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Turbulization of Shear Flows in Astrophysics G.D. CHAGELISHVILT, R.G. CHANISHVILT, AND J.G. LOMINADZE Abastumani Astrophysical Obsenatory INTRODUCTION According to a most widely spread representation (Shakura and Sun- yaev 1973; Pringle 1981; Liang and Nolan 19843, the disk accretion phe- nomenon is based on the anomalous transport of the accreting matter angular momentum outward due to turbulent viscosity. It is also the turbulence which provides such an accretion law that a bulk of released gravitational energy of the accreting matter in the disk thermalizes, and as a result, is emitted and provides the observed X-ray spectrum for some bi- na~y sources. Consequently, the investigation of the turbulence emergence in accretion disks is a problem of paramount importance (of course, to elucidate the possibility of turbuli7~tion of other shear flows in astrophysics is also important). Turbulence can be created in accretion disks by a superdiabatic pres- sure gradient across a disk (in the regions where such a gradient exists) and by differential rotation of the matter, i.e. by a shear of angular velocity In the Keplerian rotation of the matter, Q ~ r -3/2 (Shakura et al. 1978~. The superadiabatic pressure gradient resulting in thermal convection is charac- tenstic of only some parts of the accretion disk (Taylor 198Ct, Lominadze and Chagelishvili 1984; Chagelishvili et al. 19861. Therefore, in Me other parts of the disk the presence of the turbulence can be ensured only by the shear instability. The problem of instability in shear flows is treated in quite a number of works (Narayan et al. 1987). They give us a global analysis of irrotational (potential) modes in a two-dimensional compress- ible shear flow, implying that, in the presence of reDect~g boundanes, those modes can increase and that the characteristic time of the increase 55
56 AMERICAN AND SOVIET PERSPECTIVES Is several orders as high as the dynamic time in the medium. Due to the slowness of the increase of irrotational modes and to the problematical character of the theories applicability as presented in Narayan e! al. (19g7) for thin accretion disks, we have tried to develop an alternative theory of turbulization shear Dows. It is quite contrary to the above theory. Instead of the global analysis, we make a local one, far from the flow boundaries. Instead of the irrotational perturbations, we examine vertical ones. Lastly, we consider the medium to be incompressible. To gain our aim, the Dow in the accretion disk can be modeled by a plane Couette flow, without regard for boundary conditions: it can be modelled by a plane shear Dow in infinite incompressible fluid. According to the present scenano, there is a critical perturbation level in the free shear flow. If the level of initial perturbations exceeds the critical one, the flow is turbulized. We must particularly emphasize that the flow can only be turbulized by non-potential (vortex) perturbations. The fact that the turbulization comes at finite initial perturbations is explained by Me non-ordinary nature of the temporal evolution of the vortex perturbations at the linear stage. Such a non-ordinarily for some astrophysical phenomena was found for the first time in the work of Lominadze et al. (1988~. The distinguishing feature of their analysis, as well as of Goldreich and Linden-Bell (1965), is that the linear equations are integrated in a co-moving coordinate system with moving axes (X~OY~. That is to say that two local coordinate systems are used to analyze equations (see Figure 1~: a Cartesian one (XOY) with the Y-ans pointing the velocity direction and X-a~ns directed orthogonally, along the shear of the velocity, and another system, with moving axes (X~OY'). The Y~-axis is parallel to the Y-ams, and the X~-ams moves together with the unperturbed flow. (Lee relation between (x,y) and (x:,y~ ~ coordinates is given by equation (4~. The idea of the paper is as follows: equations are projected onto the Cartesian coordinate system axes. Then we substitute the variables using equation (4) so as not to change the sense of the projection of the physical quantities. That is, we do not make a physical transition from the XOY frame to the X:OY~, but a formal substitution of variables that makes the analysis easier. In fact, the linear theory equations (6~-~83 written in the terms of the new variables x and y: are uniform in regard to them Nut they are already not uniform in regard to time). So, expanding a perturbance into Fourier modes in regard to x: and ye, one can follow separately the temporal evolution of every Fourier mode. After returning to the former coordinates, we can see that besides the time variation of every Fourier mode amplitude, there is variation of its wave number along the X-a~ns in the direction of the velocity shear of the main flow (see equation (19~. There is a pecularity In the temporal evolution of the Fourier modes: the evolution law is not exponential, but power one (see equation (18~. The growth can take place
HIGH-ENERGY ASTROPHYS CS AX ~ X' _ _4 _ ~ l 57 Y Y4 FIGURE 1 The local Cartesian coordinate system XOY and the system with moving axes XlOY1 are presented. The dashed arrows show the direction of the main flow velocity Uo at different distances from the Yeats. Uo(O,Uo3~0~; Uo' = Ax; A > 0. X~-ams moves together with the shear flow. when I*-k=(t)/kyA > 0, and where knots and kit, are wave numbers of the Fourier modes under consideration along the velocity shear and the main stream velocity, respectively, but in the XOY coordinate system. A is a characteristic parameter of the velocity shear ~Jo, = Ax). When writing the wave number k=(t) we especially emphasize the time dependence to stress that the spatial scale of Fourier modes varies in time along the X-axis. We must also stress that the Fourier expansion was done in terms of the variables x, and ye (see equation (93), k= (t) and ky being the wave numbers of those Fourier modes in the XOY system. From the condition k~(t)/~3,A > 0 and equation (183 one can see that, according to linear theory, the growth time of every Fourier mode is limited. At first a Fourier mode draws on energy from the main Dow and grows; after some time k~(t) changes its sign, and We growth of the Fourier modes turns into weakening: they "return" the energy back to the main flow. It follows from the above that the possibility of turbulization of free
~8 AMERICAN AND SOVIET PE~PECT~S shear cows is ambiguous. Indeed, if you start from infinitesimal perturba- tions, you may assume that because of a limited growth time of Fourier harmonics they will not grow up to non-linearity, that is, up to such values where non-linear effects come into play. So the evolution of the perturba- tion will be purely linear, and it will disappear '~without a trace:" with no turbulization in the media. But if the level of initial perturbations in the media is not too small, then at a certain stage of increase of the Fourier harmonics, non-linear (cascade) processes come into play that can provide turbulization of the flow. LINEAR THEORY OF PERTURBATION Let us consider the linear theory of the temporal evolution of two- dimensional perturbances irk a plane, free-shear flow of incompressible fluid. 1b get the growth effect alone, let us first consider a nonviscous fluid. Allowance for viscosity won't be too difficult, so our result will be generalized for the case of a viscous fluid (see below3. Let us direct the Y-a~ns of the Cartesian coordinate system along the main how velocity UO (O,Uoy, O), and the X-axis along the shear of the flow velocity Uoy = Ax (see Figure 1~. Considenng the problem to be two-dimensional, we write the equations of continuity and motion for the perturbed quantities: ou=+Buy=O, fix by {~3~+Ax~ }u==-NIP, {it +A~;~3 }u y+Au~ (1) (2) UP (3) where us and U3, are the components of the perturbance velocity in the Cartesian coordinate system, P is the pressure perturbance normalized by the density of the matter p. Now we introduce a coordinate system with moving axes XENON. Its ongm and the Y~-a~s coincide with the respective characteristics of the XOY system; and the X:-a~s moves together with the undisturbed Dow (see Figure i). This is equivalent to the change in variables Hi =~; ye =y-Ant; to =t, (4) or
HIGH-ENER~Y ASTROPHYSICS _ = _-AX1- With the new vanables, equations (1)-(3) take on the form (A _ At1 A ) Uz + A Y = 0, A Z =-{ A -Atl by }P.' At + Auk =-{ A }P 59 (5) (6) (7) (8) As it was already noted in the introduction, the substitution of the variables (4) is not a physical transition to the new frame, because in equations (63-~8),aswellasin equations (1~-~3), the quantifies us end Al are components of the perturbation velocity in a Cartesian coordinate system. The coefficients of the initial linear equation system (1~-~3) depended on the space coordinate x. After our transformations, this inhomogeneity was changed into a time inl~omogeneity. So we can perform a Fourier analysis of equations (6~-~8) in respect to the variables x~ and yl: ( us ~ += { u=(k=~' Kim, t) ~ By ~ = /j ~k=~dly~ ~ uy~k=~'ky~'t) ie~(ik=~xi +ily~yi) (9) ~ P ) -= ~ p~k2~,ky~,t) J Substituting expansion (9) into equations (6~-~8), we obtain Okra-key Ati fun ~ by: By = 0 =-take-key Ati jP, 0tY + Au2 =-ily~P, Solving equation system (10~-~12), say, for velocity, we get u2(k k ti) = Uzbeks ky o) (k~l/kyt) + ~ U (k k t -U (k k o)[(k~l/kYl}201](1-Atlkyl/k2l) Y =1' Y1' 1 - y 21, Y1' (10) (11) (12) (13) (kxl/ky1-Atl)2 + ~, (14)
60 AMERICAN AND SOVIET PERSPECTIVES u(kxl,kyllil) = {u2 + uy}2 = [{k~l/kY1-Atl } + 1] . (15) As one can see from these formulae, only those Fourier harmonics of the initial perturbation grow the wave numbers of which satisfier the inequality k HA > 0' and the growth takes place in time t < t*, where t* _ key /kyl A (16) (17) It must be stressed that the solutions obtained here exist when the initial vorticity Is non-zero. Using notation (17), we may write solution (13) in the form UX(k2~, kin, ti) = u=(k2t, ky:, O) A2(~* t)2 + 1 (18) As one can see from this formula, for I* ~ 1 at first (as long as Aft*- t) > 1) the growth of a Fourier harmonic is rather explosive: the growth rate of the perturbations greatly increases in time. However, it is only towards the end (when Aft*-t) ~ 1 that the growth stops abruptly and (when t > t*) a weakening follows (see Figure 2a). We have performed Founer analysis of perturbances for the variables x1 and ye and followed the time evolution of Founer harmonic amplitudes. What happens to Founer harmonics in an ordinary space, be., in the Cartesian coordinate system (XOY)? Using equations (43 and (9) one can introduce a parameter that determines the characteristic linear dimension of each of the Founer harmonics along the awns X and Y at every particular instant knits = k2'-ky:Ati, by = Aye. (19) We can see from these formulae that the space scale along the X-axis of every Founer harmonics under examination varies in time. Viscosity plays an important role in many hydrodynamic flows, He more so In turbulization phenomena. For some astrophysical flows, the viscosity was considered in terms of Lominadze et al. (1988) in the work of Fridman (1989~. Using the results of the latter work, we can easily
(t) kx '1 HIGH-ENERGY ASTROPHYSICS O (kX1, k%2, t) - Jim . l \! (a) (b) t 61 FIGURE 2 (a) Time evolution of the amplitude of a perturbation velocity Fourier ilaI-mOIliG The graph is plotted in accordance tenth equation (15~. (by lime dependence of the spatial scale along the X-a~ns of Fourier harmonics increasing at initial time ~2~0~A > 0~. The graph is plotted in accordance with equation (19~. generalize our solutions to a viscous fluid too. For instance, ~ the case of a viscous fluid equation (13) takes the form u~(k=t, kyt,ti) = u2(k=t, key 0) kin' + k2 em {_u ~ [k2(t) ~ ky2] dti, (20) where z, = pip is the kinematic viscosity. Naturally, osmosis impedes perturbation grown. For every specific z, one can find Me minimum characteristic dimension of perturbations (maximal k-~/~) at which Fourier harmonics can increase.
62 AMERICAW AND SOVIET PERSPECTIVES QUALITATIVE SCENARIO OF TURBULIZATION Let us now describe the processes which can lead, we presume, to turbulization of shear flows. Of course, such a process represents an aggregate of linear and non-linear phenomena. Let us assume that the linear phenomena proceed in accordance to the theory exposed in the previous section. Then any initial perturbation can be presented as a set of Fourier harmonics. For the following discussion it is convenient to introduce a notion of a k-plane (kooky). Without a loss of generality, we shall also hold that A > 0. Let us follow linear dynamics of Fourier harmonics. In addition to amplitude variations of Fourier harmonics, their characteristic scales vary in the direction of the velocity shear: along the X-axis (see equation (19~. This variation on the kooky plane means a drift of Fourier harmonics in the direction shown by the arrows in Figure 3. The Fourier harmonics that satisfy condition k=(O)/ky < 0 (k=~/ky < O), weaken. The Fourier harmonics that satisfy the condition k2 (0~/ky > O at first draw on energy from the main flow and increase, while skim decreases. Each Fourier harmonics grows till k=(t) becomes zero (see Figure 2a and 2b). When it becomes zero, knots reverses sign, as it follows from the linear theory, and the growth changes to weakening: the Fourier harmonic returns the energy back to the main flow. Hence it is clear that if non-linear phenomena do not come to play, any initial perturbations will disappear without leaving a trace. A turbulence cannot exist without a permanent energy input from the main flow to perturbations. According to equation (16), when A > O. this takes place if there are Fourier harmonics with wave numbers corresponding to the first and third quarters of the k=Ok3, plane (see Figure 3) which we shall refer to as a "growth area." But because of the drift of Fourier harmonics from the "growth area," the linear mechanism cannot provide a permanent input of energy from the main flow to the perturbation. Actually, if at me zero time (t = 03, in addition to w~ken~g Fourier harmonics, we have a packet of increasing Fourier harmonics (the packet that fills the cross-hatched area in Figure 3), the packet will drift in the k=Ok3, plane (! ,11 ,111), so that ~ due time (t = to) it will come to be outside the "growth area." Thus, later on the "growth area" will be empty and there will be nothing to draw energy with from the main flow. What can the non-linear phenomena result in? They take no part in energy exchange with We main Dow (Joseph 1976~. But they redistn~ute energy among the Fourier harmonics. So the non-linear phenomena may return a part of the energy of the above packet back to the "growth area," as a result of the decay processes (k' ~ k" = k, see Figure 3~. Then Fourier harmonics which are able to take energy from the main flow reappear and drift in the kooky plane. It will be easily understood that if there is a
HIGH-ENERGY ASTROPHYSICS t_~`' hi\! I,, , , ~ 1 , , l, t t t\ 1 1 1 ~ 1 1 1 ~ 1 It I ~ ~ ~ , Al - ' rig ~ ~I ~I ~ I/ ~ 1\ t = 0 /l ~ l\\~ ~t=t'i /} ~ ~ T\~l 1 ~ ~1 A> 0 63 t =r FIGURE 3 The short arrows show the directions of the drift of Fourier harmonics described in the linear theory in the k space (when A > 0~. A drift of Fourier harmonic packet filling cross-hatched area I in the k space ~ HI ~III) is shown. After some time (t = tat, the packet comes to be outside of the "growth area" (k~(t"^A < 0~. An _ _ _ example of the decay process k' + k" = k which can return a part of the perturbation energy back to the "growth area" (k~(t)/KyA > 0) is given. , permanent non-linear return of a part of perturbations to the "growth region," the now will be turb~li~ed. Thus the comparative analysis of linear and non-linear mechanisms gives us a general idea of the dynamics of Fourier harmonics and energy redistribution among them. What can be observed at small initial perturba- tions? Because of the linear drift, the Fourier harmonics will be driven out of the "growth area," and because of the smallness of the amplitude, the decay processes will be weak and will not be able to resist the permanent linear theory drift. As a result, the perturbation will be damped without inducing turbulization in the media. The higher the amplitude of initial perturbations, the stronger the effects of non-linearity. Finally, at a cer- tain amplitude (which, of course, must depend on the initial perturbation spectrum and the value of the viscosity coefficient), the decay processes will be able to compensate the linear drift of the Fourier harmonics and ensure a permanent "drawing" of energy from the main flow and lead to the turbulization of the media. So, according to the present scenario, Were
64 AMERICAN AND SOVIET PERSPECTIVES is a critical level for initial perturbations in the free shear flow. If the level of initial perturbations exceeds the critical one, the flow gets turbulized. The local theory of turbulization of a plane Couette flow presented here can be applied not only to accretion disks, but to other shear flows in astrophysics, such as galactic disks, planetary rings, protostellar nebulae and possibly, to some rotating stars as well We think that beyond astrophysical applications, this theory is a ray of hope for explanations of turbulization of some "earthly" hydrodynamic flows which are steady to infinitesimal perturbations but are turbulized by finite ones. Of course, our theory is qualitative and needs more rigorous mathematical formulation in order to be quite reliable: a theory of weak and strong turbulence is to be developed. We have developed a theory of weak turbulence, but the extent of calculations does not allow us to include it into the present paper. Here we only give ache final equation of the theory of weak turbulence: Seek _k Amok _ 2k~kyA id, ~ 7,{~2 ~ E21~ = k2 ~ k2~,tT',`~24~y,~,¢-A ~ T y ~ / dk'dk"~(k-k'-kin. (kit_ k=) k k ky2ky'2{[t~b(k"'ky''t)-¢(k',ky,t)] {Ex,Ex,, - 2k,2 k92 [~(k",ky/,t)-~b(k=,ky,t)]ExEx,'}. (21) Here Ek is the energy density of a Fourier mode of a perturbation with fixed k, and ?{ = k2 arctg 11 + k~/ky(`ks/ky - At) RE~:RENCES Chagelishvili, G.D, RG. Chanishvili, and J.G. Lommadze. 1986. Proceedings of the Joint Varenna-Abastumani Intemational School and Workshop. Sukhumi, USSR, (ESA SP-251), p. 563. Fridman, NM. 1989. P~s'ma Astran. Zb., in press. Goldreich, P., and D. Lynden-Bell. 1965. MNRAS 130:125. Joseph, D.D. 1976. Stability of Fluid Motions. Springer Verlag, New YorL I iang, E.P., and P.L. Nolan. 1984. Space Sal. Rev. 38:353. Lominadze, J.G., and G.D. Chagelishvili. 1984. Astron. Zh. 61:290. Lominadze, J.G, G.D. Chagelishvili, and RG. Chanishvili. 1988. Pis'ma Astron. Zh. 14:856. Narayan, R., and P. Goldreich. 1987. MNRAS 288:1. Mingle, J.E. 1981. Ann. Rev. Astron. and Astrophys. 19:137. Shakura, N.I. and RA Sunyaev. 1973. Astron. and Astrophy~ 24:337. Shakura, N.I., RN Sunyaev, and S.S. Zilitinkevich.1978. Astron. and Astrophys 62:179.
