Advection of Magnetic Fields in Accretion Disks: Not So Difficult After All
aa r X i v : . [ a s t r o - ph ] J a n Accepted for publication in The Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 03/07/07
ADVECTION OF MAGNETIC FIELDS IN ACCRETION DISKS: NOT SO DIFFICULT AFTER ALL
David M. Rothstein and Richard V. E. Lovelace Accepted for publication in The Astrophysical Journal
ABSTRACTWe show that a large-scale, weak magnetic field threading a turbulent accretion disk tends to beadvected inward, contrary to previous suggestions that it will be stopped by outward diffusion. Theefficient inward transport is a consequence of the diffuse, magnetically-dominated surface layers ofthe disk, where the turbulence is suppressed and the conductivity is very high. This structure arisesnaturally in three-dimensional simulations of magnetorotationally unstable disks, and we demonstratehere that it can easily support inward advection and compression of a weak field. The advected fieldis anchored in the surface layer but penetrates the main body of the disk, where it can generate strongturbulence and produce values of α (i.e., the turbulent stress) large enough to match observationalconstraints; typical values of the vertical magnetic field merely need to reach a few percent of equipar-tition for this to occur. Overall, these results have important implications for models of jet formationwhich require strong, large-scale magnetic fields to exist over a region of the inner accretion disk. Subject headings: accretion, accretion disks — galaxies: jets — magnetic fields — MHD — X-rays:binaries INTRODUCTION
Early theoretical work on accretion disks arguedthat a large-scale magnetic field (of, for exam-ple, the interstellar medium) would be dragged in-ward and greatly compressed by the accreting plasma(Bisnovatyi-Kogan & Ruzmaikin 1974, 1976; Lovelace1976). Figure 1 illustrates this concept by showing asketch of an ordered magnetic field threading an accre-tion disk, in which inward advection has caused the mag-netic field lines to bunch together into an “hourglass”shape. This was thought to be a simple mechanism forgenerating dynamically significant fields in the inner disk.In the present paper, we revisit this issue, build-ing off the recent work of Bisnovatyi-Kogan & Lovelace(2007). Our motivation for doing so is that in the in-tervening years, the early theoretical arguments havebeen challenged. More detailed models of turbu-lent disks suggested that a large-scale, weak mag-netic field such as that shown in Figure 1 in factwill diffuse outward rapidly (van Ballegooijen 1989;Lubow, Papaloizou, & Pringle 1994) if the turbulentmagnetic diffusivity and turbulent viscosity are of sim-ilar order of magnitude, as they are expected tobe (Parker 1971; Bisnovatyi-Kogan & Ruzmaikin 1976;Canuto & Battaglia 1988)—the turbulence responsiblefor driving the accretion also leads to enhanced recon-nection of the large-scale radial field across the thicknessof the disk, thereby causing the vertical field to diffuseaway. This cast doubt on the idea that weak fields couldbe dragged inward and compressed by advection. Atthe same time, it was known that the angular momen-tum loss to magnetohydrodynamic (MHD) outflows froma disk threaded by a sufficiently strong large-scale fieldcould more than offset the outward diffusion and lead to Department of Astronomy, Cornell University, Ithaca, NY14853-6801; [email protected] NSF Astronomy and Astrophysics Postdoctoral Fellow Departments of Astronomy and Applied and Engineer-ing Physics, Cornell University, Ithaca, NY 14853-6801;[email protected] a rapid, implosive increase of the field in the central re-gion of the disk (Lovelace, Romanova, & Newman 1994).However, it seemed to be the case that growth of a strongmagnetic field “from scratch,” due to continual advectionof a weak field, was impossible in a thin disk. Althoughthis conclusion has been occasionally challenged (e.g.,Ogilvie & Livio 2001), it is still generally accepted, whichhas led to the recent suggestion that special conditions(extremely nonaxisymmetric regions of strong field in anotherwise weakly-magnetized disk) are required for thefield to be advected inward (Spruit & Uzdensky 2005).At the same time, recent three-dimensional MHDsimulations have been performed that allow this issue tobe addressed computationally. These simulations resolvethe largest scales of magnetorotational turbulence andtherefore self-consistently include the turbulent viscosityand diffusivity (without having to prescribe their values a priori ). Most simulations performed to date haveinvestigated conditions in which the accreting matterdoes not contain any net magnetic flux and whereno magnetic field is supplied at the boundary of thecomputational domain. However, in one simulation,weak poloidal flux injected at the outer boundary wasclearly observed to be dragged into the central regionof the disk, leading to the buildup of a strong centralmagnetic field (Igumenshchev, Narayan, & Abramowicz2003). A similar process, albeit transient, may occur insimulations without a net magnetic flux; there, radialstretching of locally poloidal field lines in the initialconfiguration often leads to large-scale poloidal fieldsand jet structures in the inner disk (e.g., Hirose et al.2004; De Villiers et al. 2005; Hawley & Krolik 2006;see also the discussion in Igumenshchev et al. 2003and McKinney & Narayan 2007 and especiallythe simulations of McKinney & Gammie 2004 andBeckwith, Hawley, & Krolik 2007, which explore theeffect of different initial field geometries on the forma-tion of jets). The extent to which any of the advectionof magnetic field lines seen in numerical simulationsrequires the presence of a thick disk or nonaxisymmetric Rothstein & Lovelace
Fig. 1.—
Sketch of the magnetic field threading an accretiondisk, showing the increase of the field due to its assumed inwardadvection with the gas (as proposed in early theoretical models). conditions is unclear.In light of these numerical results, we return to thequestion of inward advection of magnetic fields in thispaper, allowing for the possibility that the disk is thinand axisymmetric and asking once again whether advec-tion of a weak field is possible under these conditions.The mechanisms we discuss here can occur in sufficientlyionized regions of any accretion disk; although they areperhaps most widely applicable to disks around blackholes (where the large-scale magnetic field arises entirelywithin the accreting plasma), they are relevant for disksaround many other types of accreting objects as well.The organization of this paper is as follows. In §
2, weanalyze the advection of a large-scale field in an accre-tion disk and point out the importance of the verticalstructure of the disk, which was not taken into accountin most previous studies. Based on an earlier suggestion(Bisnovatyi-Kogan & Lovelace 2007), we show that thethin, highly conducting surface layer of the disk, whereturbulence is suppressed, allows a large-scale magneticfield to be advected inward and compressed. In §
3, weargue that the resulting magnetic flux through the mainbody of the disk (due to the large-scale field being ad-vected inward) can produce values of the turbulent α parameter that are in accord with observational data.This is in contrast with numerical simulations of turbu-lent disks without a net imposed magnetic flux, whichare unable to generate large enough turbulent stress. Fi-nally, in §
4, we derive detailed conditions on the fieldstrength, geometry and ionization fraction that are re-quired for the field to be advected inward and show thatthese are typically weak constraints. Conclusions of thiswork are summarized in § MAGNETIC FIELD ADVECTION AT THE SURFACE OFAN ACCRETION DISK
The evolution of the magnetic field B in an accretiondisk (averaged over the short timescales of the turbu-lence) is assumed to be described by the induction equa-tion, ∂ B ∂t = ∇ × ( v × B − η ∇ × B ) , (1)where v is the plasma velocity, η = c / (4 πσ ) is the mag-netic diffusivity, c is the speed of light, and σ is the con- disk B σ( ) z/h r i h turbulent (cid:13) body of(cid:13) disknonturbulent(cid:13) coronaboundary(cid:13) layer Fig. 2.—
Sketch of the disk and instantaneous poloidal magneticfield lines considered in this work. The toroidal field componentis not shown. The inset shows a rough illustration of the verticalprofile of the conductivity σ ( z ) in units of the coronal value σ ( h ).At the base of the corona ( z = h ), turbulence is suppressed and theconductivity is very high; therefore, if the material in this regionadvects inward with the main body of the disk, the large-scalemagnetic field will be advected inward as well. ductivity. We assume a disk with half-thickness H . r in cylin-drical coordinates. The main body of the disk is turbu-lent, and we take the effective diffusivity to be η ∼ ν ,where ν is the turbulent viscosity. The turbulence iswidely thought to be due to the magnetorotational in-stability (Balbus & Hawley 1991, 1998; Velikhov 1959;Chandrasekhar 1960), which roughly occurs when themagnetic energy density is less than the thermal energydensity. We therefore assume a weak magnetic field suchthat this condition holds in the main body of the disk.However, the time-averaged magnetic field is not ex-pected to vary strongly across the disk thickness, owingto the buoyancy of the field and the condition ∇ · B = 0(in more physical language, the field is not influencedby the vertical gravity that keeps disk material con-fined near the equatorial plane). Thus, the mass den-sity of the gas will typically decrease with height z morerapidly than the time-averaged magnetic field strength,and at a height ∼ H above the midplane, the magneticenergy density will become strong enough compared tothe thermal energy density that turbulence will be sup-pressed. The boundary between the turbulent and non-turbulent regions is likely to be “fuzzy” owing to the leak-age of some magnetic flux through the disk surface (e.g.,Galeev, Rosner, & Vaiana 1979), but at a certain height,the plasma will become completely nonturbulent. In thispaper, we will use the terms “base of the nonturbulentregion” and “surface layer of the disk” interchangeably;however, it should be noted that we are explicitly defin-ing these regions to be above the boundary layer andtherefore fully a part of the nonturbulent corona (seeFigure 2). Note that in a turbulent disk, there can also be an additionalterm in equation (1) that we have not included here, which wouldrepresent the contribution of a turbulent dynamo to the growthof the large-scale field. We ignore this term because we are onlyinterested in the growth of magnetic field due to advection, andtherefore any local, dynamo-generated field that may be producedis “extra” to that which we discuss in this section. dvection of Magnetic Fields in Accretion Disks 3This suppression of turbulence above a weakly-magnetized disk has been observed in a vari-ety of MHD simulations (e.g., Miller & Stone 2000;De Villiers, Hawley, & Krolik 2003; Hirose et al. 2004;McKinney & Gammie 2004; Fromang & Nelson 2006),including those with radiation (Hirose, Krolik, & Stone2006) and even those in which the radiation pressure iscomparable to the gas pressure (Krolik, Hirose, & Blaes2007). However, MHD simulations of fully radiation-dominant disks (Turner 2004) are less clear, and the ap-plicability of our work in this case requires further anal-ysis. Nonetheless, even above a radiation-dominated re-gion of the disk, we expect that the turbulence will besuppressed in many situations; we discuss this issue fur-ther in § h ∼ H near the disk surface causes this layer to become highlyconductive; the diffusivity will decrease from its turbu-lent value in the main body of the disk ( η ∼ ν ∼ cm s − for typical parameters) to the Spitzer value as-sociated with electrons scattering off of ions, given by η S ∼
200 ( T s / keV) − / cm s − , where T s is the surfacetemperature. This suggests that the second term on theright hand side of equation (1) can be ignored in the up-per disk layers. Specifically, the relative importance ofthe two terms (advection compared to diffusion) at anypoint in the disk is determined by the local magneticReynolds number Re m = Hu r /η , where u r is the localradial speed and H is the relevant length scale (here wemake the reasonable assumption that the time-averagedmagnetic field does not vary significantly in the radial di-rection on length scales shorter than H ). We can there-fore use the Shakura & Sunyaev (1973) disk solution tofind that a typical value of the magnetic Reynolds num-ber at the surface of the disk, where turbulence is sup-pressed, is given by Re m ∼ αm / ˙ m / ˆ r − / f − / ∗ (cid:0) H/r (cid:1) U s . (2)Here, α ≤ m isthe mass of the central object in solar masses, ˙ m is theaccretion rate in units of the Eddington luminosity di-vided by the speed of light squared, ˆ r is the radius inunits of the Schwarzschild radius, f ∗ . r that depends on the stress at theinner boundary of the disk (e.g., Agol & Krolik 2000),and U s is the ratio of the radial speed at the disk surfaceto that in the main, turbulent body of the disk. This lastterm can be smaller than unity, but not small enough toprevent the conclusion that, typically, Re m ≫ Re m ≈ H/r (assuming the magneticfield is not strong enough to affect the accretion speed)and diffusion of the magnetic field therefore dominatesover advection (Lubow et al. 1994; Lovelace et al. 1994;Heyvaerts, Priest, & Bardou 1996). In actuality, other non-ideal MHD effects, in particular Hallelectromotive forces, may be more important than the Spitzer dif-fusivity, as we show in § We can easily demonstrate that advection in the sur-face layer of the disk is able to support the overall growthof magnetic field. If we integrate equation (1) over a cir-cular surface r ≤ r that covers the top side of the disk( z = h , where h is the height at which turbulence is firstsuppressed), we can take η ≈
0, and Stokes’ theoremtherefore implies that d Φ p dt = r I dφ ( v zh B rh − v rh B zh ) | r = r , (3)where Φ p is the poloidal magnetic flux through this sur-face and the h subscript indicates that the quantity isevaluated at z = h . If the right hand side of this equa-tion has the same sign as Φ p , the magnetic flux interiorto radius r will grow.Assuming axisymmetry (or, alternatively, treating sub-sequent quantities as being appropriately averaged overazimuth) and taking z > v rh < ( B rh /B zh ) v zh , (4)provided that magnetic field with the appropriate polar-ity is available to accrete.Although turbulent stress cannot contribute directly tothe accretion at z = h , coupling between the main, tur-bulent body of the disk and the surface (as well as angu-lar momentum loss to a wind or jet) will tend to produce v rh <
0. Also, simulations indicate that for an MHDoutflow or jet ( v zh ≥ B rh /B zh ≥ § entire inward-accreting portion of the disk, so that nocurrents are allowed to accrete inward. Even a smallsliver of nonturbulent (i.e., highly conducting) materialthat advects inward at the surface layer can support themagnetic field, even though it may only contain a smallfraction of the disk’s mass. A related issue was noticedby Ogilvie & Livio (2001), who argued (on mathematicalgrounds) that the relevant radial velocity for magneticfield advection is one that has been weighted by 1 /η andaveraged over height. Here, we present a physical modelfor the behavior of η with height and show that in a diskwhere the magnetic diffusivity is due to turbulence, thecontrast between the diffusivity inside and outside theturbulent region is likely to be so sharp that the condi-tion for magnetic flux growth reduces to equation (4),which is satisfied in many parts of a typical accretiondisk. EFFECT OF THE ADVECTED MAGNETIC FIELD ONTHE TURBULENT α PARAMETER
Given the apparent ease with which a large-scale mag-netic field can advect inward in an accretion disk, it isnatural to consider the influence of this magnetic field onthe disk dynamics. In particular, in this section we dis-cuss how the advected magnetic field might be expectedto affect the turbulence in the main body of the disk, em-bodied in the α parameter of Shakura & Sunyaev (1973). Rothstein & LovelaceSince the turbulence is thought to be magnetic in natureand in particular due to the magnetorotational instability(MRI; Balbus & Hawley 1991, 1998), the effect is likelyto be a significant one.King, Pringle, & Livio (2007) have recently pointedout that observationally-determined values of α , basedprimarily on studies of outbursts in dwarf novae andX-ray transients, tend to lie in the range ∼ . − . α is a measure of the turbulent magneticstress scaled by the thermal pressure in the main bodyof the disk, it is clear that a significant amount ofturbulent magnetic energy must exist in these accretiondisks during the outburst phase. King et al. (2007)noted a potential puzzle, which is that numericalsimulations of the MRI in which the instability isallowed to develop entirely based on a local seed field(i.e., where there is no externally-imposed magneticflux through the computational region) tend to givesaturation values of the stress that are much toosmall to match the observations, with typical values α ∼ .
01 regardless of the strength of the seed field(Hawley, Gammie, & Balbus 1996; Balbus & Hawley1998; and note that Pessah, Chan, & Psaltis 2007 andFromang & Papaloizou 2007 have shown that eventhese values may be significant overestimates, due tonumerical resolution effects).However, in simulations with an externally-imposedvertical magnetic field, the turbulent stress due tothe MRI depends critically on the seed field strength.In particular, α is found to increase with the netimposed vertical field B z (Hawley, Gammie, & Balbus1995; Balbus & Hawley 1998). Shearing box simu-lations suggest a rough empirical relationship α ∼ π ( β z,ext ) − / , where β z,ext is the ratio of the ther-mal pressure in the disk to the magnetic pressure ofthe externally-imposed vertical field (this is a sim-plified version of a more general equation found inPessah et al. 2007, which is based on MRI simulations bySano et al. 2004 but also agrees with the earlier resultsof Hawley et al. 1995). Thus, values of α ∼ . − . β z,ext ∼ − ∼ −
6% of the equipar-tition field strength (i.e., β z,ext is ∼ − α with numerical simulations ofthe MRI. Also, as noted by King et al. (2007), theobservationally-determined values of α are weighted av-erages over the entire accretion disk. Thus, it is certainlypossible that advection could lead to much larger fieldstrengths in a particular region. The field strengths inquiescent disks are similarly unconstrained.An interesting effect of the dependence of α on thestrength of the large-scale magnetic field is that the vis-cous and thermal timescales in the disk (which dependinversely on α ) should change with time, in response tothe history of magnetic field advection. This process mayexplain some of the wide range of variability on manydifferent timescales seen in accreting black holes, in par-ticular in the bright X-ray binary GRS 1915+105 (e.g.,Belloni et al. 2000), where the various modes of variabil-ity seem to repeat in a semi-regular pattern over a periodof months to years that has been suggested to be a sig-nature of magnetic processes (Tagger et al. 2004).A large-scale magnetic field may be expected to haveother effects on an accretion disk besides those discussedabove. If the field is strong enough, it can begin to affectthe disk dynamics directly (through removal of angularmomentum via a wind or jet); we will consider the casewhere the advected field builds up to dynamically signif-icant values in a future paper. However, the importantpoint we make in this section is that even when the large-scale field is dynamically weak, it can have a significanteffect on the disk dynamics indirectly , through its influ-ence on the turbulent stress in the main body of the disk. DETAILED ANALYSIS OF THE CONDITIONS FORMAGNETIC FIELD ADVECTION In §
2, we showed that advection of a large-scale mag-netic field will dominate over diffusion in the nonturbu-lent surface layer of an accretion disk and that, if equa-tion (4) is satisfied, the advection can lead to a concen-tration of magnetic flux in the inner region of the disk.In this section, we present a more rigorous analysis ofthe conditions under which equation (4) is satisfied. In § v rh < § § § § Forces Acting on the Nonturbulent Surface Layerof the Disk
For an axisymmetric accretion disk in which thespecific angular momentum profile is time-independent(which can be true even in a time-dependent disk if or-bits are nearly circular), a general equation for the localradial velocity is v r = − (cid:20) ∂∂r (cid:0) r T rφ (cid:1) + ∂∂z (cid:0) r T φz (cid:1) + r ρv z ∂v φ ∂z (cid:21) rρ ∂∂r ( rv φ ) . (5)This expression is obtained by combining the conser-vation of mass and conservation of angular momentumequations for a magnetized fluid. Here, ρ is the massdensity and T rφ and T φz are components of the stresstensor, including both large-scale magnetic and small-scale turbulent stresses. If we evaluate this equation at z = h (the height in the disk where turbulence is firstsuppressed) and if the gravitational and centrifugal forcesare assumed to balance in this region (i.e., if we assumecircular orbits in Newtonian gravity), then to first orderin h/r , v rh ≈ ∂∂r (cid:0) r B r B φ (cid:1) h + ∂∂z (cid:0) r B φ B z (cid:1) h πrρ h v K + 3 hr v zh , (6)where v K is the Keplerian velocity on the equatorialplane. The first term represents the effect of the stressdue to the time-averaged magnetic field, and the secondterm represents a centrifugal effect that drives outflowingmaterial away from the inner disk. Competition betweenthese two processes determines the vertical profile of theradial velocity in the nonturbulent region of the disk.The analysis leading up to equation (6) is quite general,and we therefore make liberal use of it in the followingsections. The only exception to its generality is the as-sumption of circular orbits, which may not be valid if thebase of the nonturbulent region occurs at a height h in thedisk where the density is so low that radiation pressureor magnetic forces begin to become important in the ra-dial momentum equation. This is effectively a constrainton the magnetic field strength, and we therefore discussit further in § not assume circular orbits (provided that thedisk is stationary) and that the derivation leading fromequation (5) to (6) will still be roughly valid providedonly that the spatial derivatives of the azimuthal veloc-ity v φh in the nonturbulent region are of the same orderof magnitude as their Keplerian counterparts. Conditions on the Magnetic Field Geometry
A straightforward way to derive a condition on themagnetic field geometry is to combine equations (4) and(6). If we do this and assume that the stress due tothe time-averaged magnetic field removes angular mo-mentum from the disk surface (i.e., attempts to drag thesurface layer inward with the main body of the disk) in any amount , then a sufficient condition for growth ofmagnetic flux in the inner disk is (cid:18) B rh B zh − hr (cid:19) v zh & , (7) provided that magnetic field with the appropriate polar-ity is available to accrete.When equation (7) is satisfied, magnetic flux growthcan occur through a combination of radial and verticaladvection at the surface of the disk. However, we areprimarily interested in radial advection, which is the only sustainable way in which magnetic field lines anchored inthe surface layer can build up flux in the inner disk. Inparticular, when equation (4) is satisfied, we can identifythree regimes of interest:1. If v rh < ≤ ( B rh /B zh ) v zh , magnetic flux growthoccurs through a combination of inward radial ad-vection and vertical advection at the surface of thedisk.2. If v rh < ( B rh /B zh ) v zh <
0, magnetic flux growthoccurs via inward radial advection, even though itis partially opposed by vertical advection.3. If 0 < v rh < ( B rh /B zh ) v zh , magnetic field is ad-vected radially outward at the surface of the disk,but magnetic flux growth still occurs in the innerdisk because of vertical advection at z ≈ h .We are primarily interested in the first two regimes,where magnetic flux growth occurs at least partially dueto inward radial advection, and where equation (7) doesnot necessarily apply. Thus, for the rest of this section,we ignore the third regime and derive more stringent con-ditions that specifically guarantee inward radial advec-tion.As discussed in §
2, the first regime is likely to bemore relevant than the second (Ustyugova et al. 1999,2000), and we therefore consider it now, returning tothe second regime at the end of this section. We thushave ( B rh /B zh ) v zh ≥ v rh < H B ≡ [ ∂ ln ( B φh B zh ) /∂z ] − as the scale height of thevertical magnetic stress, we find that a sufficient condi-tion is − B φh B zh & v zh h v r i H B H (cid:18) ρ h ρ hr (cid:19) ˙ M Ω K r , (8)where ρ is the mass density on the equatorial plane,˙ M is the local mass accretion rate, Ω K ≡ v K /r is theKeplerian angular velocity, and h v r i ≡ ˙ M / (4 πrρ H ) isan appropriately height-averaged inward radial velocityin the main body of the disk (i.e., the standard radialvelocity in a one-dimensional vertically-integrated diskmodel). In interpreting this equation, it is instructive tonote that ( ˙ M Ω K /r ) / ≈ . × m − / ˙ m / ˆ r − / G isa fiducial field strength, but a maximum one, since h/r and especially ρ h /ρ can be very small parameters. Equation (8) is a “sufficient” condition for magneticfield advection in the sense that it makes the conserva-tive assumption that large-scale magnetic stresses in the Although we can generally assume h ∼ H within a factor ofa few, the distinction between those two heights must be retainedwhen evaluating the mass density, and it is important to use thecorrect value ρ h which appears in equation (8). This is becausethe mass density typically falls off very sharply with height, andthus ρ h may be many orders of magnitude smaller than both ρ H and ρ . Rothstein & Lovelacevertical direction are the only way in which the surfacelayer can be dragged inward. In particular, it does notinclude the large-scale B r B φ stress at the disk surface,which transports angular momentum radially and alsotends to give v rh < h as the height in the disk where MRIturbulence is first suppressed allows us to write ρ h ρ ∼ B zh πp , (9)where B zh ≈ | B zh | + ( H/r ) | B φh | represents a magneticfield strength that is roughly equal to the time-averagedvertical field B zh , and we define p ≡ ρ ( H Ω K ) ; for aweak field, p is roughly equal to the thermal pressureon the equatorial plane of the disk. Combining equations(8) and (9) and assuming h ∼ H , we find that a sufficientcondition for inward advection of magnetic fields is − B φh B zh B zh & H B H v zh v K , (10)which has no direct dependence on the field strength;as long as the geometry is favorable, arbitrarily weakmagnetic fields can provide enough stress to drive inwardradial advection at the surface layer of the disk, and thefields will therefore be advected inward and compressed.The essential physical point is simply that the magneticfield must be strong compared to the gas at z ≈ h (inorder to suppress the MRI), so it is therefore able to driveaccretion at this location, regardless of how weak it is inan absolute sense. If we ignore the sign of B φh B zh , equation (10) is rel-atively trivial to satisfy. For example, we can estimatethe ratio of vertical magnetic stress to energy densityon the left hand side of equation (10) that might arisenaturally in a disk (i.e., without an externally-imposedseed field) by looking at numerical simulations of theMRI. We focus on the work of Miller & Stone (2000),who studied a vertically-stratified disk in the shearingbox approximation and tabulated the properties of themagnetic field in the nonturbulent corona above thedisk. We find typical values of | B φh B zh | / B zh & . ∼ α ( H/r ) ( H B /H ) | v zh | / h v r i when the large-scale field is dynamically weak (or, alternatively, H B /r times the ratio of v zh to the disk sound speed). Thisis clearly a very small number if we make the approxi-mation that H B . H (i.e., that the scale height of the This statement is independent even of the Spitzer diffusivity;our expression for v rh in equation (6) means that we can rewritethe condition Re m ≫ ˛˛ ( H/H B ) B φh B zh / B zh + v zh /v K ˛˛ ≫ − m − / ˙ m − / ˆ r / f − / ∗ ` H/r ´ − . Clearly, this equationwill be satisfied in almost any accretion disk provided that equation(10) is not pathologically close to an equality. More importantfor our purposes, there is no dependence on the magnetic fieldstrength; even when microscopic effects such as Spitzer diffusivityare taken into account, arbitrarily weak fields appear capable ofadvecting inward along the surface layer of a fully ionized accretiondisk (but see § twisted toroidal field can be comparable to or smallerthan that of the mass density); the validity of this ap-proximation is discussed in Appendix B. Intuitively, theapproximation H B . H may be thought of as arisingfrom the presence of a voltage source (Keplerian shear)that is applied in the radial direction, with the result-ing current confined to flow in a region above ∼ h (theheight at which the plasma first becomes highly conduc-tive) but below ∼ a few × H (the height at which the massdensity becomes low enough so that orbits are no longercircular and therefore the applied voltage is significantlyreduced).The ease with which equation (10) can be satisfied sug-gests that not only will we have v rh < | v rh | ∼ h v r i . In fact, if we startwith equation (6) and go through the same analysis asabove but require v rh ≤ − h v r i rather than v rh <
0, theninstead of equation (10) we obtain − B φh B zh B zh & H B H h v r i H Ω K (cid:20) hr v zh h v r i (cid:21) , (11)where the right hand side is typically ∼ αH B /r for adynamically weak field. Like equation (10), this condi-tion is modest assuming the disk is thin, and thus wemay expect that advection of magnetic fields proceeds atthe same speed as turbulent accretion in the main bodyof the disk. In fact, equation (11) suggests that advec-tion of magnetic fields in the surface layer could proceed faster than the disk accretion speed, but as we discuss inAppendix B, this is unlikely to be sustainable.The only qualification to what we have said so far con-cerns the sign of B φh B zh . In particular, equation (8)shows that B φh B zh . ∂ ( B φ B z ) h /∂z <
0, which states that the large-scale magnetic field mustremove angular momentum from the nonturbulent sur-face layer. This is a strict requirement for disks in which v zh ≥
0. We expect that ∂ ( B φ B z ) h /∂z < B r and B φ oddfunctions of z and B z an even function), which is often as-sumed for the large-scale magnetic field advected inwardin an accretion disk (see Figure 1). However, in a regionof the disk with a quadrupole-type field symmetry (with B r and B φ even functions of z and B z an odd function),as may occur when the large-scale field extending outof the disk is generated primarily by magnetorotationalturbulence (e.g., Brandenburg et al. 1995), some regionswill likely have ∂ ( B φ B z ) h /∂z >
0. In these regions, an-gular momentum will not be removed vertically from thesurface of the disk, and inward radial advection of themagnetic field may be difficult to sustain. Correspond-ingly, we find that one of the simulations described indetail in Miller & Stone (2000) has ∂ ( B φ B z ) h /∂z < ∂ ( B φ B z ) h /∂z < B rh /B zh ) v zh <
0, and inward radial advection re-quires v rh < ( B rh /B zh ) v zh in order to overcome verticaladvection and produce a concentration of magnetic fluxin the inner region of the disk. If the disk is thin, with3 h/r ≪ | B rh /B zh | , a sufficient condition for this to oc-cur can be obtained by replacing h/r with − B rh / B zh inequation (8) and propagating this change through subse-quent expressions. In particular, equation (10) becomes − B φh B zh B zh & H B H (cid:18) − B rh B zh (cid:19) v zh H Ω K , (12)where we have ( − B rh /B zh ) v zh > | B rh /B zh | ∼ v zh is not too close to the sound speed. Conditions on the Magnetic Field Strength andIonization Fraction
From the analysis of the previous subsection, an arbi-trarily weak seed field threading an accretion disk shouldbe able to advect inward along the disk surface. But whatreally happens for arbitrarily weak fields? Are there fieldstrengths below which some of our underlying assump-tions in this paper break down?An important assumption in this paper is that the re-gion above the disk is nonturbulent and, therefore, highlyconducting; it is this region in which the magnetic fieldcan advect inward. Clearly, a nonturbulent region islikely to exist somewhere above an accretion disk, butthe question is whether it occurs at a low enough heightto be treated as the “surface layer” of the disk, as we doin this paper. Equation (9) and the usual assumptionthat the mass density decreases with height much morerapidly than the magnetic energy density suggests thatthe nonturbulent region should occur within a few scaleheights, even for a very weak seed field on the equatorialplane. However, if the magnetic energy density beginsto drop off rapidly with height, the turbulence may notbe suppressed until a very large distance above the disk.This may be what happens in the radiation-dominatedsimulations of Turner (2004), where the magnetic energydensity begins to fall off at z & H , and there is no clearevidence for a nonturbulent region anywhere within thesimulation domain (which extends out to z ∼ H ).It is difficult to predict when this type of behavior willoccur, but when it does, our assumption that the nontur-bulent region occurs “within the disk” may break down.In particular, orbits may not be circular, so that equa-tion (6) is no longer strictly valid. Considering radialforce balance and using equation (9), we find that if thedisk is sufficiently thin, magnetic forces are unlikely to be strong enough to disrupt circular orbits at the base ofthe nonturbulent region (although they may certainly doso higher up in the corona); the condition for magneticforces to be negligible at z ≈ h is B zh /B h ≫ ( H/r ) and B zh / | B rh B zh | ≫ H/r , both of which are easilysatisfied by the Miller & Stone (2000) simulations (here B zh ≈ B zh was defined in § | B rh B zh | can be approximated as ∼ H ). Radiation pressure istherefore the only realistic concern; in order for orbitsto remain circular, we require ρ h v K to dominate overthe radiation pressure. Assuming the temperature at z ≈ h is given by the effective surface temperature of aShakura & Sunyaev (1973) disk, we can use equation (9)to derive B zh ≫ − m − / ˙ m / ˆ r − / f / ∗ (cid:0) H/r (cid:1)
G (13)as the condition for circular orbits, where ˆ r ≡ ˆ r/ and m ≡ m/ (i.e, we have scaled the fiducial value to thatwhich would occur at a distance of 10 Schwarzschildradii from a supermassive black hole of mass 10 M ⊙ ).Limits on the magnetic field strength resulting fromequation (13) are plotted in Figure 3 for typicalShakura & Sunyaev (1973) accretion disks with α ≈ − , which we take to be a worst-case lower limit for theturbulent stress (note in any case that the dependence on α is very weak). In fact, α is not independent of the fieldstrength; using the relation α ∼ . B / πp betweenturbulent magnetic stress and turbulent magnetic energydensity B / π in the main body of the disk (which is a ro-bust result of MRI simulations; e.g., Hawley et al. 1995;Sano et al. 2004; Blackman, Penna, & Varniere 2006),we can rewrite equation (13) as a constraint on the fieldgeometry: B zh B ≫ × − (cid:18) αρ g cm − (cid:19) − m − ˙ m ˆ r − f ∗ . (14)Limits on this ratio for typical Shakura & Sunyaev(1973) accretion disks are plotted in Figure 4, again as-suming α ≈ − . In the radiation-dominated region ofthe disk, ρ ∝ α − , so the right hand side of equation(14) has no dependence on the magnetic field strength,whereas in the gas pressure-dominated region, the de-pendence occurs through α − / , which we have fixed tothe assumed worst-case value. Thus, although equation(14) technically represents a joint limit on the equatorialplane field strength and the vertical magnetic geometry,the condition on the geometry is more important.In a conservative analysis, equation (13) can be com-pared to the large-scale magnetic field that might be sup-plied to the outer disk (from, say, the interstellar mediumor a companion star) in order to determine whether ornot orbits are circular in this region. However, equation(14) may be a more appropriate expression if we allowfor the possibility that buoyant rising of magnetic fieldsfrom the turbulent disk into the nonturbulent corona canaffect the magnitude of B zh ; for example, the simulationsof Miller & Stone (2000) have B zh /B & .
02 (even with-out any net imposed vertical field) and thus should easilysatisfy equation (14).As mentioned in § Radius (Schwarzschild radii)10 -6 -4 -2 M i n i m u m R equ i r ed V e r t i c a l M agne t i c F i e l d ( G au ss ) Radius (Schwarzschild radii)10 -6 -4 -2 M i n i m u m R equ i r ed V e r t i c a l M agne t i c F i e l d ( G au ss ) Radius (Schwarzschild radii)10 -6 -4 -2 M i n i m u m R equ i r ed V e r t i c a l M agne t i c F i e l d ( G au ss ) M O • m • = . M O • m • = M O • m • = . M O • m • = Fig. 3.—
Minimum values of the vertical magnetic field B zh thatare required in order for the large-scale field to overcome radiationpressure and be advected inward in the nonturbulent surface layerof an accretion disk. Constraints are shown for black hole massesof 10 M ⊙ and 10 M ⊙ and dimensionless accretion rates between˙ m = 0 . m = 10 (note that the Eddington accretion ratecorresponds to ˙ m = ǫ − , where ǫ is the radiative efficiency ofthe disk and ǫ ≈ .
06 for a simplified treatment of accretion ontoa Schwarzschild black hole). We assume a Shakura & Sunyaev(1973) disk with f ∗ = 1 at the inner boundary and α = 10 − as a worst-case lower limit for the turbulent stress. Solid linesshow regions in which the Shakura & Sunyaev (1973) assumptionsabout pressure and opacity sources in the disk are good to at least50%; dotted lines are used to connect through transition regionsin which the Shakura & Sunyaev (1973) solution breaks down. the constraints on the magnetic field discussed here asfundamental lower limits. On the other hand, one mightreasonably expect that the effect of radiation pressure isto drive material radially outward from the hot inner diskand thereby prevent the nonturbulent surface layer andits associated magnetic field from accreting; in that case,circular orbits can indeed be viewed as a strict require-ment. Conversely, other mechanisms that might producenoncircular orbits, such as advection-dominated flowsat either high or low accretion rates (Abramowicz et al.1988; Narayan & Yi 1994, 1995), tend to enhance the in-ward radial velocity and therefore do not prevent inwardadvection of magnetic fields from occurring. In thesecases, equation (10) should have v K replaced by the ac-tual azimuthal velocity v φ , which makes the equationmore difficult to satisfy but does not change our overallconclusions.Even if orbits in the nonturbulent region are circular,other constraints may come about that could affect theability of this layer to advect magnetic fields inward. Inparticular, our discussion so far has implicitly assumedthat the disk is fully ionized. We do not discuss partiallyionized disks in depth in this paper, but we point out twoimportant issues. First, the diffusivity associated withelectrons scattering off of neutrals (as well as other non-ideal MHD effects) may become more important than theelectron-ion Spitzer diffusivity, and second, there maynot be enough free electrons in the nonturbulent surfacelayer of the disk to support the advected magnetic field.We consider non-ideal MHD effects first. FollowingPandey & Wardle (2007), a more general version of equa-tion (1) which includes the relevant non-ideal MHDterms is ∂ B ∂t = ∇ × (cid:20) v ′ × B − πηc J + ρ n ( J × B ) × B ρ ρ i ν in c (cid:21) , (15) Radius (Schwarzschild radii)10 -10 -8 -6 -4 -2 M i n i m u m R equ i r ed B z h2 / B Radius (Schwarzschild radii)10 -10 -8 -6 -4 -2 M i n i m u m R equ i r ed B z h2 / B Radius (Schwarzschild radii)10 -10 -8 -6 -4 -2 M i n i m u m R equ i r ed B z h2 / B M O • m • = . M O • m • = M O • m • = . M O • m • = Fig. 4.—
Same as Figure 3, but for the minimum values of theratio B zh /B (between the energy density of vertical magneticfields at the disk surface and turbulent fields on the equatorialplane) that are required in order for advection of the large-scalemagnetic field to overcome radiation pressure. where v ′ = v − J /n e e is the plasma velocity modified bythe Hall drift, J ≡ ( c/ π ) ∇ × B is the current density, n e is the electron number density, e is the proton charge, ρ n and ρ i are the neutral and ion mass densities, and ν in is the ion-neutral collision frequency. The last two termsin equation (15) represent the effects of Ohmic diffusion(i.e., the scattering of electrons off of ions and neutrals)and ambipolar diffusion, respectively.We use equations in Draine, Roberge, & Dalgarno(1983) and Balbus & Terquem (2001) to estimate theimportance of the non-ideal MHD effects. For simplic-ity, we assume a priori that the plasma is weakly ion-ized and that the electron and ion number densitiesare the same. We take “worst-case” approximationsof quantities involving the magnetic field (for example, |∇ × B | h ∼ B h /H ), assume that the plasma tempera-ture is given by the effective surface temperature of aShakura & Sunyaev (1973) disk, and make use of equa-tion (9) to substitute for the density when needed. Fidu-cial values are given for a distance of 10 Schwarzschildradii from a 10 M ⊙ black hole, where the surface tem-perature is of order 2,000 K and thus the disk shouldbe weakly ionized. For diffusive processes, we can calcu-late an effective magnetic Reynolds number Re m in thenonturbulent surface layer of the disk (see § Re m ≫
1, advection of the magnetic field will dominate.For Ohmic diffusion, we find that this condition can beapproximately written as n i n n ≫ × − m − / ˆ r − / D Ohm , (16)where n i /n n is the ionization fraction at the base ofthe nonturbulent region (i.e., the ratio of ion to neu-tral number densities) and we have defined a parameter D Ohm ≡ (10 α ) − ˙ m / f / ∗ (cid:0) H/r (cid:1) − U − s which con-tains terms of lesser importance. Here, U s ≡ | v rh | / h v r i is likely to be ≈ § n i n n (cid:18) B zh G (cid:19) ≫ − m − ˆ r − / D amb , (17)where D amb ≡ (10 α ) − f ∗ (cid:0) H/r (cid:1) µ ′ | B h / B zh | U − s and µ ′ ≡ µ (1 + m n /m i ); here, µ is the mean mass perdvection of Magnetic Fields in Accretion Disks 9particle expressed in units of the proton mass and m n /m i is the ratio of neutral to ion masses.We therefore see that ambipolar diffusion imposes ajoint constraint on the ionization fraction and magneticfield strength that is generally much more importantthan the limit imposed by Ohmic diffusion; this is a di-rect result of the low densities expected in the nontur-bulent surface layer that we consider in this paper. Fora partially ionized disk that is able to overcome Ohmicdiffusion and satisfy equation (16), magnetic fields canadvect inward, but equation (17) shows that strongerinitial seed fields than those indicated in equation (13)and Figure 3 (for the constraint imposed by circular or-bits) are generally required. However, in these applica-tions it is important to keep in mind that the ionizationfraction that appears in the above equations is calcu-lated at the surface of the disk (as opposed to the maindisk body), where interstellar cosmic rays may inflate theionization rate above the values that one would usuallyexpect (Gammie 1996).From equation (15), we see that the Hall effect, whichoccurs due to the drift of the magnetic field with respectto the ions as it is carried along by the electrons, may alsobe important, not only for partially ionized disks but alsofor fully ionized ones. However, the Hall effect does notcause diffusion of the magnetic field and therefore doesnot necessarily oppose inward advection. In fact, if theradial component of the current density is positive at thesurface of the disk, Hall drift enhances inward advectionrather than opposes it; this occurs when ∂B φh /∂z <
0, acondition which is already met by the dipole-type fieldsdiscussed in § n i n n (cid:18) B zh G (cid:19) ≫ × − m − ˆ r − / D Hall , (18)where D Hall ≡ (10 α ) − f ∗ (cid:0) H/r (cid:1) − µ | B h / B zh | U − s .This equation also applies for a fully ionized disk if wereplace n i /n n by ≈ /
2. Note that the Hall effect isgenerally more important in a fully ionized accretion diskthan the electron-ion Spitzer diffusivity discussed in § n i n n (cid:18) B zh G (cid:19) & × − " µ (cid:0) H/r (cid:1) | B h / B zh | m / ˙ m / ˆ r / f / ∗ (19)is required to produce enough current to support themagnetic field, which is generally a weaker constraintthan that imposed by ambipolar diffusion. (Note thatstronger magnetic fields are actually easier for the non-turbulent surface layer to support; this is because strongfields suppress turbulence at a lower height above thedisk, where the electron density is larger.) The Outcome of Advection
In summary, we find that weak, large-scale magneticfields can be advected inward in the surface layer of anaccretion disk. The most important condition necessaryfor this is ∂ ( B φ B z ) h /∂z < § B z ∝ r − / in the gas pressure-dominated regionof a Shakura & Sunyaev (1973) disk in order to satisfythe requirement for advection to continue; this limit be-comes B z ∝ r − / in the inner, radiation-dominated re-gion (where H is approximately constant). This growthseems difficult to sustain, which suggests the interest-ing possibility that an advected magnetic field may tem-porarily “stall” at some large radius when it becomes tooweak to advect inward, and advection will only continueonce enough magnetic flux has built up at this location.On the other hand, as we have seen in discussing equa-tion (14), even if the advected field does not increase instrength fast enough on its own, it may be reasonableto assume that the local dynamo can produce strongenough surface fields (through magnetic buoyancy) tomeet these conditions, and thus advection will continuein either case.As discussed in §
1, a sufficiently strong magnetic fieldthreading the disk can lead to inward radial advectionof the field driven by the main body of the disk too,due to extraction of angular momentum from the mainbody of the disk to a wind or jet (Lovelace et al. 1994).However, the field strengths required for this process aremuch larger; the field must drive accretion in the densebody of the disk (rather than in a low-density surfacelayer). In fact, advection of magnetic fields in the surfacelayer is always more efficient, provided that the disk isthin and the vertical field is subequipartition.Although advection in the main body of the disk musttake place fast enough to overcome turbulent diffusion,advection in the nonturbulent region is limited by themuch smaller speeds associated with the microscopic dif-fusivity. Magnetic fields can advect inward in the non-turbulent surface layer even when | v rh | ≪ h v r i , and theadvection can therefore take place on timescales longerthan a viscous timescale. This will lead to a gradualbuildup of field in the inner disk. In the case of an-gular momentum extraction from the main body of thedisk, advection of magnetic fields always occurs at least as fast as the viscous timescale, and sustained advec-tion requires that the magnetic field be strong enough tocompletely overwhelm turbulent diffusion; otherwise, asteady state will be reached in which only a small con-centration of magnetic field develops in the inner disk.0 Rothstein & LovelaceThus, we may expect a disk to experience gradual ad-vection of large-scale magnetic fields on long timescalesin the surface layer, with occasional brief bursts of fastor implosive accretion (Lovelace et al. 1994) associatedwith the presence of a strong magnetic field in a localregion of the disk that extracts angular momentum fromthe main disk body.It is interesting to note that advection of magneticfields in the surface layer is more efficient for thin disksthan thick ones, as can be seen, for example, from equa-tion (8). This is because the surface layer is tightly cou-pled to the main body of the disk rather than being partof a detached corona. On the other hand, turbulent ad-vection of magnetic fields in the main body of the diskis more efficient if the disk is thick (since Re m ≈ H/r ,as discussed in § CONCLUSIONS
This paper reanalyzes the advection of a large-scale,weak magnetic field in an accretion disk. We consider thevertical structure of the disk, which strongly influencesthe vertical profile of the conductivity, as pointed out byBisnovatyi-Kogan & Lovelace (2007). In the thin, diffusesurface layers of the disk, the magnetic energy density islarge enough compared to the thermal energy densitythat magnetorotational turbulence is suppressed. As aconsequence, magnetic field lines threading the surfacelayer can be advected inward with the main body of thedisk, without being opposed by turbulent diffusion.No special conditions are required for the field to be ad-vected inward except that it meet the rather modest con-straints in §
4. The required field strengths are relativelyweak, and the primary constraint on the field geometryis simply that it must help the nonturbulent surface layeraccrete inward (i.e., the vertical magnetic stress must ex-tract angular momentum from this layer, in virtually anyamount). This can be accomplished either via couplingbetween the main, turbulent body of the disk and thesurface, or via a wind or jet. The simplest way for thiscondition to be met is if the accretion flow is providedwith a weak, large-scale vertical seed field threading theouter region of the disk (which could come from the inter-stellar medium or a companion star), although in somecases, the proper geometry may be attained entirely as aresult of fields produced via the local magnetorotationaldynamo. Once a weak, large-scale field with the propergeometry is in place, the field will be advected inwardalong the disk’s surface layer and strengthened as it iscompressed along with the accretion flow.The presence of a large-scale magnetic field anchoredin the surface layer will drive strong magnetorotationalturbulence in the main body of the disk, which can con-sequently produce values of the α parameter (i.e., theturbulent stress) large enough to match observationalconstraints. We find that typical vertical fields on theorder of a few percent of equipartition are required forthis to occur. Because the field is advected inward andanchored in the surface layer, there is no need to worryabout maintaining the required vertical fields via in-ternal magnetorotational fluctuations (as suggested byPessah et al. 2007). We propose that long-term changesin α (in response to the history of magnetic field ad- vection) should be explored as a possible source of thelong-term evolution in the variability patterns seen inthe light curves of X-ray binaries such as GRS 1915+105(see, e.g., Tagger et al. 2004).The mechanisms we discuss in this paper are relevantto many different kinds of accreting objects. In the outerpart of the disk (far away from the central star or blackhole), our work should be applicable provided only thatthe disk is sufficiently ionized (see § azimuthally-averaged strength and azimuthally-averaged geometry that roughly meets thecriteria in § ACKNOWLEDGEMENTS
We thank G. S. Bisnovatyi-Kogan, M. M. Ro-manova, and I. G. Igumenshchev for valuable discus-dvection of Magnetic Fields in Accretion Disks 11sions. D. M. R. is supported by an NSF Astronomyand Astrophysics Postdoctoral Fellowship under awardAST-0602259. The work of R. L. was supported in part by NASA grants NAG5-13220 and NAG5-13060 and byNSF grant AST-0507760.
APPENDIX A. PHYSICAL CONDITIONS AT THE BASE OF THE NONTURBULENT REGION ABOVE AN ACCRETION DISK
In this section we derive equation (9), which defines the base of the nonturbulent surface layer of the accretion diskand is used throughout the main body of the paper.We are interested in conditions at z = h , the height in the disk where the magnetorotational instability (MRI)is first suppressed. The fundamental condition for suppression of the MRI is that the Alfv´en speed must be largeenough so that fluid elements linked by the magnetic field will be drawn back together faster than orbital shear drivesthem apart; in a WKB (small wavelength) analysis, MRI modes with wavenumber k are found to be suppressed when k · v A & Ω K , where v A is the local Alfv´en speed (e.g., Balbus & Hawley 1998) and we have assumed circular orbits(relaxing this assumption would simply replace Ω K with the actual local angular velocity of the disk). If we take theWKB approximation to its limit and make the usual assumption that the MRI will be completely suppressed when nounstable wavelength fits within the disk (e.g., Balbus & Hawley 1991) and further consider that the largest wavelengthsthat fit within the disk are of order ∼ (a few) × H in the vertical direction and ∼ (a few) × r in the azimuthal direction,then applying this condition to the material at z = h shows that MRI turbulence will be suppressed when( v Az ) h + Hr ( v Aφ ) h & H Ω K , (A1)where ( v Az ) h ≡ B zh / √ πρ h and ( v Aφ ) h is defined equivalently. Rearranging this equation gives, approximately, ρ h . B zh πH Ω K , (A2)where the equality holds at z ≈ h (the height at which turbulence is first suppressed), but the equation also appliesmore generally if h is redefined to be any height within the nonturbulent region. Here, we have defined a modifiedvertical magnetic field strength B zh ≡ | B zh | + ( H/r ) | B φh | . In the main body of the paper, we generally assume B zh ≈ | B zh | , but the full expression should be used when the toroidal field is extremely strong ( B zh ≈ | B zh | appearsto be a good approximation for the magnetic fields seen in the simulations of Miller & Stone 2000, however).If we define p ≡ ρ ( H Ω K ) , equation (A2) becomes ρ h ρ . B zh πp , (A3)which is the equivalent of equation (9). If the large-scale magnetic field is weak enough so as to not significantlyaffect the dynamics in the main body of the disk (in particular, if thermal pressure supports the disk verticallyagainst gravity), then p which appears in equation (A3) should be interpreted as the thermal pressure. However, theassumption of weak fields is not required, and equation (10), which is derived from equation (A3), applies when thefield is strong as well as when it is weak. Only in the parts of the main body of the paper where we combine equation(9) with the Shakura & Sunyaev (1973) solution are we assuming that the large-scale magnetic field is dynamicallyweak.Equation (A3) is not the same as assuming that the magnetic and thermal energy densities are comparable at z ≈ h ,which is sometimes quoted as the condition for suppression of the MRI. We note, however, that the condition onthe Alfv´en speed is more fundamental, and the condition on the magnetic energy density is merely derived from itunder a specific set of circumstances in the main body of the disk. In fact, analytical studies of both stratified andunstratified disks generally suggest that the condition k · v A & Ω K is most important for MRI suppression, regardlessof the overall magnetic energy density (Blaes & Balbus 1994; Gammie & Balbus 1994; Kim & Ostriker 2000). Thus,we believe that equation (A3) is correct.In fact, what is most clear from the above MRI studies is that the stability criteria in each coordinate directionare roughly independent; for example, a strong toroidal field does not significantly affect the most unstable verticalwavelengths. Thus, if we were to suppose that the MRI is suppressed when the local magnetic and thermal energydensities are equal, it would be reasonable to assume that the appropriate condition is B zh & πp h (i.e., that it involvesthe vertical rather than the total magnetic energy density), in which case we would derive ( p h /p ) . B zh / πp ratherthan equation (A3) as the condition for MRI suppression; here, the subscripts have their usual meanings. We thereforesee that the two possible conditions are essentially the same for a disk that is dominated by gas pressure up to z ≈ h ;one would simply need to modify equation (A3) by introducing a ratio of temperatures T /T h on the right hand side,which is generally no more than a factor of a few. Only if the disk is radiation-dominated at z ≈ h is there a significantdifference between the two conditions for suppression of MRI turbulence. In the case of a radiation-dominated disk,our use of equation (A3) means that we are assuming the MRI may be suppressed in the surface layer of the disk evenwhen the magnetic pressure is weaker than the radiation pressure there.2 Rothstein & Lovelace B. MAGNETIC FIELD ADVECTION IN QUASI-STATIONARY ACCRETION DISKS
Our work in this paper is concerned with accretion disks that are fundamentally time-dependent. In particular,we are studying situations in which a large-scale magnetic field is being dragged inward and causing the magneticflux in the inner region of the disk to change in accordance with equation (3). Such disks are complicated to studyanalytically, and for this reason we used order of magnitude approximations to reach some of our conclusions in § H B , the scale height of the vertical magnetic stress).In this section, we will give arguments for these approximations by considering the simplified case of a “quasi-stationary” disk. By this we mean a disk that is allowed to vary on the long (viscous) timescales at which magneticfield advection takes place but which is assumed to quickly adjust its structure on shorter timescales (in response tothe changing magnetic flux) so that in most respects it can be treated as stationary.We begin by evaluating the φ component of the induction equation (1) at z = h (where the magnetic diffusivity isnegligible) in an axisymmetric disk, which gives ∂B φh ∂t = ∂∂z ( v φ B z − v z B φ ) h − ∂∂r ( v r B φ − v φ B r ) h . (B1)If we assume locally circular orbits in Newtonian gravity and evaluate spatial derivatives of the orbital velocity v φ tofirst order in h/r , we can combine equation (B1) with the condition ∇ · B = 0 to derive ∂B φh ∂t ≈ −
32 Ω K (cid:18) B rh + hr B zh (cid:19) − ∂∂z ( v z B φ ) h − ∂∂r ( v r B φ ) h . (B2)The first term in this equation represents the production of azimuthal field via Keplerian shear and is generally themost important effect; it leads to the creation of B φ on a characteristic timescale of ∼ Ω − K (i.e., the orbital timescale).Therefore, in accordance with our quasi-stationary approximation, we can assume that the disk quickly adjusts itsvalue of B φh in response to the magnetic field advection so that ∂B φh /∂t is negligible on our timescales of interest.We can therefore rewrite equation (B2) as ∂B φh ∂z ≈ −
32 Ω K v zh (cid:18) B rh + hr B zh (cid:19) − B φh ∂ ln v zh ∂z − B φh r (cid:20) v rh v zh ∂ ln ( v rh B φh ) ∂ ln r (cid:21) . (B3)We can simplify this equation by assuming that ρv z is constant with height so that there is no net buildup of mass inthe vertical outflow. This is a reasonable assumption for thin disks for the timescales of interest. Thus, ∂ ln v zh /∂z = − ∂ ln ρ h /∂z . Numerical simulations show that this quantity is, in turn, well-estimated by ∼ H − throughout theatmosphere of the disk, even in time-dependent systems in which radiation and magnetic fields help to provide verticalsupport against gravity (Hirose et al. 2006; Krolik et al. 2007). Equation (B3) then becomes ∂B φh ∂z ∼ − Ω K v zh (cid:18) B rh + hr B zh (cid:19) − B φh H (cid:18) Hr (cid:20) v rh v zh ∂ ln ( v rh B φh ) ∂ ln r (cid:21)(cid:19) . (B4)If we combine this equation with the definition of H B , use ∇ · B = 0, and then ignore terms of order ∼ H/r (assumingthat the logarithmic radial derivatives are of order unity, which should be true except at boundary regions where themagnetic field structure changes dramatically), we obtain H B ∼ H f B − f B , (B5)where f B ≡ ( v zh /H Ω K ) ( − B φh /B rh ) is typically a positive number in any magnetic field geometry (due to Keplerianshear). We therefore see that unless v zh approaches the sound speed, we have H B . H for typical generic fieldgeometries. This corresponds to our approximation in the main body of the paper. Furthermore, since v zh is simplythe initial speed at which material is launched off the disk surface (before it enters any jet acceleration region), we donot expect it to be large; a reasonable estimate might be v zh ∼ h v r i , where h v r i ∼ α ( H/r ) H Ω K is the speed at whichmaterial in the main body of the disk accretes inward. In that case, we would have H B ≪ H , which is even morefavorable for inward advection.The above analysis shows that the vertical magnetic stress potentially available at z = h in an accretion disk ismore than enough to drag the surface layer inward. In fact, if we consider equation (11) in light of these results, theimmediate suggestion is that the surface layer can advect inward much faster than the main body of the disk. Theproblem with this conclusion, however, is that the above analysis only considered the atmosphere on its own, withoutregard to the main body of the disk below it. A large shear between the main body of the disk and the surfacelayer is unlikely to be stable. Although the physics in the interface between the turbulent body of the disk and thenonturbulent region above it is complicated, we can estimate the effects of a large vertical shear in v r in the case ofideal MHD. We use the r component of equation (1), which is ∂B rh ∂t = ∂∂z ( v r B z − v z B r ) h , (B6)so that there is a term B zh ∂v rh /∂z which tends to produce B rh on a typical timescale ( ∂v rh /∂z ) − . Thus a largevertical shear in v r will rapidly change the radial magnetic field—and thereby the vertical stress in accordance withdvection of Magnetic Fields in Accretion Disks 13equation (B4)—in the appropriate direction for the shear to be reduced. This suggests that very small values of H B are not sustainable, and while it is difficult to predict the exact behavior, we expect that in the general case thebase of the nonturbulent surface layer will advect inward at a similar speed as the main body of the disk. Note thatturbulent stresses may also play a role in this region in ensuring that the disk and surface layer advect inward together(similar to a “friction” term). We do not consider their effect here except to note that they are probably smallerthan the large-scale magnetic stresses. (Recall that the large-scale magnetic energy density is by definition com-parable to the thermal pressure at z = h , while the turbulent vertical stresses are likely to be smaller by a factor of ∼ α .).)