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Copy for friends not for distributionEffective stress tensor for accretion around compact astrophysical objects. . .PACS numbers:T. Mishonov, ∗ Y. Maneva, † and T. Hristov ‡Department of Theoretical Physics, Faculty of Physics,University of Sofia St. Kliment Ohridski,5 J. Bourchier Boulevard, BG-1164 Sofia, Bulgaria(Dated: June 17, 2005)I. INTRODUCTIONMatter accreting onto a compact object redistributes radially its angular momentum, dissipates gravitational energyand forms a powerful radiation source through physical processes that still cannot be regarded as understood.Accreting plasma’s temperature and pressure predict molecular viscosity that is orders of magnitude too small toaccount for the observed radiation intensity. Thus the origin of a large effective viscosity in the magnetized plasmaof the accretion discs is a significant and yet unsolved problem.Frictional release of angular momentum is believed to be an important element in star formation as well. In our solarsystem, for instance, the large mass concentrated into the sun carries only two percents of the angular momentum,while all the rest is associated with the planetary motion. Next to the Galileo assertion “Eppur si muove” (and yet itdoes move), for half a century we still face the question “why does the Sun not move (rotate) faster”. Presumably, atan early stage of the system formation the proto-planetary disc has acted as a brake which slowed down the rotationof most of the disc’s mass, so it could cluster and “ignite” as a star. Broadly, such a mechanism of angular momentumrelease appears responsible for the formation of compact astrophysical objects and possibly determines the universein the way that we observe it today. In this context, a central open question still exists of what physical processproduce the friction forces that facilitate the stars formation.The purpose of this work is to demonstrate a strong energy dissipation in a shear flow of magnetized plasma basedon a simple model. Despite the process’ interdisciplinary nature, we will pursue a description that is built on firstprinciplesphysics. Our leading hypothesis is that in a shear flow of the almost inviscid plasma the Alfven waves getamplified and later have their energy thermalized, i.e. “lasing” of Alfven waves is the basis of dissipation in accretiondiscs. We select an approach which is statistical rather than fluid-mechanical, by employing a kinetic equation forthe spectral density (proportional to the square of their amplitude) of the Alfven waves. In the spirit of the quantummechanics the square of the amplitude is proportional to the number of particles, which we will call “alfvons”. Herethe kinetic equation describes the dynamics of the alfvons population. The lasing of the media is analogous to thedynamics of ecological systems where a fast population growth is followed by a resettlement and high death-rate.Similarly here, the energy transferred from the shear flow first increases the alfvons density and is later dissipated,thus effectively raising the plasma’s viscosity and resistivity. Our goal is to deduce the kinetic equation for the spectraldensity of the Alfven waves as well as its solution, and analyze the physical factors driving the process. Standardmethods of quantum mechanics (quasi-classical approximation for short waves and perturbation theory to account forthe small initial viscosity of the plasma), will be considered as adequate. As an initial step, we employ a simplifiedtreatment of the turbulence in accretion discs — the Burgers approach and show that the turbulence’s role in a modelapproximation may be reduced to the source term in the kinetic equation. We also assume that it is the turbulencethat triggers the Alfven waves, which are later amplified by the shear flow. There are many examples where wavescan be amplified but often conditionally one can mention lasers and in general case lasing processes in some unstablemedium.The considered model problem involves convective instability and turbulence in the heated disc plasma and describesa self-sustained scenario for heating in accretion discs. The velocity fluctuations serve as random forcing that initiatesthe Alfven waves, which are then subjected to a large amplification. The energy carried by these alfvons is thentransformed to heat trough molecular viscosity and Ohmic resistivity, thus also creating friction forces. As the heatgeneration is more intense in the middle of the disk’s thickness, the consequent temperature difference drives theconvective instability and the turbulent convection. Thus the process sustains itself and the gravitational energytransforms into heat.


2II.WAVE KINEMATICS IN SHEAR FLOWSTo describe locally the motion of the accreting fluid, we choose the z axis along the velocity and the x axis indirection of the velocity’s gradient. Thus for the velocity field we haveU (0)z = Ax, U (0)x = U (0)y = 0. (1)The superscript (0) refers to an equilibrium laminar shear flow whose perturbations will be studied.A selected small element of the fluid, is carried by the flow so that its z coordinate is a linear function of timez at (t) = Axt + z at (0). (2)Under such drift the coordinates x, y remain unchanged, i.e. x at (t) = const, y at (t) = const. For hydrodynamicaldescription we will use flow-following (Lagrangian) coordinates˜x = x, ỹ = y, ˜z = z − Axt. (3)These tilde variables ˜r are the Cartesian coordinates of the “tagged” atoms at the initial moment z at (0) = z at (t)−Axt,they express the initial position of the fluid element. The change of the variables (3), however, does not alter theprojection in the xy plane. As the initial position of the “tagged” atoms is fixed we can consider the tilde coordinatesas “frozen” in the fluid.Let us now consider a plane wave in the shear flow with amplitude ∝ exp(ik·r). The requirement of phase invariancesets the transformation law˜k· ˜r = k · r (4)˜k x = k x + Atk z , ˜k y = k y , ˜k z = k z , (5)which can be validated by a substitution of (3) and (5) into (4).As the initial tilde coordinates are related to the frozen initial position of the atoms the wave vector in tildecoordinates is also “frozen”, ˜k = const. This determines the evolution of the wave-vector in Cartesian coordinatesk x (t) = ˜k x − At˜k z , k y = ˜k y , k z = ˜k z . (6)This time dependence of the wave vector k(t) has purely kinematic origin and it is nor related to the dispersion ofthe waves. Such phenomenon is well-known in the acoustics of moving media, but it can affect even non-propagatingspatial structures with “tagged” atoms. Even in this static case with exactly zero frequency the general formula forthe wave-vector evolution Eq. (6) is applicable.In the next section we will apply these kinematic relations to the Alfvén waves. Consider the Alfvén velocity V Adefined by external magnetic field B 0 , magnetic pressure p Band density of the fluid ρp B= B2 02µ 0= 1 2 ρV 2 A, (7)and the shear parameter A with dimension of frequency. For a kinematic description it is convenient to introducedimensionless wave vectors K = (V A /A) k, ˜K = (V A /A) ˜k, as well as dimensionless time τ = At; L A ≡ V A /A is theunit length. The connection between the Eulerian components of the wave-vector and the components in the tildeLagrangian system Eq. (6) reads asK x (τ) = ˜K x (0) − τ ˜K z ,K y (τ) = ˜K y (0), K z (τ) = ˜K z (0). (8)With appropriate choice of the initial time we can further set ˜K x (0) = 0. Here we have used that the wave-vectorin the Lagrangian (tilde in our notations) coordinate system is constant, ˜K(τ) = const. Lagrangian coordinates arealso known as flow-following coordinates, which in this particular flow preserve the wave vector (similarly, a plane of“tagged” atoms is moved but not deformed). For matrix presentation of the considered relations see Appendix A.


3III.DYNAMICS EQUATIONSConsider the laminar component of the velocity field V(t, r) of an incompressible flow of accreting plasma withconstant density ρ = const. In the presence of an external magnetic field B(t, r) the velocity V(t, r) evolves accordingto the Navier-Stokes equation with a Lorentz forceρ(∂ t + V · ∇)V = −∇p + η∆V + j × B + F, (9)where p is the pressure and η is the viscosity [8]. The term F(t, r) phenomenologically describes the Reynolds andMaxwell stresses associated with the turbulence. Later, when we analyze the evolution of perturbations in the shearflow, we will return to this term.The current j is given by Ohm’s lawj = σ(E + V× B) , (10)where σ ≡ 1/ϱ is the electrical conductivity and (E + V× B) expresses the effective electric field acting on the fluid;the velocity V ≪ c is nonrelativistic.For low frequencies j ≫ ε 0 |∂ t E|, so it is possible to use the magnetostatic approximationWe substitute here the current from Eq. (10) and obtainrot B = µ 0 j . (11)E = −V × B + ν m rot B, (12)where ν m ≡ ε 0 c 2 ϱ is the magnetic viscosity.The evolution of the magnetic field is governed by the other Maxwell equation rot E = −∂ t B, thus∂ t B = rot (V × B) − ν m rot rot B. (13)For incompressible flow div V=0, and taking into account that div B=0, the equation above takes the formwhere we have used the general relation(∂ t + V·∇)B = (B·∇)V + ν m ∆B , (14)rot(V × B) = VdivB + (B · ∇)V − BdivV − (V · ∇)B.We will use the dynamics equations Eqs. (9,14) to analyze propagation of magnetohydrodynamic waves in a shearflow. Let the velocity V be a sum of equilibrium shear velocity U (0) (r) and a small perturbation u(t, r) for which weare going to derive linearized wave equationThe same representation we suppose for the magnetic field and the pressureV = U (0) + u. (15)B = B 0 + B ′ , p = p 0 + p ′ . (16)As mentioned in section II, the z-axis of the coordinate system we choose along the shear velocity and x-axis alongthe velocity gradientU (0) = (0, 0, Ax). (17)For this choice the vorticity rot U (0) = (0, −A, 0) is along y-axis.The differential rotation in the accretion disk along with the freezing in of the magnetic force lines stretches theselines and eliminates the cross-flow magnetic field. In the present work we will consider a magnetic field in the plasmaparallel to the shear flowWe will consider a general perturbation caseB 0 = (0, 0, B 0 ). (18)u(t, r) = (u x , u y , u z ), B ′ (t, r) = (B ′ x, B ′ y, B ′ z). (19)


Neglecting the small quadratic terms (u · ∇)u and (u · ∇)B ′in the substantial derivatives in Eqs. (9,14) the linearizedevolution equations read⎛(∂ t + Ax∂ z ) ⎝ u ⎞ ⎛xu y⎠ = − ∇p′uρ− ⎝ 0 ⎞ ⎛0 ⎠ + 1 ⎝ F ⎞xF y⎠z Auρx F z⎛+ B 0⎝ ∂ ⎞zB ′ x − ∂ x B ′ z∂µ 0 ρ z B ′ y − ∂ y B ′ ⎠z+ ν k ∆u, (20)0⎛ ⎞ ⎛B ′ x(∂ t + Ax∂ z ) ⎝B ′ ⎠y = B 0 ∂ z⎝ u ⎞xu y⎠B ′ uzz+AB ′ xe z + ν m ∆B ′ . (21)For the density of the random force that models the turbulence we assume a white noise correlator〈F(t 1 , r 1 )F(t 2 , r 2 )〉 = ˜Γρ 2 δ(t 1 − t 2 )δ(r 1 − r 2 )11, (22)parameterized by ˜Γ. In such a way we derive in Eulerian coordinates r = (x, y, z) a system of partial differentialequations. The transition to Lagrangian ˜r = (˜x, ỹ, ˜z) variables Eq. (3), however, reduces it to an ordinary differentialequations system. To Eq. (3) we add also ˜t ≡ t and perform the change of variablesu(t, r) = u(˜t,˜r), B ′ (t, r) = B ′ (˜t,˜r), F(t, r) = F(˜t,˜r), (23)supposing that (˜t,˜r) are 4 independent variables as (t, r) are. From Eq. (3) one can easily derive the rules for thechange of variables in the derivatives∂ t = ∂˜t − A˜x∂˜z, ∂ x = ∂˜x − A˜t∂˜z , ∂ y = ∂ỹ, ∂ z = ∂˜z . (24)The main advantage of transition to Lagrangian variables is that the substantial time derivative ∂ t + Ax∂ z = ∂˜t ofthe evolution equations Eq. (20) no longer depends on the spatial coordinates. For a system with constant (spaceindependent) coefficients the solutions of the equations are plane waves. That is why using the phase invarianceEq. (4) one can seek a solution in the formu = u˜k(˜t) exp(i˜k · ˜r) = u˜k(t) exp(ik(t) · r), (25)B ′ = B ′˜k(˜t) exp(i˜k · ˜r) = B ′˜k(t) exp(ik(t) · r), (26)where u˜k(˜t) = u˜k(t) and B ′˜k(˜t) = B ′˜k(t) are the time dependent amplitudes of the plane waves. For now we supposethat the variables are complex-valued.For clarity we will perform the change of variables gradually and will analyze the result for individual terms. Firstwe will change the variables only in the substantial time derivatives so that the system Eq. (20) takes the form⎛⎝ u ⎞ ⎛x∂˜t u y⎠+ ⎝ 0 ⎞ ⎛0 ⎠+ 1 ⎝ F ⎞xF y⎠ =u z Auρx F z⎛− ∇p′ρ + B 0⎝ ∂ ⎞zB ′ x − ∂ x B ′ z∂µ 0 ρ z B ′ y − ∂ y B ′ z⎠+ ν k ∆u, (27)0⎛ ⎞ ⎛B ′ x⎝ ∂˜t B ′ y⎠=B 0 ∂ z⎝ u ⎞xu y⎠+ AB xe ′ z + ν m ∆B ′ , (28)B ′ uzz∂ x B ′ x + ∂ y B ′ y + ∂ z B ′ z = 0, (29)∂ x u x + ∂ y u y + ∂ z u z = 0. (30)4


To emphasize the physical meaning we have used Lagrangian variables, however one can start with the simple relationwhich both automatizes and ensures the separation of variables( { [(∂ t + Ax∂ z ) u(t) exp i (˜k x − At˜k z )x + ˜k y y + ˜k]})z z{ [= exp i (˜k x − At˜k z )x + ˜k y y + ˜k]}z z d t u(t).For the plane-wave in Eq. (25) the substantial derivativesafter linearization take the formD t ≡ ∂ t + V · ∇ (31)[D t u˜k(t) exp(ik(t) · r) ]]≈ exp(ik(t) · r)[d˜t u˜k(t) + u z,˜k(t)e z , (32)D t[B ′˜k(t) ]exp(ik(t) · r) ≈ exp(ik(t) · r)d˜t B′˜k(t). (33)As mentioned earlier, in Eqs. (32, 33) we have neglected the quadratic terms, which describe wave-wave interaction.Now we can substitute here the right-hand side of the supposed plane form of the waves from Eq. (25), wherethey are presented in Eulerian coordinates. The latter are more convenient to calculate the gradients ∇ = ik(˜t) andLaplacians ∆ = −k 2 (˜t) but we have to remember that according to Eq. (6) the wave-vectors k(˜t) = (A/V A )K(τ =A˜t)are time dependentk(˜t) = (˜k x − A˜t˜k z , ˜k y , ˜k z ), k 2 (˜t) = k 2 x + k 2 y + k 2 z. (34)Additionally, to obtain a system of ordinary differential equations for the amplitudes of the magnetohydrodynamicwaves, one can eliminate the pressure as it is done in Appendix A. We suppose plane waves for the velocity andmagnetic fieldwhere we have introduced several notations that will be used lateru(t, r) = u˜k(t) exp(ik(t) · r), (35)B ′ (t, r) = B ′˜k(t) exp(ik(t) · r), (36)u˜k(t) = −iũ˜k(t) = −iV A ⃗υ˜k(τ), (37)B ′˜k(t) = B 0 b˜k(τ). (38)For brevity the Lagrange wave-vector indices ˜k will be further suppressed, for example, taking the real part fromEqs. (35, 36) we haveu(t, r) = V A ⃗υ(τ) sin(k(t) · r), (39)B ′ (t, r) = B 0 b(τ) cos(k(t) · r). (40)For the real dimensionless amplitudes ⃗υ and b after some algebra given in the appendix we derive the system ofequations:5d τ υ x= K zK xK 2 υ x − K z b x − ν ′ kK 2 υ x + gf x , (41)d τ υ y = K zK yK 2 υ x − K z b y − ν kK ′ 2 υ y + hf y , (42)d τ b x = K z υ x − ν mK ′ 2 (τ)b x , (43)d τ b y = K z υ y − ν mK ′ 2 (τ)b y , (44)υ zb z= − K xK zυ x − K yK zυ y , (45)= − K xK zb x − K yK zb y , (46)


6whereg(τ) =√K 2 y + K 2 zK(τ)√K2, h(τ) = x (τ) + Kz2 , (47)K(τ)K x = −τK z , K y = const, K z = const, (48)√√K(τ) = Kx 2 + Ky 2 + Kz 2 = Ky 2 + (1 + τ 2 )Kz 2 , (49)ν k ′ ≡ A VA2 ν k , ν m ′ ≡ A VA2 ν m , Γ = ˜ΓVA 2 , (50)AVν ≡ ν k + ν m , ν ′ ≡ ν k ′ + ν m, ′ (51)〈f x (τ 1 )f x (τ 2 )〉 = 〈f y (τ 1 )f y (τ 2 )〉 = Γδ(τ 1 − τ 2 ), (52)and V is the typical volume of the system.In the simplest case of zero shear A = 0 and dissipation ν k = ν m = 0 we have a system with constant coefficientswhich gives in dimensionless form the dispersion of Alfvén waves [8] [LL, VIII, sec. 69, problem] (for details onderivation in our variables see the Appendix)ω A (k) = V AB 0|k · B 0 | − i 2 νk2 , V A ≡ B 0√µ0 ρ , (53)V gr (k) = ∂ω A(k)∂k= V A sign(k · B 0 ) B 0B 0. (54)The wave-wave interaction is small and negligible only if the dimensionless velocity and variation of magnetic filedare sufficiently smallu˜k(t)⃗υ = i , b = B′˜k(t) , υ 2 , b 2 ≪ 1. (55)V A B 0To reveal qualitative properties related to shear induced amplification of the waves, here we will concentrate onthe two-dimensional case k y = 0 which gives a complete separation of the variables. For this special case K y = 0 thedynamic equations take the formwhered τ υ x = α(t)υ x − K z b x−ν kK ′ 2 (τ)υ x + g(τ)f x (τ), (56)d τ υ y = −K z b y − ν kK ′ 2 (τ)υ y + f y (τ), (57)d τ b x = K z υ x − ν mK ′ 2 (τ)b x , (58)d τ b y = K z υ y − ν mK ′ 2 (τ)b y , (59)υ zb z= − K xK zυ x , (60)= − K xK zb x , (61)α(τ) ≡K x(τ)K zKx(τ) 2 + Kz2 = − τ1 + τ 2 , (62)K 2 (τ) = (1 + τ 2 )Kz 2 1, g(τ) = √ . (63)1 + τ2We will start our analysis with the case of short wavelengths in a dissipationless and a fluctuation-free regime.For highly conducting plasma ν m ≈ 0 with negligible kinematic viscosity ν k ≈ 0 (conditionally, say superfluid andsuperconducting plasma) the system above yieldsd τ υ x = α(τ) υ x − K z b x , (64)d τ b x = K z υ x . (65)


We differentiate Eq. (64) with respect to time, neglect for |K z | ≫ 1 the d τ α term, and substitute d τ b x from Eq. (65)which impliesϋ − α(τ) ˙υ + K 2 z υ ≈ 0, (66)where υ ≡ υ x and the dot stands for the time τ derivative. In original variables this equation readswhere the frequency of the Alfvén wavesdepends on the angled 2 t u + γ x,˜k s(t)d t u + x,˜k ω2 Au x,˜k≈ 0, (67)ω A (k) = V A |k z | = V A k| cos ϕ|, (68)7ϕ(t) = arccos k · B 0kB 0= arctan k x(t)k z= − arctan τ (69)between the external magnetic field and the wave-vector. In such a way we find the geometrical meaning of thedimensionless time.The effective friction coefficient γ s (t) in the oscillator equation Eq. (67) is presented by the dimensionless functionα(τ)γ s (t) = Aγ s (τ), (70)γ s (τ) ≡ −α(τ) =τ1 + τ 2 = 1 sin 2ϕ.2(71)Without any restrictions we have considered ˜k x = 0 and K x = 0 because it is related only to the choice of the initialtime Eq. (34).Next step is to determine the conditions that ensure small dissipation rate. Ohmic resistivity and viscosity producean additional friction coefficient in the effective oscillator equation Eq. (67) (confer with [LL, VIII, sec. 69, problem])γ ν = (ν k + ν m )k 2 (t) = Aγ ν , (72)γ ν (τ) = (1 + τ 2 ) a =acos 2 ϕ , (73)a ≡ ν ′ K 2 z = (ν k + ν m )k 2 z/A. (74)For the shear A to have significant influence on the waves it is necessary that the wave decay rate is smaller thanthe wave vector’s rate of rotation due to the sheara = γ ν(ϕ = 0)A≪ 1, (75)i.e. the parameter a has to be small.In such a way taking into account the small dissipation in the oscillator equation (67) we have to substitute theshear attenuation with the total attenuationsγ = γ s (ϕ) + γ ν = Aγ = A(γ s + γ ν ). (76)The dimensionless attenuations γ s and γ ν are more convenient for further analysis of the kinetics of Alfvén waves.From Eq. (67) we derive equation for attenuation of the averaged effective energy for a fictitious particle withcoordinate u x,˜kE eff (t) = 1 2of the oscillator [LL (I,25.5), ibid. (I, sec. 51, problem 2)]()˙u 2 + x,˜k ω2 Au 2 x,˜k(77)d t E eff ≈ −γ(ϕ)E eff . (78)Here the time dependence is implicitly included by the dependence of the attenuation on the angle ϕ, Eq. (69).


To derive the approximate oscillator equation Eq. (66) we assumed that the alteration of the wave vector componentsK(τ) and respectively the variation of α(τ) (or γ s (t) in Eq. (67)) is negligible compared to the change of the velocityυ and its derivatives and therefore all terms containing K ˙ x (τ) and ˙α(τ) have been omitted. Such approximationcorresponds to the WKB approximation in quantum mechanics where we have fast-oscillating amplitude and slowlychanging coefficients. The approximate equation Eq. (67) will be the starting point for further analysis.We will illustrate the WKB method for calculation of the attenuation on the simplest possible example of systemEq. (64) and Eq. (65) or equation Eq. (66). Substitution in this equation of amplitudes ∝ exp(λτ) gives thecharacteristic equationD(λ) = P (λ) + V (λ) = 0, (79)P (λ) = λ 2 + K 2 z , d λ P (λ) = 2λ, V (λ) = γ s λ. (80)Supposing γ s to be a small perturbative correction we have in the first Newton iterationIn such a way for the energy of the wave we obtainP (λ 0 ) = 0, λ 0 = −iK z , (81)λ ≈ λ 0 − V (λ 0)d λ P (λ 0 ) = −iK z − 1 2 γ s. (82)E(τ) ∝ | exp(λτ)| 2 = exp(−γ s τ) = exp(−γ s t). (83)This procedure applied to the more general system of equations Eqs. (56, 58) gives after a time differentiation thekinetic equation for the averaged energy of the wave Eq. (78) with total decrement Eq. (76).For the case of small dissipation the damping of the waves can be calculated directly as a ratio of dissipated powerdivided by the energy. Initially, we have to take the real part of the wave variables and then we have to average overthe period of the oscillations∑ 3i,k=1 〈(∂ iu k + ∂ k u i ) 2 〉whereγ ν = 〈j(t, r) · E(t, r)〉 + η 2∂ k ≡ρ2 〈u2 (t, r)〉 + 12µ 0〈B ′ 2 (t, r)〉Eq. (84) is an alternative way to derive Eq. (72) for the case of small shear., (84)∂∂x k, (x 1 , x 2 , x 3 ) = (x, y, z). (85)8IV.KINETICS OF ALFVÉN AMPLITUDES IN WKB APPROXIMATIONIn the spirit of quantum mechanics the number of particles is proportional to the square of the amplitude. In thissense using the dimensionless amplitude of the velocity υ one can introduce the mean number of “alfvons”n(τ) = 〈(Re υ) 2 〉, (86)where the time τ average is for the dimensionless period of the Alfvén waves 2πA/ω A = 2π/|K z |. The volume densityof the wave energy is proportional to the number of “alfvons”E = 2 ρ 2 〈(Re u x) 2 〉 = ρV 2 An. (87)Here we took into account that averaged kinetic energy of the fluid is equal to the averaged energy of the magneticfield which plays the role of the elastic energy of these transversal waves. In such an interpretation the equation forthe time derivative of the wave energy Eq. (78) can be considered as Boltzmann equation for the number of “alfvons”d τ n(τ) = −γ(τ)n(τ) + w(τ), (88)γ = γ s + γ ν , (89)γ s =τ1 + τ 2 , (90)γ ν = (1 + τ 2 )a. (91)


In order not to interrupt the explanation the derivation of the turbulence induced income term w will be given in thesection V, see Eq. (157).One can easily check that the solution of the kinetic equation Eq. (88) reads∫ τ[ ∫ τ]n(τ) = w(τ ′ ) exp − γ(τ ′′ )dτ ′′ dτ ′ . (92)−∞τ ′Let us first analyze in Eq. (88) the dissipationless regime of γ ν = 0 and zero turbulence power w = 0. Using theintegral( ∫ τ)exp − γ s (τ ′′ )dτ ′′ 1= | cos ϕ| = √ , (93)01 + τ2for the solution of the homogeneous linear equation we obtainn(τ) ≈ n 0 | cos ϕ| =n 0√1 + τ2 , (94)where the integration constant n 0 determines the spectral density of the waves with wave vector k parallel to themagnetic field B 0 .Further in the general solution Eq. (92) we have to substitute the integral( ∫ τ)√exp − γ s (τ ′′ )dτ ′′ | cos ϕ| 1 + τ=τ | cos ϕ ′ | = ′2√ . (95)′ 1 + τ2The contribution of the dissipation is given by∫ τor for the interval pointed above Eq. (92)−∫ ττ ′0γ ν (τ ′′ )dτ ′′ =γ ν (τ ′′ )dτ ′′ = awhere for simplicity we have introduced another time variableFinally the solution of the Boltzmann equation Eq. (92) takes the form(τ + 1 3 τ 3 )a, (96)[−(1 + τ 2 )u + τu 2 − 1 3 u3 ], (97)u ≡ τ − τ ′ > 0. (98)∫ ∞1n(τ) = √ w(τ − u) √ 1 + (τ − u) 21 + τ20{× exp −a[(1 + τ 2 )u − τu 2 + 1 ]}3 u3 du. (99)In order to evaluate this integral we will assume wave-vector independence of the random turbulent noise w(K(τ)) ≈const. For the real physics of accretion disks the bare viscosity is evanescent and we have to analyze the above integralon the u variable for very small values of a ≪ 1. This means that for values of |τ| of order of 1 we have to take intoaccount very large values of the u ≫ 1 variable. For large u in the integrant we have to make the approximations√1 + (τ − u)2 ≈ u, (100)9(1 + τ 2 )u − τu 2 + 1 3 u3 ≈ 1 3 u3 , (101)and for the integral in the above Eq. (99) we derive according to Eq. (94) the τ-independent evaluationn 0w∫ ∞( ) 2/3 3 ≈ v −1/3 exp(−v)dv (102)a≈0( 3a) 2/3 ( ) 2 [Γ 1 + τO(a 1/3 ) + τ 2 O(a )]2/3 ,3


10where we have made the substitutions( ) 2/3v = au3 33 , du = v −2/3 dv. (103)aIn such a way for the low dissipation limit we derived an explicit expression for the number of “alfvons” propagatingalong the magnetic fieldn 0 ≈ 32/3 Γ ( 23)w= 32/3 Γ ( 23)A 2/3 w. (104)(ν ′ Kz 2 ) 2/3 (ν h + ν m ) 2/3 k 4/3The last term is expressed via the physical variables of the initial problem, this is the central result of our analysis ofthe kinetics of Alfvén waves. In order to calculate the effective viscosity for this waves we have to perform statisticalaveraging of the dissipated power, but before this in the next section we will analyze in short the influence of weekturbulence on the magnetohydrodynamic equations.zV. WHITE NOISE TURBULENCEA. Langevin-Burgers noise for Alfvén wavesIn Langevin-Burgers approach to the turbulence the random noise is described as a white one for which the followingcorrelation may be used〈F(t 1 , r 1 )F(t 2 , r 2 )〉 = ˜Γρ 2 δ(t 1 − t 2 )δ(r 1 − r 2 )11, (105)The dimension of ˜Γ shows that this correlation is convenient not for the forces, but rather for the accelerationsWe make a Fourier transform of the acceleration[˜Γ] = (acceleration) 2 × (volume) × (time). (106)F(t, r) = ∑ k∫F k (t) =for which we apply periodic boundary conditionsF k (t) exp(ik · r), (107)F(t, r) exp(−ik · r) dx3V(108)F(t, x + L) = F(t, x) (109)where L is a characteristic length for the system V = L 3 . This sets the following restrictions for the wave vectorswhere the numbers can take only integer values(k x , k y , k z , ) = 2π L (n x, n y , n z ), (110)n x , n y , n z = 0, ±1, ±2, ±3, . . . (111)In this case the difference between two wave vectors is ∆k = 2π/L and for great lengths the sum turns into anintegration1 ∑∫=VkThe correlation for the coefficient in the Fourier series is〈F ∗ p(t 1 )F k (t 2 )〉 = 11 ˜Γρ 2d 3 k(2π) 3 . (112)V δ(t 1 − t 2 )δ p,k , (113)


where δ p,k is the symbol of Kronecker. Taking into account the upper relation for the x component of the accelerationwe have〈F ∗ k,x(t 1 )F k,x (t 2 )〉 = ˜Γρ 211V δ(t 1 − t 2 ). (114)In our work we are interested in the presence of the white noise in the magnetohydrodynamic system (56) - (58)causing the primary birth of the Alfvén waves. That is why we want to know the correlation for the dimensionlessdensity of the external forcewhere〈f ∗ (τ 1 )f(τ 2 )〉 = Γδ(τ 1 − τ 2 ), (115)f(τ) ≡ F k,x(t)ρAV A, Γ = ˜ΓV 2 A AV (116)and the parameter Γ is dimensionless.This dimensionless random force has to be taken into account as an external force in the right part of Eq. (64)which now takes the formand analogously Eq. (66) readsor introducing the dimensionless displacementwe obtaind τ υ x = α(τ) υ x − K z b x + f(τ) (117)ϋ = α(τ) ˙υ − K 2 z υ + ˙ f(τ), (118)x(τ) =∫ τ−∞υ(τ ′ )dτ ′ (119)ẍ = −γ s (τ)ẋ − K 2 z x + f(τ), (120)So we arrived at the necessity to analyze the behavior of classical harmonic oscillator with white random noise as anexternal force.B. Kinetic equation for the kinetic energyOur goal in this section is to derive an explicit and physically grounded expression for the casual force, depictingthe noises in the Boltzmann equation (??). We shall start our deduction by consideration of the kinetic equation for aBrawn particle, moving with velocity v under the influence of a casual force f(t) across a given medium with frictioncoefficient γmd t v = −mγv + f(t). (121)For the sake of simplicity from now on we shall examine the propagation of the particle in x direction, so that weshall be interested only in the projections of the velocity and the outer force along this axis, which we shall designatewith v and f. The upper ordinary differential equation may be easily solved with the help of the Euler methodv(t) = C(t) exp(−γt). (122)We replace the velocity from Eq. (121) with the given expression and obtain another ordinary differential equationwith separable variables for the unknown function C(t)d t C(t) exp(−γt) = f(t)m , (123)


12whose solution obviously is∫ tf(t 1 )C(t) = v 0 +t 0m exp(γt 1)dt 1 . (124)The integral constant v 0 here has its usual meaning of initial speed. The explicit form of the velocity becomes( ∫ t)f(t 1 )v(t) = exp(−γt) v 0 +t 0m exp(γt 1)dt 1 . (125)As we are curious about what happens to the mean kinetic energy, we have to strike the average of the square of thevelocity. For this purpose we need to know the correlation between the noises. We suppose that it has the simplestpossible form (i.e. we use white noise)〈f(t 1 ), f(t 2 )〉 = Γδ(t 1 − t 2 ), (126)which in the calculus of probability corresponds to independent events. We take into consideration that the causalforce has zero mean value 〈f(t)〉 = 0 and for the average of the square of the velocity we obtain〈v 2 (t) 〉 = exp(−2γt)( ∫ t ∫ t)× v0 2 Γδ(t 1 − t 2 )+t 0 t 0m 2 exp(γ(t 1 + t 2 ))dt 1 dt 2 =exp(−2γt)(v 2 0 +∫ tt 0)Γm 2 exp(2γt 1)dt 1 .We are now ready to calculate the mean kinetic energy of the Brawn particle, roaming in a certain volume urged bythe casual outer force and impeded by the forces of friction(127)〈E kin (t)〉 = m 2〈v 2 (t) 〉 =mv022 exp(−2γt) + Γ∫ t2m exp(−2γt) exp(2γt 1 )dt 1 =t 0mv022 exp(−2γt) + Γ4mγ {1 − exp(−2γ(t − t 0))} . (128)We would like to find a more convenient way to write down this expression. To do so, we introduce the initial kineticenergy E 0 = E kin (t 0 ) and define another average energy E ≡ . In this case the average kinetic energy looks like(〈E kin (t)〉 = E 0 exp − t )+ Eτ EΓ4mγ{1 − exp(− (t − t 0)τ E)}, (129)where τ E ≡ 12γconcurs with the relaxation time in the atomic physics and Γ is the analogue to the width of theLorentz function. In order to derive the kinetic equation we differentiate Eq. (129) with respect to time and obtaindt 〈E kin (t)〉 = − 1τ E(+ E exp(− tτ E)E 0 exp(− (t − t ))0). (130)τ EIn this way we came to the well-known Boltzmann equation for the variation of the average kinetic energyddt 〈E kin(t)〉 = − 1 (〈Ekin (t)〉 − E ) . (131)τ EThe first term here is responsible for the energy expenditure and consequently is set by the dissipation function,whereas the second stands for the energy income (i.e. a positive power), caused by the effect of noises.Now let us make a short analysis of the result. According to the Boltzmann theory in the equilibrium statistical


physics (after a very long period of time, t → ∞) the mean value of the kinetic energy equals half the absolutetemperature, multiplied by the Boltzmann constant, which we will designate with Θ〈E kin (t)〉 = m 2〈v2 〉 = 1 2 k BT = (132)12 Θ = E = Γ4mγHereby, we can deduce the connection between the dissipation coefficient γ and the fluctuations Γ13(133)Γ = 2mΘγ, (134)which is a special case of the fluctuation-dissipation theorem.Our idea is to apply the terms from the equilibrium statistics in the non-equilibrium case, therefore we shall use Θin stead of k B T (a non-equilibrium analogue to the equilibrium statistics temperature). If we mark the energy losseswith W the fluctuation-dissipation theorem readsW = E τ E= γΘ =Γ2m . (135)Now let us consider the motion of a Brawn particle as a diffusion, for which the the role of concentration is played bythe probability to find the particle at a given place in the volume of the fluid. The diffusion equation is∂ t n(x, t) = D∆n(x, t), (136)where n(x, t) stands for the concentration. The square of the average distance, which the particle passes for an intervalof time t is given by [6], sec. (60)〈x2 〉 ∫= x 2 n(x, t)dx = 2Dt. (137)According to the Einstein correlation (for a detailed derivation see appendix F), the diffusion coefficient D is proportionalto the mobility b and the temperature in the equilibrium statistical physics, which under our assumption maybe expressed asD = bΘ. (138)The mobility may be derived from the motion equation of the particle, roaming under the influence of a time independentouter force fIn the stationary case the outer force has to balance the frictionHereby for the mobility we obtainmd t v = −mγv + f. (139)v dr = bf. (140)b = 1mγ(141)and the diffusion coefficient takes the formD = Θ mγ . (142)Our next step is to consider the equation of motion for a harmonic oscillator with random noise


14Brownian motion of harmonic oscillator ...C. Oscillator under a white noisemẍ = −mω 2 ẋ + f(t). (143)We assume classical quantities (i.e. [x, p] = 0). It is convenient to examine the motion equation in the phase space(of the radius-vectors and impulses). For the purpose we introduce complex variablesc = mωx + ipc ∗ = mωx − ip, (144)where p = mẋ is the particle impulse. The energy of the oscillator (proportional to the number of particles) can beexpressed via these variables in the following wayE = 12m c∗ c = p22m + 1 2 mω2 x (145)We are interested in the explicit form of the mean power W = d t 〈E(t)〉. That is why we want to know the correlationof c and c ∗ . In order to find it first we have to determine their form as functions of frequencies, random noises andtime. The time derivative of c looks liked t c = mωẋ + iṗ = ωp + i ( −mω 2 x + f(t) )= −iω(ip + mωx) = −iωc(t) + if(t). (146)For this ordinary differential equation we seek a solution in the form of plane wave and obtain( ∫ t)c(t) = C 0 + i f(t 1 ) exp(iωt 1 )d t1 exp(−iωt)t 0( ∫ t)c ∗ (t) = C 0 − i f(t 2 ) exp(−iωt 2 )d t2 exp(−iωt). (147)t 0Now it is not difficult to calculate the correlation〈c ∗ (t)c(t)〉 = |C 0 | 2∫ t ∫ t+ 〈f(t 1 )f(t 2 )〉 exp(iω(t 1 − t 2 ))d t1 d t2 . (148)t 0 t 0We consider white noise for which 〈f(t 1 )f(t 2 )〉 = Γδ(t 1− t 2 ). Then Eq. (148) turns into〈c ∗ (t)c(t)〉 = |C 0 | 2 + Γ(t − t 0 ). (149)Therefore for the energy of the oscillator shaking under the random noise we have〈E(t)〉 = E 0 + Γ2m (t − t 0), (150)where the initial energy is E 0 = 12m |C 0| 2 . It became clear that the conducted power isW =Γ2m . (151)The equation of motion for a free particle readsD. Stochastic heating of a free particlem ˙v = f(t), (152)


where f(t) stands for the random (under our consideration white) noise. Hereby, after integration, for the velocityvector we findv(t) =∫ tt 015f(t)m dt 1 + v 0 , (153)where v 0 is the initial velocity. Using this explicit expression for the velocity the average kinetic energy reads as〈E(t)〉 = m 〈v 2 (t) 〉2= 1 ∫ t ∫ t〈f 1 (t)f 2 (2)〉 dt 1 dt 2 + m 〈 〉v22m t 0 t 02 0 . (154)Here we substitute apply the correlation for white noise 〈f 1 (t)f 2 (2)〉 = Γδ(t 1the average kinetic energy becomes− t 2 ) and integrate over the time. The〈E(t)〉 = E 0 + Γ2m (t − t 0). (155)Finally a time differentiation gives the power of the white noise acting on the free particleW = d t 〈E(t)〉 =Γ2m . (156)Actually this general result does not depends on the potential in which the particle is moving. This constant poweris applicable if the typical time scale of the particle motion is much bigger than the characteristics times of the noiseand we can approximate it as white one.E. General theorem for stochastic heating generated by white noiseThe power does not depend on the potential:w =˜Γ2AV 2 A V . (157)Formerly this income term describing the birth of “alfvons” from the sea foam that is to say the turbulent noisewas postulated in the Boltzmann equation Eq. (88).VI.STATISTICS OF ALFVONSAccording to Eq. (6) we havedk x = k z dτ (158)Calculated for the special case of k y = 0 heating to be replaced for an order estimation by the isotropic approximationin two dimensional (2D) (k y , k z )-plotQ(k z ) → Q(k 2D) (159)dk y dk z = d ( πk 2 2D)(160)


16VII.HYDRODYNAMIC PHENOMENOLOGY FOR EFFECTIVE STRESS TENSORWhen we apply the Boltzmann approach in calculation of the spectral densities we have to use the WKB approximationto find the proper frequencies of the dynamic system and analyze the most rapidly growing mode or the mostslowly fading one. As a rule this leads to numerical calculation. In order to do it analytically we have concentratedour attention on the special occasion of small dissipation when we may consider the attenuation of the Alfvén wavesperturbatively, supposing that we have slightly perturbed by the dissipation oscillations Aγ ν (τ) ≪ ω A . For a negligibleviscosity this approximation has a wide area of applicability for wave vectors less than the maximum wavevector k < k c which corresponds to Debye frequency from the physics of phonons for instance. In the present workwe will restrict ourselves with a model evaluation for the benefit of the kinetic approach to the spectral density of the“alfvons”.In this section we will treat the question for the total heating due to the dissipation process in Alfvén waves. Thatis to say that we will sum the effects of all slow magnetohydrodynamic modes without any restrictions of the angle ϕbetween the wave vectors and the external magnetic field. In order to obtain the total conducted volume density ofthe power Q tot first we have to set the boundaries for the wave vectors which are determined from the condition forexistence of Alfvén wavesγ ≪ ω A , γ = Aa(1 + τ 2 ). (161)As we know from the kinetics (see Eq. (8)) the integration with respect to k x can be reduced to integration withrespect to time∫ ∫dk x = k z dτ. (162)Then the critical wave vectors and respectfully the critical time will be fixed from these frequencies for which inequality(161) turns into equalityτ c 2 = k ck z,k c = V Aν , ν = ν k + ν m . (163)For greater times τ > τ c there will be no waves as they would simply attenuate over their period and for greater wavevectors k z > k c there will be no media at all. The average energy in our model evaluation (94) is given byn 0E = 2p Bn, n = √ , n 0 = 32/3 Γ ( 23)w. (164)1 + τ2 a 2/3The dissipative function is determined by the mean energy multiplied by the attenuation coefficientQ = γ ν E = 2an 0 p BA √ 1 + τ 2 . (165)To find the total mean power conducted into heat we have to sum over all permitted wave vectors. We shall dothat consecutively, begging with integration with respect to time. For evanescent viscosity there are practically norestrictions in time, therefore by modulus the critical time is enormous |τ c | ≫ 1. Thus for the time integration wehave∫ τc √1 + τ2dτ ≈ k c, (166)−τ ck zfrom where for the time integrated power we obtain∫Qdk x = 2an 0 k c p BA. (167)This expression for the heat depends only on the z component of the wave vector as we have assumed that k y = 0.However we believe that the same result can be derived for waves spreading in y direction. That is why in order tointegrate the dissipative function in the entire impulse space (i.e. to account the influence of the y mode which wasformerly put to zero) we make an isotropic approximation and introduce a two dimensional wave vector k 2 = k2 2D x + ky2for kinetic description. In this case the integration with respect to k z and k y is replaced with integration over thesurfacedk y dk z = d(πk 2 2D ) (168)


17and we may put k 2 z → k 2 2D in the expression for a a = νk2 2DA . (169)Thereby for the volume density of the total heating we have∫ d 3 kQ tot = V(2π) 3 Q = V35/3 Γ ( )232 4 π 2 A 2/3 wν 1/3 p Bkc 11/3 , (170)where we have integrated in the wave-vector domain for which the waves exist ω A > γ ν or k < k c . Using the explicitform of w (refer to Eq. (157)) we obtainQ tot = 35/3 Γ ( )2 5/33˜ΓVA2 5 π 2 A 1/3 ν p . (171)10/3 BWe would like to point out that the result gives a qualitative explanation to the pressing problem with the missingviscosity in quasars and AGNs, as it is obvious that the slightest the magnetohydrodynamic viscosity is the greatestthe produced power will be. Furthermore we can tell that small shear rate also leads to intensive heating. In this waywe have shown that in our opinion the presence of turbulent noise in the beginning gives birth to Alfvén waves whichare further amplified by the shear flow till they reach a certain maximum amplitude from which on their energy startsdissipating under the influence of the kinematic and magnetic viscosities, thus turning into heat.According to Newton the conducted heat is proportional to the square of the velocity gradient. This means thatwe can introduce an effective viscosity η eff for whichThus for the shear stress tensor we findQ tot = η eff A 2 = σ s A. (172)σ s = 35/3 Γ ( )2 5/33˜ΓVA2 5 π 2 A 4/3 ν p . (173)10/3 BWe may now make an analogy with Shakura-Sunyaev [26] and lay the stress tensor to be proportional to the pressure(in our case the magnetic one) σ s = α ω p Bwherefrom for the proportionality coefficient we derive5/3 ˜ΓVAα ω = C ωA 4/3 ν , C 10/3 ω = 35/3 Γ ( )232 5 π 2 C α , (174)where the constant C α is inserted to absorb the inaccuracies due to our model approximations.effective Reynolds number of the problemIntroducing theand dimensionless Burger parameterR = L AV Aν= 1 ν ′ , L A = V A /A (175)Γ = A2˜ΓV 5 A(176)we can rewrite our result asσ = 1 2 ρV 2 Aα ω , α ω = C ω Γ R 10/3 . (177)In the derivation we essentially used a parameter to be small even for k z approaching the wave-vector cutoff k c .Physically this means that the shear rate should be big enough in order the rotation to be faster then dampingω A (k c ) = γ ν (k c ) ≪ A. (178)This relation means that the effective Reynolds number should be small R = 1/ν ′ ≪ 1. Finally we derived that theeffective viscosityη eff = σ A = 35/3 Γ ( )2 5/33˜ΓVA2 5 π 2 A 7/3 ν p , p = 1 10/3 B B2 ρV A 2 (179)is shear rate dependent and the magnetized turbulent plasma is a highly non-Newtonian fluid.


18VIII.PERSPECTIVES, DISCUSSION AND CONCLUSIONSThe most essential characteristic of the derived result Eqs. (104, 171) is the presence of the viscosity in the denominator.The expression for the dissipative power in case of zero viscosity formally becomes infinite. Physically thismeans that even extremely slight turbulence, given by the Burgers parameter Γ, can lead to the appearance of significanteffective viscosity. The discussed mechanism requires simultaneous presence of magnetic field and turbulence -ingredients which are pointed out practically in all present works. As we have analyzed the waves in an incompressiblefluid the gas pressure turned out to be irrelevant to the shear stress. Therefore we consider that in Shakura-Sunyaevphenomenology the gas pressure p should be replaced with the magnetic one p B. Thus for the momentum transportwe haveσ ϕr = α ω p Bfor disk, σ zr = α ω p Bfor column, (180)which is negligible modification to the essential theory. The detailed calculation of the dimensionless parameter α ωrequires complete analysis of the kinetics for all modes in the shear flow of the magnetized plasma. This is undoubtedlya very complicated task and it is worth making a qualitative assessment of the origin of the effective viscosity in plasmawith evanescent initial one. In this case the system is extraordinary sensitive to the initial conditions. The energy ofan inconsiderable perturbation in a remote initial moment of time |τ in | ≫ 1 enhances |τ in | times in the moment τ = 0when the wave vector is parallel to the magnetic field. In this sense for τ in → −∞ there comes the “butterfly effect”- an infinite amplification of the waves intensity. The most important characteristic of the shear flow in magnetizedplasma is the “lasing” phase when the wave draws energy from the shear flow and the wave vector rotates towardsthe magnetic field. Without dissipation this increment reaches gigantic scales. The amplification stage is convertiblyreplaced with a phase of diminution of the wave energy when for τ > 0 the angle between the wave vector and themagnetic field increases. In creation of effective stress tensor and heating the big wave-vectors up to ultraviolet cut-offk c apparently dominate but one can easily evaluate that even for wave vectors belonging to the cones with angles π/4around the magnetic field the small dissipation caused by the Ohm resistance and the viscosity leads to considerabledissipation for which in the final result evanescent bare the viscosity is in denominator. Complete analysis of thesystem, of course, requires detailed numerical integration. Despite of this we can give a qualitative explanation thatthe plasma heating and the arising of a huge effective viscosity is caused by the laser amplification of the convectiveinstability. This is a rather universal mechanism playing significant role in our solar system in the diminution of theangular momentum of our sun, but also it is the mechanism responsible for the gigantic energy conducted in quasarsand AGNs. In this way we have found a qualitative answer to the question why do the most powerful sources of light inthe universe shine - namely because of the exponential instability of Alfvén waves and the slow magnetohydrodynamicones in the shear flow. [15] Langevin approach; Accretion disk theory was first developed as a theory with the localheat balance, where the whole energy produced by a viscous heating was emitted to the sides of the disk.The used geometry B = (0, 0, B 0 ) corresponds, for example, to the accretion column above the magnetic poles ofa neutron star but we believe that qualitative properties of the solution reveal what could happen in the general casewhen we have a shear flow in magnetized turbulent plasma.The used Langevin-Burgers approach to treat turbulence was also unspecific. We can modify it including, forexample, wave-vector dependence of the noise functions Γ(k) in order to simulate the spectral densities of the velocitypulsations for the corresponding scale. Some parameter related to spectral density of the turbulency indispensablywill participate in the final result for the stress tensor. Moreover we have to include self-sustained turbulence in arealistic global model for the accretion discs or accretion streams. In this direction we are not giving a pret-a-porteprescription. The purpose of our work is only to give an idea how the turbulence, magnetic field, and shear flow playtogether in the most powerful source of the light in the universe (for the mechanism of this phenomenon we do notknow more than the girlfriend of H. Bete on the energy source of stars) and simultaneously of very common situationsas creation of the solar system for example.∗ E-mail: mishonov@phys.uni-sofia.bg† E-mail: maneva@phys.uni-sofia.bg‡ E-mail: Tihomir.Hristov@jhu.edu1 L.D. Landau and E.M. Lifshitz, Mechanics, vol. 1 of Course on Theoretical Physics (Moskow, Nauka Press, 1988), (inRussian).2 L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, vol. 2 of Course on Theoretical Physics (Moskow, NaukaPress, 1988), (in Russian).3 L.D. Landau and E.M. Lifshitz, Quantum Mechanics - Non-relativistic Theory, vol. 3 of Course on Theoretical Physics(Moskow, Nauka Press, 1989), (in Russian).


194 L.D. Landau and E.M. Lifshitz, Quantum Electrodynamics, vol. 4 of Course on Theoretical Physics (Moskow, Nauka Press,1989), (in Russian).5 L.D. Landau and E.M. Lifshitz, Statistical Physics, Part I, vol. 5 of Course on Theoretical Physics (Moskow, Nauka Press,1976), (in Russian).6 L.D. Landau and E.M. Lifshitz, Fluid Mechanics, vol. 6 of Course on Theoretical Physics (Moskow, Nauka Press, 1986), (inRussian).7 L.D. Landau and E.M. Lifshitz, Theory of Elasticity, vol. 7 of Course on Theoretical Physics (Moskow, Nauka Press, 1987),(in Russian).8 L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, vol. 8 of Course on Theoretical Physics (Moskow,Nauka Press, 1982), (in Russian), page 314, Eq. (65.4).9 L.D. Landau and E.M. Lifshitz, Statistical Physics, Part II, vol. 9 of Course on Theoretical Physics (Moskow, Nauka Press,1988), (in Russian).10 L.D. Landau and E.M. Lifshitz, Physical Kinetics, vol. 10 of Course on Theoretical Physics (Moskow, Nauka Press, 1988),(in Russian).11 D. Macdonald et al. in Black Holes: The Membrane Paradigm, eds. K. Thorne, P. Price and D. Macdonald, (London, YaleUniversity Press, 1986, Chap. IV, Sec. C).12 I.D. Novikov, K.S. Thorne in Black Holes, eds. C. DeWitt and B.S. DeWitt (Gordon and Breach, New York), (1973).13 J..E. Pringle, Annu. Rev. Astron. Astrophys. 19, 137, 1981.14 R. Feynman, R. Leighton and M. Sands, Feynman Lectures on Physics, vol. 2, (London, Addison-Wesley), (1963).15 T.M. Mishonov et al., “Kinetics and Boltzmann kinetic equation for fluctuation Cooper pairs”, Phys. Rev. B 68, 054525(2003); T.M. Mishonov and D.C. Damianov, Superlattices and Microstructures 21, 467 (1997); Czech. J. Phys. 46 (Suppl.S2), 631 (1996). T.M. Mishonov, A.I. Posazhennikova and J.O. Indekeu, Phys. Rev. B 65, 064519 (2002)16 G.S. Bisnovatyi-Kogan, “Accretion Disc Theory: From the Standard Model Until Advection”, astro-ph/9810112 (1998).17 K. Menou, “Viscosity Mechanisms in Accretion Discs”, astro-ph/0009022 (2000).18 T. Hristov, C. Friehe and S. Miller, “Wave-coherent fields in air flow over ocean waves: identification of cooperative behaviourburied in turbulence” Phys. Rev. Lett. 81, 23, 5245 (1998).19 T. Hristov, C. Friehe and S. Miller, “Dynamical coupling of wind and ocean waves through wave-induced air flow” Lettersto nature 422, 55 (2003).20 T. Hristov, “Perturbations evolution in shear flow”, BAS, Space Research Institute, (1992), (in Bulgarian).21 G.S. Bisnovatyi-Kogan, “Accretion disks around black holes with account of magnetic fields”, astro-ph/0406466 (2004).22 G.S. Bisnovatyi-Kogan in “Accretion”, Physical Encyclopedia (Moscow, Nauka Press, 1988), (in Russian); Y.B. Zeldovichand I.D. Novikov, Theory of gravity and evolution of stars (Moscow, Nauka Press, 1971), (in Russian).23 H. Bondi, MN R.A.S., 112, 195 (1952).24 H. Bondi and F. Hoyle MN R.A.S., 104, 273 (1944).25 D. Rouan, “A massive black hole at the very centre of the galaxy”, Europhisics News 35, 5 (2004).26 N.I. Shakura and R.A. Sunyaev, A&A 24, 337 (1973).27 N.I. Shakura, Astro. Zh. 49, 921 (972) (Sov. Astron. 16, 756 (1973)).28 E.N. Parker, “Mechanism for magnetic enhancement of sound-wave generation and the dynamical origin of spicules”, Astrophys.J. 140, 1170 (1964).29 M. Schwarzschild, “On noise arising from the solar granulation”, Astrophys. J. 107, 1 (1948).30 S.A. Balbus and J.F.Hawley, Rev. Mod. Phys. 70, 1 (1998).31 J. Frank, A. King and D. Raine, Accretion Power in Astrophysics, (Cambridge Univ. Press, 1992).32 S. Kato, J. Fukue and S. Mineshige, Black-Hole Accretion Disks (Kyoto: Kyoto Univ. Press) (1998).33 C.W. Gardiner, Handbook of Stochastic Methods (for Physics, Chemistry and Natural Sciences), (New York, Springer-Verlag,1985).34 A. Einstein, Ann. Phys. (Leipzig), 17, 549 (1905); ibid 19, 289 (1906), 19, 371 (1906).35 P. Langevin, Comptes. Rendues Ac. Sci. Paris, 146, 530 (1908).36 W. Sutherland, Phil. Mag. 9, 781 (1905).37 A. Pais “Subtle is the Lord . . . ” The Science and the Life of Albert Einstein, (Oxford University Press, 1982), sec. 5c;‘Raffiniert ist Herrgott aber boshaft ist er nicht.’ A. E.38 H. Nyquist, Phys. Rev. 32, 110 (1928).39 I.V. Artemova, G.S. Bisnovatyi-Kogan, I.V. Igumenshchev, I.D. Novikov, “On the Structure of Advective Accretion Disksat High Luminosity”, astro-ph/0003058 (2000).40 I. Ciufolini and J.A. Wheeler, Gravitation and Inertia, (Princeton University Press, 1995), p. 204, Picture 4.5; H.C. Fordet al. “Narrowband HST images of M87: Evidence for a disk of ionized gas around a massive black hole”, Astrophys. J.Lett. 435, L27-L30 (1994); R.J. Harms et al. “HST FOS spectroscopy of M87: Evidence for a disk of ionized gas around amassive black hole”, Astrophys. J. Lett. 435, L35-L38 (1994).41 G. Chagelishvili, R. Chanishvili, L. Filipov, T.S. Khristov, J. Lominadze, “Amplification of Alfven Waves in Free ShearFlows”, Adv. Space Res. v. 11, N8, pp.(8) 61-(8) 65, 1991.42 G.D. Chagelishvili, R.G. Chanishvili, T.S. Khristov, and J.G. Lominadze, Phys. Rev. E 47, 366 (1993).43 A.D. Rogava, S.M. Mahajan, G. Bodo, and S. Marsaglia, “Swirling astrophysical flows – efficient amplifier of Alfvén waves”,Astronomy & Astrophysics A&A, 399, 421-431 (2003).


20APPENDIX A: MATRIX PRESENTATION FOR LAGRANGE-EULER TRANSFORMATIONSThe transitions between Lagrange and Euler coordinates for the space vectors and wave-vectors are very importantfor the present paper. Explanation in the text could be clear enough and can be easily reproduced but for convenienceof the reader we are giving these relations in a transparent self-explainable matrix form:〈˜k|˜r〉 = 〈k|r〉, 〈˜k| = (˜kx ˜ky ˜kz), (A1)⎛ ⎞〈˜k| = (˜kx ˜ky) ( ˜kz = kx k y)k z⎝ 1 0 00 1 0⎠ ,τ 0 1⎛⎝˜x ⎞ ⎛|˜r〉 = ỹ⎠ = ⎝ 1 0 0⎞ ⎛0 1 0⎠⎝ x ⎞ ⎛y⎠ , |r〉 = ⎝ x ⎞y⎠ ,˜z −τ 0 1 zz⎛|k(t)〉 = ⎝ k ⎞ ⎛xk y⎠ = ⎝ 1 0 −τ⎞ ⎛ ⎞˜k x0 1 0 ⎠ ⎝˜k y⎠ ,k z 0 0 1 ˜k z⎛⎝ 1 0 0⎞ ⎛0 1 0⎠⎝ 1 0 0⎞0 1 0⎠ = 11,−τ 0 1 τ 0 1⎛⎝ 1 0 −τ⎞ ⎛0 1 0 ⎠ ⎝ 1 0 τ⎞0 1 0⎠ = 11.0 0 1 0 0 1(A2)(A3)(A4)(A5)(A6)APPENDIX A: DYNAMIC EQUATIONS WITH RANDOM NOISEHere we will simplify the acceleration equation of the fluid Eq. (27). Our purpose is to eliminate the pressure p ′and wave components of the velocity u z and magnetic field B z ′ parallel to the external magnetic filed B 0 e x . For thewave components of the velocity and magnetic field we assume plane waves space dependence ∝ exp(k · r) for whichthe gradient can be substituted by the wave vector ∇ = ik(˜t). In the obtained ordinary differential equations partialtime derivative is substituted by the full time derivative ∂˜t = d˜t ; further we will not make difference between timevariables t = ˜t. Additionally using the notations ũ = iu˜k(t), ˜F = iF˜k(t), and β = B 0 /µ 0 ρ equation Eq. (27) togetherwith divu(t, r) = 0 and divB ′ (t, r) = 0 equations can be rewritten asd˜tũx = k xp ′d˜tũy = k yp ′ρ − βk zB ′ x + βk x B ′ z − νk 2 ũ x + ˜F xρ ,ρ − βk zB ′ y + βk y B ′ z − νk 2 ũ y + ˜F yρ ,d˜tũz = k zp ′ρ − Aũ x − ν k k 2 ũ z + ˜F zρ ,k x ũ x + k y ũ y + k z ũ z = 0,k x B ′ x + k y B ′ y + k z B ′ z = 0.(A1)(A2)(A3)(A4)(A5)


Here the wave vector indices ˜k are omitted. From the first 3 dynamic equation Eqs. (A1, A2, A3) we express thepressureIn the last equation we plug ũ z expressed from Eq. (A4)p ′ρ = d˜tũx + βk z B ′ x − βk x B ′ z + νk 2 ũ x − ˜F x /ρk x= d˜tũy + βk z B ′ y − βk y B ′ z + νk 2 ũ y − ˜F y /ρk y21(A6)(A7)= d˜tũz + Aũ x + νk 2 ũ z − ˜F z /ρk z. (A8)ũ z = − k xũ x + k y ũ yk z(A9)and Eq. (A8) now readsp ′ρ= − 1 kz2 (k x d˜tũx + k y d˜tũy) + Aũ xk z− ν kk 2kz2 (k x ũ x + k y ũ y ) − ˜F z.ρk z(A10)The equality between p ′ /ρ expressed from this equation Eq. (A10) and Eqs. (A6, A7) gives the system of equationwhich does not contain p ′ and ũ z( )1 + k2 xkz2 d˜tũx + k xk ykz2 d˜tũy= A k xũ x − βk z B x ′ + βk x B z′ k( z)−ν k k 2 1 + k2 xkz2 ũ x − ν k k 2 k xk ykz2 ũ y+ 1 (˜F x − k )x ˜Fz , (A11)ρ k zk y k xkz2 d˜tũx +Analogously to the ũ z we can express B ′ z from Eq. (A5)(1 + k2 yk 2 z)d˜tũy= A k yũ x − βk z B y ′ + βk y B z′ k z( )−ν k k 2 k yk xkz2 ũ x − ν k k 2 1 + k2 ykz2 ũ y+ 1 (˜F y − k )y ˜Fz . (A12)ρ k zwhich givesB ′ z = − k xB ′ x + k y B ′ yk z(A13)( )−k z B x ′ + k x B z ′ = −k z 1 + k2 xkz2 B x ′ − k xk yBky, ′ (A14)z( )−k z B y ′ + k y B z ′ = − k yk xB x ′ − k z 1 + k2 yk z kz2 B y.′(A15)


Substitution of this details in the equation for acceleration gives( )1 + k2 xIn matrix for it looks simplerThen using the matrix inversionk 2 z= A k xk zũ x − βk z(1 + k2 xk 2 z−ν k k 2 (1 + k2 xk 2 z= A k y k y k xũ x − βk zk z⎛⎝ 1 + k2 xkz2⎛k y k xk 2 z⎝ 1 + k2 xkz2k y k xk 2 z)d˜tũx + k xk ykz2 d˜tũyB x ′ k x k y− βk zkz2 B y′)ũ x − ν k k 2 k xk y+ 1 ρ(k y k xkz2 d˜tũx +(B x ′ − βk zk 2 z(˜F x − k x1 + k2 yk 2 zk 2 z1 + k2 yk 2 zũ y22)˜Fz , (A16)k z)d˜tũy)B ′ y( )−ν k k 2 k yk xkz2 ũ x − ν k k 2 1 + k2 ykz2 ũ y+ 1 (˜F y − k )y ˜Fz . (A17)ρ k zk x k yk 2 z1 + k2 y= Aũ xk zk x k yk 2 zk 2 z1 + k2 yk 2 z⎞( )⎠ d˜tũx + k z βB x ′ + ν k k 2 ũ xd˜tũy + k z βB y ′ + ν k k 2 ũ y(kxk y)+ 1 ρ⎞⎠−1( ˜Fx)−˜F ( )z kx.˜F y ρk zk y(A18)= 1 ( )k2y + kz 2 −k x k yk 2 −k y k x kx 2 + kz2 , (A19)where k 2 = kx 2 + ky 2 + kz, 2 and simple relation( ) ( ) ( )k2y + kz 2 −k x k y kx−k y k x kx 2 + kz2 = kkz2 kxy k y(A20)the dynamic equations take the explicit form( )d˜tũx + k z βB x ′ + ν k k 2 ũ xd˜tũy + k z βB y ′ + ν k k 2 ũ y( )k z kx= Aũ xk 2 − ˜F z k zk y ρk 2(kxk y)(A21)(A22)+ 1 ( (k2y + kz) 2 ˜F )x − k x k y ˜Fyρk 2 −k y k x ˜Fx + (kx 2 + kz) 2 ˜F . (A23)yNow p ′ , ũ z , and B ′ z are excluded and system contains only ũ x , ũ y , B ′ x, and B ′ y. We have a partial separation of thevariables because equations for ũ x and B ′ x does not include corresponding y-variables. In the equation for ũ y thex-component of the velocity ũ x participate as an external force like the components of the turbulent noise ˜F x , ˜F y , ˜F z .This property allows to solve the x- and y-system sequentially. And finally to express z-variables from the x and yones. Let us analyze ũ x equationd˜tũx = A k zk yk 2 ũx − k z βB ′ x − ν k k 2 ũ x + g · ˜F, (A24)


23whereg(k) = 1 k 2 (k2y + k 2 z, −k x k y , −k x k z),(A25)g 2 = k4 y + k 4 z + 2k 2 yk 2 z + k 2 x(k 2 y + k 2 z)(k 2 ) 2 = k2 y + k 2 zk 2 . (A26)In the considered shear flow k x = −k z τ and for large enough dimensionless times |At| we have the asymptoticg 2 (τ) ≈ k2 y + k 2 zk 2 z1, for |τ| ≫ 1. (A27)τ2In Langevin-Burgers model for turbulence the Fourier components of the density of external force ˜F x , ˜Fy , ˜Fz arerandom white noise variables. For this white noise is essential only the dispersion and taking into account theforce-force correlator Eq. (105) we can make the replacementwhich givesg · ˜F → gF x , 〈F x (t 1 )F x (t 2 )〉 = ˜Γρ 2V δ(t 1 − t 2 ),(A28)Analogous equation for the ũ y variable readsd˜tũx = A k zk xk 2 ũx − k z βB ′ x − ν k k 2 ũ x + gF x . (A29)whered˜tũy = A k zk yk 2 ũx − k z βB ′ y − ν k k 2 ũ y + hF y , (A30)Dividing Eqs. (A29, A30) by AV A and taking into account thatwe obtain the system in dimensionless variableswhereh(k) = 1 k 2 (−ky k x , k 2 x + k 2 z, k y k z),(A31)h 2 = k4 x + k 4 z + 2k 2 xk 2 z + k 2 y(k 2 x + k 2 z)(k 2 ) 2 = k2 x + k 2 zk 2 . (A32)⃗υ = ũV A, Kb = βAV AkB ′(A33)d τ υ x = K zK xK 2 υ x − K z b x − ν ′ kK 2 υ x + gf x , (A34)d τ υ y = K zK yK 2 υ x − K z b y − ν ′ kK 2 υ y + hf y , (A35)f x ≡F x, f y ≡ F y, Γ = ˜ΓρAV A ρAV A VAVA2 (A36)〈f x (τ 1 )f x (τ 2 )〉 = 〈f y (τ 1 )f y (τ 2 )〉 = Γδ(τ 1 − τ 2 ). (A37)The system Eqs. (A34, A35) is completed by Eqs. (58, 59) which remain the sameandd τ b x = K z υ x − ν mK ′ 2 (τ)b x , (A38)d τ b y = K z υ y − ν mK ′ 2 (τ)b y , (A39)υ zb z= − K xK zυ x − K yK zυ y ,= − K xK zb x − K yK zb y .(A40)(A41)


24Those are the general linearized equations of magnetohydrodynamics for our problem.We for the special case of K y = 0 have to substitute− K xK z= τ, K 2 = (1 + τ 2 )K 2 z , K z = const,g =(A42)1√ , h(τ) = 1, (A43)1 + τ2and the system takes the formd τ υ x = − τ1 + τ 2 υ x − K z b x−ν ′ kK 2 z (1 + τ 2 )υ x +f x(τ)√1 + τ2 ,(A44)d τ υ y = −K z b y − ν ′ kK 2 z (1 + τ 2 )υ y + f y . (A45)Taking into the definition α(τ) = −τ/(1 + τ 2 ) we return to Eqs. (56, 57).As a test example let us finally analyze the textbook case of negligible viscosity and big enough wave-vector K z .The systems for υ x , b x and υ y , b y have approximate time dependent solution ∝ exp (−i|K z |τ) . This means that bothx- and y-polarized modes have dispersion of Alfvén wavesω(k) = ω A = V A |k z | − i 2 νk2 , ω A ≫ A, νk 2 . (A46)This imaginary term may be obtained as the first Newton correction in the method described above (see Eq. (82)).Let us mention that attenuation given by the small imaginary part of the frequency is an isotropic function of thewave vector, cf. Ref. 8, sec. 69, problem.For both branches the oscillation of velocity u are transversal to the wave-vector k. For y-mode the velocityoscillation u is perpendicular to the k-B 0 -plane, this is the true Alfvén wave. For x-mode which we are analyzingin the paper displacement of the fluid and the magnetic force lines lie in the k-B 0 -plane. For finite compressibilityx-mode is hybridized with the sound, that is why often it is called slow magneto-sound wave even if in low magneticfields p B≪ p its dispersion is given by the Alfvén wave dispersion ω A ≈ V A |k · B 0 |/B 0 .APPENDIX A: DISSIPATIVE FUNCTION AND ATTENUATION COEFFICIENT FOR ALFVÉN WAVESIn order to derive the dissipative function for Alfvén waves spreading in a highly conducting plasma, we have toknow the energy and its variation in time due to kinematic and magnetic friction. In this section we are interestedonly in the average dissipation, that is why it is convenient to work with averaged (with respect to the wave’s phase)quantities. The density of electromagnetic energy is given byE Ω≈ B22µ 0, (A1)where we have used that for highly conducting media the electromagnetic energy E 2 /2ε 0 ∝ 1/σ 2 is negligible comparedto the magnetic one. What is important for us is the wave part both the magnetic and the velocity field, as theconstant part does not play any role in the dynamics of the system. Rewritten in terms of the dimensionless wavecomponents the expression for the energy density looks likeE Ω≈ b 2 p B, (2)where the magnetic pressure p B= B0/2µ 2 0 has already been introduced in the text above. Applying the virial theoremfor the Alfvén waves the distribution of the kinetic energy density to the total one may be given byE kin = υ 2 p B. (3)Q tot = Q kin + Q Ω . (4)


25d t V = (d˜t − iAxk z)(U (0) + u) (5)[( )]ρu · d˜t u = ρ 1 + k2 xkz2 u x d˜t u x + u y d˜t u y . (6)d˜t = Ad τ ,⃗υ = iu , k(˜t) = A K(τ), ν ′ = A V A V A VA2 ν. (7)ρu · d˜t u = −ρAV 2 A⃗υ · d τ ⃗υ = −2Ap B⃗υ · d τ ⃗υ (8)( )[⃗υ · d τ ⃗υ = υ x 1 + K2 x Kx K z+ υ y(K 2 z1 + K2 yK 2 z]K 2 υ x − K z b x − ν kK ′ 2 υ x + gf x) [Ky ]K zK 2 υ x − K z b y − ν kK ′ 2 υ y + hf y(9)We take the average value of the upper expression and only the quadratic terms remain. Therefore for the kineticpart of the dissipative function we have[( ) ( )〈ρV · d t V〉 = −2Ap B1 + K2 x Kx K zKz2 K 2 − ν kK ′ 2 〈υx〉2( ) ]−1 + K2 yK 2 zν ′ kK 2 〈υ 2 y〉(10)For K y = 0 this yields〈Q kin 〉 = −2Ap BK 2 z()(1 + τ 2 ) 2 ν k〈υ ′ x〉 2 + (1 + τ 2 )ν k〈υ ′ y〉2−2Ap Bτ〈υ 2 x〉. (11)[( )]ρu · d˜t u = ρ 1 + k2 xkz2 u x d˜t u x + u y d˜t u y . (12)d˜t = Ad τ ,⃗υ = iu , k(˜t) = A K(τ), ν ′ = A V A V A VA2 ν. (13)In the particular case when K y = 0 this givesQ Ω = j · E = j2σ = (rotB)2µ 2 0 σ (14)= 2p Bν m (rotb) 2 = −2Ap Bν ′ m(K × b) 2 . (15)Q Ω = −2Ap Bν ′ mRewritten in terms of dimensionless time this yields[(Kx 2 + Ky)b 2 2 y + 1](K2Kz2 x + Ky2 ) 2b 2x . (16)Q Ω = −2Ap Bν ′ mK 2 z (1 + τ 2 ) [ (1 + τ 2 )b 2 x + b 2 y]. (17)


In this way having in mind thatfor the average density of the total power dissipated in the fluid we have〈υ x 〉 = 〈b x 〉 , 〈υ y 〉 = 〈b y 〉 (18)Q tot = −2Ap Bν ′ K 2 z (1 + τ 2 )((1 + τ 2 )υ 2 x + υ 2 y) + 2Ap Bτυ 2 x. (19)As we have shown above the average density of the total energy may be given byE tot = 2p B[(1 + τ 2 )υ 2 x + υ 2 y]. (20)It is easy to show now that the attenuation coefficient takes the following formγ = − Q tot= A(1 + τ 2 )ν ′ Kz 2 + O( 1 ). (21)E tot τ26APPENDIX A: DYNAMICS EQUATIONSLet us consider the movement of incompressible homogeneous plasma in the presence of magnetic field.evolution in time is given by the system [8]Theρ(∂ t v + v·∇v) = −∇p − 1 µ 0(B × rot B) + η∆v + F,(A1)∂ t B + (v·∇v) = (B·∇)v + 1 ∆B. (A2)µ 0 σWe presume small variations from the stable current (v = U 0 + u , B = B 0 + B ′ , p = p 0 + p ′ ). Then using theostensible form of the velocity in shear flows and the assumptions for the direction of the external magnetic field andvelocity (B 0 = B 0 e z , U 0 = Axe z ), we findAnalogously, for the mixed multipliers we obtainand(v · ∇)v = (U 0 + u) · ∇(U 0 + u) == (U 0 · ∇)U 0 + (u · ∇)U 0 + (U 0 · ∇)u == Au x e z + Ax∂ z u. (A3)(v · ∇)B = (U 0 + u) · ∇(B 0 + B ′ ) == (U 0 · ∇)B 0 + (u · ∇)B 0 + (U 0 · ∇)B ′ =(B · ∇)v = (B 0 + B ′ ) · ∇(U 0 + u) == (B 0 · ∇)U 0 + (B ′ · ∇)U 0 + (B 0 · ∇)u == Ax∂ z B ′ , (A4)= B 0 ∂ z u + AB ′ xe z , (A5)∇p = ∇p ′ , ∆B = ∆B ′ , ∆v = ∆u.The time derivative of both the variables is expressed only by their varying components∂ t v = ∂ t u , ∂ t B = ∂ t B ′ .Having in mind the above mentioned assumptions the vector product of the magnetic filed and its rotation may berewritten in the formB × rot B = B 0 × rot B ′ =⎛⎞(B 0 ) y (rotB ′ ) z − (B 0 ) z (rotB ′ ) y⎝(B 0 ) z (rotB ′ ) x − (B 0 ) x (rotB ′ ) z⎠ =(B 0 ) x (rotB ′ ) y − (B 0 ) y (rotB ′ ) x⎛⎝ −B ⎞ ⎛0(rotB ′ ) yB 0 (rotB ′ ) x⎠ = B 0⎝ ∂ ⎞xB ′ z − ∂ z B ′ x∂ y B ′ z − ∂ z B ′ ⎠y.00(A6)(A7)(A8)


27Consequently, Eqs. (A1) - (A2) turn intoρ(∂ t u + Ax∂ z u + Au x e z ) = −∇p ′⎛+ η∆u + 1 ⎝ ∂ ⎞zB ′ x − ∂ x B ′ z∂µ 0 ρ z B ′ y − ∂ y B ′ z⎠+ F, (A9)0∂ t B ′ + Ax∂ z B ′ = B 0 ∂ z u + AB xe ′ z + 1µσ ∆B′ . (A10)

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