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The stabilizing effect of viscosity onmagnetic wind flows from accretiondiscsChristopher Graham Campbellaa School of Mathematics and Statistics, Newcastle University,Herschel Building, Newcastle upon Tyne NE1 7RU, UK.Published online: 31 Mar 2014.

To cite this article: Christopher Graham Campbell (2014) The stabilizing effect of viscosity onmagnetic wind flows from accretion discs, Geophysical & Astrophysical Fluid Dynamics, 108:3,350-362, DOI: 10.1080/03091929.2013.870167

To link to this article: http://dx.doi.org/10.1080/03091929.2013.870167

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Geophysical and Astrophysical Fluid Dynamics, 2014Vol. 108, No. 3, 350–362, http://dx.doi.org/10.1080/03091929.2013.870167

The stabilizing effect of viscosity on magnetic wind flows fromaccretion discs

CHRISTOPHER GRAHAM CAMPBELL∗

School of Mathematics and Statistics, Newcastle University, Herschel Building, Newcastle upon TyneNE1 7RU, UK

(Received 5 August 2013; in final form 25 November 2013; first published online 31 March 2014)

Magnetically channelled winds are believed to be a feature of most accretion discs. It has been shownthat such flows can remove significant amounts of angular momentum from the disc and make a majorcontribution to driving the inflow. For a suitable range of poloidal magnetic field bending, only a smallfraction of the disc mass is lost in the wind flow, so most material reaches the inner region of thedisc. However, discs driven purely by such a process are prone to a field-bending instability whichcan lead to runaway mass loss. It is shown here that a small amount of disc viscosity can quenchsuch an instability and allow steady disc-wind models to be constructed. The effects of perturbationsto the coupling between the radial and vertical structures are allowed for, with the thermal balancehaving particular relevance. Runaway increases in field bending are prevented by increases in the disctemperature and magnetic diffusivity mainly caused by viscous dissipation.

Keywords: Accretion discs; Magnetic winds; Stability

1. Introduction

1.1. Observational background

Wind flows are believed to be a feature of a wide range of disc-accreting systems. Theseinclude active galactic nuclei (AGN), T Tauri stars, X-ray binaries and cataclysmic variables.Absorption feature indicative of outflowing gas are observed in AGN (Weymann et al. 1991,Crenshaw et al. 1999, Ferrari 2004). In Tauri stars, outflows are mainly characterized by opticalemission line spectra (Reipurth and Bally 2001). Jets have been observed from X-ray binaries(Fender et al. 2003, Migliari and Fender 2006) and most of the strong features observed inthe UV spectra of cataclysmic variables are believed to be due to wind outflows (Long andKnigge 2002). The frequently observed collimation features suggest the presence of large-scalemagnetic fields, which can channel the wind flow and lead to centrifugal driving.

1.2. Magnetic disc-wind theory

Theories of magnetically influenced wind flows from accretion discs have been developed toexplain their sources and consequences. The pioneering paper of Blandford and Payne (1982)

∗Email: christopher.campbell@ncl.ac.uk

© 2014 Taylor & Francis

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Viscous stabilization of magnetic disc winds 351

showed that for a sufficiently strong magnetic field inclined beyond a critical angle, a ther-mally driven magnetically channelled wind flow can develop from the disc surfaces. Pressuregradients accelerate material to reach a sonic point, beyond which centrifugal force drives thewind flow away from the disc. Such a wind is effective at removing angular momentum fromthe disc and this can drive the inflow. This can occur with a small amount of mass loss viathe wind, so most material reaches the inner region of the disc. It was assumed that a suitablemagnetic field is advected inwards from the surrounding medium. The model was developedas an alternative to the viscous disc, since the origin of accretion disc turbulence was notknown at the time. It was subsequently discovered that magneto-rotational instabilities cangenerate turbulence in discs and, together with the radial shear, can lead to a self-sustainingdynamo. Disc dynamos can generate a dipole-symmetry magnetic field suitable for launchingand channelling wind flows.

Most of the studies since Blandford and Payne (1982) have been numerical simulations ofthe large-scale structure of the wind flow, treating the disc as a boundary condition for thewind source. There have been a large number of such simulations, with particular emphasis onthe production of jets from the inner region of a disc in relation to observations of flows fromprotostars and active galactic nuclei. Early work parametrized the mass loss rate from the discsurfaces and investigated the consequences of this in the presence of a large-scale magneticfield (e.g. Ferrari 2004). The flow solutions of main interest are those which are initiatedby vertical gradients in the thermal pressure with the magnetically channelled flow passingthrough a sonic point (essentially the same as the slow magnetosonic point) and subsequentlythrough Alfvén and fast magnetosonic points. Later studies set up an initial structure andallowed the disc/wind system to evolve. Fendt and Cemeljic (2002) and Casse and Keppens(2002) investigated the effects of a disc magnetic diffusivity, η, on the structure of the centraljet, particularly in relation to its collimation far from the disc. Larger values of η tend to leadto lower amounts of collimation.

More recent simulations have considered the effects of an anisotropic form of η in the disc.Parametrized forms for the poloidal and toroidal diffusivities, ηp and ηφ , have been usedand allowance made for turbulent motions to extend further vertically before decaying. Thesimulations of Zanni et al. (2007), Tzeferacos et al. (2009) and Sheikhnezami et al. (2012)indicate that these effects may favour evolution to a steady flow. The poloidal diffusivityaffects the wind launching process, with higher values of ηp giving lower wind mass fluxes.The toroidal diffusivity influences the angular velocity of the wind beyond the disc. Fendtand Sheikhnezami (2013) investigated the effects of an asymmetric η distribution in relationto observed asymmetries in some jet flows. Murphy et al. (2010) found that the numericalresolution near the disc surface can significantly affect the mass loss rates attained, so highresolution is necessary in this region to obtain reliable results.

Most studies of disc/wind systems have considered the disc magnetic diffusivity to be dueto turbulence, and have used a parametrized form of η to represent this. Wardle and Konigl(1993) argued that ambipolar diffusion will be important in protostellar accretion discs, andconsidered this mechanism in detail. As with most other studies, a large-scale magnetic fieldof undetermined origin was assumed to be present in the disc. In order to obtain sufficientpoloidal field bending to launch the wind, a strong field had to be considered. This supressedthe magnetorotational instability and led to strong vertical compression of the disc, exceedingthat due to stellar gravity. This led to supersonic values for the inflow speed.

The stability of a purely magnetic wind-driven disc was investigated by Lubow et al. (1994).Their simple model indicated that such a system would be unstable. A small increase in theinflow speed led to increased poloidal magnetic field bending which lowered the sonic point

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352 C. G. Campbell

and caused an increased rate of mass loss in the wind. This increased the angular momentumloss which in turn resulted in a further increase in the inflow speed. Hence, unstable mass lossresults due to increasing field bending. Such an instability was confirmed by a more detailedmodel in Campbell (2009). Konigl and Wardle (1996) and Konigl (2004) argued that thereis a branch of solutions with large magnetic compression of the disc giving a certain relationbetween the disc height and inflow speed which does not lead to a field-bending instability.However, as noted by Ferreira and Casse (2004), such solutions have supersonic inflow speedsand are unlikely to be sustainable. Li (1995) had also found solutions with large magneticcompression leading to supersonic inflow speeds, and expressed doubts about their stability.

Most models and the original Blandford and Payne (1982) analysis ignored viscosity, whichis unlikely to be justified in a realistic disc. A turbulent magnetic Prandtl number, Np = ν/η,can be defined where ν and η are the turbulent viscosity and magnetic diffusivity, respectively.It is most likely that Np is not ignorably small, so magnetic and viscous stresses will both playa part in angular momentum transport. Detailed models show that self-consistent disc-windsystems result for Np ∼ 1 (Campbell 2000, 2003, 2005).

1.3. The present paper

The model of Campbell (2000) employed vertical averaging of the disc structure and was shownin Campbell (2001) to be stable. However, this did not account for the coupling between thevertical and radial structures. The analysis of Campbell (2003) solved for the radial and verticalstructures and the present paper investigates the stability of this model. This allows inclusionof the effects of perturbing the magnetic compression of the disc, which changes the windmass loss rate. Perturbation of the thermal coupling can also be accounted for, since a detailedsolution of the thermal equations is incorporated, this being particularly important since itrelates to changes in η which may stabilize the system. In Campbell (2005), the magnetic fieldstructure was found for the inner part of the wind flow, well within the Alfvén surface, andmatched to the dynamo field at the disc surface. This is incorporated in the present analysis.

In Section 2, the essentials of the model of Campbell (2003) are presented and discussed,including the magnetic, dynamical and thermal problems. The key wind equations are alsopresented. In Section 3, the time-dependent equations describing the angular momentumproblem are formulated, allowing for the coupling between the radial and vertical structuresand the dependences on the accretion rate. Section 4 derives the linearized equations forperturbations to the angular momentum balance, then uses these to analyse global and localstability of the disc-wind system. The nature of the quenching mechanism is considered. InSection 5, the results are summarized and discussed.

2. The steady disc-wind model

The basic equations used in Campbell (2003) are presented here, together with the key featuresof their solution. The thin nature of the disc allows certain terms to be ignored in the fullequations, but the essential terms are retained. The large-scale magnetic field is generated by adynamo process in the disc, this acting in the regime in which the magneto-rotational instabilityis operable and generates the turbulence causing the α-effect (e.g. Campbell 1997).

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Viscous stabilization of magnetic disc winds 353

2.1. The disc equations

Cylindrical coordinates (�, φ, z) are used, centred on the star of mass M and radius R. Amagnetic field of dipole symmetry is generated in the disc by a simple αω-dynamo. Thepoloidal and toroidal components of the induction equation give

v� Bz + η∂ B�

∂z= αBφ, (1)

η∂2 Bφ

∂z2= −�Ω ′

K B� , (2)

where η is the magnetic diffusivity, α(�) a poloidal field generating function and

ΩK =(

G M

� 3

)1/2

. (3)

The magnetic force has a small effect in the radial component of the momentum equation, sothe angular velocity in the main body of the disc is close to a Keplerian form, as in the case ofa thin viscous disc. Equation (1) expresses the balance between the inward radial advection ofpoloidal magnetic field, its diffusion and its creation via the turbulent α-effect, while (2) givesthe balance between the diffusion of toroidal field and its generation by the shearing of radialfield.

The vertical equilibrium is given by

Ω2K z + 1

ρ

∂

∂z

(P + B2

φ

2μ0

)= 0, (4)

which expresses the balance of the compressional forces of stellar gravity and the toroidalmagnetic pressure gradient against the expansion force of the thermal pressure gradient. Themagnetic pressure due to the radial field is ignorable, since B2

φ � B2� . The thermal pressure

obeys the equation of state

P = Rμ

ρT, (5)

where R is the gas constant and μ is the mean molecular weight.The angular momentum equation can be written as:

�ρv�

d

d�(� 2ΩK) = ∂

∂�

(ρν� 3Ω ′

K

)+ 1

μ0

∂

∂z

(� 2 Bφ Bz

), (6)

giving the rate of change of material angular momentum due to the action of viscous andmagnetic torques. The viscosity is taken to have the standard parametrized form

ν = εvcsh, (7)

where cs is the isothermal sound speed, h is the disc height and εv < 1. The viscous torqueleads to an outward radial transfer of angular momentum through the disc, while the magnetictorque causes a vertical transfer of angular momentum from the disc into the magneticallychannelled wind flow. The continuity equation is

1

�

∂

∂�(�ρv� ) + ∂

∂z(ρvz) = 0. (8)

For suitable wind launching angles, only a small mass loss occurs through the disc surfaces,so most mass reaches the inner region of the disc near the stellar surface.

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354 C. G. Campbell

The divergence of the radiative energy flux is related to the viscous and magnetic dissipationsby

∂ FR

∂z= ρν(�Ω ′

K)2 + η

μ0

(∂ Bφ

∂z

)2

. (9)

The radiative diffusion equation gives

FR = − 4σB

3κρ

∂

∂z(T 4), (10)

where σB is the Stefan–Boltzmann constant and κ is the Rosseland mean opacity. A Kramersform is used for the opacity, so

κ = KρT −7/2, (11)

with K a constant.

2.2. The wind equations

Material is lost from the disc surface, with thermal pressure gradients accelerating the gas from asubsonic flow up to the sonic point. The magnetic field channels the flow and significant angularmomentum is lost from the disc when the Alfvén point lies well beyond the disc surface, sodistortions of the poloidal magnetic field are small. The magnetic stress at the disc surface isrelated to the Alfvén coordinate �A by

1

μ0�s Bφs Bzs = − tan is

(1 + tan2 is)1/2m� 2

A ΩK(�s), (12)

where the subscript “s” denotes surface values, m is the wind mass flux through the sonic pointand is is the angle the poloidal magnetic field makes with the horizontal. Hence,

tan is = Bzs

B� s. (13)

The nearly force-free magnetic field is straight in the region between the disc and sonic point,to a good approximation (Campbell 2005). At the sonic point, the gravitational and centrifugalforces tangential to the poloidal magnetic field balance and the wind mass flux is given by

m = aρs exp

[−(

Ω2K h2

2a2

tan2 is

(3 − tan2 is)+ 1

2

)], (14)

where the wind is taken to be isothermal with sound speed a. The Alfvén coordinate �A isgiven by

� 2A = k1 I

2π I1

∣∣ f′′φ (1)

∣∣ M

m

(1 + tan2 is

)1/2tan is, (15)

where M is the accretion rate and k1 is a constant, defined below.

2.3. Solution of the equations

The toroidal magnetic field can be expanded in the separable form

Bφ(�, z) = Bφs(�) fφ(ζ ), (16)

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Viscous stabilization of magnetic disc winds 355

where ζ = z/h. The induction equation can then be solved to yield fφ(ζ ), satisfying suitableboundary conditions for a dipole symmetery field. The poloidal field bending is related to thebalance between diffusion and radial advection by

tan is = 1

I

η

|v� |h , (17)

where

I =⟨∂ B�

∂z

⟩/(∂ B�

∂z

)z=0

, (18)

with the angled brackets denoting a vertical average.Other quantities can be expressed in separable forms, and equations follow for the radial and

vertical structures of the disc. The radial equations are algebraic and can be solved analytically,while non-linear integro-differential equations arise for the vertical structure and these mustbe solved numerically. The vertical equilibrium equation leads to

B2φs

2μ0= k1Ω

2K h2ρc, (19)

where the separation constant k1 gives a measure of the amount of compression of the disc bythe magnetic pressure. The thermal equations yield

32

45

σB

K

T 15/2c

ρ2c h

= k2

I2Ω2

K νΣ, (20)

where

Σ = 2∫ h

0ρdz, (21)

I2 is a dimensionless integral related to the vertical dependence of ρ and k2 is a thermalseparation constant.

The constants k1 and k2 can be determined from photospheric surface conditions at the windbase (Campbell 2003). This gives dependences on M of

k1 = C1 M7/12 and k2 = C2 M17/6, (22a,b)

where C1 and C2 are constants. These relations allow coupling between the horizontal andvertical structures to be accounted for in the perturbation analysis to investigate the disc-windstability.

3. The time-dependent equations

The stability of the steady angular momentum balance is investigated here. Perturbations tothis balance change on a timescale much longer than the dynamical, thermal and magnetictimescales. Hence, a perturbation from angular momentum balance evolves through quasi-steady states and time derivatives can be ignored in all the equations except the continuityequation. Noting that v� has a weak z-dependence, except close to the disc surface, verticalintegration of the continuity equation gives

∂Σ

∂t= − 1

�

∂

∂�(�Σv� ) . (23)

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356 C. G. Campbell

Vertical integration of the angular momentum equation yields

v� (�, t) = − 3

� 2ΩKΣ

∂

∂�(� 2ΩKμ) + 4

μ0

Bφs Bzs

ΩKΣ, (24)

where

μ = νΣ. (25)

The surface magnetic stress Bφs Bzs/μ0 can be expressed in terms of μ and v� . The inductionequation leads to

Bφs Bzs = Nα

K 3f

′′φ (1) tan is B2

φs, (26)

where

Nα = αh

η(27)

is a magnetic Reynolds number and K 3 is the dynamo number. Using (17) and (19) to eliminatetan is and B2

φs in (26) yields

1

μ0Bφs Bzs = Nα

Np

∣∣ f′′φ (1)

∣∣K 3 I

k1

I1Ω2

K

μ

v�

, (28)

with the magnetic Prandtl number

Np = ν

η. (29)

The substitution of (28) in (24) gives

v� = − 3

� 2ΩKΣ

∂

∂�(� 2ΩKμ) + SΩK

k1μ

v� Σ, (30)

where

S = 4Nα

Np

∣∣ f′′φ (1)

∣∣K 3 I I1

. (31)

Equations (23) and (30) give the time-dependent angular momentum equation.The thermal equation (20) together with (27), the vertical equilibrium and the parametrized

form of ν given by (7) leads to

μ = C� 15/14k1/72 Σ10/7, (32)

where C is a constant. Eliminating M in (22a,b) via

M = 2π� |v� |Σ, (33)

yields

k1 = 2πC1�7/12|v� |7/12Σ7/12 (34)

and

k2 = 2πC2�17/6|v� |17/6Σ17/6. (35)

Using these expressions to eliminate k1 in (30) and k2 in (32) gives

|v� | = 3

� 2ΩKΣ

∂

∂�(� 2ΩKμ) + S

� 7/12ΩKμ

|v� |5/12Σ5/12, (36)

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Viscous stabilization of magnetic disc winds 357

with

μ = C� 31/21|v� |17/42Σ11/6, (37)

where S and C are constants. The incorporation of k1 and k2 to derive (36) and (37) willallow for the effects of perturbing the coupling between the radial and vertical structures to beaccounted for.

4. Stability analysis

4.1. Perturbation of the angular momentum equation

Taking Eulerian perturbations in (23) gives

∂

∂t(δΣ) = 1

�

∂

∂�

[� (Σsdδ|v� | + |v� |sdδΣ)

], (38)

where the subscript ’sd’ refers to the steady state. Perturbation of (36) and (37) leads to

δ|v� | = − 7

10

(1 − 29

12U

)(1 + 7

10U

)−1 |v� |sd

μsdδμ (39)

+ 77

20

(1 + 7

10U

)−1 1

� 2ΩKΣsd

∂

∂�

(� 2ΩKδμ

)and

δΣ = 7

10

(1 + 1

84U

)(1 + 7

10U

)−1Σsd

μsdδμ (40)

− 17

20

(1 + 7

10U

)−1 1

� 2ΩK|v� |sd

∂

∂�(� 2ΩKδμ),

where

U = S� 7/12ΩKμsd

|v� |17/12sd Σ

5/12sd

. (41)

It can be shown that

U = (Tm/Tv)sd

1 + (Tm/Tv)sd, (42)

where Tm and Tv are the magnetic and viscous torques, per unit radial length. The ratio(Tm/Tv)sd is essentially constant through the bulk of the disc, and hence U is constant. Itis noted that U has the range 0 < U < 1.

Using (39) and (40) to eliminate δ|v� c| and δΣ in (38) leads to

7

10

∂

∂t(δμ) − 17

40

νsd

� |v� |sd

∂

∂t(δμ) − 17

20

νsd

� |v� |sd�

∂

∂�

(∂

∂t(δμ)

)

= νsd

�

∂

∂�

[17

10

� |v� |sd

νsdUδμ + 3

�ΩK

∂

∂�

(� 2ΩKδμ

)]. (43)

Noting that

ν

� |v� | = I Np tan ish

�� 1, (44)

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358 C. G. Campbell

it follows that the second time derivative term in (43) is small relative to the first. The thirdtime derivative term is retained to allow for the case of short radial length scale perturbationsto be considered. The equation then reduces to

∂

∂t(δμ) − 17

14

νsd

� |v� |sd�

∂

∂�

(∂

∂t(δμ)

)(45)

= 45

7

νsd

�

[(1 + 17

45U

)∂

∂�(δμ) + 2

3�

∂2

∂� 2(δμ)

],

where

U = � |v� |sd

νsdU. (46)

This equation incorporates perturbations to the viscous and magnetic torques, including theeffects of changes of poloidal magnetic field bending due to changes in the inflow speed andmagnetic diffusivity. Perturbations to the thermal balance are also accounted for, includingviscous and magnetic dissipation.

The field-bending instability of Lubow et al. (1994) results from a positive perturbation in|v� | which causes an increase in the bending of the poloidal magnetic field, via the advectionterm in the induction equation. This lowers the sonic point which leads to an increase in thewind mass flux, m, and hence increases the rate of loss of angular momentum. Hence, |v� | isfurther increased and an instability results. However, this argument neglects perturbations to thethermal balance and the dependence of the magnetic diffusivity on temperature. These effectsare included in the present analysis; in particular, perturbations to the viscous and magneticheating and radiative transfer are allowed for. The effect of the increased temperature on themagnetic diffusivity is investigated. A sufficiently large positive value of δη can relieve fieldbending and prevent the instability. The wind flow can adjust on a dynamical timescale andthe Alfvén point position changes in response to changes in disc quantities, via the relation(15). This is investigated globally and locally below.

4.2. Global stability

Firstly, the case in which δμ varies on a radial length scale of ∼ � is considered. The secondtime derivative term in (45) is then small relative to the first and hence the equation reduces to

∂

∂t(δμ) = 45

7

νsd

�

[(1 + 17

45U

)∂

∂�(δμ) + 2

3�

∂2

∂� 2(δμ)

], (47)

The unperturbed viscosity coefficient can be expressed as:

νsd = 3

2

εv

ε

N 2α

K 3

(h

�

)2

sd

� 2ΩK. (48)

The perturbation δμ can be written as:

δμ = q(�)eσ t . (49)

Using (48) and (49) in (47) yields

xd2q

dx2+ a

dq

dx− bσ x1/2q = 0, (50)

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Viscous stabilization of magnetic disc winds 359

with x = �/R,

a = 3

2

(1 + 17

45U

)and b = 7

45

1

Np

K 3

Nα

(�

h

)2

sd

(R3

G M

)1/2

. (51a,b)

The inner and outer edges of the disc are taken to be at x = 1 and x = xD, respectively.Equation (50) can be written in the self-adjoint form

d

dx

(xa dq

dx

)− bσ xa−1/2q = 0. (52)

Then, for homogeneous boundary conditions, σ is real and perturbations will be stable forσ < 0. Multiplying (52) by q and integrating through the disc gives

bσ

∫ xD

1xa−1/2q2dx =

[xaq

dq

dx

]xD

1−∫ xD

1xa(

dq

dx

)2

dx . (53)

Maintenance during perturbation of the standard conditions of μ → 0 as � → R andμ → constant for � � R gives q(1) = 0 and q ′(xD) = 0 and hence

σ = −1

b

[∫ xD

1xa(

dq

dx

)2

dx

](∫ xD

1xa−1/2q2dx

)−1

. (54)

This gives σ < 0 corresponding to stability through the disc.

4.3. Local stability

The stability of local perturbations can be considered by expressing δμ as

δμ = Aeik� est , (55)

where A, k and s are constants with k� � 1. The second time derivative in (45) mustnow be retained, since it contains a larger radial derivative than in the foregoing global case.Substitution of (55) in (45) yields

s = −30

7

νsd

� 2

[k2� 2 − 3

2

(1 + 17

45U

)ik�

] [1 − 17

14

U

Uik�

]−1

. (56)

Calculating σ = Re{s} gives

σ = −30

7

νsd

� 2k2� 2

[1 + 51

28

(1 + 17

45U

)U

U

][1 +

(17

14

U

U

)2

k2� 2

]−1

. (57)

Since σ < 0, this shows that local perturbations are stable and gives their decay rate.

4.4. The stabilization mechanism

The magnetic diffusivity η is related to the inflow speed v� and the wind launching angle is

by

η = 2

3

I 2 K 3

NαΩK

v2� tan2 is. (58)

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360 C. G. Campbell

It follows that

δη = 2ηsd

|v� |sdδ|v� | + 4ηsd

sin 2isδis. (59)

Also,

cs =(R

μTc

)1/2

= Np

εvI |v� | tan is. (60)

Hence, δTc is related to δ|v� | and δis by a similar equation to (59). For δ|v� | > 0 and δis > 0(59) gives δη > 0 and (60) yields δTc > 0. This corresponds to an increase in |v� | releasingbinding energy and heating the disc to give δTc > 0. Then δη > 0 and δis > 0 are consistentwith a reduction in poloidal magnetic field bending due to an increase in magnetic diffusivity.Hence, the field-bending instability is quenched. The wind structure can adjust on a dynamicaltimescale, with the Alfvén point position changing in response to the perturbation in m. Hence,the wind can adjust through quasi-steady states on the longer angular momentum adjustmenttime.

It is noted that the perturbation in η, which relieves poloidal field bending, results fromthe increase in temperature caused by increased dissipation. Since viscous dissipation isconsiderably larger than magnetic dissipation (see Campbell 2003), it is the presence ofviscosity which mainly causes the increase in η via an increase in Tc and hence quenchesthe field-bending instability.

5. Summary and discussion

The stability of the detailed magnetic wind-disc model of Campbell (2003) has been con-sidered, incorporating the solution of Campbell (2005) for the inner wind flow region whichmatches to the dynamo-generated disc field. This model allows the coupling between theradial and vertical structures to be accounted for. In particular, the vertical compression of thedisc due to magnetic pressure and the thermal balance affect the stability of the magneticallychannelled wind. Previous investigations indicate that a field-bending instability could causerunaway mass loss in a magnetic wind-driven disc. However, it is argued here that the magneticPrandtl number is unlikely to be ignorable in any realistic disc model and hence viscous effectsmust be accounted for.

The magnetic diffusivity, η, and the viscosity, ν, are believed to have a turbulent origin inaccretion discs, and they are both expected to be temperature dependent. It is shown here thata small increase in the disc inflow speed has an associated increase in the temperature andin η. This relieves the increase in poloidal magnetic field bending that results from increasedadvection and so prevents an unstable increase in the wind mass loss rate. Such a quenchingmechanism operates in a global and local situation, allowing stable mass loss from the mainbody of the disc.

Equations (54) and (57) show that only a small amount of viscosity is needed to causedecay of the perturbations and stabilize the wind flow. The parametrized form of ν, given by(7), can be compared with values estimated from simulations of turbulence generated by themagneto-rotational instability. Brandenburg et al. (1995) gives 0.01 � εv � 0.1. Such valueslead to quenching of the field-bending instability. It is noted that the numerical simulationsdiscussed in Section 1 indicate that diffusive discs can approach a steady or quasi-steadystate. The simulations also indicate that higher values of η lead to smaller wind mass fluxes

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Viscous stabilization of magnetic disc winds 361

(e.g. Zanni et al. 2007; Sheikhnezami et al. 2012), which is consistent with the stabilizingmechanism found here in which increases in η relieve poloidal field bending. Although not allthe simulations explicitly include viscosity in the disc, some numerical viscosity is likely tobe present (e.g. Murphy et al. 2010). This could be consistent with the general tendency of thesimulation solutions to approach a steady state.

Acknowledgement

The author is grateful to the referee for helpful suggestions which led to improvements in thepresentation of the paper.

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