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Angular Momentum Transport in CV Accretion Disks Mem. S.A.It. Vol. 83, 594 c SAIt 2012 Memorie della Angular momentum transport in CV accretion disks O. Regev1;2 and O. Umurhan3;4 1 Dept. of Physics, Technion - Israel Institute of Technology, Haifa 32000, Israel 2 Dept. of Astronomy, Columbia University, New York, NY 10027 e-mail: [email protected] 3 Dept. of Applied Mathematics, University of California Merced, Merced, CA 95343 4 Dept. of Astronomy, City College of SF, SanFrancisco, CA 94112 e-mail: [email protected] Abstract. A physical mechanism for enhanced angular momentum transport (AMT) out- wards is needed for the theoretical modeling of accretion disks (ADs) in e.g. Cataclysmic Variables (CVs). It is clear that ordinary microscopic viscosity is out of the question - the time scale of transport it dictates is many orders of magnitude longer than the one deduced from mass accretion rate consistent with observations. It is however reasonable that tur- bulent transport can provide the needed timescale value. Astrophysicists have persistently looked for an linear instability in ADs, which can ultimately lead to turbulence. In this con- tribution we dispute the ”common knowledge”, accepted during the last two decades or so, that the magneto-rotational instability (MRI) is the physical agent that destabilizes thin ADs and ultimately drives turbulence and AMT. We also suggest, thus, that we do not yet have at our disposal any better model for AMT than the30 year old α model Key words. Binary stars: accretion disks – MHD instabilities: MRI – Transport: angular momentum 1. Introduction ity. In doing so, thin AD, for the first time, could be quantitatively modeled in a variety of Prendergast & Burbidge (1968) were the first astrophysical settings. In turn, this has brought to propose the existence of thin ADs, to explain about significant progress in our understanding some galactic X-ray sources. Similar objects of these important systems. The value and na- were also proposed in CVs. Greatly enhanced ture of the non-dimensional viscosity param- (turbulent) viscosity was invoked for making eter α (see below) still remains to be a the- the model viable. A few years later Shakura oretical unknown and phenomenological ap- & Sunayev (1973) and Lynden-Bell & Pringle proaches have usually been employed to deter- (1974) successfully bypassed the specific lack mine it. of understanding of turbulence (a situation that is largely still with us) by employing physically No definitive candidate hydrodynamical motivated parametrization of turbulent viscos- instability had been convincingly identified in ADs and widely accepted as a possible Send offprint requests to: O. Umurhan initial driver of turbulence before (and af- Regev: Angular momentum transport 595 ter...) Balbus & Hawley (1991) found that ployed, for example, by Hawley et al. (1995) weak magnetic fields destabilize differential (H95), but the very recent calculations of Lesur Keplerian rotation. The linear MRI (Velikhov & Longaretti (2007) included explicit viscosity 1959, Chandrasekhar 1960), was shown to op- ν and resistivity η in a an SB high-resolution erate under conditions characterizing rotation- numerical simulation. Systematically changing ally supported magnetized ”cylindrical accre- the Reynolds number (R) and the magnetic tion disks”. Prandtl number mPr ≡ ν/η, in some range, This suggestion has inspired intensive re- these authors aimed at uncovering the trends search activity on the linear MRI and its non- and scalings of relevant physical properties, linear development. It has been investigated for notably angular momentum transport, with the various conditions, geometries, and boundary relevant non-dimensional number(s). conditions (BC). Beyond the linear analysis it The most recent calculations in the SB has been approached, almost exclusively, by framework with relatively high resolution have methods of computational (magneto)fluid dy- been reported by Fromang & Papaloizou namics. The complexity of the nonlinear devel- (2007) (FP) and by Fromang et al. (2007) opment became readily apparent quickly ap- These studies (both with PBC) focused on the parent and, as a consequence some understand- issue of convergence (i.e., numerical resolu- ing of the nonlinear transition and saturation tion) in ideal SB calculations and on (R) and processes, is indispensable, if the aim is to mPr dependencies of the transport in dissipa- improve on the more than 30 years old phe- tive conditions. nomenological α-model. At the same time King, Pringle & Livio The purpose of this contribution is to as- (2007) (KPL), pointed out the discrepancy (of sess the viability of the most common approxi- at least one order of magnitude) between the mation used in numerical studies - the shear- values of α inferred from observations and the ing box (SB). The review articles by Balbus estimates of this parameter based on (mainly & Hawley (1998) and Balbus (2003) contain the early) SB numerical simulations, suggest- a comprehensive list of references, of which ing that the effective α decreases (almost lin- only few will be referred to here. We shall also early) with the value of the imposed Bz. They present, in the last section, a semi-analytical conjectured that perhaps only zero net flux (ZF) study of the saturation of the MRI, in an exper- calculations should be used. imental simplified setup, which has been used in experimental efforts to observe the MRI and its nonlinear development in the laboratory 2.1. The problematics of the SB (e.g. Liu, Goodman & Ji 2006 and references approximation therein). The essence of the SB approximation is in its local approach, that is, the resulting equa- 2. Numerical calculations of the MRI tions for the perturbations on a steady base nonlinear development flow are approximately valid in a small re- gion (a Cartesian box) about a typical point in Most of the existing SB calculations (at least the disk. The global MHD base flow in an al- of the early ones) employ a base state with most Keplerian accretion disk is not only ro- a constant vertical magnetic field and numer- tating and strongly sheared, but it is also inho- ically follow the development of perturbations mogeneous, non-isotropic (endowed with non- on this state. This situation, usually referred to zero gradients in density and other physical as fixed net flux (FF), seems to be the most ap- variables and these gradients have very differ- propriate for disks threaded by external mag- ent scales in different directions) and swirling netic fields . The majority of existing numerical (streamlines are curved). works of this sort have used ideal (that is, non- Umurhan & Regev (2004) followed a sys- dissipative) MHD SB equations with periodic tematic derivation of the approximation us- boundary conditions (PBC) in the manner em- ing asymptotic scaling arguments with the pur- 596 Regev: Angular momentum transport pose of quantifying the approximations that length and the integral scale, denote them as ` are made leading to the SB (see Appendix A and refer to them interchangeably. This identi- of that paper). Two defining nondimensional fication has an obvious intuitive physical basis. numbers were stressed in their study- (the In a disk, which is strongly non-isotropic, typical disk height h0, in units of r0, atypical two disparate injection scales seem a priori to radius) and δ (the box size in the same units). appear - the horizontal one `h∼< r0 and the verti- For details see that paper. cal one `h∼< r0 (both given here in dimensional We stress that virtually all numerical cal- units). It is however reasonable to identify ` culations that utilize the SB approximation, in with the smaller of the two, because this is the all its variants, employ PBC in which the pe- size of largest eddies. A quantitative measure riodicity in the radial direction is sheared (see of ` can be ”phenomenologically” determined H95). These sheared-periodic-BC are equiva- by the relevant wave-number at which the dis- lent to enforcing that all perturbation quanti- turbance kinetic energy Ek(k) spectrum peaks. ties to be triply (or in the case of a 2-D calcu- The fact that the scale at which resistivity and lation, doubly) periodic in the sheared coordi- viscosity come into play need not be equal nates system. It is also important that our cri- makes also the determination of the dissipation tique of the SB scheme applies to both FF and scale not unique. Using the Spitzer values we ZF conditions. get typically mPr 1 ( = 6 × 10−29T 4/ρ ∼ 5 × 10−5 c.g.s, for CV disk conditions), i.e., kν > kη where kν and kη respectively represent 2.1.1. Relevant length scales the dissipation scale wave-numbers for viscos- ity and resistivity. Phenomenologically, the in- In a typical accretion disk, r0 is the scale on which curvature terms appear and this is ertial range is considered to be the k region, also the scale of any underlying radial struc- in which Ek(k) exhibits a power-law behav- ture gradients. Vertical stratification obviously ior. Thus, the inertial range in the MHD tur- appears on the vertical pressure scale-height, bulence of a disk can be considered as includ- ing wave-numbers roughly in the range −1 which is = h0=r0 1 in our notation and units. In studies of turbulence it is customary k kη. For a fairly detailed discussion of the −1 MHD turbulence length scales and relevant ref- to consider the injection scale, ` ≡ kin , on which the process of energy injection into the erences see BI. In any case, possible nonlinear system is effected and the dissipation scale, interactions between the short vertical wave- −1 length unstable modes with modes (possibly ≡ kd , on which microscopic viscosity (and in MHD also resistivity) are operative. These also horizontal ones) whose scale is larger than are linked by the inertial range of scales. Since the SB scale cannot be captured in a SB sim- ulation.
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