aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Astrophysics and Space ScienceDOI 10.1007/s ••••• - ••• - •••• - • Acoustic α -disk Ya. N. Istomin c (cid:13) Springer-Verlag •••• [email protected]
Abstract
It is shown that the turbulent flow of acous-tic waves propagating outward from the inner edge ofthe disk causes the accretion of the matter onto thecenter. The exponential amplification of waves takesplace in the resonance region, ω = ( n ± ω isthe frequency of the acoustic wave, n is its azimuthalwave number, Ω( r ) is the angular frequency of rotationof the disk. The effect is similar to the inverse Landaudamping in a collisionless plasma. Energy comes fromthe energy of rotation of the disk. That leads to de-crease of the disk angular momentum and to accretionof the matter. The value of the accretion rate dM/dt is˙ M = πrc s Σ ( c s /v φ ) W . Here c s is the speed of soundof the disk gas, v φ is the Keplerian rotation velocity,Σ is the surface density of the disk, W is total power ofthe acoustic turbulence, W ≃ R ∞ dω P n ≥ (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω, n ), | Σ ′ | ( ω, n ) is the spectral power of turbulence. Thepresented picture of accretion is consistent with the ob-served variations of X-ray and optical radiation fromobjects whose activity is associated with accretion ofgas onto them. Keywords accretion, accretion disks
As is well known, the problem of disk accretion is thatfor the Keplerian rotation, v φ ∝ r − / , the specific an-gular momentum rv φ increases with the distance to the Ya. N. Istomin P.N. Lebedev Physical Institute, Leninsky Prospect 53, Moscow119991, Russia Moscow Institute Physics and Technology, Institutskii per. 9,Dolgoprudnyi, Moscow region, 141700, Russia center. In order for matter to fall onto the center, an-gular momentum dissipation required. Taking into ac-count the gas viscosity gives the necessary dissipation.Let us consider the stationary accretion of a viscous gas.Then the φ components of the Navier-Stokes equationyields the relationΣ v r r ∂∂r ( rv φ ) = η s ∂∂r r ∂∂r ( rv φ ) . (1)Here Σ = R ρ ( r, z ) dz is the surface gas density, and η s is the surface viscosity, η s = R η ( r, z ) dz . Sub-stituting from the continuity equation the relationΣ v r = − ˙ M / πr , we obtain the solution of (1) for η s = const ( r ) in the form of power-law functions, v φ ∝ r σ , σ = − , σ = 1 − ˙ M / πη s . For a disk closedto the Keplerian one, v φ ≃ r − / , σ = − /
2, we findthe value of the accretion rate˙ M ≃ πη s . (2)From this relation it follows that the rate of accretionis proportional to the gas viscosity η . Without gas vis-cosity there is no accretion. If we use the value of theclassical viscosity of an ionized gas (plasma) η = 0 . (cid:18) T eV (cid:19) / (cid:18) c (cid:19) g/s cm ( T is the plasma temperature and Λ c is the Coulomblogarithm), then it turns out that classical viscosityis not able to provide the necessary rate of accretionof the typical value of ˙ M ≃ − M ⊙ /y ≃ g/s for observed galactic sources, whose activity is as-sociated with gas accretion. Therefore, in Shakira(1972), Shakura & Sunyaev (1973) the disk model wasproposed in which an anomalous viscosity was intro-duced. It was assumed that the component of the ten-sor of viscous tensions π rφ , which is responsible for the radial transfer of the angular momentum, is pro-portional to the gas pressure p , π rφ = − αp . Mod-els of accretion disks, based on this assumption, arecalled α -models, and disks are called α -discs. Since π rφ = − η ( ∂ ( rv φ ) /∂r ) /r , then for the Keplerian diskthe introduction of the coefficient α is equivalent tointroduction of the anomalous viscosity η = 2 αpr/v φ .Putting the thermal velocity of the gas, p = ρv T / h = rv T /v φ , we obtain η = αρv T h . This corresponds to the kinematic viscos-ity ν = αv T h . Interpretation of this expression is asfollows: there is the anomalous viscosity caused by theturbulence of the gas flow. Then ν = v t l t /
3, wherequantities v t and l t are the characteristic velocity andscale of the turbulence respectively. Assuming l t = h and v t = αv T /
3, we obtain the required expression forthe turbulent viscosity.Thus, the classical (collisional) viscosity can not pro-vide large accretion flows, it is necessary to introduce aturbulence. For a gas disc, in which one can neglect theinfluence of the magnetic field, turbulence is an acous-tic turbulence, i.e. superposition of acoustic waves withrandom phases. In addition to the collisional dissipa-tion, whose effect on the dynamics of the gas disk, aswe have seen, is small, there exists a collisionless dis-sipation mechanism, the well-known example of whichis the Landau damping (Landau (1946)). The Landaudamping is due to the resonance interaction. Acousticwaves in the disc also experience resonance with the az-imuthal rotation. Excitation (or absorption) of wavesin resonances leads to their growth (or attenuation),i.e. to appearance of the collisionless dissipation. Thisis the subject of this work. In the second section we findresonances, in the third and fourth sections we calcu-late the behavior of acoustic waves in resonances, thencalculate the rate of the accretion of a gas due to theresonant interaction.
Let us consider the motion of a gas in a thin accretiondisk. Therefore, it is convenient to introduce a sur-face density of the matter Σ( t, r, φ ) through the usualdensity ρ ( t, r, φ, z ) by the relation Σ = R ρ ( t, r, φ, z ) dz .Here z is the coordinate orthogonal to the plane of thedisk, and coordinates r and φ are the cylindrical coor-dinates in the plane of the disk. In steady state thesurface density Σ does not depend on time and alsoon the azimuthal angle φ , Σ = Σ ( r ). In an arbi-trary perturbed state, the surface density is the sumof Σ and disturbances Σ ′ ( t, φ, r ), Σ = Σ + Σ ′ . Wewill consider disturbances are not very large, Σ ′ < Σ . In turn, the velocity of the matter in a disk hastwo components, v φ ( t, r, φ ) and v r ( t, r, φ ). The sta-tionary velocity v φ is the Keplerian rotation veloc-ity, v φ ( r ) = ( GM/r ) / , where G is the gravitationalconstant, M is the mass of the central object. Veloc-ities also have perturbations, v φ = v φ ( r ) + v φ ( t, r, φ ), v r = v r ( t, r, φ ). We also introduce the Keplerian fre-quency of rotation, Ω( r ) = ( GM/r ) / . As well asthe surface density, we introduce the surface pressureof the gas, P = R p ( t, r, φ, z ) dz . The pressure and thegas density in the disk is connected by the equationof state, p = p ( ρ ). Introducing the speed of sound, c s = ∂p/∂ρ | ρ = ρ , the surface pressure can be repre-sented as P = P ( r ) + ¯ c s ( r )Σ ′ ( t, r, φ ). The value of¯ c s ( r ) is the value of the square of the sound velocityat a certain middle point ¯ z . The perturbed quanti-ties v r ( t, r, φ ) , v φ ( t, r, φ ) , Σ ′ ( t, r, φ ) can be representedas expansions( v r , v φ , Σ ′ ) = ∞ X n = −∞ π Z ( v r ( r, ω, n ) ,v φ ( r, ω, n ) , Σ ′ ( r, ω, n )) exp {− iωt + inφ } dω. Equations of the ideal hydrodynamics for the two-dimensional velocity v r ( r, φ ) , v φ ( r, φ ) have the form ∂v r ∂t + v r ∂v r ∂r + v φ r ∂v r ∂φ − v φ r = − GMr − ∂P∂r ,∂v φ ∂t + v r ∂v φ ∂r + v φ r ∂v φ ∂φ + v r v φ r = − r Σ ∂P∂φ , (3) ∂ Σ ∂t + 1 r ∂∂r (Σ rv r ) + 1 r ∂∂φ (Σ v φ ) = 0 . Substituting quantities Σ = Σ + Σ ′ , v φ = v φ + v φ , v r into the first two equations of the system (3)and linearizing, we get v r ( r, ω, n ) = − i ¯ c s Σ ( ω − n Ω) ∂ Σ ′ /∂r − n ΩΣ ′ /r [ ω − ( n − ω − ( n + 1)Ω] ,v φ ( r, ω, n ) = − ¯ c s Σ Ω( ∂ Σ ′ /∂r ) / − n ( ω − n Ω)Σ ′ /r [ ω − ( n − ω − ( n + 1)Ω] . (4)In deriving equations (4), we neglected the deriva-tive ( ∂ ln ¯ c s /∂r ) in comparison with the derivative( ∂ ln Σ ′ /∂r ) since the first quantity is of the order of r − , while the second is of the order of the inversewavelength of acoustic waves λ , λ ≃ c s /ω << r . We seethat the gas velocity strongly increases near resonances,called Lindblad resonances. And for fixed values of ω and n we have two resonant surfaces, Ω − = ω/ ( n − +1 = ω/ ( n + 1). However, velocities v r , v φ arenot turn to infinity at the resonance, since poles in ex-pressions (4) are not at real values of r , but at com-plex values, since to the frequency ω it is necessary -disk 3 to add a small positive imaginary value, ω → ω + i ω = ( n ± ∝ r − / . For an arbitrary depen-dence Ω( r ), resonances have the following general form ω = n Ω ± κ , where κ is, so-called, epicyclic frequency κ = (2Ω d ( r Ω) /dr ) /r . Under the Keplerian rotation κ = Ω. Substituting the expressions for the perturbed veloc-ities (4) into the continuity equation, we obtain theequation describing density waves ∂∂r (cid:20) r ¯ c s ( ω − n Ω) ∂ Σ ′ /∂r − n ΩΣ ′ /r [ ω − ( n − ω − ( n + 1)Ω] (cid:21) + n ¯ c s Ω( ∂ Σ ′ /∂r ) / − n ( ω − n Ω)Σ ′ /r [ ω − ( n − ω − ( n + 1)Ω] + r ( ω − n Ω)Σ ′ = 0 . As before, neglecting the derivative ∂ ln(¯ c s ) /∂r in com-parison with the derivative ∂ ln Σ ′ /∂r , but leaving thederivative ∂ Ω /∂r , which is important for resonances,we get ∂ Σ ′ ∂r − r nω − ( n − ω − ( n − ω − ( n + 1)Ω] ∂ Σ ′ ∂r + (5) (cid:20) − n r + 3 n Ω r ( ω − n Ω) ω − ( n − [ ω − ( n − ω − ( n + 1)Ω] +[ ω − ( n − ω − ( n + 1)Ω]¯ c s (cid:21) Σ ′ = 0 . Outside resonances, | ω − n Ω | >> Ω, the equation (5)gives the usual dispersion equation for acoustic oscilla-tions. In the quasiclassical approximation, k r r >> ′ ∝ exp { i R k r dr } , we have k = k r ( r ) + k φ = k r ( r ) + n r =[ ω − ( n − ω − ( n + 1)Ω]¯ c s . We consider not global oscillations of the disk, butsmall-scale turbulence, that is k φ r >> n >>
1. Thenfar from resonances ω = ( n ± ω − n Ω) / ¯ c s . Whereinthe dispersion equation has the usual form of acous-tic waves propagating in a medium rotating with thefrequency Ω. It is interesting to note that if a wavepropagates from internal central regions to its reso-nance ( ω < n Ω , ω = n Ω − k ¯ c s ), then its group velocity, ∂ω ( r, k ) /∂k r = − ¯ c s k r /k , is antiparallel to the phase velocity. While passing through the resonance region, ω > n Ω , ω = n Ω + k ¯ c s , the group velocity becomesparallel to the phase velocity, ∂ω ( r, k ) /∂k r = ¯ c s k r /k .It means that the wave propagating from the center tothe periphery before resonance has the negative radialwave vector, k r <
0. Then, having passed the reso-nance, the wave vector k r becomes positive. Approach-ing resonances ω ≃ ( n ± k r passes through zero. Taking into account the azimuthalwave vector k φ = n/r , resonances ( k r = 0) are slightlyshifted, their positions are determined by the relations ω = Ω n ± c s v φ ! / . Since for a thin disk, ¯ c s << v φ , this displacement canbe neglected, and therefore the term n /r in Eq. (5)can be omitted.Such a ’strange’ behavior of the acoustic wave in theinner region ( ω < n Ω) is due to the fact that the waveenergy density E = Σ v / E is equal to E = 12 Σ ( v r + v φ ) + Σ ′ v φ v φ . If the first term is always positive,12 Σ ( v r + v φ ) = ¯ c s Σ | Σ ′ Σ | + ¯ c s Σ n r ( ω − n Ω) | Σ ′ Σ | +¯ c s Σ n Ω r ( ω − n Ω) | Σ ′ Σ | + ¯ c s Σ
16 Ω ( ω − n Ω) | Σ ′ Σ | , and under conditions n >> , ¯ c s << v φ is equal to¯ c s Σ | Σ ′ Σ | , then the summandΣ ′ v φ v φ = ¯ c s Σ n Ω ω − n Ω | Σ ′ Σ | is negative in the inner region ω < n Ω. As a result, theexpression for the wave energy density has the form E = ¯ c s Σ ω + n Ω ω − n Ω | Σ ′ Σ | . (6)In deriving the expression for the wave energy den-sity, we used formulas (4) for the radial and azimuthalvelocities v r and v φ , where the denominator was re-placed by ( ω − n Ω) , and also used the expression k r = ± ( ω − n Ω) / ¯ c s , which is valid far from resonances.Here one also need to keep in mind that the product a (Σ ′ ) means ( a Σ ′ Σ ′∗ + a ∗ Σ ′∗ Σ) /
4, where the sign ( ∗ )denotes a complex conjugation. Thus, we see that for acoustic waves with ω < n Ω max their energy is negative in the inner region Ω( r ) >ω/n, r > r min , Ω max = Ω( r min ). This means thatsuch waves are easily excited in a dissipative medium byso-called dissipative instability (Mikhailovskii (1974)).This is because any dissipation leading to decrease ofthe total energy means growth of the amplitude of thewave of negative energy. Famous Landau damping, which was discovered by himin equilibrium plasma (Landau (1946)), means a colli-sionless phenomenon due to the resonant interactionof the wave with particles. It leads to wave damping.However, in a non-equilibrium medium, for example,a beam of fast particles in plasma, certain waves will,on the contrary, grow. Decrement is replaced by incre-ment. Thus, the term ’inverse’ means not decrease, butincrease of the wave amplitude, and this phenomenonhas the same collisionless character like damping.Near resonances ω = ( n ± k r , vanishing at resonance points,becomes purely imaginary inside the interval betweenthem, k r <
0. This occurs in the range r − < r < r +1 ,where values of r ± are determined the relations ω =( n ± r ± ). For n >>
1, points r ± are located closeto the point r = r , ω = n Ω( r ), r ± = r (1 ± / n ).As a result, we have k r = − ω ¯ c s ( r − r − )( r +1 − r ) r . The purely imaginary value of the radial wave vectormeans amplification or attenuation of waves when theypass through resonance region r − < r < r +1 . The gain(or attenuation) A is equal to | A | = exp Λ , Λ = ± Z r +1 r − | k r | dr = (7) ± πω c s r ( r +1 − r − ) = ± π ωr ¯ c s n . For not too large n , n < ( ωr / ¯ c s ) / , the wave greatlychanges its amplitude, passing through the resonanceregion. For large values of n , the distance betweenthe points r ± becomes less than the characteristicwavelength in the resonance region, and the resonantlayer becomes transparent for the wave. We are in-terested in the case of opacity. In order to determinewhether amplification or attenuation of waves happens,it is necessary to solve the equation (5) in the reso-nance region r − < r < r +1 . The problem is posed as follows: a wave with the negative radial wave vec-tor, k r <
0, but with the positive group velocity, dω/dk r > , k r ≃ ( ω − n Ω) / ¯ c s , is incident onto thelayer r − < r < r +1 from the left side, i.e. from in-ner regions of the disk. Part of a wave can be reflectedfrom the layer, part passes through the layer and be-gins to propagate into the outer region r > r +1 , k r > r > r +1 there exist only the past wave exists, whereasin the region r < r − there are both the incident andthe reflected wave. We start, naturally, from the region r > r +1 . In the neighborhood of r ≃ r +1 the equation(5) takes the form ∂ Σ ′ ∂r − r − r +1 (cid:18) ∂ Σ ′ ∂r − nr Σ ′ (cid:19) + 3 ω ¯ c s n r − r +1 r Σ ′ = 0 . (8)From this equation we see that the characteristicsize λ of the change of the wave amplitude Σ ′ is λ ≃ r (¯ c s n/ ω r ) / . Then, under parameters n < ( ωr / ¯ c s ) / we can neglect the term (2 n/r )Σ ′ in thisequation. Finally we obtain ∂ Σ ′ ∂r − r − r +1 ∂ Σ ′ ∂r + 3 ω ¯ c s n r − r +1 r Σ ′ = 0 . Introducing the dimensionless coordinate x = (3 ω r / ¯ c s n ) / ( r − r +1 ) /r , we obtain the equa-tion ∂ Σ ′ ∂x − x ∂ Σ ′ ∂x + x Σ ′ = 0 . (9)The required solution of the equation (9) for x > ′ = axH (1)2 / (cid:18) x / (cid:19) . (10)Here the function H (1) is the Hankel function of thefirst kind. It describes a wave traveling in the posi-tive radial direction. Its amplitude at x = 0 is equal Fig. 1
The scheme of the passage of the acoustic wave,incident from the left, through the resonant layer. Thereare also the reflected and the past waves. The past waveexponentially increases. Semicircles above the poles showthe rule of their bypass. -disk 5 to − ia / Γ(2 / /π . At large distances from the res-onance, x >
1, the solution goes to the quasiclas-sical wave with the radial wave vector k r ∝ x / ,Σ ′ = a (3 /π ) / x / exp { i (2 x / / − π/ } . Then, atthe distance r − r ≃ r /n << r , when the influ-ence of resonances ω = ( n ± k r = ( ω − n Ω) / ¯ c s .Now we need to analytically continue the solution(10) into the region r < r +1 , x <
0. Recall that thepole ω + i − ( n + 1)Ω( r ) = 0 is not on the real axis r , but under it in the complex plane ( Re ( r ) , Im ( r )),since d Ω /dr <
0. Thus, the bypass of the point r = r +1 must be realized counterclockwise in the up-per half-plane Im ( r ) > x = | x | exp( − iπ ), where | x | ∝ r +1 − r > H (1)2 / goes tothe McDonald function K / , H (1)2 / (2 x / / , x <
0) = − (2 i/π ) exp( − iπ/ K / (2 | x | / /
3) (Abramowitz & Stegun(1964)). Therefore, the solution in the region r − l < r 3. Thus, in the region of the reso-nance the wave ceases to oscillate, and begins to de-cay exponentially when moves away from the point r = r +1 and approaches the point r = r − . It isnecessary to match the solution obtained with a so-lution near other resonance point r = r − l . By the sameway as before, we introduce the dimensionless coordi-nate y = (3 ω r / ¯ c s n ) / ( r − r − ) /r and transform theequation (5). We get ∂ Σ ′ ∂y − y ∂ Σ ′ ∂y − y Σ ′ = 0 . Solutions of this equation are both the MacDonaldfunction, yK / (2 y / / yI / (2 y / / K / that exponentially falls with | x | , it is necessary tochoose the function I / exponentially growing with y ,Σ ′ = byI / (cid:18) y / (cid:19) . (12)Equating the asymptotic values of Σ ′ (11,12) and itsderivatives with respect to the radius r at some point r ∗ , r − l < r ∗ < r +1 , which turns the middle of the seg-ment ( r − , r +1 ) , r ∗ = ( r +1 + r − l ) / r , we obtain the connection between amplitudes a and bb = 2 a exp( iπ/ 6) exp (cid:18) − / ωr ¯ c s n (cid:19) . Finally, we extend analytically the solution of (12)into the region r < r − , y < 0. As we have alreadyestablished, it is necessary to put y = | y | exp( − iπ )in the expression (12), then I / (2 y / / , y < 0) =exp( iπ/ J / (2 | y | / / 3) (Abramowitz & Stegun, 1964).Therefore, the solution in the region r < r − isΣ ′ = − b iπ/ | y | (cid:18) H (1)2 / ( 23 | y | / ) + H (2)2 / ( 23 | y | / ) (cid:19) . (13)Here the Hankel function of the first kind, H (1)2 / , de-scribes a wave incident onto a resonant layer from theinner region of the disk r < r − . It has a negativewave vector, k r < 0. Using the connection betweenquantities a and b , we find that the amplitude of thepast wave a exponentially increases in comparison withthe amplitude of the incident wave − b exp( iπ/ / 2. Itsamplification is A = exp( i π (cid:18) / ωr ¯ c s n (cid:19) . We note that the exponent obtained is in good agree-ment with the expression for Λ (7) found from thequasiclassical approximation, 2 / / . π/ 3. In theregion r < r − , there is also a reflected wave de-scribed by the Hankel function of the second kind, H (2)2 / , k r > , dω/dk r < 0. Its amplitude is equal tothe amplitude of the incident wave.Thus, we see that acoustic waves having not too largeazimuthal numbers n , n < ( ωr / ¯ c s ) / , passing thoughresonant points, experience exponential growth. Sincethe value of r itself depends on n , Ω( r o ) = ω/n , thenthe resonance amplification condition is as follows n < n ∗ = (cid:18) v φ max ¯ c s (cid:19) / (cid:18) ω Ω max (cid:19) / . Values with the index ’max’ correspond to their valuesat the inner edge of the disk. On the other hand, for aresonance to exist, it is necessary to be n > ω/ Ω max .Thus, a high level of turbulence can be for waves withazimuthal wave numbers lying in the range ω Ω max < n < (cid:18) v φ max ¯ c s (cid:19) / (cid:18) ω Ω max (cid:19) / , (14)which is possible for not too large frequencies, ω <ω ∗ = Ω max ( v φ max / ¯ c s ). Waves of this frequency band, ω < ω ∗ , and from the azimuthal wave number region, n < n ∗ , should initially have huge amplitudes, Σ ′ ≃ Σ , since for them the coefficient of the amplification, | A | = exp Λ , Λ > 1, is exponentially large. It is clearthat their nonlinear interaction, decays and fusions ofdifferent harmonics, as well as the inverse influence ontothe rotation profile of the disk Ω( r ), will lead to theformation of a wide range of turbulence with the mostprobable power-law distribution over frequencies ω andwave numbers n , | Σ ′ / Σ | ( n, ω ) ∝ ω − β n − γ .It should be noted that, just as in a plasma withthe inverse Landau damping (for example, beam in-stability), the energy of the waves is drawn from theenergy of motion of the matter. In our case it is fromthe rotation of the disk. Under the Keplerian rota-tion, d Ω( r ) /dr < 0, the amplification of acoustic waves,propagating out, should lead to slow down of the ro-tation of the inner layers of the disk, i.e. equalizingof the angular velocity of rotation. So, for solid rota-tion, Ω( r ) = const , the amplification effect is absent,and for d Ω( r ) /dr > Re ( r ) , Im ( r )) from the upper half-plane to the lowerhalf-plane. Slowing down of the rotation of the disk,associated with the excitation of acoustic waves, leadsto the decrease of the angular momentum of the diskand to possibility of an accretion. In an acoustic wave propagating from the inner edgeof the disk to the periphery, dω/dk r > , k r = ( ω − n Ω) / ¯ c s , the matter moves in the radial direction withthe velocity proportional to the wave amplitude Σ ′ . Farfrom resonances, the radial velocity is (see the equation(4)), v r = − i ¯ c s Σ (cid:20) ∂ Σ ′ /∂r ( ω − n Ω) − n ΩΣ ′ /r ( ω − n Ω) (cid:21) . The mass intersecting a circle of radius r per unit timeequals to˙ M = r Z π dφv r ( r, φ )Σ( r, φ ) = r Z π dφ ( v r Σ ′ + v (2) r Σ )= ˙ M + ˙ M . Here, the velocity v r is the radial velocity of the matter(4), proportional to the first power of the amplitudeΣ ′ . The radial velocity v (2) r is the second-order velocity, proportional to the square of the amplitude Σ ′ . Wefirst calculate the value of ˙ M .˙ M ( ω ) = r π Z π dφ Z dω ′ X n,n ′ v r ( ω ′ , n )Σ ′ ( ω − ω ′ , n ′ ) e i ( n + n ′ ) φ = r Z dω ′ X n v r ( ω ′ , n )Σ ′ ( ω − ω ′ , − n ) . Since Σ ′ ( − ω, − n ) = Σ ′∗ ( ω, n ), we obtain˙ M ( ω ) = r Z dω ′ X n v r ( ω ′ , n )Σ ′∗ ( ω ′ − ω, n ) . Here the sign ’ ∗ ’ means complex conjugation. The ra-dial gradient ∂ Σ ′ /∂r is equal to ik r Σ ′ = i ( ω − n Ω)Σ ′ / ¯ c s .Recall that in the inner region, ω < n Ω, the radial wavevector is negative for a wave propagating in a positivedirection, dω/dk r > 0. As a result, we have˙ M ( ω ) = r ¯ c s Σ Z dω ′ X n (cid:20) in Ω¯ c s r ( ω ′ − n Ω) (cid:21) Σ ′ ( ω ′ , n )Σ ′∗ ( ω ′ − ω, n ) . We will now assume that the acoustic waves are a ran-dom turbulent field, i.e. quantities Σ ′ contain randomphases. In this case averaging h ... i over realization of arandom field gives h Σ ′ ( ω ′ )Σ ′∗ ( ω ′ − ω ) i = 2 π | Σ ′ | ( ω ′ ) δ ( ω ) . The quantity | Σ ′ | ( ω ′ ) is the spectral density of turbu-lence. Since the spectral density is an even function ofarguments, | Σ | ( − ω, − n ) = | Σ | ( ω, n ), then we can re-strict ourselves only to positive frequencies, ω > 0, andto positive azimuthal wave numbers, n > 0. Thus, thevalue of ˙ M = R ˙ M ( ω ) dω , that is the part of the ac-cretion rate, is positive for waves propagating outwardfrom internal areas of the disk, and is equal to˙ M = 2 πr ¯ c s Σ Z ∞ X n ≥ (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω ′ , n ) dω ′ . In order to calculate the second part of the accretionrate ˙ M = r R π dφv (2) r Σ it is necessary to find theradial velocity of the second order v (2) r . Equations forsecond-order quantities v (2) r , v (2) φ , Σ (2) follow from the -disk 7 system of equations (3): ∂v (2) r ∂t + Ω ∂v (2) r ∂φ − v (2) φ + ¯ c s Σ ∂ Σ (2) ∂r = − v r ∂v r ∂r − v φ r ∂v r ∂φ + v φ r − (cid:18)Z ρ dz Z ρ ′ ∂ρ ′ ∂r ∂c s ∂ρ dz − Z ρ ′ dz Z c s ∂ρ ′ ∂r dz (cid:19) ; ∂v (2) φ ∂t + Ω ∂v (2) φ ∂φ + 1 r ∂ ( r Ω) ∂r v (2) r + ¯ c s r Σ ∂ Σ (2) ∂φ = − v r r ∂ ( rv φ ) ∂r − v φ r ∂v φ ∂φ − (cid:18)Z ρ dz Z ρ ′ r ∂ρ ′ ∂φ ∂c s ∂ρ dz − Z ρ ′ dz Z c s r ∂ρ ′ ∂φ dz (cid:19) ; ∂ Σ (2) ∂t + Ω ∂ Σ (2) ∂φ + 1 r ∂∂r (cid:16) r Σ v (2) r (cid:17) + Σ r ∂v (2) φ ∂φ = (15) − r ∂∂r ( r Σ ′ v r ) − r ∂∂φ (Σ ′ v φ ) . Here we are interested in the radial velocity ¯ v (2) r , whichdoes not depend on the time t and on the azimuth angle φ , and is the only one which gives contribution to thesecond part of the accretion rate M . From the secondequation of system (15) it follows that¯ v (2) r = − v r ∂ ( rv φ ) ∂r / ∂ ( r Ω) ∂r . (16)We see that the mean radial velocity of the second order¯ v (2) r appears as compensation of acceleration of the mat-ter in the azimuthal direction caused by the quadraticaction of velocities v r , v φ of the acoustic wave. Sub-stituting values of velocities v r , v φ from (4) into theexpression (16), we obtain˙ M = − ˙ M − π ¯ c s r Σ (cid:18) ¯ c s v φ (cid:19) Z ∞ X n ≥ n Ω ( ω − n Ω) (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω, n ) dω The expression obtained agrees with the relationΣ ¯ v (2) r = − Σ ′ v r + const ( r ) / πr , which follows fromthird equation of the system (15). Therefore, ˙ M =2 πr Σ v (2) r = − ˙ M + const ( r ). Since in our calculationswe have neglected derivatives of slowly varying quanti-ties ¯ c s , Σ with respect to the radius r , then they canbe considered as constants. Finally we have˙ M = − π ¯ c s r Σ ¯ c s v φ Z ∞ X n ≥ n Ω ( ω − n Ω) (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω, n ) dω. (17) Here the sign minus means that acoustic waves propa-gating outward from the inner edge of the disk, where k r = ( ω − n Ω) / ¯ c s , induce the opposite motion of thematter of the disk, i.e. its accretion. This is due to thefact that the wave with fixed values of ω and n , prop-agating in the positive direction r , increases its energy(6), since the Keplerian rotation velocity Ω decreaseswith increasing of r . This is true as for the inner region,where ω < n Ω and the wave has the negative energy,and in the external one, ω > n Ω, where the energy ispositive. Thus, carrying out of acoustic waves outsideshould be accompanied by decrease of the energy of thematter of the disc. Since the energy per unit mass ofthe matter is negative and is equal to − v φ / ∝ r − for the Keplerian rotation, then decrease of the energymeans motion toward the center. It should be notedthat acoustic waves propagating from the external edgeof the disk to the center, k r = − ( ω − n Ω) / ¯ c s , cause theoutflow of the matter, ˙ M > 0. The expression for ˙ M inthis case has the same form as (17) but with the signplus.Let us determine the quantity entering into expres-sion (17) as the effective dimensionless power of acousticturbulence W , W = Z ∞ X n ≥ n Ω ( ω − n Ω) (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω, n ) dω. Then the expression for the accretion rate (17) be-comes | ˙ M | = πr ¯ c s Σ (¯ c s /v phi ) W . Therefore, com-paring the formula (17) with the expression (2) andintroducing the thickness of the disk, h = r ¯ c s /v φ ,one can define the turbulent kinematic viscosity, ν = h ( h/r )¯ c s W/ , | ˙ M | = 3 πν Σ . Thus, the character-istic scale and the turbulent velocity are quantities l t = h, v t = ¯ c s ( h/r ) W respectively, ν = l t v t / 3. Thisdetermines the value of α , α = hr W = hr Z ∞ X n ≥ n Ω ( ω − n Ω) (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( ω, n ) dω. (18)We see that in the case of acoustic turbulence, the pa-rameter α is uniquely determined by the level of turbu-lence W . It should be noted that the presence of thedenominator ( ω − n Ω) in the expression (18) for W does not mean that the resonance gives infinite con-tribution to the integral (18). Expressions obtainedare in the quasiclassical approximation, which does notwork in the region k r ≃ 0. We used the expression k r = ± ( ω − n Ω) / ¯ c s . In fact, the resonance occursin region between points r − , ω = ( n − r − ), and r +1 , ω = ( n + 1)Ω( r +1 ). The distance between points r +1 − r − = ∆ r = 4 r/ n << r determines the min-imum value of the wave vector, k min ≃ ∆ r − . Thismeans that | ω − n Ω | > c s /v φ ) n Ω / Summation over azimuthal numbers n can be re-placed by integration, and the expression for W canbe reduced to the form W = ∞ Z dω − Z ∞ dn ( n − ω/ Ω) ∂∂n (cid:20) n (cid:12)(cid:12)(cid:12) Σ ′ Σ (cid:12)(cid:12)(cid:12) ( n, ω ) (cid:21) . The integral in the right-hand side is the integral in thesense of the principal value and is the Hilbert transfor-mation. It is a power-law function of ω/ Ω with the sameexponent as the integrand function ∂n | Σ ′ | ( n ) /∂n if itis a power-law function of the argument n . Assuming | Σ ′ / Σ | ( n, ω ) = W ( ω ) n − γ , we obtain W = (2 − γ ) cot( πγ ) Z ∞ W ( ω ) (cid:16) ω Ω (cid:17) − γ dω, γ > . The accretion rate onto the star ˙ M a is determined bythe value of | ˙ M | at r = r min ,˙ M a = πr ¯ c s Σ (¯ c s /v φ ) W (cid:12)(cid:12)(cid:12) r = r min . (19)Since, as we see, W ∝ Ω γ − , for stationary accretion itis necessary to establish such distribution of the spec-trum of the acoustic turbulence | Σ ′ / Σ | ( r ) over the ra-dius r , so that the product r ¯ c s Σ (¯ c s /v φ ) Ω γ − W ( r )remains constant. Such dependence of the turbulencepower on the radius for stationary accretion indicatesthat, generally speaking, the quantity α = ¯ c s W/v φ isnot constant along the disk.The picture of turbulent accretion presented herecan be to approved with observations of variations inthe X-ray flux from accreting sources. It is natural toassume that the change in the radiation flux is pro-portional to the rate of the accretion of the matter˙ M a ( ω ). Then we see that the observed power-law spec-trum ˙ M a ( ω ) ∝ ω − β ′ reflects the power-law spectrum ofacoustic waves | Σ ′ / Σ | ( ω ) ∝ ω − β , β = β ′ + 1 − γ . Fordifferent sources the value of β ′ is of the order of unity.So for SS433 β ′ = 1 . f from 10 − Hz to 10 − Hz both in the X-ray and inthe optical ranges of the spectrum of electromagneticwaves (Revnivtsev et al. (2006)). The study of the propagation of acoustic waves in arotating medium has been the subject of many stud-ies since the 1970s (Goldreich & Linden-Bell (1965),Goldreich & Tremaine (1978), Drury (1980), Drury(1985), Papaloizou & Pringle (1984), Papaloizou & Pringle(1985), Papaloizou & Pringle (1987), Narayan, Goldreich & Goodman(1987), Glatzel (1987a), Glatzel (1987b)). The present interpretation of the interaction of waves witha resonant layer is as follows. The r − < r < r +1 layeris a barrier for acoustic waves propagating from theinner regions of the disk to the resonant region. Theradial wave number becomes purely imaginary insidethe layer. The wave cannot propagate there, and theregion r − < r < r +1 is declared forbidden for acousticwaves. They can experience there only the subbarriertunneling onto the allowed domain r > r +1 . There-fore, the amplitude of the wave transmitted throughthe layer is exponentially small compared to the ampli-tude of the incident wave, and the reflected wave canslightly increase only. This conclusion is mainly basedon the consideration of a model problem in which themotion of a liquid (gas) is considered in the frame ro-tating with the angular frequency Ω( r ) (see, for ex-ample, papers of Goldreich & Tremaine (1978) andNarayan, Goldreich & Goodman (1987)). Here Ω( r )is the rotation frequency of the disk in the middle ofthe resonant layer. Further, the unperturbed rotationfrequency Ω is assumed to be linearly falling with thecoordinate r − r , and the cylindrical coordinates ( r, φ )are replaced by Cartesian coordinates ( r, y ) , y = rφ .Moreover, the equation (5) for acoustic waves is simpli-fied, only the first and the last terms in left hand sideof it remain, ∂ Σ ′ ∂r + [ ω − ( n − ω − ( n + 1)Ω]¯ c s Σ ′ = 0 . (20)For a linear dependence of the rotation frequency onthe coordinate r, Ω − Ω( r ) ∝ − ( r − r ), the equation(20) reduces to the standard equation ∂ Σ ′ ∂z + (cid:20) z − C (cid:21) Σ ′ = 0 . (21)Here the coordinate z is, z = (3 ωr / ¯ c s ) / ( r − r ) /r and the constant C is, C = ωr / c s n . The equa-tion (21) is the parabolic cylinder differential equa-tion. It has two fundamental solutions, U ( C, z ) and U ( C, − z ). The first, U ( C, z ), exponentially falls pass-ing through the region − C / < z < C / , another,U (C, -z), independent of U ( C, z ), grows exponentiallyduring the transition from z < − C / to z > C / .The dependence of solutions U ( C, z ) , U ( C, − z ) on thecoordinate z are shown on Figure 2. We dwell hereonto mathematics in details since its misunderstand-ing leads to mistakes. Using real independent functions U ( C, z ) , U ( C, − z ) one can build two independent com-plex functions E ( z ) and E ∗ ( z ) (Abramowitz & Stegun(1964)), E ( C, z ) = k − / U ( C, z ) + ik / U ( C, − z ) , -disk 9 k = (1 + τ − ) / − τ − , τ = exp( − πC ); E ∗ ( C, z ) = k − / U ( C, z ) − ik / U ( C, − z ) . It should be noted that πC = | Λ | (7). Functions E ( z ) , E ∗ ( z ) are convenient to use because for large ar-guments z , | z | >> 1, they become propagating qua-siclassical waves, E ( C, z ) ∝ exp( iz / E ∗ ( C, z ) ∝ exp( − iz / E, E ∗ canbe represented as a superposition of these two. In par-ticular, the handbook (Abramowitz & Stegun (1964))gives the relationship between waves E ( z ) , E ∗ ( z ) , E ∗ ( − z ), E ∗ ( C, z ) − (1 + τ ) / E ( C, z ) = − iτ E ∗ ( C, − z ) . For small values of τ , i.e. large values of πC , the rightside in this ratio is much less than the left. AuthorsNarayan, Goldreich & Goodman (1987) on the basis ofthis relation made the following conclusion. The quote(page 10): ’This equation has a particularly transparentphysical interpretation. It say the ingoing wave E ∗ ( z )of unit amplitude interacts with the forbidden regionaround corotation to produce a transmitted wave of am-plitude τ and a reflected wave of amplitude (1 + τ ) / .’Well, this is the solution falling with z . And where isa growing solution conjugated to it? After all, the realfunction U ( C, − z ), corresponding to U ( C, z ), grows ex-ponentially. Let’s choose another triple of solutions: E ( z ) , E ( − z ) , E ∗ ( − z ). E ( z ) is the wave that passesthrough the resonant layer in the positive direction of z , E ( − z ) is the wave incident on the layer on the left, and E ∗ ( − z ) is the wave reflected from the layer. The rela-tion between these waves is not difficult to obtain usingexpressions of E ( z ) , E ∗ ( z ), defined above, in terms of U ( z ) , U ( − z ). We get E ( C, z ) = iτ − h ((1 + τ ) / E ∗ ( C, − z ) − E ( C, − z ) i . (22)We see that the transmitted wave E ( z ) is τ − timeslarger than the incident one E ( − z ). This is the expo-nentially growing solution. But it is not written in thehandbook Abramowitz & Stegun (1964). And its inter-pretation does not match the above quote. The relation(22) just corresponds to the solution obtained above inthis paper. Formally, mathematics does not make a se-lection between a falling and a growing solutions, theyare equivalent. How to choose a solution correspond-ing to the physical problem? It is necessary to use thephysical principle of causality, which says that only thepast affects the present. In order for the contribution ofthe past not to be infinite, it is necessary to make theamplitude of the plane wave exponentially small in the far past. This is achieved by adding a small positiveimaginary part to the real frequency ω , ω → ω + i z = − C / and z = 2 C / . Since the transmittedwave exists in the region z > C / , we analyze thesolution of the equation (21) near the point z = 2 C / .Introducing the variable u = C / ( z − C / ), we obtainthe Airy equation, ∂ Σ ′ ∂u + u Σ ′ = 0 . We need to take a wave propagating in the positivedirection, so we choose the Hankel function of the firstkind with index 1/3,Σ ′ = u / H (1)1 / (cid:18) u / (cid:19) , u > . (23)The point u = 0 is a branch point, and the continuationof the solution of the (23) onto the region u < u = 0. In the upperhalf-plane of the complex u , u = exp( − iπ ) | u | , or inthe lower half-plane, u = exp( iπ ) | u | . In the first case,the function H (1)1 / (2 u / / 3) will become the MacDonaldfunction K / (2 | u | / / , u < 0, and in the second case- into the modified Bessel function I / (2 | u | / / , u < 0. Since according to the causality principle, on whichwe have already discussed above, the pole lies in thelower half-plane when d Ω /dr < 0. Then it should bepassed above. And we get an exponentially growingsolution. The transmitted wave amplifies, and this isthe amplification of the acoustic wave passing throughthe resonance from left to right, from the region of rapidrotation to the region of the slower rotation. This is theLandau effect of reverse damping, the wave energy isdrawn from the energy of disk rotation. In the oppositecase, d Ω /dr > 0, the pole lies on the top, the waveattenuates. This is the Landau damping. - - x - - U Fig. 2 Fundamental solutions of the parabolic cylinderdifferential equation (21). The solution U ( x ) (solid line)falls exponentially, the solution U ( − x ) (dashed line) growsexponentially in the region − < z < 2. Here C = 1. Thus, the interpretation of the passage of acousticwaves through resonance as the subbarrier tunneling ,which still exists until now, is not correct, it was basedon incorrectly understanding mathematical formulas.In addition, in later works (Tsang & Lai (2008),Tsang & Lai (2009)), equations of the type (20) tak-ing into account small derivatives d (ln(Ω / Σ)) /dr ≃ r − << λ − were analyzed by authors to identifythe effects of superreflection and instability of acousticwaves during their passage through the resonant layer.Small increments were found depending on the mag-nitude and sign of the derivative d (ln(Ω / Σ) /dr . How-ever, for some reason not specified by the authors, so-lutions in the domains r − < r < r +1 and r < r − were obtained by continuing solutions from the domains r > r +1 and r − < r < r +1 , respectively, replacing r − r +1 ( r > r +1 ) by ( r +1 − r ) exp( iπ )( r < r +1 ) and re-placing r − r − ( r > r − ) by ( r − − r ) exp( iπ )( r < r − ),respectively (see formulas (26,30,31) of Tsang & Lai(2008)). This corresponds to a bypath of the pole inlower half-plane, which in our case contradicts the Lan-dau’s rule of bypath of a pole for d Ω /dr < We have shown that the collisionless dissipation of theangular momentum of a disk leads to the accretion ofmatter onto the center. This requires the existence of aturbulent flow of acoustic waves propagating from theinternal edge of the disc outwards. Passing throughthe resonance regions, ω = ( n ± α here is the turbu-lence power W (18), which has a quite definite physicalmeaning. The value of W can be calculated for a con-crete system in which an accretion disk is formed. Wecan also connect variations of radiation in the X-ray andthe optical ranges ˙ M ( ω ) with properties of an accretiondisk.Finally, it should be noted that in the presence ofa magnetic field in an ionized disk, when the Alfv´envelocity exceeds the acoustic velocity, it is necessary to consider a magnetosonic turbulence. However, theresonance in the region r − < r < r +1 will be of thesame nature, since it is associated only with the neigh-bourship of the frequency of the wave ω to the angularvelocity of the disk rotation Ω( r ). Specifically, whathappens with a magnetosonic turbulence in a magne-tized disk requires special consideration. This work was supported by Russian Foundation forBasic Research, grant number 17-02-00788. -disk 11 References Abramowitz, M., Stegun, I. 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