ON THE NATURE OF ANGULAR MOMENTUM TRANSPORT IN NONRADIATIVE ACCRETION FLOWS Steven A. Balbus and John F. Hawley

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1 The Astrophysical Journal, 573: , 2002 July 10 # The American Astronomical Society. All rights reserve. Printe in U.S.A. ON THE NATURE OF ANGULAR MOMENTUM TRANSPORT IN NONRADIATIVE ACCRETION FLOWS Steven A. Balbus an John F. Hawley Deptartment of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903; sb@virginia.eu,jh8h@virginia.eu Receive 2002 January 25; accepte 2002 March 19 ABSTRACT The principles unerlying a propose class of black hole accretion moels are examine. The flows are generally referre to as convection-ominate an are characterize by inwar transport of angular momentum by thermal convection an outwar viscous transport, vanishing mass accretion, an vanishing local energy issipation. In this paper, we examine the viability of these ieas by explicitly calculating the leaingorer angular momentum transport of axisymmetric moes in magnetize, ifferentially rotating, stratifie flows. The moes are estabilize by the generalize magnetorotational instability, incluing the effects of angular velocity an entropy graients. It is explicitly shown that moes that woul be stable in the absence of a estabilizing entropy graient transport angular momentum outwar. There are no inwar-transporting moes at all, unless the magnitue of the (imaginary) Brunt-Väisälä frequency is comparable to the epicyclic frequency, a conition requiring substantial levels of issipation. When inwar-transporting moes o exist, they appear at long wavelengths, unencumbere by magnetic tension. Moreover, very general thermoynamic principles prohibit the complete recovery of irreversible issipative energy losses, a central feature of convection-ominate moels. Dissipationless flow is incompatible with the increasing inwar entropy graient neee for the existence of inwar-transporting moes. Inee, uner steay conitions, issipation of the free energy of ifferential rotation inevitably requires outwar angular momentum transport. Our results are in goo agreement with global MHD simulations, which fin significant levels of outwar transport an energy issipation, whether or not estabilizing entropy graients are present. Subject heaings: accretion, accretion isks black hole physics convection instabilities MHD turbulence 1. INTRODUCTION 749 Originally evelope to be powerful luminosity sources, black hole accretion moels are now confronte by an embarrassing plethora of unerluminous X-ray emitters, the best known of which is the Galactic center source Sgr A* (Melia & Falcke 2001). These low-luminosity objects are thought to be prime caniates for a class of theoretical accretion moels that has been intensively stuie in recent years, an that we shall refer to generically as nonraiative accretion flows. Accretion generally requires significant energy loss, an the absence of raiative losses in these flows means that etermining the ultimate fate of the gas is less than straightforwar. Much of the recent interest in nonraiative flows was sparke by the work of Narayan & Yi (1994), who examine a series of one-imensional, self-similar, steay accretion moels referre to as avection-ominate accretion flows, or ADAFs. In these moels, an viscosity allows angular momentum transport an the resulting flows are quasi-spherical an substantially sub-keplerian. Dissipative heating increases towar the flow center, creating an inwarly increasing entropy profile which we shall refer to as averse. Nonraiative accretion flows are amenable to numerical simulation. Hyroynamical simulations carrie out with large assume values (>0.3) show some similarities to ADAFs (Igumenschev & Abramowicz 1999, 2000), but smaller values of le to flows with substantial turbulence, smaller than anticipate net inwar mass accretion rates, an ensity istributions that are far less centrally peake than in ADAFs (Stone, Pringle, & Begelman 1999; Igumenschev & Abramowicz 1999, 2000). Both inwar an outwar mass fluxes were observe at ifferent times an in ifferent locations within the flow, an they nearly cancelle. These finings were given the following interpretation by Narayan, Igumenschev, & Abramowicz (2000, hereafter NIA), Quataert & Gruzinov (2000), an Abramowicz et al. (2002, hereafter AIQN). The averse entropy graient triggers an instability, an significant levels of convection result; hence, the global solutions were calle convectionominate accretion flows (CDAFs). The next step in the argument is key: invoking the finings of other hyroynamical simulations (Stone & Balbus 1996; Igumenschev, Abramowicz, & Narayan 2000), the angular momentum transport generate by the convective turbulence was claime to be inwar. This inwar transport by convection was envisione to be great enough to cancel the primary outwar angular momentum transport by whatever process the viscosity was moeling presumably the magnetorotational instability, or MRI (Balbus & Hawley 1991). In the CDAF scenario, the vanishing of the angular momentum flux implies that the R component of the stress tensor responsible for accretion also vanishes (NIA, AIQN). This has the further consequence that there is essentially no mass accretion an no issipation, espite the presence of vigorous turbulence throughout the bulk of the flow. The only region where there is any mass accretion in the moel is at the very inner ege of the flow. All of the energy release associate with this small net accretion is transporte outwar to infinity by the surrouning convective flow, which is maintaine with no further issipative losses. For this reason, CDAFs are put forth as natural caniates to explain uner-luminous X-ray sources. These moels have been elaborate upon, becoming influential

2 750 BALBUS & HAWLEY Vol. 573 an wiely cite. Since black hole accretion moels are central to our unerstaning of much of X-ray astronomy, the theoretical founations for CDAFs eserve careful scrutiny. In this work, we carry out an explicit analysis of magnetize, rotationally supporte gas in the presence of an averse entropy graient. We fin that CDAF moels have two major inconsistencies. First, locally unstable isturbances with averse entropy graients o not generally transport angular momentum inwar in magnetize fluis. Rather, they generally transport angular momentum outwar. Qualitatively, their behavior is inistinguishable from stanar MRI moes. This latter point has been emphasize elsewhere (Hawley, Balbus, & Stone 2001; Balbus 2001), but here we emonstrate it quantitatively by explicitly calculating the leaing orer angular momentum transport associate with unstable WKB moes. Convective moes transport angular momentum outwar when magnetic tension is significant an inwar only for the very longest wavelength (global-scale) isturbances, where magnetic tension forces are negligible. Inee, for a given wavenumber, the irection of angular momentum transport is less a matter of whether it is estabilize by convection or rotation an more a matter of the nature of backgroun meium: is it effectively magnetize or not? This is the crucial issue. The secon ifficulty is more irect an funamental, affecting magnetic an nonmagnetic moels alike. By relying upon issipate heat energy to trigger a convective instability that supposely reners the flow issipation-free, CDAFs run afoul of thermoynamic principles. If the source of the free energy is ifferential rotation, the irection of angular momentum transport must be outwar. This is a serious inconsistency. The issipation is quite significant an is inee essential if convectively unstable entropy profiles are to be sustaine. We are le to a much more stanar picture of the ynamics of turbulent accretion flows, though one at os with the tenets of CDAF theory. The turbulent stress tensor in magnetize ifferentially rotating gas oes not vanish. There is vigorous local turbulent issipation. There is mass accretion. The near cancellation of instantaneous inwar an outwar mass fluxes is a property of any turbulent flow with large rms fluctuations an not a superposition of the contributions from two istinct sources of mass flux with opposite signs. In the following sections, we present the etails of the angular momentum calculation showing outwar transport (x 2), an explanation of important thermoynamic inconsistencies evient in CDAF theory (x 3), an a brief review of numerical simulations an a concluing summary (x 4). 2. RADIAL ANGULAR MOMENTUM TRANSPORT 2.1. Local WKB Moes Consier a isk with raially ecreasing outwar entropy an pressure graients. We use stanar cylinrical coorinates ðr;;zþ. The square of the Brunt-Väisälä frequency (N 2 ) is thus negative an tens to estabilize. In what follows, it is convenient to work with the positive quantity N 2 N 2 ln P 5=3 > 0 : The isk is ifferentially rotating with ecreasing outwar angular velocity ðrþ an epicyclic frequency 2 ¼ 4 2 þ 2 ln R ¼ 1 R 4 2 R 3 R > 0 : ð2þ A vertical magnetic fiel B ¼ Be z threas the isk. Its associate Alfvén velocity is v 2 A ¼ B2 =4, where is the gas ensity. Axisymmetric WKB plane wave isplacements of the form ðr; Z; tþ ¼ expðikz i!tþ ; ð3þ where k an! are, respectively, the vertical wavenumber vector an the angular frequency an lea to the ispersion relation (Balbus & Hawley 1991) where ~! 4 þ ~! 2 ðn 2 2 Þ 4 2 ðkv A Þ 2 ¼ 0 ; ð4þ ~! 2 ¼! 2 ðkv A Þ 2 : ð5þ Let ¼ i!. Then, the unstable branch of the ispersion relation (eq. [4]) is 2 ¼ ðkv A Þ 2 þ 1 2 N2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðn 2 2 Þ 2 þ 16 2 ðkv A Þ 2 : ð6þ It is straightforwar to show that these unstable moes must have ðkv A Þ 2 < N 2 2 ln R ¼ N2 þ 2 ln R ; ð7þ an that the maximum growth rate is max ¼ N 2 ln 2 þ 4 2 ln R ; ð8þ which is attaine for wavenumbers satisfying " # ðkv A Þ 2 max ¼ 2 1 ðn2 2 Þ : ð9þ 2.2. Stress Calculation The angular momentum flux is irectly relate to the R component of the stress tensor T R ¼ ðv R v v AR v A Þ ; ð10þ where enotes an Eulerian perturbation an v A ¼ p B ffiffiffiffiffiffiffiffi : ð11þ 4 For the local WKB moes we consier here, the angular momentum flux is RT R. Hence, the sign of the transport is simply the sign of T R. The neee expressions can be written own immeiately from equations (2.3c 2.3g) of Balbus & Hawley (1991). In terms of, they are " v v R ¼ðv 2 R Þ ðkv A Þ 2 # ln 2 D 2 ln R 2 2 ; ð12þ

3 No. 2, 2002 ANGULAR MOMENTUM IN ACCRETION FLOWS 751 v A v AR ¼ðv AR Þ 2 2 D ¼ðv RÞ 2 kv 2 A 2 D ; where ð13þ D ¼ 1 þ ðkv AÞ 2 2 : ð14þ These equations are general beyon our simple example, holing in the presence of both vertical an raial entropy graients. Note that there is no explicit epenence upon N; the only epenence upon N anywhere is through the growth rate. The Maxwell stress (eq. [13]) must always be positive. For a given growth rate, angular momentum transport is completely etermine by rotation an magnetic tension. Figure 1 shows the stability an angular momentum transport properties of an unmagnetize Keplerian isk in the k 2 -N 2 plane. The vertical scale is immaterial: when N >, convectively unstable moes are triggere at all wavenumbers. There is nothing special about the physics of convection per se with regar to inwar angular momentum transport. In the absence of a magnetic fiel, any axisymmetric isturbance governe by the above equations woul transport angular momentum inwar. This is why hyroynamic simulations consistently fin inwar transport. Matters change raically when a magnetic fiel is present. By combining equations (6), (12), an (13), we may calculate the full range of wavenumbers that transport angular momentum outwar: ðkv A Þ 2 > 1 ln ln R 2N 2-2 : ð15þ It is apparent that unless N is sustaine at values in excess of =2 1=2, every moe, convective or otherwise, will transport angular momentum outwar. The content of equation (15) is shown graphically in Figure 2. Clearly, the ominant irection of angular momentum transport is now outwar. Note that the sense of transport is outwar, even for those wavenumbers that satisfy the conition 2 ln R < ðkv AÞ 2 < N 2 þ 2 ln R ; ð16þ an are unstable only because of the presence of an averse entropy graient. These wavelengths occupy region A in the figure, specifically the shae wege above the otte line ðkv A Þ 2 ¼ 3 2. These wavelengths woul otherwise be stable to the pure MRI. They are estabilize only because of the existence of averse entropy graients, yet all region A moes transport angular momentum outwar. Physically this is because the large magnetic tension, which regulates angular momentum transport in this regime, woul orinarily be a strongly stabilizing agent. The averse entropy graient estabilizes but oes not alter the outwar flow of angular momentum. Region B enotes the shae area below region A, but above the unshae wege comprising regions C an D. The B region is the wavenumber omain of outwar transporting moes that woul be unstable in the presence of the pure MRI. Their growth rate is increase by the presence of finite N 2. The curve showing the most rapily growing wavenumber (eq. [9]) lies in region B, as shown, but eventually leaves this region at larger values of N 2 (see below). The inwar transporting moes are confine to the unshae narrow wege at the bottom of the iagram, corresponing to long wavelengths. These moes have only a small magnetic tension force, ðkv A Þ Region C consists of moes with N 2 < 2. Comparison with Figure 1 shows that region C moes are estabilize only by the MRI (i.e., they are stable in hyro isks) but are nevertheless associate with inwar transport. Finally, region D ientifies the moes that woul be unstable in a purely hyroynamic system that likewise transports angular momentum inwar. In this narrow omain, both magnetic tension an rotational stabilization are smaller than the averse entropy graient. The curve enoting the most rapily growing wavenumber enters region D for values of N 2 in excess of 4 2. Fig. 1. Region of instability in the k 2 -jn 2 j= 2 plane for an unmagnetize isk with a Keplerian rotation profile. The system is unstable for all wavenumbers when jn 2 j > 2, an the sense of transport is inwar. Fig. 2. Region of instability in the ðkv A =Þ 2 -jn 2 j= 2 plane for a magnetize isk obeying a Keplerian rotation law. (Other rotation profiles lea to qualitatively similar iagrams.) Hatche region: omain of outwar transport. Soli line labele max : wavenumber of maximum growth rate as a function of N 2. The thin wege at the bottom correspons to the inwar transport region. See the text for the efinitions of regions A through D.

4 752 BALBUS & HAWLEY Vol. 573 The region of inwar transport is very small inee, nonexistent for N 2 < 0:5 2, an it woul be surprising if these marginally vali WKB moes effectively halte angular momentum transport in nonraiative flows. In the next section, we show that this is, in fact, impossible in any system where the seat of free energy is ifferential rotation. For the present, it is useful to have an estimate (or a boun) of the size of N 2 one expects in nonraiative flows. The entropy equation for a monotomic gas is 3 2 P ln P 5=3 ¼ Q þ ; ð17þ t where Q þ represents issipative heating. This will generally not excee the total energy buget available from ifferential rotation, T R = ln R, an may be much less. In a one-imensional approximation, we therefore expect 3 2 Pv r r ln P 5=3 T R ln R : ð18þ Let us work near the equatorial plane an switch to cylinrical raius R. The inwar rift velocity is relate to the stress tensor by (Balbus & Hawley 1998) v R T R R ; ð19þ which leas to 1 ln P 5=3 1 R 2 R 3 P ln R ; ð20þ an N 2 ¼ 3 P ln P 5=3 1 ln P 2 5 R R 5 ln R R : ð21þ Generally, when ifferential rotation increases, the pressure graient ecreases, an vice versa. For a Keplerian profile, N 2 = 2 0:6 ln P= ln R. In such a isk, pressure graients are small, an N 2 = 2 is likely to be less than unity. If a significant fraction of free energy goes into generating a magnetic fiel that is carrie off from the isk (i.e., not into issipative fiel reconnection), then N 2 coul be appreciably smaller. The point here is that a well-efine boun on the rate of energy issipation limits the size of N THEORETICAL IMPLICATIONS To the extent that an MHD flui escription is vali, the stability of black hole accretion flow is regulate by the generalize Høilan criteria presente in Balbus (1995, 2001). The notion of a convective moe is ambiguous, as evience in Figure 2. A perturbation in a given backgroun flow is best characterize simply by its wavenumber. The important physical point is not to categorize an stuy convective moes, but to unerstan that magnetic tension in an accretion flow prouces outwar angular momentum transport Turbulence an Irreversibility The issipative properties of CDAFs are very striking, an, in light of the results of x 2, they merit reexamination. In a CDAF (NIA, AIQN), outwar angular momentum is riven by a viscous-like primary instability in the flui, an the energy from the ifferential rotation is issipate as heat. This heating creates an averse entropy graient which, in turn, generates a convective-like seconary instability. When both the primary viscous an seconary convective instabilities are active, their angular momentum transport is equal an opposite, an the total angular momentum flux rops to zero, as oes the volume-specific energy issipation rate Q þ. This means that the convective instability recovers the issipate heat (which originally prouce the convective instability) an returns it to the flui in the form of work (inwar angular momentum an outwar energy fluxes). Clearly, this is a violation of the secon law of thermoynamics, whether the system is hyroynamical or magnetohyroynamical: the onset of convection is cause by irreversible heat issipation, an this energy cannot be fully recovere in the form of work. NIA ientify T R ð22þ ln R as the funamental expression for the volume-specific, issipative energy-loss rate. But the funamental efinition of the energy loss must be in terms of the issipation coefficients themselves. With v equal to the viscous iffusivity, B the resistivity, an J the fluctuating current ensity, the energy issipation rate per unit volume is Q þ X h v jrv i j 2 þ B J 2 i ; ð23þ i where the sum is over vector components. These losses never vanish in a turbulent cascae, an a seconary source of turbulent fluctuations can never fully recover the energy that is issipate in the process of triggering the creation of the same source. It amounts to reversing iffusion, a thermoynamic impossibility. This effect is not small. In a realizable nonraiative flow, none of the entropy generate via issipation ever leaves the system (except in an outflow), for entropy can only increase. The CDAF escription ignores all of the entropy prouction in the flow, entropy which woul be require to prouce the vigorous convection riving angular momentum inwar. Recall that N nees to excee =2 1=2 for there to be any unstable moes transporting angular momentum inwar, an even this minimum threshol alreay requires substantial levels of energy issipation to be present. The escription of a CDAF as a issipation-free flow (or even nearly so) lacks a funamental self-consistency. The sources an sinks for turbulent fluctuations may be rea off from Balbus & Hawley s (1998) equation (89): T R ln R þ Prv Qþ : ð24þ The expression (eq. [22]) happens to be equal to issipative energy losses Q þ uner strictly efine conitions that may be inferre from the above; steay, local turbulence in which the work one by pressure is negligible. This is, in fact, a goo escription of the behavior of turbulence in magnetize, ifferentially rotating flows. An immeiate consequence is that the time-average value of T R must be >0, i.e., T R ¼ 1 Q þ > 0 : ð25þ ln R The unavoiable presence of issipation compels a net out-

5 No. 2, 2002 ANGULAR MOMENTUM IN ACCRETION FLOWS 753 war angular momentum flux in any time-steay magnetize system in which the free energy source is ifferential rotation. 4. COMPARISON WITH SIMULATIONS The point of contact of CDAFs with numerical simulations is the power-law scaling of the ensity, ðrþ, where r is spherical raius. The argument runs that since the CDAF energy flux is conserve (no issipation), v 3 r 2 is a constant. With v r 1=2, this gives r 1=2 as well. By way of contrast, constancy of the mass flux vr 2 woul give a much steeper r 3=2 power law, which is associate with an ADAF solution. The power-law scaling r 1=2 has been interprete as evience in support of convection-ominate flows. Since gas accreting into a black hole is almost certainly magnetize at some level, there is very little to be gaine by hyroynamical simulations: the stability an transport properties of magnetize an unmagnetize gases are simply too ifferent from one another. In general, MHD simulations have not been supportive of CDAFs (Stone & Pringle 2001; Hawley, Balbus, & Stone 2001; Hawley & Balbus 2002). The one exception cite is that of Machia, Matusmoto, & Mineshige (2001). This simulation follows the evolution of a magnetize torus in a Newtonian potential. Convective motions are observe in the subsequent accretion flow. However, these authors raw no istinction between what they refer to as convective motions an turbulence in general. Turbulence is, of course, inevitable in an MHD accretion flow; it is harly the efining signature of a CDAF. The only quantitative basis for the claime agreement with CDAFs was the observation that at one point in time, the ensity exhibite a r 1=2 power-law behavior. However, the flow was a highly time-epenent, the r epenency evolve, an a ifferent power law emerge at the en of the simulation. Even if r 1=2 scaling were unambiguously extracte from a simulation, there is no compelling association with thermal convection. Uner steay conitions, all nonraiative flows, regarless of their level of issipation, have a conserve mass flux an a conserve total energy flux. When the flow is turbulent, to aress the behavior of these fluxes, one must speak in terms of correlations of the flow quantities. In the presence of fluctuations, it is common to have weak correlations in the mass flux an much stronger correlations in the energy flux; inee, orinary waves isplay this sort of behavior. The scaling r 1=2 coul emerge, for example, if the correlation between pressure an velocities in the energy flux was strong, an the magnitue of all velocity fluctuations scales by the local virial velocity. Neither the presence or absence of thermal convection nor the rate of energy issipation is relevant. In any case, any similarity between theory an simulation in a raial power-law inex is not an answer to funamental ynamical an thermoynamical inconsistencies. Verification shoul run in the opposite irection: in the course of proucing MHD turbulence, any numerical simulation shoul show, on average, a positive value for T R an significant levels of energy issipation. The notion of a convectively unstable moe is ill-efine in magnetize, ifferentially rotating systems. When the magnitue of the Brunt-Väisälä frequency N is less than =2 1=2, all unstable moes transport angular momentum outwar. If N is maintaine above this, equation (15) an Figure 2 show that a small range of wavelengths on the largest scales transport angular momentum inwar. (It is, of course, not at all surprising that sufficiently large values of N can be chosen to rener the ynamics of ifferential rotation less important than convection.) For N <, the inwar-transporting moes are estabilize by magnetic tension; for larger N values, the estabilization is by convection. Small wavenumber convective moes (region A in Fig. 2) always transport angular momentum outwar, an overall transport remains overwhelmingly outwar for values of N. This is a very serious ifficulty for CDAF moels, which rely on a significant level of inwar angular momentum transport engenere by convectively-riven moes. Moreover, an most importantly, the claims of vanishing stress an vanishing energy issipation are inconsistent with funamental thermoynamic principles. They are also inconsistent with the nee to maintain N >=2 1=2, a requirement for the existence of any inwar-transporting unstable moes. The central feature of CDAF theory, a vanishing of the angular momentum flux, is not possible if the seat of free energy is ifferential rotation. Characteristic power-law scalings extracte from numerical simulations are not evience for low issipation flow. Black hole accretion features far more conventional an familiar flui ynamics. Unerluminous sources are likely to be a consequence of low ensities an lower-than-anticipate temperatures, as oppose to issipation-free turbulence. MHD turbulence leas to outwar angular momentum transport an to a positive R stress component, irreversible issipative heating, mass accretion, an significant mass outflow, as well. These are all manifest in numerical MHD simulations. The evelopment of the spectral an energetic properties will test these moel flows more stringently. M. Begelman, O. Blaes, an C. Terquem provie helpful comments on a preliminary raft of this paper, as i R. Narayan, though he is in isagreement with our conclusions. This research was supporte by NSF grant AST an NASA grants NAG an NAG Abramowicz, M. A., Igumenschev, I. V., Quataert, E., & Narayan, R. 2002, ApJ, 565, 1101 (AIQN) Balbus, S. A. 1995, ApJ, 453, , ApJ, 562, 909 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, , Rev. Mo. 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