turbulence and Predictability in Geophysical Fluid Dynamics and Climatic Dynamics, North-Holland, Amsterdam, 1985. B. Coppi, Phys. Lett. A 11 (1964) 226. K. Schindler, J. Geophys. Res. 70 (1974) 2803. J.F. Drake, Y.C. Lee, Phys. Fluids 20 (1977) 134. J.F. Drake, R.G. Kleva, Phys. Rev. Lett. 66 (1991) 1458. R.G. Kleva, J.F. Drake, F.L. Waelbroeck, Phys. Plasmas 2 (1995) 23. J.F. Drake, in: P. Song, B.U.O. Sonnerup, M.F. Thomsen (Eds.), Physics of the Magnetopause, Geophys. Monogr. Ser., Vol. 90, Amer. Geophys. Union, Washington, DC, 1995. A.A. Galeev, Phys. Fluids 21 (1978) 1353. J. Wesson, Nucl. Fusion 30 (1990) 2545. F. Porcelli, Phys. Rev. Lett. 66 (1991) 425. M. Ottaviani, F. Porcelli, Phys. Rev. Lett. 71 (1993) 3802. B.K. Shivamoggi, J. Plasma Phys. 58 (1997) 329. B.K. Shivamoggi, Phys. Lett. A 292 (2001) 134. N. Al-Salti, B.K. Shivamoggi, Phys. Plasmas 10 (2003) 4271. A.Y. Aydemir, Phys. Fluids B 4 (1992) 3469. L.E. Zakharov, B. Rogers, S. Migliulo, Phys. Fluids B 5 (1993) 2498. E. Cafaro, D. Grasso, F. Pegoraro, F. Porcelli, A. Saluzzi, Phys. Rev. Lett. 80 (1998) 4430. C. Meneveau, K.R. Sreenivasan, Nucl. Phys. B 2 (1987) 49. J.W. Armstrong, B.J. Rickett, S.R. Spangler, Astrophys. J. 443 (1995) 209. W.M. Elsa¨sser, Phys. Rev. 79 (1950) 183. W.H. Matthaeus, P. Dmitruk, L.J. Milano, Phys. Plasmas 9 (2002) 2440. H. Politano, A. Pouquet, V. Carbone, Europhys. Lett. 43 (1998) 516. D. Biskamp, E. Schwarz, Phys. Plasmas 8 (2001) 3282. K. Nakayama, Astrophys. J. 556 (2001) 1027. N. Kukharin, S.A. Orszag, V. Yakhot, Phys. Rev. Lett. 75 (1995) 2486. H.P. Furth, J. Killeen, M.N. Rosenbluth, Phys. Fluids 6 (1963) 459. H. Politano, A. Pouquet, P.L. Sulem, Phys. Plasmas 2 (1995) 2931. E. Marsch, C.Y. Tu, J. Geophys. Res. 95 (1990) 8211. M.L. Goldstein, D.A. Roberts, W.H. Matthaeus, Annu. Rev. Astron. Astrophys. 33 (1995) 283. T.C. Halsey, M.H. Jensen, L. Kadanoff, I. Proccacia, B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82. M. Nelkin, Phys. Rev. A 42 (1990) 7226. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959. B.K. Shivamoggi, Ann. Phys. 270 (1998) 263. R. Benzi, L. Biferale, G. Paladin, A. Vulpiani, M. Vergassola, Phys. Rev. Lett. 67 (1991) 2299. A.N. Kolmogorov, Dokl. Akad. Nauk SSR 32 (1941) 16.