The effects of Prandtl number on black hole accretion flows

Available online at www.sciencedirect.com

New Astronomy Reviews 51 (2008) 814–818 www.elsevier.com/locate/newastrev

The effects of Prandtl number on black hole accretion flows Steven A. Balbus *, Pierre Lesaffre Laboratoire de Radioastronomie, E´cole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris, Cedex 05, France Available online 18 March 2008

Abstract Recent numerical simulations of MHD turbulence, under very different driving conditions, and by several different investigators, all indicate a sensitivity of the rms fluctuations to the ratio of the microscopic viscosity to resistivity. This dimensionless quantity is known as the magnetic Prandtl number Pm. In general, standard astrophysical accretion disks are characterized by Pm  1 throughout their radial extent, while low luminosity accretors (e.g. Sag A*) have Pm  1. Here, we show that standard a models of black hole accretion disks have a transition radius, measured in tens of Schwarzschild radii, at which the flow goes from Pm  1 to Pm  1. Moreover, this transition may well be dynamically unstable, leading to a sort of two-phase ‘‘Prandtl number medium” We advance the idea that this is the physical reason underlying the change in the accretion properties of the inner regions of Keplerian disks, leading to a truncation of the cool disk (Pm  1) and the onset of hot, low density gas flow (Pm  1). Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The standard model for compact X-ray sources of thin disk accretion onto a central source has long been known to be inadequate. These sources have a hard X-ray component that cannot be produced by a composite thermal spectrum that is expected from a disk. Instead, it seems to come from a very hot, low density gas. Jean-Pierre Lasota has been an important contributor to our understanding of how accretion in compact object binary systems could lead to such a flow. In particular, Jean-Pierre, Marek Abramowicz and Ramesh Narayan have all played leading roles in advancing the concept of what is commonly known as an ADAF, for ‘‘Advectively Dominated Accretion Flow” (see Narayan et al., 1998; for an historical review). We prefer the term RIAF, for ‘‘Radiatively Inefficient Accretion Flow”, since it involves less theoretical interpretation, so we hope that Jean-Pierre (who coined the term ADAF) will indulge us! By whatever name it is known, in an important class of black hole accretion models a significant portion of

*

Corresponding author. E-mail addresses: [email protected] (S.A. Balbus), pierre.lesaffre@ lra.ens.fr (P. Lesaffre). 1387-6473/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2008.03.010

the inner accretion flow is dominated by this poorly radiating gas. An interesting and as yet not well-understood issue associated with these models is the question of how accretion can start via a relatively cool a disk, but then at some point make a transition to a RIAF. One standard model holds that the inner portion of the disk is thermally evaporated by coronal gas (Meyer and Meyer-Hofmeister, 1994). In this contribution, we will offer an alternative explanation for accretion flow transitions that is based on fundamental properties of MHD turbulence. We believe that the ratio of the microscopic viscosity to resistivity, otherwise known as the magnetic Prandtl number Pm, could well prove to be a critical quantity for understanding black hole accretion. 2. Magnetic Prandtl number 2.1. Plasma regimes Accretion onto black holes of an ionized plasma can occur in one of three physical regimes, which I will refer to as (i) collisionless; (ii) dilute; and (iii) collisional. In all three of these, the global scale of the flow L is much larger than the ion gyro-radius rg, otherwise the flow would be effectively unmagnetized. What separates the regimes is

S.A. Balbus, P. Lesaffre / New Astronomy Reviews 51 (2008) 814–818

where in the asymptotic ordering the collisional mean free path k lies: rg  L  k rg  k  L

ðcollisionlessÞ ðdiluteÞ

ð1Þ ð2Þ

k  rg  L

ðcollisionalÞ

ð3Þ

In the midplane of classical Shakura and Sunyaev (1973) disks, one is comfortably in the collisional regime, but a few scale heights above the midplane, the gas is dilute. This has potentially interesting consequences for the stability of the stratification of this gas (Balbus, 2001), but we focus here on the collisional gas near the midplane. This gas will generally be fully ionized, and the dissipation coefficients most important in regulating the MHD turbulence will be the electrical resistivity g and kinematic viscosity m. Because the gas is collisional, we will adopt the Spitzer values 3:9  1012 g ’ cm2 s1 T 3=2

ð4Þ

and m ¼ 2:3  1016

T 5=2 2 1 cm s q

ð5Þ

where T is the temperature in K, and q the mass density in grams per cm3. The magnetic Prandtl number Pm is given by Pm ¼

m T4 T4 ¼ 5:9  1029 ¼ 2:5  105 g q nH

ð6Þ

where nH is the number density of hydrogen (cosmic abundances assumed). Note the extreme sensitivity to T. 2.2. Pm behavior in numerical simulations and disk models There is now a significant number of numerical simulations of MHD simulations all of which find that an increase in Pm leads to an increase of the rms fluctuation levels of the turbulence. Schekochihin et al. (2004, 2005) investigated whether dynamo activity was present in MHD turbulence driven at small scales in a box with hard wall boundary conditions. They found a dramatic change in behavior in the neighborhood of Pm = 1. For values in excess of unity, dynamo activity was clearly observed, whereas for values less than unity it was not. Very recently, low Pm dynamos have in fact been found, but with growth much smaller than their high Pm counterparts (Iskakov et al., 2007; Schekochihin et al., 2007). MRI applications have been looked at by Fromang et al. (2007) (zero mean field present), and by Lesur and Longaretti (2007) (net vertical field present). In the latter case, the dominant component of the Maxwell Stress tensor behaved monotonically with Pm, showing no sign of saturating for 0.12 < Pm < 8. Since the Prandtl number clearly is an important parameter in numerical simulations of MHD turbulence, a natural question to ask is what is the behavior of Pm in classical

815

a models? Recently, Pierre Henri and SAB looked at this question (Balbus and Henri, 2008). Adopting the bound-free form of Kramer’s opacity and ignoring radiation pressure, the calculation can be done by taking values straight out of a standard textbook (Frank et al., 2002). The mass density q in cm g m3 in such a model is given by 11=20

_ 16 q ¼ 3:1  108 a7=10 M

15=8

ðM=M  Þ5=8 R10

ð7Þ

_ 16 is the mass accretion rate in units of 1016 g s1, where M M/M is the central mass in solar units, and R10 is the cylindrical radius R in units of 1010 cm. The temperature is 3=10

_ 16 ðM=M  Þ T ¼ 1:4  104 a1=5 M

1=4

3=4

R10

K

ð8Þ

This gives a Prandtl number of 9=8 _ 13=20 Pm ¼ 9  105 a1=10 M ðM=M  Þ3=8 R10 16

ð9Þ

Typical disk Prandtl numbers are therefore very small, and insensitive to scaling with a. Transitions from low to high Pm, if they occur at all, will occur in the inner disk regions. A quantity of interest is evidently Rcr, the radius at which Pm = 1 1=3 _ 26=45 ðM=M  Þ cm Rcr ¼ 2:5  106 a4=45 M 16

ð10Þ

At least for solar mass objects, the region of interest is on scales of tens of Schwarzschild radii (RS). With RS = 2GM/ c 2, Rcr _ 26=45 ¼ 8:5a4=45 M ðM=M  Þ2=3 16 RS

ð11Þ

It is immediately apparent that only objects smaller than 10–100 RS are of interest to us. Black holes and neutron stars do very nicely; any other standard astrophysical object that might host a disk would be far too large to have a Pm transition in the flow. Finally, let us scale the mass accretion rate with M. If we _ 2 assume that the source luminosity L is a fraction  of Mc and a fraction d of the Eddington luminosity LEdd ¼ 1:26  1038 ðM=M  Þ erg s1 then Rcr 4=45 26=45 8=9 ¼ 59ða2 M=M  Þ ðd=Þ l RS

ð12Þ

where a2 is a in units of 0.01. The ratio d/ is just the mass accretion rate measured in units of the Eddington value _ Edd ¼ LEdd =c2 . This shows that the critical radius at which M the Prandtl number transition occurs, when measured in units of RS, is remarkably insensitive to the central mass. In general we find that Rcr varies roughly between 10 and 100 RS. In principle, the low Pm region could in some cases extend all the way down to 2  3RS, particularly for larger AGN masses. For example, iron line observations of the well-studied Seyfert galaxy MCG-60-30-15, (Fabian et al., 2002) suggest the presence of an ordinary Keplerian-like disk down to 3RS, and the Pm transition hypothesis must accommodate this. No transition should also be a possibility.

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Why should there be a Prandtl number dependence for the scale of the fluctuation amplitudes? The speculation put forth almost a decade ago in Balbus and Hawley (1998) appears to be consistent with recent results. The idea is that much of the field dissipation in a turbulent fluid is accompanied by strong velocity gradients at the microscopic resistivity scale. If the viscous scale is well below the resistive scale, corresponding to Pm  1, then these strong velocity gradients offer no viscous stresses to resist the magnetic field dissipation (reconnection?) process. But if viscous scale is much larger than the resistive scale, the cost for the field to dissipate is prohibitive if large velocity changes are needed. The viscous stresses keep the fluid locked up. This means that a turbulent cascade of magnetic field would be stopped at the (relatively large) viscous scale with nowhere to go but back up! Indeed, numerical simulations show both greater field coherence as well as larger field strengths at high Pm. If this is the correct explanation, it suggests that the Prandtl number dependence should be most pronounced at values of Pm near unity. If, for example, the resistive scale is much smaller than the viscous scale, whether it is 103 or 106 smaller would appear to be of little consequence: viscosity will almost certainly be the dominant dissipation process. (This assumes, of course, that the basic collisional character of the flow is the same for both values of Pm. If we are comparing a collisional with a collisionless system, matters need not be so straightforward.) On the other hand it is liable to make a great deal of difference whether Pm is 5 or 0.2, as the basic dissipation mechanism changes over this regime. Hence, the need to search for saturation effects in numerical simulations. 3. Behavior of the flow at the Pm transition radius

dT ¼ Wy 2  CðT Þ dt

ð14Þ

Eqs. (13) and (14) comprise our model dynamical system for the disk.

3.1. A dynamical systems model What might be the consequences of a Pm transition in the flow? In particular, should we expect the flow to passively transit from one Pm regime to another? To address this question directly is daunting, since it involves changes in state of a turbulent MHD fluid. Instead, we will use a ‘‘dynamical systems” approach that has been fruitful in other problems too complex for a direct assault. The idea is to study a well-posed set of coupled nonlinear differential equations that are chosen to mimic the key properties of the original system. Unforeseen connections and consequences emerge from such an analysis, and these can be predictive for the original system. Two key properties of turbulence in an accretion disk that we wish to retain are that of linear instability and nonlinear saturation. Here, the nonlinear saturation is directly related to Pm. Consider the equation dy ¼ ðc  Ay 2 Þy dt

where y represents an RMS fluctuation of a turbulent amplitude. (For example, a magnetic field fluctuation.) c is the linear growth rate, which is modified by the nonlinear saturation term Ay2 for sufficiently large amplitudes in y. An equation of the form (13) is known as a Landau equation (e.g. Drazin and Reid, 1981). For our particular application, c is related to the shearing rate in the disk and is insensitive to the disk state, but A is related to Pm and is therefore expected to be highly temperature sensitive when Pm  1. In this regime, the larger the value of Pm, the larger the saturation amplitude, y 20 ¼ c=A. Since Pm grows with temperature T, A(T) is regarded as a generally decreasing function of T, dA/ dT < 0. Away from Pm = 1, A(T) should be relatively flat, decreasing rapidly through the transition. (The density dependence in Pm is ignored here, but would not change the fundamental behavior.) To illustrate a point of principle, we leave the explicit details of A(T) unspecified, except at the end of this section, where a specific example is shown. The temperature of the disk is determined by turbulent heating and radiative cooling. If the source of the turbulent heating is the differential rotation and the energy extracted from the differential rotation is all locally dissipated, then the heating rate is TR/dX/dlnR, the product of the radial-azimuthal component of the stress tensor and the local shear. We shall adopt these standard assumptions. Since TR/ is quadratic in the fluctuation amplitudes, we will write the heating rate as Wy2, where W is a temperature-independent constant. The cooling rate is written as C(T), a function of T to be specified. The equation of heat balance is taken to be

ð13Þ

3.2. Stability The fixed points for our system of equations are y0 and T0. They satisfy AðT 0 Þy 20 ¼ c

Wy 20 ¼ CðT 0 Þ

ð15Þ

or, eliminating y0, 1 CðT 0 Þ ¼ AðT 0 Þ Wc

ð16Þ

The system fixed points correspond to the intersections of the curve 1/A(T) with a function directly proportional to C(T). The Pm dependence of A suggests that 1/A will be slowly varying with T for low Pm, rise sharply for P  1, and then once again slowly varying at higher T (larger Pm). In Fig. 1, we have represented the curve for 1/A(T) as a sort of step-function, flat on either end, but sharply rising in the middle. C(T) is the smoothly rising curve. There could be as many as three points of intersection of these

S.A. Balbus, P. Lesaffre / New Astronomy Reviews 51 (2008) 814–818

817

C(T)

1/A(T)

Fig. 1. A schematic plot of the functions C(T) and 1/A(T), with T increasing along the abscissa. (The factor of Wc is set to unity.) Points of intersection representing the fixed points of the dynamical system (13) and (14) are shown. These correspond to both dynamical and thermal equilibrium. The two end points are stable, representing viable disk states, but the middle point, corresponding to the Pm = 1 transition, is unstable.

two curves, corresponding to three equilibria. We will now show (Balbus and Lesaffre, in preparation) that for the system (13) and (14), the middle equilibrium point in Fig. 1 is unstable, while the two end points are stable. We examine small departures, denoted by d, from the fixed points of Eqs. (13) and (14). All quantities are assumed to have the time dependence exp(st). The linearized perturbation equations are ðs  c þ 3Ay 20 Þdy ¼ AT y 30 dT

ð17Þ

ðs þ C T ÞdT ¼ 2Wy 0 dy

ð18Þ

where the subscript T on C and A denotes differentiation with respect to temperature. These equation can be satisfied only if   C T AT s2 þ sðC T þ 2cÞ þ 2cC þ ¼0 ð19Þ C A where Eq. (15) has been used to eliminate y0. The real part of s is negative, and the system stable, if and only if the two criteria C T þ 2c > 0;

C T AT þ >0 C A

ð20Þ

are satisfied. The first of these is a slightly modified version of the Field (1965) criterion for a direct thermal instability, but the last is something new. It establishes the claim made at the end of the last paragraph, viz. that instability is present if 1/A increases more rapidly with temperature than C. Fig. 1 is reminiscent of the classical two-phase model of the interstellar medium, where the same type of curve topology gives rise to two stable phases in pressure equilibrium (the endpoints), and an unstable phase at an intermediate temperature (Field et al., 1969). This is driven by the classical Field (1965) thermal instability. As noted above, our model also exhibits thermal instability if C falls too rapidly with temperature, but this is not the physics behind

Fig. 2. The phase space diagram of the system of equations (13) and (14). In this computation, C(T) is a simple linear function and A(T) is an arctangent function. T and y are non-dimensionalized. Fixed points are shown as plus signs: A1 and A2 are stable; I2 is unstable. The arrows indicate the direction of increasing time. The dotted and dashed line are the separatrices.

the Pm instability. To understand the latter, the role of fluctuations must be brought to the fore. Here is what is going on. Imagine a small departure from a disk equilibrium state (zero growth of the fluctuations; dissipative heating balancing radiative losses), with 1/A a sharply rising function of temperature. A small increase in T would then lower A and increase the level of the dynamical y fluctuations. This, in turn, increases the level of heating (Wy2), and if the cooling C(T) does not rise sufficiently rapidly with T, the temperature will rise yet more. This further increases the level of the y fluctuations, and a runaway is present. The same argument holds in reverse for a temperature decrease. A specific realization of the instability is presented in Fig. 2, which displays the trajectories of the system of Eqs. (13) and (14) in the (T, y) plane. We use a linear cooling function C(T), and an arctangent function for A(T) at the Pm = 1 transition, which mimics a step-function. Any starting point in the plane evolves toward one of the two stable fixed points A1 (cooler and less turbulent) and A2 (hotter and more turbulent). 4. Discussion While our dynamical systems approach is surely in many ways a great simplification of MHD turbulence, the Pm instability has a certain simple inevitability to it that makes it quite plausible. It remains to be explored, however, under what conditions the Pm saturation is more sensitive to temperature than is the effective cooling function C(T). In this view, a Prandtl number transition in a disk corresponds to a fundamental change in the quality of the accretion flow. It is a universal phenomenon, in the sense that all MHD turbulent fluids undergo a transition when the ratio of T4/q exceeds a certain well-defined threshold.

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By way of contrast, in thermal evaporation models, it is the coexistence of a corona that causes the transition. It is argued that the juxtaposition of hot, low density gas against the cooler disk gas will thermally evaporate the latter (Meyer and Meyer-Hofmeister, 1994). But the column density of the hot corona in these models is only a tiny fraction of that of the disk it is supposed to be evaporating. For example, in the specific application discussed by Meyer and Meyer-Hofmeister, the corona has a column density of 6  103 g cm2. The disk, by way of contrast, is a factor of a thousand times larger: 8 g cm2. Despite this remarkable discrepancy, the calculation supposes that the temperature in the corona will remain constant over the course of the evaporation, an assumption that surely needs to be refined. Much more likely, in our view, is that any thermal interaction that may be present between the disk and the corona will endanger the latter, not the former. Of the basic sensitivity to Pm of MHD turbulence there can be little doubt. Whether this translates to the eruptive change in accretion we are suggesting here is, however, far from certain. But this is precisely the sort of question that can, and should be, addressed numerically. First, the current round of local box simulations (shearing or otherwise) needs to be extended to find saturation effects of high and low Pm. This will require resolving three well-separated scales, two of them dissipative. This is a still a stretch from a numerical point of view, marginal at 5123, more comfortable at 10243. But Pm saturation should in principle be observable in two dimensions, which would allow considerably finer resolution studies. Second, there is a need for simulations with a temperature dependence included for the viscosity and the resistivity. For an ionized plasma, we know that both of these coefficients are highly temperature sensitive, but with opposite senses: the viscosity rises with T, the resistivity drops. These scalings are at the heart of what seems to be unstable Pm behavior. This requires a careful treatment of heating and cooling processes (something not yet highly developed), but it does not require particularly high resolution. These simulations are needed to establish the existence of, and if present, the nature of, the Pm instability. In particular, local shearing box simulations would be well suited

for studying initial value problems when (theoretically) two possible stable end states exist. The ultimate goal would be to carry though a global disk simulation that could follow the evolution of the Pm transition radius. In principle, an unstable transition could evolve into an effective discontinuity, or exhibit more complex, limit cycle behavior, as seen in classical models of dwarf novae. Such a simulation, with four distinct scales (disk radius, disk height, resistive and viscous scales) is currently beyond the resources of a single investigator, but is just at the limit of what the world’s largest clusters are capable of. Fortunately, Prandtl number studies are still in their infancy, and there is much that can still be learned without a huge investment of computational resources. Understanding MHD turbulence promises gainful employment for some time to come. References Balbus, S.A., 2001. ApJ 562, 909. Balbus, S.A., Hawley, J.F., 1998. Rev. Mod. Phys. 70, 1. Balbus, S.A., Henri, P., 2008. APJ 674, 408. Balbus, S.A., Lesaffre, P., in preparation. Drazin, P.G., Reid, W.H., 1981. Hydrodynamic Stability. Cambridge University Press, Cambridge. Fabian, A.C., Vaughan, S., Nandra, K., Iwasawa, K., Ballantyne, D.R., Lee, J.C., De Rosa, A., Turner, A., Young, A.J., 2002. MNRAS 335, L1. Field, G.B., 1965. ApJ 142, 531. Field, G.B., Goldsmith, D.W., Habing, H.J., 1969. ApJ 155L, 149. Frank, J., King, A., Raine, D., 2002. Accretion Power in Asrophysics. Cambridge University Press, Cambridge. Fromang, S. et al., 2007. A&A. 476, 1123. Iskakov, A.B., Schekochihin, A.A., Cowley, S.C., McWilliams, J.C., Proctor, M.R.E., 2007. Phys. Rev. Lett. 98, 208510. Lesur, G., Longaretti, P.Y.-L., 2007. Available from: 0704.2943 [astro-ph]. Meyer, F., Meyer-Hofmeister, E., 1994. A&A 288, 175. Narayan, R., Mahadevan, R., Quataert, E., 1998. In: Abramowicz, M.A., Bjornsson, G., Pringle, J.E. (Eds.), Theory of Black Hole Accretion Disks. Cambridge University Press, Cambridge, p. 148. Schekochihin, A.A., Cowley, S.C., Taylor, S.F., Maron, J.L., McWilliams, J.C., 2004. ApJ 612, 276. Schekochihin, A.A., Haugen, N.E.L., Brandenberg, A., Cowley, S.C., Maron, J.L., McWilliams, J.C., 2005. ApJ 625, L115. Schekochihin, A.A., Iskakov, A.B., Cowley, S.C., McWilliams, J.C., Proctor, M.R.E., Yousef, T.A., 2007. New J. Phys. 9, 300. Shakura, N., Sunyaev, R., 1973. A&A 24, 337.