A&A 513, A47 (2010) Astronomy DOI: 10.1051/0004-6361/200913169 & c ESO 2010 Astrophysics

Low magnetic-Prandtl number flow configurations for cold astrophysical disk models: speculation and analysis

O. M. Umurhan1,2

1 Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK e-mail: [email protected] 2 Astronomy Department, City College of San Francisco, San Francisco, CA 94112, USA Received 24 August 2009 / Accepted 28 January 2010

ABSTRACT

Context. Simulations of astrophysical disks in the shearing box that are subject to the magnetorotational instability (MRI) show that activity appears to be reduced as the magnetic Prandtl number Pm is lowered. It is therefore important to understand the reasons for this trend, especially if this trend is shown to continue when higher resolution calculations are performed in the near future. Calculations for laboratory experiments show that saturation is achieved through modification of the background shear for Pm 1. Aims. Guided by the results of calculations appropriate for laboratory experiments when Pm is very low, the stability of inviscid disturbances in a shearing box model immersed in a constant vertical background magnetic field is considered under a variety of shear profiles and boundary conditions in order to evaluate the hypothesis that modifications of the shear bring about saturation of the instability. Shear profiles q are given by the local background Keplerian mean, q0, plus time-independent departures, Q(x), with zero average on a given scale. Methods. The axisymmetric linear stability of inviscid magnetohydrodynamic normal modes in the shearing box is analyzed. Results. (i) The stability/instability of modes subject to modified shear profiles may be interpreted by a generalized Velikhov criterion givenbyaneffective shear and radial wavenumber that are defined by the radial structure of the mode and the form of Q. (ii) Where channel modes occur, comparisons against marginally unstable disturbance in the classical case, Q = 0, shows that all modifications of the shear examined here enhance mode instability. (iii) For models with boundary conditions mimicing laboratory experiments, modified shear profiles exist that stabilize a marginally unstable MRI for Q = 0. (iv) Localized normal modes on domains of infinite radial extent characterized by either single defects or symmetric top-hat profiles for Q are also investigated. If the regions of modified shear are less (greater) than the local Keplerian background, then there are (are no) normal modes leading to the MRI. Conclusions. The emergence and stability of the MRI is sensitive to the boundary conditions adopted. Channel modes do not appear to be stabilized through modifications of the background shear whose average remains Keplerian. However, systems that have non- penetrative boundaries can saturate the MRI through modification of the background shear. Conceptually equating the qualitative results from laboratory experiments to the conditions in a disk may therefore be misleading. Key words. accretion, accretion disks – magnetohydrodynamics (MHD) – instabilities – protoplanetary disks

1. Introduction the order of 10−6 at best and that the MRI may be operating only in the disk’s corona as its ionization fraction is sufficient to merit After almost 40 years of investigation the source of the anoma- an MHD description there rather than near the disk midplane lous transport in accretion disks still remains an open question. (see Balbus & Henri 2008, and references therein). Thus, the fate As it is mostly assumed that the transport is a reflection of some of the MRI in disk environments, in which the magnetic Prandtl underlying turbulent state, identification of its source mecha- number is characteristically small, remains to be settled. nism and process has been the focus of much research activity. It is not clear why turbulent transport appears to vanish in A leading candidate mechanism is the magnetorotational insta- numerical experiments when P weakens. One possibility may bility (Balbus & Hawley 1991). m be that under those conditions the MRI cannot grow sufficiently The magnetic Prandtl number (Pm), i.e. the ratio of a fluid’s to excite a secondary transition that would, in turn, open the way viscosity to its magnetic diffusivity, appears to play an important to a turbulent cascade. Theoretical considerations of the MRI for role in the ability of the MRI to drive a fully-developed turbulent restrictive configurations like laboratory experiments with cylin- state. Recently resolved simulations of the MRI in a shearing drical geometry predict that a low Pm sheared fluid undergoing box environment (e.g. Lesur & Longaretti 2007; Fromang et al. the MRI settles onto a pattern state whose momentum transport α 2008) show the appearance of a turbulent state for moderately scales as ∼(Pm) where α ≥ 1 – in other words, saturation of the high Reynolds numbers (Re) when Pm is an order 1 quantity or instability is sensitive to the microscopic viscosity of the fluid. higher. However, the amount of transport delivered by the MRI This counterintuitive result is rationalized by noting that a weak appears to depend on Pm: as this quantity decreases the vigor of fluid viscosity means that it takes very little effort to readjust the the turbulent state decreases and if Pm drops below some critical underlying flow profile and establish a new shear in which the value, then the turbulence appears to vanish altogether. instability cannot operate. The stronger the viscosity, then, it be- Estimates of the properties of cold astrophysical disks, like comes more difficult for the fluid to shut off the shear and the protoplanetary disks, show that their characteristic Pm’s are on instability may operate unabatedly during the fluid’s nonlinear Article published by EDP Sciences Page 1 of 16 A&A 513, A47 (2010) development and eventual cascade to turbulence via secondary instabilities (e.g. parasitic instabilities, Xu & Goodman 1994). −1/2 In the idealized quasi-linear study of Knobloch & Julien Keplerian Profile: V ~ r (2006) it was shown that in the limit of very large hydrodynamic and magnetic Reynolds numbers (i.e. Re  1andRem  1re- ff spectively) the azimuthal velocity profile tips over to shut o Instability Modified the instability by establishing a new velocity profile which re- Mean Velocity Profile duces the shear throughout the domain. Their analysis shows that in the limits where Re →∞and Rem →∞the resulting weaker than net velocity profile tends toward zero shear. Examinations of the Keplerian shear Mean Rotational Velocity: V thin-gap Taylor-Couette system in the low Pm limit (Umurhan stronger than et al. 2007a,b) show that once the system reaches saturation, the Keplerian shear resulting velocity field within large mid-portions of the exper- imental domain is characterized by weakened shear. However Radial Distance: r these regions, where the tendency toward instability is reduced, Fig. 1. Qualitative depiction of a hypothesized mean rotational velocity are sandwiched by regions of strengthened shear, where the ten- profile for a low magnetic Prandtl number disk suggested by Jamroz dency toward instability is enhanced. Nonetheless, the aggregate et al. (2008). The dashed line shows the usual Keplerian velocity profile configuration results in a profile that is stable against the MRI describing a rotationally supported disk. The mean velocity profile re- or any secondary instability and the system settles onto non- sulting from the saturation of the MRI in the low Pm limit is speculated turbulent pattern state. Equally relevant is that in this pattern to consist of alternating regions of low and high shear (black lines). The state the amplitudes of all fluid quantities, except the modified mean shear over the extent of the disk follows the general Keplerian profile. shear, scale as some power of Pm. By comparison the modi- fication of the shear is an order 1 quantity. Thus, in the limit where Pm becomes very small only the modification to the back- The range of questions that figure prominently are: (i) what ground shear appears as a noticeable response in the idealized happens to the unstable mode leading to the MRI when there are studies of the laboratory setup. departures of the steady shear from the background Keplerian Because resolved shearing box numerical experiments of state? (iii) How does the nature or existence of unstable modes high Re and low Pm flows appropriate for cold astrophysical depend upon the radial boundary conditions of the shearing box disks is currently out of reach, speculation at this stage is jus- (e.g. whether it is open or periodic in the radial direction)? tifiable. In particular, what may be the effective azimuthal ve- (ii) Can one infer a possible reason or explanation for why turbu- locity profile of high Re and low Pm disks (or their sections)? lence seems to vanish in numerical experiments of the shearing As Ebrahimi et al. (2009) recently note, quasi-linear satura- box when Pm is small? tion of the shear profile might be an unreasonable expectation This work is organized as follows: in Sect. 2 the equations given that the strong gravity tends to restore the shear pro- of motion are laid out and the framework within which the sta- file within disks. However, the theoretical results of the thin- bility analysis is discussed. Section 3 develops the profiles for gap Taylor-Couette system and the quasi-linear studies, together the steady state for given modifications of the background shear with the objection of Ebrahimi et al., point to a possible hybrid while in Section 4 the normal mode stability analysis is devel- scenario: what if the velocity profile of a low Pm disk is on aver- oped and the governing ODE is derived including a statement of age Keplerian but locally exhibits alternating zones of very weak the generalized Velikhov criterion (Velikhov 1959). Section 5 (or no) shear and very strong shear? This hypothetical configu- presents the results of the stability analysis for a variety of ration of the azimuthal velocity profile, an example of which is boundary conditions and shear profiles and their forms. The im- depicted in Fig. 1, might very well be driven into place by the plications of the results are discussed in the final section. MRI under very low Pm situations. The recent studies of Julien & Knobloch (2006) and Jamroz et al. (2008) argue for the rele- 2. Equations of Motion vance of this scenario. This study is an examination of the axisymmetric linear The small shearing box equations (SSB, Lesur & Longaretti stability of incompressible ideal magnetohydrodynamic distur- 2007;Regev&Umurhan2008) are adopted which are the in- bances in a shearing box threaded by a constant vertical mag- compressible incarnation of the usual shearing box equations netic field and characterized by a variety of mean velocity pro- (Goldreich & Lynden-Bell 1965). Their usage is justified as files, including an extreme instance of the one appearing in compressibility is an inessential feature for the MRI. The box Fig. 11. Such configurations might be representative of real as- is moving in a rotating frame moving with the local Keplerian ¯ 2 trophysical disks characterized by very small values of Pm. velocity VK . The length scales of the box are measured by a size L¯ which is much smaller than the radial scale of the disk R¯ . Guided by the low Pm results of Knobloch & Julien (2005) 0 and Umurhan et al. (2007a,b), the analysis undertaken in this On these scales, the background Keplerian shear appears as a study is inviscid. The only low P effect that is included here is linear Couette flow profile. Time units are scaled by the local m Ω¯ ¯ the possibility that the fundamental shear profile is significantly rotation time of the disk (R0), as measured at the fiducial ra- ¯ modified. This altered shear profile may be easily included into dius R0, implying that the velocities of interest are scaled by ¯ = ¯Ω¯ ¯ the equations of motion as any radially dependent barotropic U0 L (R0). Given this, the disparity of the ratio of the local steady shear profile is a permitted equilibrium flow solution of disk soundspeed to the Keplerian speed (since the gas is rela- ¯ the small shearing-box equations. tively cold), and the assumption that the scale L is much less than the vertical scale height of the disk, the flow dynamics have the character of incompressible rotating magnetic Couette flow. 1 Extreme in the sense that regions of sharp shear are represented by delta-functions in the analysis herein. See also discussion in Sect. 6. 2 Overbars over quantities refer to their dimensional values.

Page 2 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles

Magnetic fields are scaled by a reference field scale B¯ 0 and the linear stability analysis that follows only the steady state total density isρ ¯. These quantities form a velocity defined to be the vorticity field, denoted by q, plays a role and it is defined as Alfven speed via U¯ 2 ≡ B¯ 2/4πρ¯. Furthermore the Cowling num- A 0 = + , ber is defined q q0 Q (7) ¯ 2 where Q(x) ≡−∂ U/Ω . A variety of functional forms for Q will UA x 0 C≡ , be examined. The only restriction on Q will be that its domain U¯ 2 0 integral is bounded, i.e., which may be considered as related to the inverse-square of the “β-parameter” of plasma physics. This is not to be confused with < ∞, Qdx the parameter β used later on in this work to designate radial D wavenumbers. The non-dimensionalized SSB equations of mo- where D symbolizes the domain under consideration. In many tion are cases this integral will be zero. 1 ∂ + u ·∇u + Ω z × u = − ∇p + Ω2q xx + CJ × B, ( t ) 2 0 ˆ ρ 2 0 0 ˆ (1) 4. Linear theory ∇·u = 0, (2)

∂t B = ∇×u × B, (3) Axisymmetric infinitesimal disturbances about the steady state shaped by q(x) are introduced into the governing equations of J = ∇×B, (4) motion (1−4) revealing, in which u is the velocity vector, p is the pressure, ρ is the con- ∂ u − 2Ω v = −∂ p /ρ + CB b + CB ∂ b , (8) stant density (which is 1 in these units). Ω0 is the local rotation t 0 x 0 z z z z x rate but in the units used here it is also 1. The parameter q0 de- ∂tv + (2 − q)Ω0u = CBz∂zby, (9) fines the amplitude of the local Keplerian shear and is formally ∂ w = −∂ /ρ + C + C ∂ , given by t z p 0 Bzbz Bz zbz (10) ∂ u + ∂ w = , R¯ ∂Ω¯ x z 0 (11) ≡− · , q0 ∂ b = B ∂ u, Ω¯ ∂R¯ t x z z (12) R¯0 ∂tby = −Ω0qbx + Bz∂zv , (13) / whichisequalto32 for rotationally supported Keplerian disks. ∂ = ∂ w , tbz Bz z (14) This is the value assumed for q0 throughout the remainder of this work. The magnetic field is given by the vector B and the in which u ,v w and p denote perturbations in the radial, az- 3 current J and is governed by Ampere’s Law . imuthal vertical velocities and pressures, respectively, while

bx,y,z represent perturbations in the magnetic field components. q x 3. Steady state It should be recalled that ( ) is unspecified at this stage. However the existence of a steady state is assumed for a given It is assumed that the fluid is of constant density ρ0 and the reasonable q(x) profile. Normal mode perturbations are assumed fluid is threaded uniformly throughout by a constant vertical which are periodic in the vertical direction. A perturbation vari- , , σ + magnetic field, i.e. B = Bz ˆz. The current source of this back- able f (x z t) is thus expressed via the ansatz f (x)exp( t ground field is taken to be external to the domain. As stated in ikz)whereσ is the temporal response while k is the vertical the Introduction the steady velocity field that represent depar- wavenumber. The incompressibility condition (11) means that tures to the Keplerian shear are taken as given and represented a streamfunction ψ may be defined such that, by U(x). The total mean velocity is written as V(x) which is the u = ∂zψ ,w≡−∂xψ . sum of the Keplerian portion −q0Ωx and the departures U. Thus the steady state configuration requires satisfying the radial mo- Similarly the expression (12)and(14) allow for the definition mentum balance which becomes of a flux function Φ in which the radial and vertical components

− Ω − Ω + = −∂  + Ω2 , of the perturbed magnetic field are expressed via bx = ∂zΦ and 2 0( q0 0 x U) x 0 2 0q0x (5) bz ≡−∂xΦ . in which the total mechanical and magnetic pressure is given by Utilizing these definitions, the linearized perturbation equa- tions are reduced to a single one for the streamfunction (wherein 1  = p /ρ + CB2, and henceforth the primes are dropped), 0 0 0 2 z ∂2ψ − κ2ψ = , x 0 (15) where p0 is the steady state mechanical pressure field. For the model analyzed here where Bz is constant, the steady pressure in which field p0 is related to the azimuthal velocity departures by ⎛ ⎞ ⎜ ω2σ2 − Ω2 ω2 ⎟ ⎜ 2q az ⎟ ∂ = Ω ρ. κ2 ≡ k2 ⎝⎜1 + 0 ⎠⎟ ; ω2 ≡CB2k2, (16) x p0 2 0U ¯ (6) σ2 + ω2 2 az z ( az) The above expression represents a radially geostrophic state. ω Note that the solutions to p are smooth and well behaved so where az is the Alfven frequency and the spatially dependent 0 epicyclic frequency is defined by ω(x)2 ≡ 2(2 − q)Ω2. For later long as V˜ is continuous across the domain. In the axisymmetric 0 usage κ is rewritten in the alternate form, 3 ⎛ ⎞ Note that by its construction (3) contains the statement that the time 2 2 2 ⎜ 4Ω σ 2Ω q ⎟ evolution of the divergence of the magnetic field is zero. If initially κ2 = k2 ⎝⎜1 + 0 − 0 ⎠⎟ . (17) ∇·B = 0 everywhere, then it remains identically zero subsequently. σ2 + ω2 2 σ2 + ω2 ( az) az Page 3 of 16 A&A 513, A47 (2010)

The solutions to this system is governed by several parameters 5. Results including the vertical wavenumber k, the Cowling number, C, the radial size of the domain (appearing below) L and the form 5.1. Classical MRI modes and channel solutions: a review and amplitude of the deviation shear profile Q. Taken as a rep- The classical system is recovered for Q = 0. This means that κ resentation of a small disk section it is assumed that there exists is independent of x and a minimum vertical wavenumber k for disturbances since it is 0 ⎛ ⎞ not reasonable to consider fundamental disturbances whose ver- ⎜ Ω2σ2 Ω2 ⎟ ⎜ 4 2 q0 ⎟ tical extent is much larger than the disk height itself (Curry et al. κ2 → κ2 ≡ k2 ⎝⎜1 + 0 − 0 ⎠⎟ · (22) 0 σ2 + ω2 2 σ2 + ω2 1994, and see discussion). ( az) az Before proceeding to these results it is important to keep Solutions of (15) are sought on a periodic domain L and are in mind a number of matters. First, Knobloch (1992)showed given in general by that for vertical field configurations like that considered here, together with boundary conditions in which individual distur- ψ = A cosh(κ x + φ), (23) bances or their combinations are zero on the boundaries, the 0 σ character of the normal mode response of the MRI is that is where A and φ are constants set by the boundary conditions either real (growing/decaying exponential modes) or imaginary ψ + / = 4 of the system. The periodicity condition that (x L 2) (oscillating modes), but never complex . A version of the argu- ψ(x − L/2) then means that ment leading to this conclusion is given in Appendix A. Thus when the character of the fluid response is analyzed, σ2 will be 1 − cosh [Lκ0] = 0. (24) taken as ∈ Reals. The relevance of this observation is that as a function of the system’s parameters, a mode becomes unstable which has the solution κ L = 2niπ where n is any integer includ- 2 0 by passing through σ = 0, sometimes known as exchange of ing zero. This condition recovers the incompressible limit of the stabilities (Chandrasekhar 1961). As such, in some instances the classical dispersion relationship of the ideal case (e.g. Acheson stability analysis of this system will focus on identifying which &Hide1973; Balbus & Hawley 1991) where specifically, σ2 = mode becomes marginal (i.e. 0) and under which parame- ter conditions. 2 2 2 2 2 2 2 2 2 2 2 β + k (σ + ω ) + ω k σ − 2Ω q0k ω = 0. (25) Secondly, the very same integral argument developed in n az 0 0 az Appendix A also leads to a general statement about the rela- in which ω2 ≡ 2(2 − q )Ω2 is the epicyclic frequency and where tionship between the temporal response and the structure of the 0 0 0 the radial wavenumber β is modes in the domain, n 4n2π2 β¯2 4Ω2σ2 2Ω2q¯ β2 ≡ · 1 + + 0 − 0 = 0. (18) n L2 2 σ2 + ω2 2 σ2 + ω2 k ( az) az The solutions for the temporal response, written in terms of re- β¯ is like an average radial wavenumber of the disturbance, spected pairs, is given by  ⎛ ⎞ 2 ⎜ 2 ω2 ⎟ |∂ ψ| dx ⎜ k 0 ⎟ 2 D x σ2 = σ2 (β2) ≡−⎝⎜ω2 + ⎠⎟ β¯ ≡ , (19) 0n n az β2 + 2 |ψ|2 n k 2 D dx ⎡⎛ ⎞ ⎛ ⎞⎤1/2 ⎢⎜ 2 ω2 ⎟2 ⎜ Ω2 2 ⎟⎥ ⎢⎜ k 0 ⎟ ⎜ 2 0q0k ⎟⎥ and, evidentally, β¯2 is always greater than zero.q ¯ represents the ± ⎣⎢⎝⎜ω2 + ⎠⎟ − ω2 ⎝⎜ω2 − ⎠⎟⎦⎥ · (26) az β2 + k2 2 az az β2 + k2 mode weighted average shear over the domain n n  The solutions associated with the “−” branch are termed the hy- Q|ψ|2dx D drodynamic inertial (HI) modes while those associated with the q¯ ≡ q0 + · (20) |ψ|2 + D dx “ ” are the hydromagnetic inertial (HMI) modes (viz. Acheson &Hide1973). For a given mode n, the magnetorotational insta- = ω2 > One immediate consequence of this is that ifq ¯ 0 (i.e. the mode bility occurs for the latter of these provided that 0 0and weighted average shear over the domain is zero), then there is no Ω2 2 possibility of instability. This follows from evaluating the crite- 2 0q0k σ2 ω2 − < 0. (27) rion for marginality in (18), setting to zero and revealing az β2 + 2 n k β¯2 2Ω2q¯ The background field must be sufficiently weak for the 1 + − 0 = 0. (21) 2 ω2 HMI-modes to be unstable. k az ω2 For given values of az the absolute minimum criterion re- The above expression cannot be satisfied ifq ¯ is zero. In general quired for instability to set in (i.e. for the mere possibility of 2 this relationship is not useful for direct calculations as the quan- σ > 0) is when n = 0, i.e. κ0 = 0, and the condition in (27) tities therein depend implicitly upon the eigenvalue σ.However, predicts this to happen if, it will be called upon to aid in interpreting the results that are Ω2 − ω2 > , obtained from specific direct calculations in Sect. 6. 2 0q0 az 0 (28)

4 It was also shown in that study that complex values of σ can only (Velikhov 1959; Acheson & Hide 1973). This particular special occur if the field configuration is helical. In this case the instability is of mode, in which its radial structure is uniform, is generally re- a traveling wave. ferred to in the literature as the channel mode which describes

Page 4 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles purely 2D flow with no vertical velocity. The temporal response Solutions of this are developed in a singular perturbation series in this case is given by σ2 = σ2 in which, expansion in powers of . Thus the ansatz, ⎛ ⎞ 0n 00 ⎜ ω2 ⎟ ψ = ψ + ψ + 2ψ + ··· σ2 = − ⎝⎜ω2 + 0 ⎠⎟ 0 1 2 00 az σ2 = σ2 + σ2 + 2σ2 + ··· 2 0 1 2 ⎡⎛ ⎞ ⎤1/2 2 2 2 2 2 ⎢ 2 2 ⎥ κ = κ + κ + κ + ··· (33) ⎢⎜ ω ⎟ ⎥ 0 00 01 02 ± ⎢⎜ω2 + 0 ⎟ − ω2 ω2 − Ω2 ⎥ . ⎣⎢⎝ az ⎠ az az 2 q0 ⎦⎥ (29) 2 0 Given the definition of κ2 givenin(22) it follows that κ2 = 0 00 κ2 σ2 When unstable (for the HMI-mode branch) this radially uniform 0( 0)and ⎛ ⎞ mode plays the central role in the development of MRI induced ⎜ ∂κ2 ⎟ − > ⎜ 0 ⎟ turbulence in numerical experiments. If 2 q0 0, then the κ2 ≡ ⎝⎜ ⎠⎟ σ2, 2 01 ∂σ2 1 expression for σ is σ2 00 ⎛ 0⎞ ⎛ ⎞ σ2 + ω2 = −Ω2 − ⎜∂2κ2 ⎟ ⎜ ∂κ2 ⎟ (2 q0) 1 ⎜ 0 ⎟ ⎜ 0 ⎟ 00 az 0 κ2 ≡ ⎝⎜ ⎠⎟ σ4 + ⎝⎜ ⎠⎟ σ2, (34) ⎡ ⎤1/2 02 ∂σ4 1 ∂σ2 2 ⎢ ω2 Ω2 ω2 ⎥ 2 σ2 σ2 ⎢ 4 az 8 q0 az ⎥ 0 0 ±Ω2(2 − q ) ⎣⎢1 + + 0 ⎦⎥ . (30) 0 0 Ω2 − Ω4 − 2 0(2 q0) 0(2 q0) wherein ⎛ ⎞ ⎛ ⎞ The combination expression σ2 + ω2 (which shall appear on a ⎜ ∂κ2 ⎟ 2 ⎜ ω2 Ω2ω2 ⎟ 00 az ⎜ ⎟ k ⎜ 8 az ⎟ ⎝⎜ 0 ⎠⎟ = ⎝⎜− 0 + 0 ⎠⎟ · (35) number of occasions in the following discussion) is always pos- ∂σ2 σ2 + ω2 σ2 + ω2 (σ2 + ω2 )2 itive for HMI modes and is always negative for the HM modes az az az since the expression inside the square-root operation is always σ2 = The possibility that 1 0mustbeallowed(seebelow)Thisis positive. why the next order term is also included in the expansion. With It should be noted that for Q = 0therearenolocalized nor- these expansions in mind (15) is written out for each power of . mal modes possible for a domain which is infinite in extent. Put To lowest order in another way, there is no solution of (15) for modes on an infi- →±∞ ∂2ψ − κ2 ψ = , nite domain with both (a) all quantities going to zero as x x 0 00 0 0 (36) and (b) κ is constant as given in (22). andtoorder ⎛ ⎞ ⎜ 2Ω2k2Q ⎟ 5.2. Weak shear variations on a finite domain: 0 < Q 1 2 2 ⎜ 2 0 1 ⎟ ∂ ψ1 − κ ψ1 = ⎝κ − ⎠ ψ0. (37) x 00 01 σ2 + ω2 In this section weak shear variations are considered whose aver- 0 az age are zero on the domain L. Before proceeding a caveat ought The order 2 form is relegated until later. Solutions to (36, 37) to be stated. Despite the spatially periodic nature of the shear Q, are developed in the following subsections for various boundary the analysis necessary to determine the temporal response uses conditions and restrictions at x = ±L/2. a multiple time-scale analysis. When periodic boundary condi- tions are imposed, as they are in Sect. 5.2.3, the multiple time- scale analysis is akin to a Floquet analysis. In this instance a 5.2.1. Channel conditions long-time scale behavior (in the form of correspondingly weak Channel conditions require that the perturbed radial velocity is σ corrections to the temporal response ) must be invoked in or- zero at the boundaries x = ±L/2. This, in turn, amounts to re- der to ensure that the resulting perturbation solutions remain pe- quiring that ikψ = 0 at these positions, or in terms of the pertur- riodic on the length scale L. For boundary conditions other than bation variables periodic ones (Sects. 5.2.1 and 5.2.2), the multiple-time scale analysis employed is a generic procedure involving perturba- ψ0 = ψ1 = 0, at x = ±L/2. tion series expansions and imposition of solvability conditions at successive perturbation orders (Bender & Orszag 1999). The The lowest order solution is given by role of the solvability conditions is to make sure that higher ∞ ∞ mπ order perturbation solutions satisfy the boundary conditions of ψ = ψ = A sin γ (x + L/2); γ ≡ , (38) 0 0m m m m L the system. m=1 m=1 The shear with zero average on the domain of length L may κ2 = −γ2 ψ be decomposed into the Fourier series expansion and where m. The general solution for 0 is written 00 ψ ∞ out as a sum of the linearly independent solutions 0m.Thein- 2nπ π Q = Q ; Q = q sin x − , (31) dex m ought not be confused with the index n which refers to 1 1 n L 2 the Fourier component of the shear profile Q appearing in (31). n=1 Stability, on the other hand, is evaluated by examining each in- where the parameter 1 measures the overall severity of the dividual eigenmode m. Thus all eigenvalues σ (and their pertur- shear. qn measures the Fourier amplitude of the shear compo- bative corrections) will be identified according to which eigen- nent n and is treated hereafter as a tunable parameter. Reference mode under examination, e.g. σ0m,σ1m etc. Note also that here to (6) indicates that this form for Q ensures that the devia- the index m does not include m = 0 as this would correspond to tion steady velocity U and, hence, the radial pressure gradients the trivial state ψ = 0. are zero at the boundaries x = ±L/2. The governing equation, κ2 Given the functional form of 0 found in (22), these relation- with Q as given above, is expressed as σ2 σ2 ships imply that 0 is equal to the functional form 0n found Ω2 2 2 2 2 2 k Q1 in (26) except with β replaced by γ ,i.e.σ = σ (γ ). The ∂2ψ − κ2ψ = − 0 ψ. (32) n m 0 0n n x 0 σ2 + ω2 stability of these modes is now a function of the mode number m az Page 5 of 16 A&A 513, A47 (2010) and, henceforth, the summation sign in (38) will be dropped and 1 it shall be henceforth understood that when reference is made ψ m to 0, a particular eigenmode mode will be analyzed. At the next order it can be seen that in order to have solutions that sat- isfy the boundary condition that ikψ1 = 0 at the two endpoints, the terms on the RHS of (37) must satisfy a particular solvability condition which will relate the correction σ1 to the variation Q1. 0 This is seen by multiplying (37)byψ0 given in (38) and integrat- ing the result across the domain from −L/2toL/25. In a general sense, then, one finds, ⎛ ⎞ ⎜ ∂κ2 ⎟ Ω2 2 ψ2  ⎜ 0 ⎟ 2 0k Q1 0m κ2 = ⎝⎜ ⎠⎟ σ2 = , (39) 01 ∂σ2 1m σ2 + ω2 ψ2  σ2 0m az 0m

0m Scaled Deviation Amplitude (n=1) in which the bracket notation is −1 −L/2 0 L/2 L/2 Radial Position: x  ≡  dx. (40) −L/2 Fig. 2. The n = 1 component of the shear profile (31). The solid line q1 = 1 and the dashed line q1 = −1. For problems in which chan- It follows that nel wall conditions are imposed the solid line corresponds to a stable σ2 = − − mΥ , deviation profile while the dashed line corresponds to an unstable pro- 1m qm( 1) (m) (41) file. The situation is reversed when fixed-pressure boundary conditions in which are imposed. Ω2(σ2 + ω2 )2 Υ(m) ≡ 0 0m az · −ω2σ2 + 2Ω2(2 + q )ω2 0 0m 0 0 az which is very similar in form to (38) except that the solution in The denominator term of Υ(m) never crosses zero for physically the domain is composed of even functions and the series includes relevant values of the parameters so that Υ > 0. the mode with m = 0, which is the classical channel mode. The For illustration consider the m = 1 mode (which is also the correction to the temporal response for all values of m  0is most unstable one) near marginality in the limit where kL is σ2 σ2 = − mΥ , very large. In order for 01 to be nearly zero it must be that 1m qm( 1) (m) (45) 2 2 ω ≈ 2Ω q0,cf.(28). It follows that the correction for this az 0 which is the negative of the solution found in (41). As Fig. 2 mode, σ11, is given approximately by the solution to indicates, this result says that shear profiles in a channel con- ω2 Ω2q σ2 ≈ az = 0 0 , figuration that promote stability (instability) play a destabilizing 11 q1 q1 (42) 2(2 + q0) (2 + q0) (stabilizing) role for configurations with fixed-pressure bound- ary conditions. where the last equality is established because the marginal mode, However the m = 0 mode is a special case which requires a σ2 ≈ 0 is under examination. The introduction of this profile 01 separate analysis. In its simplest guise, the channel mode ψ is q < 00 pattern stabilizes this otherwise marginal mode if 1 0and a constant with respect to x. Because of the periodicity of Q it destabilizes it when q > 0. The corresponding shear profile for 1 1 means that Q ψ2  = 0 and the analysis advanced up until this this example is depicted in Fig. 2. It can be seen that the modified 1 00 point to evaluate the correction σ showsthatitwillbezeroat shear is stabilizing when the shear is positive near the boundaries 1 this order. Thus given this and the fact that κ2 = 0 the solution and negative in the interior. 00 must be expanded according to, ψ = ψ + ψ + 2ψ + ··· 5.2.2. Pressure conditions 0 00 10 20 σ2 = σ2 + 2σ2 + ··· An analysis of (10)and(14) shows that the total magnetic plus 00⎛ ⎞ 20 ⎜ ∂κ2 ⎟ mechanical pressure perturbations are proportional to the radial ⎜ 0 ⎟ κ2 = 2 ⎝⎜ ⎠⎟ σ2 (46) gradient of ψ i.e., 0 ∂σ2 20 σ2 00 σ2 + ω2 az ⎛ ⎞ ⎛ ⎞ p/ρ0 + BzCbz = ∂xψ. (43) ⎜ ∂κ2 ⎟ 2 ⎜ ω2 Ω2ω2 ⎟ σ ⎜ 0 ⎟ k ⎜ 0 8 0 az ⎟ ⎝⎜ ⎠⎟ = ⎝⎜− + ⎠⎟ , (47) ∂σ2 σ2 + ω2 σ2 + ω2 (σ2 + ω2 )2 Fixing the total pressure at the boundary in a Lagrangian sense 00 az 00 az 00 az (Curry et al. 1994) is equivalent to setting to zero the above ex- = ± / where σ2 , i.e. the temporal response of the channel mode, is pression at x L 2 which, in the present case, means setting 00 6 ψ ∂xψ = 0 at those locations . The solution ψ0m is then given by givenin(29). The equations for the solution of are in succes- mπ sive order given by ψ0m = Am cos γm(x + L/2); γm ≡ , (44) L ∂2ψ = , x 00 0 (48) 5 This procedure of applying a solvability condition actually requires 2Ω2k2Q ψ 2 0 1 multiplying across by the adjoint solution of 0. However since the ∂ ψ10 = − ψ00, (49) ψ x σ2 + ω2 eigenfunctions comprising 0 and the linear operator of (36) are real 00 az ψ ⎛ ⎞ and Hermitian, the adjoint is 0. ⎜ ∂κ2 ⎟ Ω2 2 6 ⎜ ⎟ 2 k Q1 Note that this is valid since by construction the radial gradient of the ∂2ψ = ⎝⎜ 0 ⎠⎟ σ2 ψ − 0 ψ . (50) x 20 ∂σ2 20 00 σ2 + ω2 10 steady state pressure field is zero at the end points. σ2 00 az 00 Page 6 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles

σ2 The first of these says that the lowest order solution which sat- As in the previous section, the correction 1m for the channel isfies the boundary condition is a constant ψ00 = A00. The next mode (m = 0) is zero and the first non-trivial temporal response order down the solution is comes at order 2. The procedure is exactly the same as found in Sect. 5.2.2 and its correction σ2 is 2Ω2k2 20 ψ = − 0 QA , (51) ⎛ ⎞ ⎛ ⎞ 10 σ2 + ω2 00 ⎜ ∂κ2 ⎟ ⎜ Ω2 2 ⎟2 az ⎜ ⎟ ⎜ 2 k ⎟ 00 ⎝⎜ 0 ⎠⎟ σ2 = ⎝⎜ 0 ⎠⎟ (∂ Q˜)2, (56) ∂σ2 20 σ2 + ω2 x σ2 az where 00 00 ⎡ ∞  2 ⎢ with L ⎢ 2nπ ⎡ ⎤ Q = − qn ⎣⎢sin x + θn ∞   2nπ L L 2 ⎢ 2nπ π ⎥ n=1 Q˜ = − q ⎣⎢sin x + ⎦⎥· (57) ⎤ n 2nπ L 2 π ⎥ n=1 2n n ⎥ − x(−1) cos θn⎦⎥. (52) L Imposing periodic boundary conditions also implies that there are values of the coefficients of the perturbed shear Q1 which Q ∂2Q = In other words the solution is such that x Q1 which satis- can be chosen to promote stability for all the modes of the system fies the boundary condition ∂xQ|x=±L/2 = 0. In order to generate except for the channel mode. For example, for the m = 1 mode a bounded solution for ψ20 a solvability condition must be ap- and given its relative phase φ1 one can choose q2 so that the RHS pliedtotheRHSof(50), which is obtained by re-expressing ψ10 of (55) is negative. This can be done for all other modes of the in terms of its solution as given in (51) and then integrating the system as well except for the m = 0 channel mode, which is equation from across the domain and setting the result to zero: destabilized no matter what choice is made for the parameters ⎛ ⎞ ⎛ ⎞ of Q1,i.e.thesetqn. ⎜ ∂κ2 ⎟ ⎜ Ω2 2 ⎟2 ⎜ 0 ⎟ ⎜ 2 0k ⎟ ⎝⎜ ⎠⎟ σ2 = − ⎝⎜ ⎠⎟ Q QA . ∂σ2 20 σ2 + ω2 1 00 σ2 00 az 5.3. Single shear defect on an infinite domain: localized 00 disturbances Q However, given the relationship between and Q1 it follows = after an integration by parts that in fact, A profile with a single shear defect located at x 0isrepre- sented by the delta-function profile ⎛ ⎞ ⎛ ⎞ ⎜ ∂κ2 ⎟ ⎜ Ω2 2 ⎟2 ⎜ 0 ⎟ ⎜ 2 0k ⎟ Q = Q Lδ(x). (58) ⎝⎜ ⎠⎟ σ2 = ⎝⎜ ⎠⎟ (∂ Q)2. (53) 0 ∂σ2 20 σ2 + ω2 x σ2 az 00 00 The corresponding departures from the mean Keplerian shear profile U is given by Given that for HMI modes  ⎛ ⎞ − Ω / , < ⎜ ∂κ2 ⎟ = Q0 0L 2 x 0; ⎜ 0 ⎟ U (59) ⎝⎜ ⎠⎟ > 0, Q0Ω0L/2, x > 0. ∂σ2 σ2 00 ψ > If ± represents the solution of (15) respectively for x 0and the above solvability condition says something remarkable: that x < 0, then solutions which decay as x →±∞are the channel mode is always destabilized no matter what Q1 hap- ∓|κ |x ψ = A±e 0 , (60) pens to be. By contrast, values of qn may be chosen so that ± each eigenmode of the system with non-trivial radial structure where κ is as given in (22) and provided that κ2 > 0. If the latter > 0 0 (i.e. m 0) may be stably influenced by the shear Q1,yet,there condition is not met, then there are no localized normal modes is no such configuration of the disturbed shear flow that can act allowed. The coefficients of the streamfunctions in both regions to stabilize a channel mode. This result, above all others, empha- are equal in order that they be continuous across x = 0, thus sizes the important uniqueness of these modes. Also worth not- A+ = A− = A. Integrating (15) in an infinitesimal region around ing is that while the corrections to the growth rates of all modes = = x 0, and given the defect (58), imposes a jump condition for (except m 0) is proportional to , the corresponding growth the derivatives of ψ approaching either side of the defect, rate correction for the channel mode scales as 2: which means that though the channel mode is consistently destabilized by any 2Ω2k2 0 ∂ ψ + − ∂ ψ − = − Q LA. (61) periodic shear profile, its influence is markedly weaker. x 0 x 0 σ2 + ω2 0 az Putting in the solutions for the respective domains gives the 5.2.3. Periodic conditions quantization criterion Solutions of ψ that are periodic on the domain L are given by 0m Ω2k2   |κ | = 0 Q L. (62) 0 σ2 + ω2 0 ψ0m = Am sin βm(x + L/2) + φm , (54) az Solutions of the above equation are constrained to two possibili- β ≡ π/ φ where m 2m L and m is an arbitrary phase. The tempo- ties: (i) if Q < 0, then σ2 + ω2 < 0 for a normal mode solution σ2 β2 0 az ral response is given by 0m( m), as defined in (26). As in the to exist while (ii) if Q > 0, then it must be that σ2 + ω2 > 0 previous section, if attention is first given to those modes where 0 az  for there to be a normal mode. The first of these cases says that m 0, then one finds that normal modes are allowed only for those values of σ2 which are   π negative which are associated with the HI modes. The second of σ2 = q 1 (−1)m sin − 2φ Υ(m). (55) 1m 2m 2 2 m these cases are associated with the HMI modes of which the MRI Page 7 of 16 A&A 513, A47 (2010) is a possibility. Thus it is an interesting result that the MRI is a streamfunctions and their first derivatives across the domain. permitted localized normal mode disturbance only if the shear Thus, for x < −L/2andx > L/2(15)is defect is positive. If the shear defect is negative (so that there is ∂2ψ − κ2ψ = . an effective reduction of the shear inside the region), then there x 0 0 (66) are no normal modes predicted. The solutions in the region where x < −L/2isgivenby The solution of (62) is obtained by first taking the square of |κ | + / ψ = 0 (x L 2), both sides of (62) yielding, − A−e (67) 4Ω2ω2 ω2 Ω4k2 while for x > L/2 1 − 0 az + 0 = 0 Q2L2. (63) σ2 + ω2 2 σ2 + ω2 σ2 + ω2 2 0 −|κ | − / ( ) ( ) ψ = 0 (x L 2), az az az − A+e (68) However one must be careful in interpreting the solutions of κ2 > − / < together with the constraint that 0 0. In the region L 2 this simplified equation. Naively solving (63) will yield twice as x < L/2(15)is many solutions than are allowed. This overcounting is corrected ⎛ ⎞ ⎜ Ω2 2 ⎟ by requiring that only those solutions of (63) that satisfy ⎜ 2 0Q0k ⎟ ∂2ψ − ⎝⎜κ2 − ⎠⎟ ψ = 0, (69) x 0 σ2 + ω2 / σ2 + ω2 > , az Q0 ( az) 0 7 are permitted or else (62) cannot be satisfied. Solutions which has the two linearly independent solutions of (62)are, κ ψ = κ + sinh[ x],    0 A0 cosh[ x] B0 κ (70) σ2 + ω2 = 1 −ω2 ± ω4 + 4(Q2k2L2Ω4 + 4Ω2ω2 ) . az 2 0 0 0 0 0 az in which Because the term inside the radical sign is always greater than Ω2 2 2 2 2 0Q0k zero for q > 0, the “+” branch of the above expression is always κ (κ0) ≡ κ − · (71) 0 0 Σ(±)(κ ,ω ) greater than zero while the “−” branch is always less than zero. 0 0 Thus the “+” branch is the permitted solution for Q0 > 0 while The functional form the “−” branch is the allowed one for Q < 0. Incorporating this 0 σ2 + ω2 =Σ(±) κ ,ω , formally yields the expression az ( 0 0) (72)    in which σ2 + ω2 = 1 −ω2 + ω4 + 2 2 2Ω4 + Ω2ω2 . az sgn(Q0) 4(Q k L 4 az) 2 0 0 0 0 0 Σ(±) κ ,ω2 ≡ 0 0 ⎡ ⎡ ⎛ ⎞⎤ ⎤ Rewriting the terms appearing in the interior of the radical sign ⎢ 2 1/2⎥ 1 ⎢ ⎢ ⎜ κ ⎟⎥ ⎥ leads to finally,   ⎢−ω2 ± ⎣⎢ω4 + 16Ω2ω2 ⎝⎜1 − 0 ⎠⎟⎦⎥ ⎥ , (73) κ2 ⎣ 0 0 0 az 2 ⎦ ⎛ ⎞ ⎡⎛ ⎞ − 0 k 2 2 1 2 ⎜ ω2 ⎟ ⎢⎜ ω2 ⎟ k 2 ⎜ 2 0 ⎟ ⎢⎜ 2 0 ⎟ σ = − ⎝ω + ⎠ + sgn(Q0)⎣⎢⎝ω + ⎠ az 2 az 2 is introduced in order to make the following discussion more ⎤ / transparent. The ± designation references the HMI or HI modes ⎥1 2 ⎥ respectively. It is also worth noting that given the constraint − ω2 ω2 − 2Ω2q +Ω4(kL)2Q2⎦⎥ , (64) az az 0 0 0 0 that σ2 must be real, that there is an absolute maximum value κ2 allowed for 0 and this is dictated by the requirement that the which is similar in content to the classical channel mode disper- terms subject to the square root operator must remain greater sion relation (29) except for the fact that at any one time there than zero or else the value of σ2 will turn out complex. In other σ2 κ2 exists only one branch of modes as a possible normal mode so- words, given the reality of one does not expect solutions of 0 lution. The MRI mode only manifests as a normal mode when κ2 to exceed max where Q0 > 0 and its growth rate is enhanced by the defect. ⎛ ⎞ ⎜ ω4 ⎟ ⎜ 0 ⎟ κ2 = k2 ⎝⎜ + 1⎠⎟ . (74) max Ω2ω2 5.4. Symmetric shear step on an infinite domain: localized 16 0 az disturbances In all of the numerical solutions calculated below it is found that, κ2 <κ2 A finite version of the shear profile evaluated in the previous indeed, all the solutions determined satisfy 0 max. section is, Imposing the conditions that the streamfunction and its first ⎧ derivative match across the two boundaries x = ±L/2 reveals ⎪ , x < − L ⎨⎪ 0 2 ; that normal mode solutions exist provided the following condi- Q = ⎪ Q , − L < x < L ; (65) ⎩⎪ 0 2 2 tion is satisfied 0, x > L ·     2 L κ L κ tanh κ + coth κ + = 0. (75) This profile for Q has a top-hat structure and this descriptor will 2 |κ0| 2 |κ0| be used interchangeably with the expression symmetric shear ∞ The terms in the first bracket represent a quantization condition step. Evidentally the value of the integral, Qdx = Q L, −∞ 0 for odd-parity modes, which is to say that in the dimpled region is the same for both forms of Q given in (58)and(65). Solutions to (15) must be developed separately in the three regions and 7 The second of the solutions appearing in (70) is written in the form appropriately matched across the boundaries separating the re- displayed in order to make sure that both linearly independent solutions gions. Because Q is bounded it will suffice to match both the are represented even in the event κ = 0(Friedman1990).

Page 8 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles

ψ ∼ κ /κ (a) Temporal Response, Q = 0.6 (HMI Mode) 0 sinh[ x] , while the second represents even-parity modes 5 0 10 where similarly ψ0 ∼ cosh[κx] in the same region. Attention is first given to the even-parity mode in which the quantization condition, 0   σ2 10 L κ coth κ = − , 2 |κ0| −5 κ κ 10 must be satisfied for some value of 0.If were real and greater −1 0 1 than zero, then the LHS of the above relationship is always posi- 10 10 10 κ2 kL tive. Since 0 is real and positive, if the modes are to be localized, then the RHS of this expression is always less than zero. In this κ (b) Temporal Response, Q = −0.6 (HI Mode) case it means that no solution exists for real. On the other hand 0 κ2 ≤ κ = κ solutions do exist if 0. Rewriting i˜ then the quantiza- 0.85 tion condition above becomes −10 κ˜ 2 cot[˜κL/2] = , (76) σ |κ0| 0.87 now with the constraint for the existence of localized normal −10 modes is re-expressed as 0 0.5 1 1.5 2 2.5 3 2Ω2Q k2 kL κ2 κ ≡ 0 0 − κ2 > . ˜ ( 0) (±) 0 0 (77) Σ (κ0,ω0) Fig. 3. The temporal response resulting from the quantization condi- tion (76) as a function of kL (solid line) compared to the quanti- The strategy for obtaining a solution is as follows: find the values κ zation condition (62) for defect profile (dashed line). For both plots of 0 which solve (76), subject to the constraint given in (77), and ω2 = 2Ω2q with q = 3/2. a) For Q > 0 only HMI normal modes 2 az 0 0 0 0 then given the solution κ0 find σ through the relationship (72). are permitted. The agreement between the two profiles severely breaks The same procedure will be implemented for the even-parity so- down for kL exceeding 2. b) For Q0 < 0 only HI normal modes are lution below as well. Solutions of the quantization condition will allowed. The agreement between defect theory and step profile is better be sought graphically in the range of values for κ0 ∈ (0,κmax). forHImodes. Inspection shows that the constraint (77) is violated if Q0 < 0 Σ(±) > Q > Σ(±) < Σ(+) > and 0, or if 0 0and 0. Since 0 it means theory, see (28). The main result is that the temporal response of that there are no HMI normal modes as allowable solutions if the defect profile of the previous section closely resembles the Q < 0. HI normal modes are associated with the Σ(−) branch 0 temporal response of this step profile here in the limit where kL except for very spe- of solutions and since this is less than zero is small less than one. The limiting form represented by the de- cial circumstances, one can generally expect HI normal modes fect profile becomes a poor representation of the top-hat profile for Q < 0 and for them to be stable. The aforementioned spe- 0 when kL exceeds order 1 values, however, the agreement tends cial circumstance occurs if there are solutions to (76)inwhich to be much better for the HI. κ > k since, in that case, Σ(−) > 0 and HI modes may exist as 0 Additionally, for small values of kL there is always at least well for Q > 0. 0 one normal mode expected, whether it be the HI or HMI mode. The even-parity mode limits to the single-defect configura- However, as the horizontal length of the symmetric step profile tion discussed in Sect. 5.3 for small values of the parameter L. increases, the number of permitted normal modes increases as Analysis of (76) in this limit shows that, well (Fig. 4). Similar behavior is predicted for modes which are 2 κ˜ localized in the vertical direction of the disk (Liverts & Mond ≈ , 2009). As the number of permitted normal modes increases their κ˜L |κ0| κ0 values will cluster (countably as kL →∞) around the maxi- which, after restoring the definition ofκ ˜ and some rearranging, mum value becomes   ⎛ ⎞ κ2 = 2 2 0 ⎜2Ω Q k ⎟ max L ⎜ 0 0 2⎟    |κ0| = ⎝ − κ ⎠ , σ2 + ω2 0 Q0 2 az ω2 − 2Q Ω2 + (ω2 − 2Q Ω2)2 + 16Ω2ω2 . (78) ω2 0 0 0 0 0 0 0 az 4 az (±) where the definition of Σ has been restored. For values of Q0 Ω2  σ2 + ω2 κ2/ 2 Odd-parity modes are interesting since they have no analogues such that 0Q0 ( az) 0 k the above expression re- duces to in the single defect profile studied in the previous section. Examining the quantization condition for it, Ω2Q Lk2 |κ | = 0 0 ,   κ 0 σ2 + ω2 L az tanh κ + = 0, 2 |κ0| which is the quantization condition for the single-defect problem shows that there are no real solutions of κ that satisfy the con- on the infinite domain, (62). The range of values for Q0 and L κ = κ for which this limiting form agrees with the actual result of the straint. Utilizing the expression i˜, as used previously, the ω2 quantization condition takes the form quantization condition is shown in Fig. 3,where az is set to the value 2Ω2q in order to compare the temporal response against κ 0 0 κ / = − ˜ · conditions in which the channel mode is marginal in the classical tan[˜L 2] (79) |κ0| Page 9 of 16 A&A 513, A47 (2010)

0 integral of the shear over the periodic domain is zero, i.e. Q = • 0. The effects of shear defects with zero mean may be examined. −0.5 The governing equation for ψ is (a) kL = 2 π −1 2 2 2Ω k Q0Lδ(x) −1.5 ∂2ψ − κ2 ψ = − 0 , (83) x 00 σ2 + ω2 az

0 • where κ is given by • 00 • ⎛ ⎞ −0.5 ⎜ Ω2σ2 Ω2 − ⎟ Functional Values π ⎜ 4 2 (q0 Q0)⎟ (b) kL = 12 • κ2 = k2 ⎝⎜1 + 0 − 0 ⎠⎟ · (84) −1 00 σ2 + ω2 2 σ2 + ω2 ( az) az −1.5 Similar to what was done in the previous sections, one may ex- −2 σ κ Σ(±) 0 0.1 0.2 0.3 0.4 0.5 0.6 press in terms of 00 via the expression and, for the sake of completeness, this is written out explicitly, κ κ (max) 0 0 σ2 + ω2 =Σ(±) κ ,ω2 ≡ Fig. 4. Plot showing the increase of allowed normal modes with domain az ⎡ 00 00 ⎤ ω2 = Ω2 = / = . ⎡ ⎛ ⎞⎤1/2 size L. In all plots az 2 0q0, q0 3 2andQ0 0 6 (HMI mode ⎢ ⎢ ⎜ κ2 ⎟⎥ ⎥ κ/|κ | κ / 1 ⎢ ⎢ ⎜ 00 ⎟⎥ ⎥ permitted). The dashed line is ˜ 0 and solid line is cot ˜L 2. Panel a):   ⎢−ω2 ± ⎣⎢ω4 + 16Ω2ω2 ⎝⎜1 − ⎠⎟⎦⎥ ⎥ , (85) = π κ = . κ2 ⎣ 00 00 0 az 2 ⎦ kL 2 corresponds to one normal mode at 0 0 50. Panel b): − 00 k 2 1 2 kL = 12π supporting four normal modes κ0 = 0.32, 0.465, 0.54, 0.56. k 2 All modes correspond to values of σ > 0. κ0(max) = 0.576. in which the shear modified epicyclic frequency is defined as ω2 ≡ 2Ω2(2 − q + Q ). Solutions to (83) must be developed 00 0 0 0 Inspection of the condition (79) shows that for there to exist a separately for either side of x = 0 and matched. Solutions for normal modeκ ˜ must be greater than π/L as the functional form the streamfunction in which ψ(−L/2) = ψ(L/2) and ∂xψ|−L/2 = tan[˜κL/2]/κ˜ is positive for values ofκ ˜ ∈ (0,π/L). This places a ∂xψ|L/2 (i.e. that they are periodic) are given by constraint upon the minimum value of L that permits this mode    to exist and this is obtained by solving ψ = κ ∓ L , ± A cosh 00 x (86) π 2 κ˜ = , (80) ψ ∈ , ± / kL where ± is the solution for the region x (0 L 2). Continuity of the stream function is ensured by construction of the form of and, using the definition for κ found in (71) one finds that the the solution in (86). The streamfunction must show a jump in its minimum value for L must satisfy, first derivative following the same arguments found in Sect. 5.3. ⎡ ⎛ ⎞⎤ Integrating (83) in a small region around x = 0 results in the ⎢ ⎜ ⎟⎥1/2 ⎢ ω2π2 ⎜ Ω2ω2 ⎟⎥ same condition for the jump in ∂ ψ found in (61). Putting in the ⎢ 0 ⎜ 16 0 az ⎟⎥ x kLmin = ⎢ ⎜sgn(Q0) 1 + − 1⎟⎥ . (81) ψ ⎣ 4Q ⎝ ω4 ⎠⎦ forms for ± found in (86) leads to the quantization condition 0 0   2 2 L Ω k Q0L Note that the inclusion of the expression sgn(Q ) reflects how κ κ = 0 = F (±) κ ,ω2 . 0 00 tanh 00 00 (87) HMI/HI modes associate with the sign of Q so that the expres- 2 Σ(±) κ ,ω2 00 0 00 00 sion appearing above encapsulates both possibilities. Odd-parity (±) modes bifurcate into existence for L →∞as Q0 grows from It should be kept in mind that the quantities F as appearing ff κ zero and therefore it means that these modes are not expected on the RHS of (87)aredi erent functions of 00 depending upon to play a role for values of the defect which are small as this whether one is considering HMI/HI (±) modes, respectively. would entail very large values of the top-hat region. The tempo- Thus when solutions of the quantization condition are sought, ral response of these odd-parity modes are otherwise similar in separate attention will be given depending upon which type of most every respect to their even-parity counterparts and, as such, mode is of interest. Furthermore, consideration of the solutions no further analysis of their properties is pursued here. to (87) must be done by restricting attention to cases where κ00 is It should be noted that as in the previous section unstable lo- either real or imaginary. Additionally from previous arguments calized MRI normal modes (HMI-modes) can exist only if the showing that σ2 must be real, there will be a maximum real value step function in the shear is greater than zero. If the step is neg- possible for κ00 ative (indicating a region of weakened shear), then there are no ⎛ ⎞ ⎜ ω4 ⎟ HMI types of localized normal modes admitted by the configura- ⎜ 00 ⎟ κ2 ω2 = k2 ⎝⎜ + 1⎠⎟ . (88) tion and, hence, there is no possibility of normal mode unstable max 00 Ω2ω2 16 0 az disturbances. Values of κ which are greater that κ ω2 would lead to com- 00 max 00 2 5.5. Single shear defect on a periodic domain plex values of σ as an inspection of (85) shows that if κ00 does exceed this maximum value, then the term inside the square-root Next a shear defect of the form operator becomes negative which would lead to a complex value of σ2. This restriction upon the value of κ helps in finding so- Q = −Q0 + Q0Lδ(x), (82) 00 lutions for it. is considered on the periodic domain −L/2 ≤ x ≤ L/2. This The limit where Q0 → 0 recovers the classical limits dis- form is very similar to the one considered in Sect. 5.3 except the cussed in Sect. 5.1. In that case channel modes, i.e. those modes

Page 10 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles

(b) Q = 1.5 (a) Q = −1.0 (a) Q = 0.1 0 0 1 0 3 2 4 ) F(+) 0.8 L/2 1.5 κ 00 2 2 1 0.6 ) tanh( κ L/2 00 0.5 0.4 κ 00 1 κ tanh( 0 00 (+) 0 0.2 F −0.5 0 (−) Various Functions −2 0 F (−) −1 −0.2 F −1 −1.5 −4 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 κ κ κ 00 00 00 − i κ 00 (b) Q = −2.5 (c) Q = 2.0 (d) Q = 3.5 0 0 0 4 10 8 10 (+) (+) F 8 F 8 3 6 6 6 2 4 4 ) κ L/2 4 00 (−) 2 1 κ tanh( κ L/2 ) F 2 00 2 κ tanh( 00 00 Various Functions 0 (−) 0 F 0 0 −2

−1 −2 −2 −4 0 0.5 1 1.5 0 0.5 1 1.5 2 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 κ κ κ − i κ 00 00 00 00

Fig. 5. Graphical solutions for the quantization condition (87). Fig. 6. Similar to Fig. 5 except negative values of Q0 are investigated. κ = π ω2 = Ω2 = / Attention restricted to real values of 00 marking the mode which In all plots kL and az 2 0q0 and q0 3 2. Panel a):theleft becomes the classical channel mode in the limit Q0 → 0. Each plot shows that the channel mode admitted is a stable HI mode with panel depicts the three functions κ tanh κ L/2 (thick line), and F (±) κ = . σ2 = − . 00 00 00 0 92 and 5 30. The right plot in panel a) shows the first (dashed/thin lines). In all plots kL = π and ω2 = 2Ω2q . Panels a) κ = . az 0 0 two overtone modes. The first HMI mode is unstable with 00 0 69i and b) show that HMI modes are admitted as the solid curve crosses σ2 = . κ = . and 0 88 while the next HI mode admitted has 00 1 32i and the F (+) line while for the latter two it crosses the line correspond- σ2 = − . κ 2 88. Note that in these cases 00 is imaginary. Panel b):stable ing to F (−) meaning that HI modes are selected. The solution κ and κ = . σ2 = − . 00 HMI mode with 00 1 21 and 8 30. For the first two overtones, temporal response for each: a) Q = 0.1, κ = 0.27, σ2 = 0.005, κ = . σ2 = . 0 00 the HMI mode is unstable with 00 0 78i and 2 02 while for the b) Q = 0.5, κ = 0.65, σ2 = 0.14, c) Q = 2.0, κ = 1.33, σ2 = 3.36 κ = . σ2 = − . < 0 00 0 00 HI mode, 00 1 08i and 2 47. For all values of Q0 0thereis and d) Q = 3.5, κ = 1.37, σ2 = 6.73. Notice how the channel mode κ 0 00 at least one unstable HMI mode with imaginary 00 . goes from being an unstable HMI mode to an unstable HI mode as Q0 is increased. where the difference now is that the ψ disturbances are set to zero at the boundaries x = ±L/2. This is equivalent to enforcing ψ for which κ0 = 0, are the most unstable. In the current set- zero normal velocity at the boundaries. Thence, in this case is κ = = ting this corresponds to 00 0whenQ0 0. Additionally, for L 2 2 sinh κ x ∓ ω = 2Ω q the channel mode is exactly marginal for Q = 0, 00 2 az 0 0 0 ψ = ∓A , (89) ω2 ± κ / viz. the arguments leading to (28). This value for az will be as- sinh 00 L 2 sumed so that a controlled analysis may be carried out in which where just as in the previous section, ψ is the solution for the one can track how varying Q affects the stability characteristics ± 0 region x ∈ (0, ±L/2). Evaluating the jump condition at x = 0 of an otherwise marginal channel mode. leads to an analogous quantization condition like that in (87), Figures 5, 6 show the graphical solutions of (87) with some namely, details of the results. The main result is that the mode that is   2 2 2 2 identified with the classical channel mode in the ideal limit al- L Ω k Q0L Ω k Q0L κ coth κ = 0 = 0 , (90) ways exhibits exponential temporal growth no matter what am- 00 00 σ2 + ω2 ± 2 az Σ( ) κ ,ω2 plitude the shear defect takes. Figure 7 shows that the temporal 00 00 response of the classical channel mode is maximal for Q0 = 0 in which Σ(±) κ ,ω2 and κ and ω2 are exactly as they are and that as Q0 moves away from zero instability is predicted. 00 00 00 00 The mode exhibits an exponentially decaying spatial character found in Sect. 5.5. It can be seen here, as before, that there there 2 2 κ < (κ > 0) for Q0 > 0, while it is weakly oscillatory (κ < 0) for are never real solutions of 00 for HMI modes if Q0 0 and, 00 00 > Q0 < 0. This general trend persists for a wide range of kL val- likewise, no such solutions for HI modes if Q0 0. ues. As such, the results depicted in these figures can be taken In Fig. 8 the stabilizing behavior of the shear defect on to be qualitatively representative. The figures also indicate that the HMI modes is depicted for the most unstable mode of the the unstable HMI mode can also turn into an unstable HI mode system. The results from the weak shear limit discussed in if the amplitude of Q0 becomes large enough. Sect. 5.2.1 are used as a reference point to help interpret the following. From the classical theory discussed in that section it was established that the most unstable normal mode allowed is 5.6. Single shear defect on finite domain: channel boundary the one in which m = 1, in turn implying γ1 = π/L. Reference conditions to (27) shows that this mode is marginally stable when This section is concerned with the response of disturbances to k2L2 ω2 = 2Ω2q · (91) a Q configuration that is exactly like that considered in Sect. 5.5, az 0 0 k2L2 + π2 Page 11 of 16 A&A 513, A47 (2010)

Behavior of Channel Mode with Defect Amplitude Behavior of Least Stable Mode 2 2 kL = π kL = π, ω2 = 2 q 1.5 az 0 1.5

1 Classical

π Channel Mode −iκ kL = 2 Δ κ2 0.5 Hydro−inertial 00 1 1 00 Modes

• 0 Δ

0.5 kL = 4π 0.5 −0.5 Δ 0.25 0 −1 π 5 Hydromagnetic−inertial 0.3 kL = 2 Modes kL = 4π 4 0.2 σ2 2 σ 3 0.1 kL = π 2 0 1 −0.1 0 −2 −1.5 −1 −0.5 0 0.5 −3 −2 −1 0 1 2 3 Q : Shear Defect Amplitude Q : Defect Amplitude 0 0 Fig. 8. The behavior of the first unstable (HMI) mode for the problem Fig. 7. Behavior of the classical channel mode as a function of defect with channel walls (q = 3/2). The values of iκ˜ = κ and the cor- amplitude Q for kL = π and ω2 = 2Ω2q , q = 3/2, corresponding 0 00 00 0 az 0 0 0 responding temporal response σ2 are given as a function of the defect 2 2 to a marginal (σ = 0) channel mode at Q0 = 0. Top panel depicts κ 00 amplitude Q0 for three values of the parameter kL. The top panel shows as a function of Q in which the classic channel mode has κ2 = 0. κ ω2 0 00 ˜00 . The bottom panel shows the temporal response. The value of az for Bottom panel shows the temporal response which predicts exponential each curve is set so that the temporal response is marginal at Q0 = 0– growth/decay for all values Q  0. The channel mode goes from being κ = 0 details of this is described in the text. The value of ˜00 for Q0 0are a HMI mode to a HI mode at Q0 ≈ 1.87. shownbythe∇ symbol in the top panel.

(This is obtained by replacing βn with γ1 in that expression.) The response of the modes for three different values of kL are in turn, is driven into place by the MRI itself. The final con- ω2 depicted. With the above choice for az all modes are marginal dition is a stable pattern state. at Q0 = 0. For the range of parameters investigated and shown 4. In the final pattern state reached by the model laboratory κ in the figure, 00 must be imaginary in order for the quantiza- system the modified shear profile’s amplitude is independent κ of P . This is not true of the remaining fluid and magnetic tion condition to be satisfied. Thus the values for 00 are written m κ quantities which scale as Pλ with typical values λ ≈ 1/2. as i˜00 . Inspection of the results show that stability is promoted m only for values of Q0 which are negative. The range of values As Pm is made very small one would observe a fluid state of Q0 for which stability is predicted decreases for larger domain characterized by a uniform background magnetic field and sizes, i.e. as L is made larger the window in Q0 for which dis- an azimuthal flow showing deviations from a Keplerian state. turbances are stable decreases. For those ranges in Q0 in which 5. The main mode of instability and driver of turbulence in the ∈ − . , = π ω2 stability is predicted, e.g. for Q0 ( 0 5 0) for kL 2 with az numerical experiments of the shearing box is the channel given in (91), the temporal response of the other overtones was mode which is a special response with no radial structure. checked and all were found to have σ2 < 0. This mode precipitates a secondary instability which is un- derstood to lead to a turbulent cascade. There is no chan- nel mode allowed in a setup appropriate for a laboratory 6. Discussion experiment. This study is devoted to examining the axisymmetric inviscid linear response of a shearing sheet environment threaded by a The saturation hypothesis is then the following: is it possible that constant vertical magnetic field for an array of different shear the fluid response in a shearing sheet environment characterized profiles. The motivation for considering this setup are restated by very low Pm stabilizes itself by modifying the background from the Introduction: shear as suggested by the calculations appropriate for laboratory conditions? Of course, there are some constraints placed upon 1. Numerical experiments of low magnetic Prandtl number the way in which the shear may be modified in that the grav- flows appropriate for cold disk environments are outside cur- itational field predominantly “forces” there to be a Keplerian rent computational reach. profile (Ebrahimi et al. 2009). However by analogy to the sit- 2. Current numerical experiments seem to show the trend that uation for the laboratory setup, there may come into play un- as Pm is lowered so is the corresponding angular momentum dulations of the shear whose average is zero over some length transport. The reasons for this is not yet clear. scale L (e.g. Fig. 1). 3. Theoretical analysis of laboratory setups evaluating the re- The main difference between the laboratory setup and nu- sponse of the MRI subject to a vertical background field find merical calculations of the shearing box are their respective that transport is reduced with Pm. Furthermore this reduction boundary conditions. The numerical experiments adopt periodic emerges through the modification of the basic shear profile boundary conditions while the calculations for the laboratory (e.g. Taylor-Couette) inside the experimental cavity which, experiments require no normal flow conditions at an inner and

Page 12 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles outer radius and the former of these permits channel modes to 6.1. On the use of delta-functions come into play. A comparison between the responses of a single delta-function Thus, what has been done here is a simple model calcula- shear profile on an infinite domain (Sect. 5.3) and a top-hat pro- tion in which a shearing box configuration with a constant back- file for Q (Sect. 5.4) show that the results of the two config- ground vertical magnetic field is tested for stability for a num- ff urations are qualitatively similar. Although the symmetric step berofdi erent shear profiles. The perspective taken is that if function allows for more normal modes to exist as its radial ex- there exists a shear profile (say, whose average over some length tent is increased – a feature which is missed if one uses the delta- scale is zero) that is stable to the MRI, then it would strongly function prescription – the qualitative flavor of the response is suggest that low Pm flow configurations might saturate the MRI captured by its use nevertheless. by driving into place a new azimuthal flow profile which leads to saturation. Another deficiency found in using the defect prescription is that for the top-hat profile studied in Sect. 5.4 there are both What is found here is something very interesting. For flow even and odd parity normal modes possible whereas only the configurations in which no-normal flow conditions are imposed even-parity mode is admitted in the defect model. The odd-parity at both radial boundaries (Sects. 5.2.1 and 5.6) there exists a mode has no counterpart in the single defect model. It was also modification to the shear in which the most unstable MRI mode shown that the odd-parity mode may exist if the radial domain is stabilized. However, for flows in which periodic boundary of the symmetric shear step is large enough as (81) indicates that conditions (Sects. 5.2.3 and 5.5) or fixed Lagrangian pressure the minimum size of the domain√ required for this mode to exist conditions (Sect. 5.2.2) are imposed there are no modulations of has the proportionality L ∼ 1/ Q0. For small values of the de- the shear which shut off the instability. In these cases, all modifi- fect amplitude the minimum size of the shear step required for cations of the shear (with zero mean on some radial length scale) this odd-parity mode to exist is so large that one does not expect seem to enhance instability of the channel mode. For example, that this should play any physically relevant role in the dynamics if the given channel mode is marginally stable to exponential if one is interested in the response to narrow regions of altered growth, then introduction of the modified shear, no matter what shear. It is with some confidence to suppose that it is reasonable is its non-zero amplitude, seems to destabilize it as the results of to represent the dynamical response of slender regions of mod- Sects. 5.2.3 and 5.5 indicate. ified shear through the use of delta-function defects and this is This strongly implies that in those numerical experiments in the motivation and justification for its use in Sects. 5.5 and 5.6. which a shear is threaded by a constant vertical magnetic field, the shear modification/MRI-stabilization process may only ap- ply for those flow conditions in which there is no normal flow 6.2. Channel modes on either one or both radial boundaries of the system. In the small Q theory of Sect. 5.2, it was shown that channel On the other hand, for a similar magnetized fluid configu- modes exist for periodic boundary conditions and for condi- ration described by periodic boundaries which therefore permits tions in which the total pressure perturbations are zero at the channel modes to exist, modifications of the shear appears to fur- two boundaries. Conversely, if either one of the two boundaries ther destabilize the channel mode. If one were to consider a flow force the velocity perturbation to be zero there, then there is no configuration (e.g. the local Keplerian profile in a shearing box channel mode permitted as a normal mode solution. In light of threaded by a constant background field) that admits an unsta- the previous results it is important that the boundary conditions ble channel mode, then there seems to be no modification of the appropriate for disks be properly ascertained in order to assess shear with zero mean that would saturate the growth of the chan- and better understand how activity in them are driven. nel mode since all such modifications further enhances its desta- Channel modes in the classical limit have no radial struc- bilizing influence. The emergence of a pattern state leading to ture. In Sect. 5.5 the development of this mode was studied as a saturation like the one envisioned by the hypothesis would seem function of the defect amplitude Q0 and it is shown that though to be out of the question under these circumstances. Moreover the channel mode develops some amount of radial structure, this suggests that the tendency for angular momentum transport κ as evinced by the non-zero value of its wavenumber 00 , its ten- reduction with Pm, as indicated by numerical experiments of the dency to be unstable survives and is even enhanced. In Regev & shearing box, is due to some other process or processes (which Umurhan (2008) it was postulated that the channel mode might are not discussed here). disappear as a natural mode of the system if some amount of These issues deserve further investigation as the conclusions radial symmetry breaking were introduced into the governing reached here were done so utilizing very simple functional forms equations of motion, for instance, in the form of a radially vary- for the shear profile Q. Namely, the analysis performed in the ing shear profile. However, the results of this section show that previous sections either assumed weak values of Q,inorder the classical channel mode indeed persists in its existence de- to use singular perturbation theory techniques, or that Q is de- spite the introduction of a shear profile which deviates from the scribed by delta-functions (see also below). Use of these func- background constant shear state with zero average over some tions, which greatly facilitate analysis, have offered some in- length scale. These conclusions are consistent with similar re- sights into the nature of the responses of these modes. Of course, sults regarding the nature of the MRI in more global contexts further investigation would require considering more physically (Curry et al. 1994). realizable profiles and to check the robustness of the results they yield against the ones offered by use of these more simplified forms. As such the results reached here can serve as guideposts 6.3. On the existence and number of localized normal modes for further enquiries along these lines. In Sects. 5.3 and 5.4 the existence of localized modes was ex- In the following some reflections are presented on other amined. Localization is understood in this case to correspond to issues pertaining to the implications of the calculations per- modal disturbances which show exponential decay as the radial formed here. coordinate x →±∞. Thus, by construction, incoming/outgoing

Page 13 of 16 A&A 513, A47 (2010) waves are excluded from consideration8. The existence of lo- 6.4. Stabilization in general and an interpretive tool calized normal modes depends on the relative sign of the shear defect/step profile Q : HMI modes exist if the shear in the The absolute minimum condition that must be met for any nor- 0 mal mode of the system to be unstable in a configuration with defect/top-hat region is stronger than the background (Q0 > 0) < constant shear q0 is given by the Velikhov criterion (28): if the while HI modes exist if the shear is relatively weakened (Q0 ω2 0)9 – summarized here as square of the Alfven frequency ( az) is greater than the shear Ω2 by 2 0q0, then none of the HMI modes of the system can be Q0 > 0 ←→ hydromagnetic inertial disturbances, unstable. For individual modes in the same system with some β2 Q < 0 ←→ hydro − inertial disturbances. amount of radial structure the criterion is given by (27), where n 0 is the square of the radial wavenumber of the disturbance. As can ω2 Given that the former mode type leads to the MRI, it means that be seen, for fixed values of az there are two ways in which sta- if the shear in the defect zone is weakened, then there are no bility may be promoted - either by weakening the shear or by in- localized normal modes which can go unstable as only hydro- creasing the radial wavenumber. In the problem where the shear inertial modes are supported (which happen to be stable ex- is constant, these quantities are set from the outset as parameters. cept for very extreme circumstances). Of course, as an initial At the end of Sect. 4 there is a series of arguments for ar- value problem, it does not mean that there are no hydromagnetic bitrary shear profiles which lead to the identity found in (18), inertial modes permitted but, instead, the mode probably has which is like a generalized eigenvalue condition. This relation- some type of algebraic temporal growth/decay associated with it ship is the analog of the dispersion condition of the classical which cannot be described by the usual normal mode approach. limit (uniform q) contained in (25)10. Care must be adopted be- This needs to be further examined by studying the associated fore interpreting this expression as an eigenvalue condition since initial-value problem. Similar features are known to exist for ver- σ appears implicitly in the expressions forq ¯ and β¯. Nonetheless, tically localized MRI modes in disks (Liverts & Mond 2009)in a parallel analysis shows that stability is promoted if which the response of the initial value problem can exhibit ex- 2Ω2qk¯ 2  ≡ ω2 − 0 > , ponential growth with an amplitude supporting some algebraic az 0 (92) time-dependence. Growth can initially be very small and it can β¯2 + k2 take a very long time for a given disturbance to reach the same where the effective wavenumber β¯ and shearq ¯,given amplitude as expected for its counterpart normal mode. A con- in (19, 20)are, jecture is that the meaning of the absence of normal modes in   the problem considered here may have similar attributes to the |∂ ψ|2dx Q|ψ|2dx β¯2 ≡ D x , ≡ + D · features associated with the initial-value problem examined by q¯ q0 |ψ|2 x |ψ|2 x Liverts & Mond (2009). D d D d Irrespective of this outcome it should be remembered that the D is the domain and ψ is the radial eigenmode of the distur- dichotomy observed here for modes on an infinite radial domain bance in question. The stability condition for general q may be is a consequence of the imposition of exponential decay of the rationalized in the same way as the criterion is understood for disturbances. This distinction disappears once waves are allowed the classical case with the replacements, βn → β¯ and q0 → q¯. to enter and exit from the “infinite” boundaries. With these aforementioned observations and caveats in mind, Lastly it is also noteworthy that for the localized problems one may use (92) as an interpretive tool to aid in understand- considered here, when normal modes are permitted they usually ing the reasons for stability/instability in a particular problem. come as a finite set. This is in contrast to other localized flow Thus, stability is promoted if (i) for fixed effective wavenumber, problems considered elsewhere in the literature. For example, in the effective shear is reduced or (ii) for fixed effective shear the the problem of compact rotating magnetized jets in an infinite effective wavenumber is increased, or in more general terms (iii) medium considered by Bodo et al. (1989) a countably infinite if the combination results in an overall reduction of the quotient number of normal-modes are predicted for that system. This has q¯ some origin in the uniformity of both the rotation and magnetic · field profiles of the jet considered in that study. By contrast, there β¯2 + k2 are examples in geophysical flows in which the number of nor- If one considers the results concerning the problem with periodic mal mode disturbances are discrete and finite. A recent example boundaries in Sect. 5.5, then the persistent instability expected ffi can be found in the work of Gri ths (2008) where the stabil- for channel modes can be understood in terms of this criterion. ity of inertial waves are studied in a stratified rotating flow with In the classical limit where the defect amplitude is zero then β¯ strong horizontal shear. For strongly peaked forms of the poten- for the channel mode is zero. As the defect amplitude is raised tial vorticity only a finite set of normal modes are predicted for β¯ increases suggesting that this effect has a stabilizing influence, the system. Such finiteness of the number of normal modes per- e.g. see Fig. 9. However, as the defect amplitude is raised away mitted is not so unusual in systems in which background states, from zero the effective shear increases no matter what the sign like the shear or stratification, have strongly peaked functional of Q0 and, to effect, this increase always sufficiently overpowers forms like they are in the cases studied in this work. the increase in β¯2 so that the end result is further instability. In the same figure is plotted the quantity 8 Given the mathematical structure of the ode describing axisymmet- ric disturbances, the radial eigenfunctions that are solutions have either  q¯ β2 + k2 = n , (93) strictly oscillatory or exponential radial profiles. Thus the possibility of  q β¯2 + 2 decaying oscillations is ruled out. 0 0 k 9 The mode existence dependency upon the sign of Q0 was checked which, for the generalized Velikhov criterion, is a measure of the for other shear profiles (for instance, by replacing the top-hat with a ratio of the destabilizing term for the shear profile q compared truncated parabolic profile). Aside from differences in the magnitude 10 of the temporal response and the multiplicity of modes allowed, the The classical limit is obtained when the replacements β¯ → βn and aformentioned sign dependency still holds. q → q0 are made.

Page 14 of 16 O. M. Umurhan: Speculation on low magnetic Prandtl number disk velocity profiles

2 2.5

1.8 σ2 2 β2 1.6

1.4 1.5

1.2 ω/ω 0 1 1 ω/ω 0.5 Q 0 0.8 Q Q σ2 Functional Values Functional Values 0.6 0

0.4 2 −0.5 β β2 0.2

0 −1 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0

Q : Shear Defect Amplitude Q : Shear Defect Amplitude 0 0 Fig. 9. The behavior of the unstable channel mode for the problem with Fig. 10. The behavior of the least stable HMI mode for the problem with = π ω2 = = π ω2 = / 2 + β2 periodic boundary conditions examined in Sect. 5.5: kL and az channel walls examined in Sect. 5.6: kL and az 2q0 (k ) = / σ2 1 2q0 with q0 3 2. Plotted is the temporal response, , together with with β = π/L and q = 3/2. Plotted is the temporal response, σ2, ff β¯ = + ¯ 1 0 its corresponding e ective wavenumber and shear andq ¯ q0 Q together with its corresponding effective wavenumber and shear β¯ and Q¯ / ( plotted), and the stability measure for this mode 0. All values of q¯ = q0 + Q¯ (Q¯ plotted), and the stability measure for this mode /0. Q ff the shear defect 0 correspond to increasing the e ective shear to above Stability occurs for Q0 ∈ (−1.32, 0). In this window the effective Q¯ Q > q the background value ( 0) 0. This increase in ¯ always overpowers shear is less than the background value (¯q − q0 = Q¯ < 0) for only β¯2 σ2 the stabilizing influence of an increased . Note, the values of have part of this range, i.e. for Q0 ∈ (−0.5, 0). Stabilization in the range been scaled by an arbitrary factor in order to facilitate clear comparison. Q0 ∈ (−1.32, −0.5) is achieved by a sufficient increase of the effective wavenumber so that the overall effect upon the stabilization parame- 2 ter /0 is for it to be less than one. σ has been similarly scaled as in to its nominal value when the shear is only q0:for/0 > 1 the previous figure. one expects enhanced destabilization against the classical mode while for /0 < 1 comes with enhanced stabilization. In Fig. 9 this quantity is plotted for one of the channel modes dis- and vertical velocities, have saturated profiles which scale as β = cussed in 5.5. Noting that in this case n 0 (i.e. that its radial some positive power of Pm so that for Pm 1 their contri- wavenumber is zero in the ideal case) the radial profile of the butions become negligible. The basis of this assumed trend de- mode develops as the amplitude of the shear defect increases no rives from the theoretical calculations of laboratory setups dis- matter what its sign. The stabilizing rise in β¯ is accompanied cussed in the Introduction. It is therefore assumed here that a by the destabilizing increase of the effective shear, that is to say similar trend may appear for configurations in shearing boxes Q¯ > 0, so that the quotient of their respective influences results with periodic radial boundary conditions as well. The perspec- in /0 > 1, indicating instability. tive taken here with regards to those configurations is that if this Similarly plotted in Fig. 10 is the influence Q0 has upon β¯, is indeed the case, then one can (in principle) test the stability q¯, and the stability measure /0, for the single defect problem of a wide variety of shear profiles which show deviations (con- on a finite domain studied in Sect. 5.6. The presence of walls fil- strained to have a zero average over some length scale) from the ters out the channel mode and the resulting flow potentially can basic Keplerian profile – configurations which are akin to those support shear profiles which can stabilize the least stable mode. predicted from the calculations of laboratory setups. If a sta- The Figure shows that within the range of (negative) values of bility calculation shows that there exists a shear profile which the defect amplitude Q0 for which the mode does not exponen- is stable to these axisymmetric disturbances, then the shear- tially grow/decay the effective shear is less than the background modification/mode-saturation hypothesis should become a seri- shear state only for part of the stable range. It means to say, then, ous candidate explanation. in that range of parameter values in which the effective shear is However, the calculations done in this study indicate that for greater than the background shear, yet there is still stability of periodic shearing box environments there is always an instabil- the mode, stabilization is brought through an increased effective ity of the basic channel mode no matter what shear profile is radial wavenumber which over-compensates the increased desta- assumed (satisfying the aforementioned constraints). Unlike the bilization brought about by an increase in the effective shear. situation for laboratory setups, where saturation can be achieved by altering the basic shear profile, saturation (if present) in shear- 6.5. Final reflections and an implication ing box environments with periodic boundary conditions is likely not due to this mechanism. The fundamental basis of this study is the assumption that in This further implies that the nonlinear response of systems low magnetic Prandtl number flows for model environments supporting the MRI stongly depends upon the boundary condi- like that considered in this study, the only noticeable outcome tions employed which is not so surprising as many physical sys- of the development of the MRI is to alter the basic shear pro- tems in Nature exhibit this sensitivity to their boundaries. Thus file. All other quantities, such as magnetic fields and the radial it seems to this author that some care must be taken before one

Page 15 of 16 A&A 513, A47 (2010) equates the results of laboratory experiments to those physically It must be that both I1 and I2 are zero if (A.1) is to be satisfied. related/analogous systems for which the experiments are meant Supposing that b  0 then the relationship implied by I2 = 0, to represent (cf. Ji et al. 2006). This cautionary note is not to is that ⎡ ⎤ ⎡ ⎤ diminish the value of one over the other, rather, it is intended to ⎢ω2 − Ω2 ⎥ ⎢ Ω2ω2 ⎥ ⎢ 0 2 0Q⎥ ⎢4 0 az2a⎥ emphasize that there are many subtle and, at times, conflicting ⎣⎢ ⎦⎥ |ψ|2dx = ⎣⎢ ⎦⎥ |ψ|2dx. features between such systems that must be clearly understood D a2 + b2 D (a2 + b2)2 before any firm conclusions are reached. This result used in the expression for I1 lead to Ω2ω2 2 2 4 az 2 I1 = − |∂xψ| dx − k 1 + |ψ| dx. (A.4) Acknowledgements. The author thanks Marek Abramowicz and the organiz- D a2 + b2 D ers of the Asymptotic Methods in Accretion Disk Theory workshop at CAMK (May/June 2009) where the impetus for this work originated. The author However all the quantities in the above expression are always also thanks James Cho, Lancelot Kao, Olivier Gressel, Paola Rebusco, Edgar positive and, therefore, I1 can never be zero. Consequently b Knobloch and Oded Regev for their generous support and fruitful conversations σ2 during the course of this study. must be zero which, in turn, means that can only be real. With b = 0 and restoring the definition of a the equation I1 = 0 appears now as 2 2 Appendix A: Proof that σ is never complex 0 = − |∂xψ| dx D ⎡ ⎤ (15) is rewritten into the form ⎢ ω2 − Ω2 Ω2ω2 ⎥ ⎢ 2 Q 4 az ⎥ ⎛ ⎞ − k2 ⎣⎢1 + 0 0 − ⎦⎥ |ψ|2dx. (A.5) σ2 + ω2 σ2 + ω2 2 ⎜ ω2 4Ω2ω2 2Ω2Q ⎟ D az ( az) ∂2ψ − 2 ⎜ + 0 − 0 az − 0 ⎟ ψ = , x k ⎝1 ⎠ 0  σ2 + ω2 (σ2 + ω2 )2 σ2 + ω2 |ψ|2 az az az Dividing the above expression by D dx and utilizing the def- inition of κ2 given in (22) leads to the expressions appearing where ω2 = 2Ω2(2 − q ).Thenatureofσ2 will be assessed by 0 0 0 0 in (18−20). supposing that σ2 + ω2 = a + ib az References , ∈ where a b Reals. It will be shown below that b must be zero. Acheson, D. J., & Hide, R. 1973, Rep. Prog. Phys., 36, 159 The boundary conditions on ψ are one of the following pos- Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 sibilities: (i) ψ goes to zero on the domain boundary ∂D pro- Balbus, S. A., & Henri, P. 2008, ApJ, 674, 408 vided the latter is finite; (ii) ψ and its derivative go to zero on Bender, C. M., & Orszag, O. 1999, Advanced Mathematical Methods for Scientists and Engineers (Springer) the domain boundary if the boundary tends to infinite distances; Bodo, G., Rosner, R., Ferrari, A., & Kobloch, E. 1989, ApJ, 341, 631 (iii) ψ is periodic on a length scale L in the domain D. Replacing Chandrasekhar, S. 1961, Hydrodynamic and Hydromagnetic Stability (Oxford) σ2 + ω2 + ψ az with a ib in the equation for , followed by multi- Curry, C., Pudritz, R. E., & Sutherland, P. G. 1994, ApJ, 434, 206 plication by the complex conjugate of ψ (i.e. ψ∗), integrating the Ebrahimi, F., Prager, S. C., & Schnack, D. D. 2009, ApJ, 698, 233 result and applying the boundary conditions gives Friedman, B. 1956, Principles and Techniques of Applied Mathematics (New York: Wiley) + = , Fromang, S., Papaloizou, J., Lesur, G., & Heinemann, T. 2007, A&A, 476, 1123 I1 iI2 0 (A.1) Goldreich, P., & Lynden-Bell, D. 1965, MNRAS, 130, 125 Griffiths, S. D. 2008, J. Fluid Mech., 605, 115 where Jamroz, B., Julien, K., & Knobloch, E. 2008, Astron. Nachr., 329, 675 Ji, H., Burin, M., Schartman, E., & Goodman, J. 2006, Nature, 444, 343 2 I1 = − |∂xψ| dx Julien, K., & Knobloch, E. 2006, in Stellar Fluid Dynamics and Numerical D Simulations: From the Sun to Neutron Stars, ed. M. Rieutord, & B. Dubrulle, EAS Publ. Ser., 21, 81 ⎡ ⎤ ⎢ ω2 − 2Ω2Q 2 − 2 ⎥ Knobloch, E. 1992, MNRAS, 255, 25 − 2 ⎢ + 0 0 − Ω2ω2 a b ⎥ |ψ|2 , Knobloch, E., & Julien, K. 2005, Phys. Fluids, 17, 094106/16 k ⎣1 a 4 az ⎦ dx (A.2) D a2 + b2 a2 + b2 Lesur, G., & Longaretti, P.-Y. 2007, MNRAS, 378, 1471 Liverts, E., & Mond, M. 2009, MNRAS, 392, 287 and Regev, O., & Umurhan, O. M. 2008, A&A, 481, 1 ⎡ ⎤ Umurhan, O. M., Regev, O., & Menou, K. 2007a, PRL, 98, 034501 ⎢ω2 − 2Ω2Q 4Ω2ω2 2a⎥ Umurhan, O. M., Menou, K., & Regev, O. 2007b, PRE, 76, 036310 2 ⎢ 0 0 0 az ⎥ 2 I2 = bk ⎣ − ⎦ |ψ| dx. (A.3) Velikhov, E. P. 1959, J. Exp. Theor. Phys. (USSR), 36, 995 D a2 + b2 (a2 + b2)2

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