Mem. S.A.It. Vol. 83, 594 c SAIt 2012 Memorie della

Angular momentum transport in CV disks

O. Regev1,2 and O. Umurhan3,4

1 Dept. of , Technion - Israel Institute of Technology, Haifa 32000, Israel 2 Dept. of Astronomy, Columbia University, New York, NY 10027 e-mail: [email protected] 3 Dept. of Applied Mathematics, University of California Merced, Merced, CA 95343 4 Dept. of Astronomy, City College of SF, SanFrancisco, CA 94112 e-mail: [email protected]

Abstract. A physical mechanism for enhanced angular momentum transport (AMT) out- wards is needed for the theoretical modeling of accretion disks (ADs) in e.g. Cataclysmic Variables (CVs). It is clear that ordinary microscopic is out of the question - the time scale of transport it dictates is many orders of magnitude longer than the one deduced from mass accretion rate consistent with observations. It is however reasonable that tur- bulent transport can provide the needed timescale value. Astrophysicists have persistently looked for an linear instability in ADs, which can ultimately lead to turbulence. In this con- tribution we dispute the ”common knowledge”, accepted during the last two decades or so, that the magneto-rotational instability (MRI) is the physical agent that destabilizes thin ADs and ultimately drives turbulence and AMT. We also suggest, thus, that we do not yet have at our disposal any better model for AMT than the30 year old α model

Key words. Binary stars: accretion disks – MHD instabilities: MRI – Transport: angular momentum

1. Introduction ity. In doing so, thin AD, for the first time, could be quantitatively modeled in a variety of Prendergast & Burbidge (1968) were the first astrophysical settings. In turn, this has brought to propose the existence of thin ADs, to explain about significant progress in our understanding some galactic X-ray sources. Similar objects of these important systems. The value and na- were also proposed in CVs. Greatly enhanced ture of the non-dimensional viscosity param- (turbulent) viscosity was invoked for making eter α (see below) still remains to be a the- the model viable. A few years later Shakura oretical unknown and phenomenological ap- & Sunayev (1973) and Lynden-Bell & Pringle proaches have usually been employed to deter- (1974) successfully bypassed the specific lack mine it. of understanding of turbulence (a situation that is largely still with us) by employing physically No definitive candidate hydrodynamical motivated parametrization of turbulent viscos- instability had been convincingly identified in ADs and widely accepted as a possible Send offprint requests to: O. Umurhan initial driver of turbulence before (and af- Regev: Angular momentum transport 595 ter...) Balbus & Hawley (1991) found that ployed, for example, by Hawley et al. (1995) weak magnetic fields destabilize differential (H95), but the very recent calculations of Lesur Keplerian rotation. The linear MRI (Velikhov & Longaretti (2007) included explicit viscosity 1959, Chandrasekhar 1960), was shown to op- ν and resistivity η in a an SB high-resolution erate under conditions characterizing rotation- numerical simulation. Systematically changing ally supported magnetized ”cylindrical accre- the Reynolds number (R) and the magnetic tion disks”. Prandtl number mPr ≡ ν/η, in some range, This suggestion has inspired intensive re- these authors aimed at uncovering the trends search activity on the linear MRI and its non- and scalings of relevant physical properties, linear development. It has been investigated for notably angular momentum transport, with the various conditions, geometries, and boundary relevant non-dimensional number(s). conditions (BC). Beyond the linear analysis it The most recent calculations in the SB has been approached, almost exclusively, by framework with relatively high resolution have methods of computational (magneto)fluid dy- been reported by Fromang & Papaloizou namics. The complexity of the nonlinear devel- (2007) (FP) and by Fromang et al. (2007) opment became readily apparent quickly ap- These studies (both with PBC) focused on the parent and, as a consequence some understand- issue of convergence (i.e., numerical resolu- ing of the nonlinear transition and saturation tion) in ideal SB calculations and on (R) and processes, is indispensable, if the aim is to mPr dependencies of the transport in dissipa- improve on the more than 30 years old phe- tive conditions. nomenological α-model. At the same time King, Pringle & Livio The purpose of this contribution is to as- (2007) (KPL), pointed out the discrepancy (of sess the viability of the most common approxi- at least one order of magnitude) between the mation used in numerical studies - the shear- values of α inferred from observations and the ing box (SB). The review articles by Balbus estimates of this parameter based on (mainly & Hawley (1998) and Balbus (2003) contain the early) SB numerical simulations, suggest- a comprehensive list of references, of which ing that the effective α decreases (almost lin- only few will be referred to here. We shall also early) with the value of the imposed Bz. They present, in the last section, a semi-analytical conjectured that perhaps only zero net flux (ZF) study of the saturation of the MRI, in an exper- calculations should be used. imental simplified setup, which has been used in experimental efforts to observe the MRI and its nonlinear development in the laboratory 2.1. The problematics of the SB (e.g. Liu, Goodman & Ji 2006 and references approximation therein). The essence of the SB approximation is in its local approach, that is, the resulting equa- 2. Numerical calculations of the MRI tions for the perturbations on a steady base nonlinear development flow are approximately valid in a small re- gion (a Cartesian box) about a typical point in Most of the existing SB calculations (at least the disk. The global MHD base flow in an al- of the early ones) employ a base state with most Keplerian is not only ro- a constant vertical magnetic field and numer- tating and strongly sheared, but it is also inho- ically follow the development of perturbations mogeneous, non-isotropic (endowed with non- on this state. This situation, usually referred to zero gradients in density and other physical as fixed net flux (FF), seems to be the most ap- variables and these gradients have very differ- propriate for disks threaded by external mag- ent scales in different directions) and swirling netic fields . The majority of existing numerical (streamlines are curved). works of this sort have used ideal (that is, non- Umurhan & Regev (2004) followed a sys- dissipative) MHD SB equations with periodic tematic derivation of the approximation us- boundary conditions (PBC) in the manner em- ing asymptotic scaling arguments with the pur- 596 Regev: Angular momentum transport pose of quantifying the approximations that length and the integral scale, denote them as ` are made leading to the SB (see Appendix A and refer to them interchangeably. This identi- of that paper). Two defining nondimensional fication has an obvious intuitive physical basis. numbers were stressed in their study-  (the In a disk, which is strongly non-isotropic, typical disk height h0, in units of r0, atypical two disparate injection scales seem a priori to radius) and δ (the box size in the same units). appear - the horizontal one `h∼< r0 and the verti- For details see that paper. cal one `h∼< r0 (both given here in dimensional We stress that virtually all numerical cal- units). It is however reasonable to identify ` culations that utilize the SB approximation, in with the smaller of the two, because this is the all its variants, employ PBC in which the pe- size of largest eddies. A quantitative measure riodicity in the radial direction is sheared (see of ` can be ”phenomenologically” determined H95). These sheared-periodic-BC are equiva- by the relevant wave-number at which the dis- lent to enforcing that all perturbation quanti- turbance kinetic energy Ek(k) spectrum peaks. ties to be triply (or in the case of a 2-D calcu- The fact that the scale at which resistivity and lation, doubly) periodic in the sheared coordi- viscosity come into play need not be equal nates system. It is also important that our cri- makes also the determination of the dissipation tique of the SB scheme applies to both FF and scale not unique. Using the Spitzer values we ZF conditions. get typically mPr  1 ( = 6 × 10−29T 4/ρ ∼ 5 × 10−5 c.g.s, for CV disk conditions), i.e., kν > kη where kν and kη respectively represent 2.1.1. Relevant length scales the dissipation scale wave-numbers for viscos- ity and resistivity. Phenomenologically, the in- In a typical accretion disk, r0 is the scale on which curvature terms appear and this is ertial range is considered to be the k region, also the scale of any underlying radial struc- in which Ek(k) exhibits a power-law behav- ture gradients. Vertical stratification obviously ior. Thus, the inertial range in the MHD tur- appears on the vertical pressure scale-height, bulence of a disk can be considered as includ- ing wave-numbers roughly in the range −1  which is  = h0/r0  1 in our notation and units. In studies of turbulence it is customary k  kη. For a fairly detailed discussion of the −1 MHD turbulence length scales and relevant ref- to consider the injection scale, ` ≡ kin , on which the process of energy injection into the erences see BI. In any case, possible nonlinear system is effected and the dissipation scale, interactions between the short vertical wave- −1 length unstable modes with modes (possibly ≡ kd , on which microscopic viscosity (and in MHD also resistivity) are operative. These also horizontal ones) whose scale is larger than are linked by the inertial range of scales. Since the SB scale cannot be captured in a SB sim- ulation. It is conceivable that such interactions kin  kd (at least in astrophysical systems), the inertial range extends over many orders may play a role in the instability saturation and of magnitude. In MHD turbulence the energy the development of activity. cascades in 2D and 3D are both direct, i.e., from small wave-numbers to large ones, see, 2.1.2. Numerical resolution and its e.g., Biskamp (2003) (BI), and consequently relationship to dissipation the injection scale can be roughly identified with the system’s structural scale. In the classi- Astrophysical fluid systems and their magneto- cal Kolmogorov theory the scale of the largest fluid counterparts (like an accretion disk flow) eddies, which contain most of the energy, is are endowed with extremely large (R) and mag- referred to as the integral scale. This is also netic Reynolds (Rm = R × mPr) numbers and the length scale appearing in the definition of therefore numerical simulations cannot yet re- R and serves as as a measure for the scale over solve the full dynamical range of such turbu- which the turbulent fluctuations are correlated. lent flows. In other words, given the present To avoid complications we shall practically magnitude of computing power, numerical res- identify the integral scale with the correlation olution of the full spatial range of these sys- Regev: Angular momentum transport 597 tems, from the system’s energy injection scale All existing SB simulations nominally be- through the full inertial scale and down to the long to the first class, however the great dispar- dissipation scale, is still out of reach. ity between the smallest resolved scale lnum and the true dissipation scales prevent them from It has been argued in the past (e.g., BH98) being regarded as true DNS. Except for the old that perhaps a full dynamic range is actually α prescription for MRI mediated or induced not needed in the accretion disk problem, be- MHD turbulence, there is as yet no thoroughly cause the scale of the most unstable linear tested sub-grid model as might be employed MRI modes is large and in the saturated tur- in a LES (or one using some other turbulence bulent state, most of the energy resides in, model) calculation. and the angular momentum transport is done Given the doubt discussed above that nu- by, the large scale eddies. However, this state- merical dissipation by itself is sufficient in an ment is strongly contested, even for purely hy- LES calculation, it seems to be imperative to drodynamical flows. For an extensive, recent, look for a sub-grid model that is appropriate discussion on the use of unresolved hydrody- for the problem at hand. Investigating MRI in- namical codes for the study of turbulence, see duced or mediated turbulence in local DNS Grinstein, Margolin & Rider (2007). In any simulations may certainly offer in-roads into a case, the conjecture appears to be clearly un- better understanding of its local salient physi- founded for MHD turbulence with very large cal features which includes, e.g., some charac- R. Similar general conclusions also appear in terization of the emerged turbulence in terms BI. Thus, it appears that a viable estimate of α of eddy spectra and correlations. Information in a MHD turbulent disk, cannot be obtained of this sort can provide the ingredients for the from poorly resolved SB simulations. In ad- needed “turbulence model” which would go dition, it is even difficult to quantitatively as- into a faithful global accretion disk LES (or sess the energetics of MRI driven turbulence a simulation employing other turbulent flow from such simulations. The numerical investi- computational schemes). Since local simula- gations of FP and others actually show, for the tions are needed to infer the turbulent behavior FF case that increasing the resolution of simu- down to the dissipation scale, rmpC with phys- lations relying on numerical dissipation shows ically definite boundary conditions seem to be a decreasing trend in the calculated transport, not less appropriate than SB (whose periodic- suggesting that this is indeed the case. BC introduce additional difficulties and in- We may say that systems including turbu- consistencies, see below) for this purpose. Of lent thin accretion disks can be approached nu- course, such simulations must be fully resolved merically in roughly three ways: (DNS) and pushed to the maximum Re and Rm (a) Local ideal or/and dissipative calcula- values in order for them to yield this kind of tions.These are a DNS (direct numerical information. simulation)-like, maximally resolved, calcula- tions, ignoring any scale interaction and gradi- 2.1.3. Symmetry and boundary ent or boundary effects. conditions (b) Global calculations employing some physically motivated sub-grid scale turbu- The SB approximation with PBC is endowed lence model. In principle, these LES (large with a particular symmetry: it allows for shear- eddy simulations)-like calculations allow con- wise (in the Cartesian geometry of the box it fronting accretion disk models with relevant corresponds to the x-direction) invariant solu- observations (see Bisikalo’s contribution in tions. these proceedings). The x-invariance symmetry, which allows (c) Global calculations, albeit with unrealisti- for the existence of special solutions (channel cally low Reynolds numbers, - in the purpose modes) in the SB, is obviously broken by any of experimenting with the trends of the dynam- non-periodic-BC on x, even in local SB anal- ics with increasing R and/or Rm. yses and this is effected on the box scale. The 598 Regev: Angular momentum transport

78 L = π, L = 2π SB scale, appropriate for thin accretion disks 76 x y 2 satisfies  · r0 < L <  · r0. Radial symmetry 74 breaking by the above mentioned global phys- 72 ical effects are manifested on a length scale of 70 pBC 68 the order r0, but non-PBC are operative on the 66 smaller box scale. 64 We finally turn to the most important obser- 62 wBC vation about the use of PBC in numerical stud- L = 2π, L = 2π x y ies of turbulence and its bearing on SB simu- 255 lations. Clearly, space PBC are not accessible in realistic physical situations, and they only Domain Integrated Energy 250 pBC can be faithfully employed in the study of ho- 245 mogeneous turbulence when one assumes that real boundary effects are not important (Dubois 240 wBC

235 et al.1999). The caveats pertaining to the use 0 5 10 15 20 25 of periodic-BC in numerical simulations of time turbulence are discussed in detail in §7.2 of Davidson (2004) from which we only bring Fig. 1. The total energy (in arbitrary units) in the SB as a function of time (in units of Ω−1). In the up- here issues that are explicitly relevant to SB 0 per panel the size of the SB is Lx = π and Ly = 2π simulations of MHD turbulence in an accre- and the result of the calculation with PBC is com- tion disk. In order for the bulk of the turbulence pared with the one using WBC. In the lower panel not to be seriously affected by the imposed pe- the same is shown for a SB having a double shear- riodicity, the box size should satisfy L  `, wise extent. In both cases the simulations are started otherwise the ”tails” of the relevant correlation with the same initial conditions functions (of the turbulent fluctuations) are not only cut off, but also very significantly lifted (see figure 7.6 of Davidson, 2004). This cru- the domain. A particularly useful consideration cially affects the behavior of the large scale dy- of this kind, in this problem, arises from the ex- namics. If we accept the reasonable conjecture amination of the energy budget for the system that the injection scale relevant for an accre- of equations We have performed a number of numerical tion disk is indeed ` = `v∼< h0 = r0, the min- imal size of the SB must be the disk height. experiments in the purpose of examining the But then z periodic-BC cannot be used and we effects of the kind of BC employed (periodic are only left with the option of semi-global or versus wall) and of the SB size (relatively to a global simulations. In the next Subsection we relevant integral scale ` of the problem) on the shall demonstrate, with the help of a numerical evolution of the SB energy content. We have experiments, that using a SB, whose size is not chosen to focus on the total integrated energy large enough (as compared to the scale of the of the domain, as this quantity has (at least to relevant dynamical structures), and PBC, gives us) the most definite and best understood phys- rise to spurious energy fluctuations, casting a ical meaning. serious doubt on the physical significance of We shall consider the evolution of the total the simulation results. domain energy for two sets of boundary condi- tions. The first of these are the PBC, as laid out before in the paper. The second type of BC are 2.1.4. Energetics of the SB appropriate for a rotating channel and we refer In any continuum mechanics problem posed in to them as WBC They implement at the shear- a finite domain, as is the case for SB MHD nu- wise boundaries i) no normal flow ii) no nor- merical calculations of the kind discussed here, mal magnetic field flux and iii) free-slip bound- it is generally advantageous to consider inte- ary conditions. i.e. gral physical conservation statements, valid on u = 0, bx = 0, ∂xvy = 0, at x = ±Lx/2. (1) Regev: Angular momentum transport 599

WBC are periodic in the azimuthal direction. The end result of the rather lengthy asymp- Note that the equations solved are identical for totic procedure procedure. see Umurhan et both sets of BC. al. 2007b (URM), is the well-known real The description of the results can be seen Ginzburg-Landau Equation (rGLE) which, for in the figure caption. More details and the rPm  1, is whole contentent of Section 2 (which is rather schematic due to the lack of space) appear in 1 2 2 ∂T A = λA − A|A| + D∂ A. (2) Regev & Umurhan (2008). rPmC Z

Here A ia proportional to the amplitude of the 3. Saturation of the MRI perturbation, while the other symbols are con- We have also performed a weakly nonlinear stants related to the physical parameter values, analysis of the MRI near threshold for a rotat- see Umurhan et al. 2007a (UMR). ing flow in channel geometry, subject to ide- The key results of this idealized analysis alized but mathematically expedient boundary are the saturation valuse, conditions. This kind of approach is impor- √ tant because the viability of this linear insta- A → As ∼ rPm C. (3) bility as the driver of turbulence and angular- momentum transport relies on understanding This gives a fixed energy for the region in ques- its nonlinear development and saturation. By tion and the following scaling of the AMT complementing the above mentioned simula- tions, analytical methods remain useful to gain J˙ ∼ R−1, (4) further physical insight. To facilitate an ana- lytical approach we make a number of sim- plifying assumptions so as to make the sys- for rPm  1. tem amenable to well-known methods (e.g. We have also performed numerical calcu- Cross & Hohenberg, 1993) for the deriva- lations, using a 2-D spectral code to solve tion of nonlinear envelope equations (i.e. with the original nonlinear equations, in the stream- an amplitude weakly dependent on time and function and magnetic flux function formula- space). Additionally we assume a narrow chan- tion, near MRI threshold. The asymptotic the- nel geometry, i.e. the small-gap limit of Taylor- ory reflects the trends seen in these simula- Couette flow. tions. The hydromagnetic equations in cylindri- The numerical and asymptotic solutions cal rotating coordinates are applied to the developed here show that in the saturated state neighborhood of a representative point in the the second order azimuthal velocity perturba- system, the base flow is steady and incom- tion becomes dominant over all other quanti- pressible with constant pressure. The veloc- ties for rPm  1 and appears to be the primary ity V = U(x)yˆ has a linear shear profile agent in the nonlinear saturation of the MRI U(x) = −qU0 x, representing an axisymmet- in the channel. It acts anisotropically so as to ric flow about a point r0, that rotates with a modify the shear profile and results in a non- rate Ω0 defined from the differential rotation diagonal stress component (relevant for radial −q law Ω(r) ∝ Ω0(r/r0) . We restrict ourselves angular momentum transport), which behaves to a constant initial magnetic field B = B0zˆ. like ∼ 1/R. For a detailed exposition of this Cartesian coordinates x, y, z are used to repre- asymptotic work which shows that the MRI sent the radial (shear-wise), azimuthal (stream- saturates asymtotically for low mPR numbers wise) and vertical directions, respectively. The see UMR and URM. base flow is axisymmetricaly disturbed by per- This analysis is complementary to that of turbations of the velocity, u = (ux, uy, uz), mag- Knobloch and Julien (2005) who have reported netic field, b = (bx, by, bz), and total pressure, the results of an asymptotic MRI analysis for a $. developed state far from marginality. 600 Regev: Angular momentum transport

0 (a) −2 (b) 10 10

−1 10 −3 10 3 R = 5x10 R = 104 −2 4 10 R = 10 R = 5x104 −4 10 R = 5x104 −3 10 : total energy V

E 3 5 −5 −4 R = 5x10 R = 2x10 10 10 . J: momentum transport

5 −5 −6 R = 2x10 10 10 0 1 2 3 0 2 10 10 10 10 10 10 Time (in box orbits) Time (in box orbits)

Fig. 2. EV [panel(a)] and J˙ [panel(b)] as a function of time from numerical calculation (solid lines) for Reynolds numbers R = 5 × 103, 104, 5 × 104, 2 × 105. The diamonds in panel (b) show the scaling predicted by our asymptotic analysis, which also predicts a constant final value of the disturbance energy, as apparent in panel (a). Given that J˙ > 0, it means that the angular momentum transport is outwards (positive x direction). One ”box” orbit =2π in non-dimensional units

4. Summary boundaries of the system (Dubois et al. 1999; Davidson, 2004). Based on the arguments of the work described It seems to us that the logical approach to here and in Section 2 we have no choice but the accretion disk angular momentum transport to conclude that decades of numerical simula- problem should thus consist of tions of the ”nonlinear development” resulting from the MRI in disks has not brought us new 1. Mathematically and physically sound cal- and significant knowledge on AMT in accre- culations, most likely in the mrpC (mag- tion disks as are though to exist in CVs. netic rotating plane Couette) or mTC (mag- Also laboratory experiments have so far netic Taylor Couette) thin-gap setups, in not yielded any conclusive indications of MRI the purpose of learning about the detailed driven turbulence, save for the case in which local properties of MHD turbulence (its en- relatively strong azimuthal fields are applied In ergy spectrum, correlations etc.) thin accretion disks, however, strong toroidal 2. Devising a turbulence model, based on the (or any other, for that matter) magnetic fields above, so as to replace the α model. cause significant magnetic pressure and are 3. Global calculations employing the above likely to be inappropriate for accretion disks in turbulence model(s) and various boundary the thin, rotationally supported, disk limit. conditions, giving rise to accretion disk models which can be confronted with ob- It is not easy to answer the question what servations. kind of numerical scheme or which setup is best suited for the problem. In any case, it The problem of angular momentum trans- is not clear what is less appropriate for local port in accretion disks, especially if MHD tur- numerical studies of accretion disks: WBC or bulence in a non-isotropic strongly sheared PBC in the SB. We believe that the use of medium is the physical mechanism responsible PBC, especially in the radial direction, suffers for it, is extremely difficult and involved. This from serious problems unless the power in the work, as well as that of KPL and others, indi- perturbation fields is localized away from the cates that the understanding gained so far from Regev: Angular momentum transport 601

SB and other numerical simulations seems un- Davidson P. A. 2004, Turbulence: An fortunately not to be adequate enough to pro- Introduction (Oxford University Press, vide a viable physical prescription that can be Oxford) effectively used in modeling accretion disks, Dubois, T., Jarbertau, F. & Temam, R. 1999, beyond the old α-viscosity parametrization. Dynamic Multilevel Methods in Numerical We think that only an extensive and coher- Simulation of Turbulence (Cambridge ent program, taking also into account recent University Press, Cambridge) developments and ideas of transient growth, Grinstein, F. F., Margolin, R. G. & Rider, W. may shed a new light on the difficult problem J. 2007, Implicit Large Eddy Simulation: of turbulence in AD. The lesson to be learned Computing Turbulent from that failure of the MRI driven turbulence (Cambridge University Press - Cambridge) idea, is, in our opinion, that the study has to Fromang, D. & Papaloizou, J. 2007, A&A, be conducted in a collaborative and completely 476, 1113 (FP) sincere and open manner - incorporating and Fromang, D., Papaloizou, J., Lesur, G. & taking account the efforts of different groups Heinemann T. 2007, A&A, 476, 1123 and approaches. Hawley, J. F., Gammie, C. F. & Balbus, S. A. 1995, Ap.J. 440, 742 (H95) King, A. R., Pringle, J. E. & Livio, M. 2007, 5. Discussion MNRAS, 325, L1 (KPL) DIMITRI BISIKALO: In purely hydrody- Knobloch E. & Julien, K. 2005, Phys. Fluids, namic models there were suggestions of tran- 17, 094106 sient growth. But it is bounded. Liu, W., Goodman, J. & Ji, H. 2006, ApJ, 643, ODED REGEV: Not much specific is known. 306 It is hoped that some finite perturbations may Lesur, G. & Longaretti, P.-Y. 2007, MNRAS, grow to sizes, in which nonlinear interaction 378, 1471 between them may play a role. There is no def- Lynden-Bell, D. & Prigle, J. E. 1974, MNRAS, inite model. Just an idea 168, 603 Prendergast, K. H. & Burbidge, G. R. 1968, ApJ, 151, L83 References Regev, O. & Umurhan O. M. 2008, A&A 481, 21 Balbus, S. A. 2003, ARA&A, 41, 555 Shakura, N. I. & Sunyaev, R. A. 1973, A&A, Balbus S. A. & Hawley, J. F. 1991, ApJ, 376, 24, 337 214 Umurhan, O. M. & Regev O. 2004, A&A, 427, Balbus S. A. & Hawley, J. F. 1991, Rev. Mod. 855 (UR) Phys., 70, 1 Umurhan, O. M., Menou, K. & Regev O. Biskamp, D. 2003, Magnetohydrodynamic 2007a, RRL, 98. 034501 (UMR) Turbulence (Cambridge University Press, Umurhan, O. M., Regev O. & Menou, K. Cambridge) (BI) 2007b, PRE, 98. 034501 (URM) Chandrasekhar, S. 1960, Proc. Natl. Acad. Sci, Velikhov E. P., 1959, J. Exp. Theor. Phys. 46, 53 USSR, 36, 1398