Magnetorotational instability in Keplerian disks a non-local approach N. Shakura 1 K. Postnov12 D. Kolesnikov13 and G. Lipunova14

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Magnetorotational instability in Keplerian disks:

a non-local approach

N. Shakura*1, K. Postnov1,2, D. Kolesnikov1,3, and G. Lipunova1,4

1Moscow State University, Sternberg Astronomical Institute, Universitetskij pr. 13,

119234 Moscow, Russia

2Kazan Federal University, Kremlyovskaya 18, 420111 Kazan, Russia

3The Raymond and Beverly Sackler School of Physics and Astronomy,

Tel Aviv University, Tel Aviv 69978, Israel

4Max-Planck-Institut f¨

ur Radioastronomie, Auf dem H¨

ugel 69, 53121 Bonn, Germany

December 19, 2023

Abstract

We revisit the modal analysis of small perturbations in Keplerian ideal gas

ﬂows leading to magneto-rotational instability (MRI) using the non-local approach.

We consider the case of constant vertical background magnetic ﬁeld, as well as

the case of radially dependent background Alfv´

en velocity. In the case of con-

stant Alfv´

en velocity, MRI modes are described by a Schr¨

odinger-like diﬀeren-

tial equation with some eﬀective potential including ’repulsive’ (1/r2) and ’attrac-

tive’ (−1/r3) terms. Taking into account the radial dependence of the background

Alfv´

en speed leads to a qualitative change in the shape of the eﬀective potential. It

is shown that there are no stationary energy levels corresponding to unstable modes

ω2<0 in “shallow” potentials. In thin accretion disks, the wavelength of the dis-

turbance λ=2π/kzis smaller than the half-thickness hof the disk only in “deep”

potentials. The limiting value of the background Alfv´

en speed (cA)cr, above which

the magnetorotational instability does not occur, is found. In thin accretion disks

with low background Alfv´

en speed cA≪(cA)cr, the increment of the magnetoro-

tational instability ω≈ −√3icAkzis suppressed compared to the value obtained in

the local perturbation analysis.

Keywords hydrodynamics, instabilities, magnetic ﬁelds

1 Introduction

Shear ﬂows in astrophysical objects, characterized by an inhomogeneous velocity ﬁeld,

are a universal source and agent of energy transport and are closely related to phenom-

ena of turbulence (Shakura and Sunyaev 1973, Bisnovatyi-Kogan and Lovelace 2001,

*E-mail: nikolai.shakura@gmail.com, kpostnov@gmail.com

arXiv:2210.15337v2 [astro-ph.HE] 17 Dec 2023

Lipunova et al. 2018), magnetic ﬁeld generation (Boneva et al. 2021), and particle

acceleration (Somov et al. 2003).

The stability of shear hydrodynamic ﬂows with respect to small perturbations in

a magnetic ﬁeld in laboratory conditions was ﬁrst considered papers by E. Velikhov

and S. Chandrasekhar (Velikhov 1959, Chandrasekhar 1960). In the absence of mag-

netic ﬁeld, hydrodynamic instability in a rotating shear ﬂow appears when the angular

momentum decreases outward from the axis of rotation (Lord Rayleigh 1916).

Velikhov and Chandrasekhar showed that in a vertically magnetized, axisymmet-

ric, diﬀerentially rotating ﬂow with angular velocity decreasing outward, magnetorota-

tional instability (MRI) is possible.1

The theory of MRI was applied to astrophysical accretion disks in an inﬂuential pa-

per by Balbus and Hawley (1991), and it is now believed that this instability generates

turbulence in accretion disks (see review Balbus and Hawley (1998)). Nonlinear nu-

merical simulations (e.g., Hawley et al. 1995, Sorathia et al. 2012, Hawley et al. 2013)

conﬁrm that MRI can indeed sustain turbulence in accretion disks.

It is believed that for the study of MRI and analysis of its properties, a local ap-

proximation in an ideal incompressible ﬂuid is suﬃcient, where small perturbations

are taken in cylindrical coordinates r,z,φin the form of plane waves ∝ei(ωt−krr−kzz).

In this case, the diﬀerential MHD equations are transformed into algebraic equations,

the dispersion relation for perturbations is found in the form of a biquadratic equation

(Balbus and Hawley 1991, Kato et al. 1998), and the instability increment does not

depend on the magnetic ﬁeld (see Eqs. (77) and (78) below, respectively). Taking into

account non-ideal eﬀects in this approximation slightly changes the conditions for the

occurrence of MRI but leaves qualitatively the same picture (Shakura and Postnov

2015a, Zou et al. 2020).

However, already in the pioneer work of Velikhov (1959), a global analysis of

long-wave disturbances in the direction perpendicular to the plane of the main ﬂow was

carried out for ﬂows between two rotating conducting cylinders with a constant angular

momentum over radius, Ω(r)r2=const. Radial perturbations were found from the

solution of the Sturm-Liouville boundary value problem (in the VKB approximation).

It was shown that such an approach implies a critical magnetic ﬁeld, above which

instability is suppressed, and the dependence on the boundary conditions remains even

in the case of extending the outer cylinder to inﬁnity.

Due to the potential importance of MRI in the emergence of turbulence in disk

ﬂows (accretion and protoplanetary disks, gaseous disks in galaxies), extensive ana-

lytical and numerical investigations of MRI were conducted in the 1990s-2000s in the

approximation of incompressible ﬂuid in a homogeneous magnetic ﬁeld, including a

global analysis of this instability. Nonlocal analysis shows that in shear ﬂows, the dis-

persion equation for the mode ω(k) contains a term dependent on radius as ∝ −1/r2,

which is usually neglected in local modal analysis. As expected, a critical value of

magnetic ﬁeld appears in the global analysis, above which MRI are stabilized (Pa-

paloizou and Szuszkiewicz 1992, Kumar et al. 1994, Gammie and Balbus 1994, Curry

et al. 1994, Latter et al. 2015). For accretion disks around a central gravitational body,

1We consider only the case of a vertical background magnetic ﬁeld. Axisymmetric gas ﬂows with toroidal

background magnetic ﬁelds in the presence of gravity are subject to Parker instability (Parker 1966).

the obtained results depend on the choice of boundary conditions for perturbations at

the inner and outer radii of the ﬂow (Knobloch 1992, Dubrulle and Knobloch 1993,

Latter et al. 2015). The choice of boundary conditions aﬀects the discretization of local

dispersion relations (Latter et al. 2015).

Despite the considerable previous scrutiny of this problem, in this study, we inde-

pendently and comprehensively perform a non-local linear analysis of MRI in Keple-

rian accretion disks with an angular velocity Ω(r)∼1/r3/2. We derive a dispersion

equation that can be reduced to a Schr¨

odinger-like equation with an “energy” E=−k2

and an eﬀective potential U(r) consisting of two terms: an “attractive” term propor-

tional to −1/r3and a “repulsive” term proportional to 1/r2. In contrast to the local

analysis, the eﬀective potential vanishes at a point r0, which depends on the mode fre-

quency ω, the wave number kz, and the background Alfv´

en velocity. We examine in

detail both cases where the outer radius of the disk, rout, is greater or less than r0. We

numerically solve the boundary value problem for radial boundary conditions corre-

sponding to both rigid and free ﬂow boundaries. We emphasize the signiﬁcance of

the position of the ﬂow boundaries relative to the zero points of the eﬀective poten-

tial, appearing in the non-local analysis for the Sturm-Liouville boundary problem. We

demonstrate that in “shallow” eﬀective potentials, there can be a situation, depending

on the position of the inner ﬂow boundary, where MRI do not occur. Such a situation

is possible for ﬂows around normal stars with large radii. Naturally, for ﬂows around

compact objects, the potential wells are very deep, and the energy spectrum is nearly

continuous. We explicitly derive the critical magnetic ﬁeld value that suppresses MRI

and ﬁnd the dependence of the MRI increment on the background homogeneous mag-

netic ﬁeld. We consider for the ﬁrst time the case of a background magnetic ﬁeld that

varies with radius in a power-law manner. We also examine the case of an incom-

pressible ﬂuid with density depending on the radial coordinate, where the equations

for small perturbations remain the same as for a constant density, while the eﬀective

potential changes.

The structure of the paper is as follows. In Section 2, we perform the linear modal

analysis for small perturbations in an ideal ﬂuid in the form ∼f(r) exp[i(ωt−kzz)]. In

Section 2.5, we derive the algebraic dispersion equation ω(kz) and the critical Alfv´

velocity below which MRI occurs. In Section 3.2, we consider for the ﬁrst time the

MRI in the presence of radially varying vertical magnetic ﬁeld and demonstrate that in

this case, the eﬀective potential can change nontrivially. In Section 4, we compare our

results with the standard results obtained in the local modal analysis. In Appendix A,

we numerically solve the Schr¨

odinger equation for nonlocal perturbations with a con-

stant background Alfv´

en velocity in the case when the external radius of the ﬂow, rout,

is greater than the zero radius of the eﬀective potential r0. In Appendix B, we consider

the case of rout <r0, when the problem reduces to solving the standard Sturm-Liouville

problem with third-type boundary conditions.

2 Non-local modal analysis

2.1 Basic equations

We consider a diﬀerentially rotating ideal ﬂuid in homogeneous vertical magnetic ﬁeld.

Classical results were obtained in papers by E. Velikhov and S. Chandrasekhar who

studied the stability of sheared hydromagnetic ﬂows (Velikhov 1959, Chandrasekhar

1960).

Equations of motion of ideal MHD ﬂuid read:

1) mass conservation equation

∂ρ

∂t+∇(ρu)=0,(1)

2) Euler equation including gravity force and Lorentz force

∂u

∂t+(u∇)u=−1

ρ∇p− ∇ϕg+1

4πρ(∇ × B)×B(2)

(here ϕgis the Newtonian gravitational potential2),

3) induction equation

∂B

∂t=∇ × (u×B) (3)

We will consider adiabatic perturbations with constant entropy

∂s

∂t+(u∇)s=0.(4)

For such adiabatic perturbations, perturbed density variations are zero, ρ1=0, and

pressure variations in the energy equation vanish, p1=0 (see, e.g., Appendix in

Shakura and Postnov (2015b)).

We analyze the case of a purely Keplerian rotation where the unperturbed velocity

is vϕ≡uϕ,0=pGM/r,ur,0=uz,0=0. We assume that the forces caused by pressure

gradient are small and only appear in the perturbed equations.

2.2 The case of incompressible ﬂuid

Let us consider small Eulerian perturbations in an ideal incompressible ﬂuid. The

velocity components in the background undisturbed ﬂow with velocity uϕ,0will be

ur,uϕ,uz. The magnetic ﬁeld can be expressed as B=B0+b, and the pressure as

p0+p1. We will consider the poloidal background ﬁeld B0. We will seek perturbations

in the form of f(r) exp[i(ωt−kzz)], noting that time tand the coordinate zonly appear

in the system of equations through the derivative sign.

The choice of perturbations with harmonic functions in the vertical coordinate

is dictated by the nature of the problem for disk ﬂows that are conﬁned in the z-

coordinate. In such ﬂows (accretion and protoplanetary disks, gas disks in galaxies),

2In principle, one can solve the problem in Schwarzschield metric using the potential ϕ=c2

2ln 1−rg

r,

rg=2GM/c2see Shakura and Lipunova (2018).

the vertical pressure gradient is balanced by the gradient of gravitational force along

the z-coordinate, distinguishing them from laboratory ﬂows.

The integration of the unperturbed Euler equation over the z-coordinate leads to a

polytropic density distribution ρ(r,z)=ρc(1 −(z/z0)2)n, with a semi-thickness of z0=

2(n+1)Pc/(Ω2(r)ρc), where ρcand Pcare the central density and pressure, and nis the

polytropic index (n=3/2 for a convectively stable disk). For the polytropic equation of

state n=1/(γ−1), where γis the adiabatic index, and in the case of an incompressible

ﬂuid γ→ ∞ and n=0. In this limiting case, the vertical density gradient vanishes,

and the model ﬂow represents a Keplerian disk with a constant vertical density limited

by the disk’s semi-thickness h(Π-shaped density distribution). The density can vary

radially (see Section 3).

For inﬁnitesimal perturbations, the approximation of perturbations with harmonic

functions in the z-coordinate is suitable for both thin disks (h/r≪1) and thick disks

(h/r≲1), provided that the wavelength of the perturbations is smaller than the disk’s

half-thickness: λ=2π/kz<h. Therefore, the ﬁnal equations for linear perturbations in

the case of an incompressible ﬂuid, which will be derived below, are not diﬀerent from

the equations for laboratory plasmas with a constant density along the z-coordinate.

For the chosen perturbations, the continuity equation (1) for an incompressible

ﬂuid, ∇u=0, in cylindrical coordinates can be expressed as follows:

∂

∂r(rur)−ikzuz=∂ur

∂r+ur

r−ikzuz=0.(5)

It should be noted that, in the local approximation, the small term ur/rin the continuity

equation is typically neglected. In this case, the perturbations can be sought in the form

∝exp[i(ωt−krr−kzz)]. The equation of magnetic ﬁeld solenoidality, ∇B=0, can be

written in a similar manner:

∂

∂r(rbr)−ikzbz=0.(6)

The radial, azimuthal, and vertical components of the Euler equation are, respec-

tively,

iωur−2Ωuϕ=−1

ρ0

∂p1

∂r−c2

B0"∂bz

∂r+ikzbr#,(7)

iωuϕ+κ2

2Ωur=−ic2

kzbϕ(8)

(here we introduced the epicyclic frequency κ2=4Ω2+r(dΩ2/dr)≡(1/r3) d(Ω2r4)/dr

and unperturbed Alfv´

en velocity c2

A=B2

0/(4πρ0)),

iωuz=ikz

ρ0

.(9)

The three components of the induction equation, taking into account the solenoidal-

ity of the magnetic ﬁeld ∇B=0, read:

iωbr=−iB0kzur,(10)

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MagnetorotationalinstabilityinKepleriandisks:anon-localapproachN.Shakura*1,K.Postnov1,2,D.Kolesnikov1,3,andG.Lipunova1,41MoscowStateUniversity,SternbergAstronomicalInstitute,Universitetskijpr.13,119234Moscow,Russia2KazanFederalUniversity,Kremlyovskaya18,420111Kazan,Russia3TheRaymondandBeverlySackler...

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Magnetorotational instability in Keplerian disks a non-local approach N. Shakura 1 K. Postnov12 D. Kolesnikov13 and G. Lipunova14

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