Magnetic field gradient effects on the magnetorotational instability
aa r X i v : . [ a s t r o - ph . S R ] M a y Astronomische Nachrichten, 18 October 2018
Magnetic field gradient e ff ects on the magnetorotational instability Suzan Do˘gan , ,⋆ Department of Astronomy and Space Sciences, Faculty of Science, University of Ege, Bornova, 35100, ˙Izmir, Turkey. Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK.Received XXXX, accepted XXXXPublished online XXXX
Key words
Accretion, accretion discs — instabilities — MHD.The magnetorotational instability (MRI), also known as the Balbus – Hawley instability, is thought to have an importantrole on the initiation of turbulence and angular momentum transport in accretion discs. In this work, we investigate thee ff ect of the magnetic field gradient in the azimuthal direction on MRI. We solve the magnetohydrodynamic equationsby including the azimuthal component of the field gradient. We find the dispersion relation and calculate the growth ratesof the instability numerically. The inclusion of the azimuthal magnetic field gradient produces a new unstable regionon wavenumber space. It also modifies the growth rate and the wavelength range of the unstable mode: the higher themagnitude of the field gradient, the greater the growth rate and the wider the unstable wavenumber range. Such a gradientin the magnetic field may be important in T Tauri discs where the stellar magnetic field has an axis which is misalignedwith respect to the rotation axis of the disc. Copyright line will be provided by the publisher
An accretion disc with angular velocity decreasing radiallyoutward and threaded by a weak magnetic field is dynam-ically unstable (Balbus & Hawley 1991). This unstable be-havior, which is known as the magnetorotational instabil-ity (MRI), rapidly generates magnetohydrodynamic (MHD)turbulence which is probably the origin of “anomalous vis-cosity” in accretion discs. MHD turbulence driven by MRIe ff ectively transports angular momentum radially outward(Balbus & Hawley 1998).After Balbus & Hawley (1991) established the impor-tance of MRI in the structure and the dynamics of the ac-cretion discs, it was pointed out by many authors that thenon-ideal MHD e ff ects play a significant role in modifyingthe MRI. Non-ideal MHD e ff ects must be considered in sys-tems where the accretion discs are weakly ionized. For in-stance, protoplanetary discs (PPDs) are too dense and cool,so the thermal ionization process is expected to be inef-fective. Low ionization fraction makes the non-ideal MHDe ff ects important in such discs, as the plasma is not well-coupled to the magnetic field. The outer regions of dwarfnovae discs may also be weakly ionized and the non-idealMHD e ff ects are again expected to be important (Gammie1996; Gammie & Menou 1998; Stone et al. 2000).The importance of the Hall e ff ect in modifying the max-imum growth rate and the characteristic wavelength of MRIwas first pointed out by Wardle (1999). Wardle (1999) foundthat the growth rate of the instability depends on the relativesigns of the initially vertical magnetic field and the angu- ⋆ e-mail: [email protected] lar momentum of the disc, i.e. whether Ω · B is positive ornegative where Ω is the angular velocity vector of the discand B is the magnetic field vector. When the Hall currentis dominated by the negative (positive) species, the parallelcase ( Ω · B >
0) becomes less (more) unstable than the anti-parallel case ( Ω · B < ff ect in protostellar discsand found that the inclusion of the Hall e ff ect destabilizesthe disc with any di ff erential rotation law. Sano & Stone(2002), who investigated the e ff ect of the Hall term on theevolution of MRI in weakly ionized discs, showed that theHall term is important to the linear properties of the MRIand therefore must be included in models of both dwarfnova discs in quiescence, and in PPDs. Recent works (e.g.,Wardle & Salmeron 2012; Kunz & Lesur 2013; Bai 2014,2015) have also emphasized that the inclusion of Hall e ff ectmakes the properties of the MRI depend on the orientationof the net vertical field with respect to the disc rotation axisin PPDs.In addition to local analyses, several global MRI cal-culations have been performed for discs and spheres overthe last two decades. R¨udiger & Kitchatinov (2005) consid-ered the influence of the Hall e ff ect on the global stabil-ity of cool protoplanetary discs. They determined the min-imal and maximal magnetic field amplitudes and the val-ues of critical Reynolds numbers for which MRI would oc-cur. They showed that the MRI of a Kepler flow is signifi-cantly modified by the Hall e ff ect depending on the orien-tation of the magnetic field in relation to the rotation axis.Kondi´c et al. (2012, 2011) investigated the stability of theHall-MHD system for a spherical shell and determined itsimportance for newborn neutron stars. Similarly, their re- Copyright line will be provided by the publisher
S. Do˘gan: Magnetic field gradient e ff ects on MRI sults di ff er for magnetic fields aligned with the rotation axisand anti-parallel magnetic fields.For simplicity, most theoretical models assume that themagnetic axis and the rotation axis of the disc / star co-incide. This is, however, may not be the case in mostof the astrophysical systems. Misalignment can occur insystems where the central object has a dipole field witha magnetic axis misaligned with respect to its spin axis(e.g. Alencar & Batalha 2002; Symington et al. 2005). Wenote that the magnetic field geometry of PPDs may not bepurely dipole (Donati et al. 2008; Johns-Krull et al. 1999;Valenti & Johns-Krull 2004). Their structure is probablymore complicated, however it is expected that the dipolecomponent of the field dominates at distances of severalstellar radii (Johns-Krull 2007). In any case, when the mag-netic and the rotation axis are not exactly aligned, the discparticles “feel” a magnetic field gradient in both radial andazimuthal directions as their distance from the magneticpoles and the equator changes during their motion per or-bit. It has recently shown by Devlen & Pek¨unl¨u (2007) andDoˇgan & Pek¨unl¨u (2012) (hereafter DP12) that the mag-netic field gradient produced in the radial direction changesthe growth rates of the MRI significantly. The inclusionof the gradient terms produces new unstable regions andincreases the maximum value of the growth rates. In thepresent investigation, we aim to examine the e ff ect of themagnetic field gradient which is produced in the azimuthaldirection in the disc. Such a gradient can be generated bythe misalignment of the magnetic axis with respect to therotation axis of the disc.The plan of the paper is as follows: In Section 2, wedescribe the basic geometry of the problem, present thelinearized MHD equations that include the terms describ-ing the magnetic field gradient in the azimuthal directionand find the dispersion relation. In Section 3, the numeri-cal growth rates found from the dispersion relation are pre-sented both for zero and finite resistivity. The e ff ect of themagnetic field gradient on the instability is investigated. Wefinally discuss our results in Section 4. Let us begin by considering the simple geometry such thatthe disc fluid orbits a central object with a dipole field andwhere the stellar magnetic axis and rotation axis of thedisc are aligned. If we use standard cylindrical coordinates( R , φ, z ) with the origin at the disc center, then the magneticand the rotation axes coincide on the z-axis. Therefore, themagnetic equator lies in the disc plane and the magneticfield strength remains constant for the drifting fluid parti-cles at the same radial distance. The particles feel no mag-netic field gradient in the azimuthal direction during theirorbital motion. This is the case where the configuration isaxisymmetric. When the magnetic axis and rotation axis of the disc arenot aligned, the drifting particles spend half the drift period on the northern magnetic hemisphere and the other half onthe southern one. In this case, both the magnetic latitude andthe distance from the magnetic axis vary during their per or-bital motion. Therefore, the particles encounter a magneticfield gradient in the azimuthal direction due to the fact thatthe magnetic field strength B ∝ (1 + Λ ) / / R where R is the distance from the central star and Λ is the mag-netic latitude measured northwards from the magnetic equa-tor. When the particles move towards the region of the discwhich is close to the magnetic pole (equator), the magneticfield strength increases (decreases). We know that the MRIoccurs only if the magnetic field is weak (i.e. sub-thermal).Therefore, we expect to see the e ff ect of the fluctuation ofmagnetic field on the instability. If the particles experience adecrease in an already weak field, the instability is expectedto become more powerful. We should also mention that theMRI may switch o ff if the field is too weak. In this context,global calculations by R¨udiger & Kitchatinov (2005) pointout that if the external magnetic field strength does not ex-ceed 0.1 G then the MRI can not occur without the Halle ff ect of parallel fields.In a dipole field we know that the total field vector ispurely vertical at the magnetic equator, i.e. B = B z ˆ z at Λ = ◦ .Therefore, the magnetic field lines are perpendicular to thedisc plane when the magnetic axis of the central object andthe rotation axis of the disc are aligned. However, in a mis-aligned case the magnetic field threading the disc will alsohave radial and azimuthal components ( B R , B φ ) in additionto the vertical component. The magnitude of each compo-nent in the disc plane di ff ers from those they have in thealigned case. The importance of the R and φ componentsdepend on the inclination angle of the magnetic axis. In thisstudy, we consider the vertical component of the magneticfield ( B z ) only, as it is crucial for MRI. Besides, the pres-ence of a radial component causes a linear time dependencein B φ . Therefore, it is mathematically the simplest to con-sider the case of vanishing radial field (Balbus & Hawley1998). The fundamental equations are mass conservation, ∂ρ∂ t + ∇ · ( ρ v ) = ρ ∂ v ∂ t + ( ρ v · ∇ ) v = −∇ P + c J × B (2)and the induction equation, ∂ B ∂ t = ∇ × " v × B − η π c J − J × B en e (3)where v is the fluid velocity, η is the microscopic resistivityand n e is the number density of electrons. Here the current Copyright line will be provided by the publisher sna header will be provided by the publisher 3 density ( J ) is given as J = c π ∇ × B . (4)In a misaligned dipole, the azimuthal gradient in mag-netic field ( ∇ B ) is expected to generate a magnetic pressureforce which pushes the plasma particles towards the neg-ative ∇ B direction. If the frozen-in condition is satisfied,frozen-in particles bend the magnetic field lines in such away as to give them a curvature. As a result, the curvature ofthe magnetic field lines produces a magnetic tension force inthe opposite direction of the magnetic pressure force. There-fore, the magnetic tension force balances the magnetic pres-sure gradient in the equilibrium. In our stability analysis, wehave ignored the current produced by the electrons underthe influence of curvature drift. We use standard cylindri-cal coordinates ( R , φ, z ) with the origin at the disc center.We consider the local stability of a Keplerian disc threadedby a vertical field with a gradient in the azimuthal direc-tion, B = B ( φ )ˆ z . Therefore the gradient of the magneticfield is ∇ B = ( dB / Rd φ ) ˆ φ . We shall work in the Boussi-nesq limit which is frequently used in descriptions of the na-ture of accretion disc transport (e.g. Balbus & Hawley 1991;Balbus & Terquem 2001). We should note that the disc fluidis not exactly but nearly incompressible (Balbus & Hawley1998). Therefore, the velocity field in accretion discs maybe taken as ( ∇ · u =
0) for the turbulent flows. Finally, weassume that the perturbed quantities vary in space and timelike a plane wave, i.e., exp( ikz + ω t ), where k is the wavevector perpendicular to the disc and ω is the angular fre-quency. We find the linearized radial, azimuthal and verticalcomponents of equation of motion as ωδv R − Ω δv φ − ik πρ B δ B R = ωδv φ + κ Ω δv R − πρ " ikB δ B φ − ∂ BR ∂φ δ B z = ik δ P ρ − πρ ∂ BR ∂φ δ B φ = . (7)Including the finite resistivity and the Hall e ff ect, we findthe same components of the linearized induction equationas( k η + ω ) δ B R − ikB δv R + c π en e " k B δ B φ + ∂ BR ∂φ δ B z = k η + ω ) δ B φ − ikB δv φ − " d Ω dlnR + c π en e Bk δ B R = k η + ω ) δ B z + ∂ BR ∂φ δv φ − ik c π en e ∂ BR ∂φ δ B R = . (10) Here κ = Ω + d Ω / d ln R is the epicyclic frequency and Ω is the angular velocity of the disc. The Eqns. (5) - (10)yield a fifth - order dispersion relation given below: ω + k ηω + ω κ + k v A " k η Ω + − G k + k v H Ω d Ω d lnR − k v A v H Ω G + k v H + ω k η κ + k v A k η v A + − G k + k v H d Ω d lnR − k v H v A Ω G + k v H Ω + ω k v A + k v H Ω κ " d Ω d lnR + k (cid:0) v A + v H (cid:1)(cid:18) − G k (cid:19) + k η v A " κ + k v A (cid:18) − G k (cid:19) + η k v A + v H κ Ω " d Ω d lnR + k (cid:0) v A + v H (cid:1)(cid:18) − G k (cid:19) = G = dln B / R d φ represents the magnitude of magneticfield gradient. The Alfv´en and the Hall velocities are definedas v = B / πρ and v = Ω · B c / π en e . The disc becomesunstable if any of the roots of the Eq. (11) has a positive realpart. We seek the solution at a fiducial radius ( R ). In this in-vestigation, the fiducial radius corresponds to the corotationradius where Keplerian angular velocity in the disc equalsthe angular velocity of the star. We first solve the dispersionrelation for the limit η −→ η = s = ω/ Ω , X = ( k v A / Ω ) , Y = ( k v H / Ω ) , ˜ κ = κ/ Ω ,˜ G = G / k , χ ≡ v H /v A . Therefore, in the limit of zero resis-tivity ( η =
0) the dispersion relation (11) can be rewritten interms of dimensionless parameters as s + s " ˜ κ + X (2 − ˜ G ) + Y d ln Ω d ln R + Y − χ X ˜ G ! + X + Y ˜ κ ! " d ln Ω d ln R + ( X + Y )(1 − ˜ G ) = . (12)We note that Eq. (12) is reduced to Eq. (62) of Copyright line will be provided by the publisher
S. Do˘gan: Magnetic field gradient e ff ects on MRI (a) χ = 4, G = 0 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.4 0.8s (b) χ = 4, G = 0.1 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.4 0.8s (c) χ = 4, G = 0.25 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.7 1.4s Fig. 1
Growth rates found from Eq. (12) for Ω · B > ff ect of ∇ B on growth rates, we plot ourgraphs for ˜ G =
0, 0.1, 0.25 (see text for definitions). A newunstable mode comes into existence with the inclusion of ∇ B . See Table 1 for maximum values of the growth rates( s m ).Balbus & Terquem (2001) when ˜ G =
0. We can easily calcu-late ˜ G values by using the magnetic dipole formulae whichgive the magnetic field strength depending on radial dis-tance and the magnetic latitude. Assuming that the equato-rial magnetic field strength on stellar surface B ∗ = kR ≫ G as about 0.25 for moderate inclinations. We assume a Keplerian rotation, therefore the dimensionless epicyclicfrequency is ˜ κ =
1. The Hall parameter for T Tauri diskscan be estimated as follows (see also DP2012): The radialdistance where the stellar magnetic field starts to control themotion of the accreting plasma is given by Bouvier et al.(2007b) as about 7 stellar radii for B ∗ = R M = R ∗ is B M = B ∗ ( R ∗ / R M ) ≈ . ρ g = − gcm − (Alexander 2008) and n e = cm − (Glassgold et al. 2007), we find the χ valueas 4 for a typical period P ∼ d . We will use these valuesto solve the dispersion relation given by (12) numerically.The solution of Eq. (12) gives two fast and two slow modeswhich are labeled according to the magnitude of their phasevelocities, v ph = ω i / k . One of the slow modes is unstable.The graphical solutions of Eq. (12), the numerical growthrates ( s ) of the slow mode, are thus shown in (X,Y) planein Fig. 1. Positive s implies an unstable exponential growthof the mode. The regions of instability are therefore seenas “ridges” above the (X,Y) plane. We first assume that Ω · B >
0. In order to see the e ff ect of the azimuthal compo-nent of the magnetic field gradient on growth rates, we plotour graphs for weak ( ˜ G = G = G = unstable region I ( UR-I ). The maximumgrowth rate of the instability for this case has been foundto be 0.75. Fig. 1b and 1c show that the inclusion of themagnetic field gradient in the azimuthal direction producesa new unstable region in the (X,Y) plane. We refer to thisnew region of instability as the unstable region II ( UR-II ).Fig. 1b shows the growth rates for ˜ G = UR-I is 0.77. If we com-pare Fig. 1b and Fig. 1a, we see that the region of instabilitybecomes wider with the inclusion of the gradient. The max-imum value of the growth rate for
UR-II is smaller than thatof
UR-I . Fig. 1c shows the growth rates for strong gradient( ˜ G = UR-II becomes higherthan that of
UR-I when ˜ G = UR-II reaches a value of 1.72. Themaximum growth rates of the
UR-I and the
UR-II are listedin Table 1 for di ff erent values of ˜ G . The growth rates of the UR-II are a ff ected by the magnitude of the gradient morethan that of UR-I .Fig. 2 shows the growth rates for Ω · B < UR-II againspreads over a larger space in the (X,Y) plane for ˜ G = G = UR-I becomes smaller whenthe gradient is strong. When we compare Fig. 1 and Fig. 2,we see that the maximum values of the growth rates are verysimilar. However, the wavenumber range where the instabil-ity occurs is narrower in Fig. 2. The di ff erence in unstableregions between two figures is much more apparent in UR-I . Copyright line will be provided by the publisher sna header will be provided by the publisher 5 (a) χ = -4, G = 0 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.4 0.8s (b) χ = -4, G = 0.1 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.4 0.8s (c) χ = -4, G = 0.25 0 5 10 15 20 25 30 35 X -30-20-10 0 10 20 30 Y 0 0.7 1.4s Fig. 2
Growth rates found from Eq. (12) for Ω · B < ff ect of ∇ B on growth rates, we plot ourgraphs for ˜ G =
0, 0.1, 0.25 (see text for definitions). A newunstable mode comes into existence with the inclusion of ∇ B . See Table 1 for maximum values of the growth rates( s m ). Table 1
The maximum growth rates of the instability. s m χ = χ = − G = G = G = η , s + XRe m s + s ˜ κ + X XRe m + − ˜ G ! + χ X d ln Ω d ln R + ( χ − ˜ G ) X ! + s XRe m κ + X XRe m + − G ! + χ X d ln Ω d ln R + ( χ − ˜ G ) X ! + s X + Y κ ! (cid:20) d ln Ω d ln R + X (cid:18) ( χ + − ˜ G ) (cid:19)(cid:21) + X Re m (cid:16) ˜ κ + X (2 − ˜ G ) (cid:17) + X Re m + κ χ Re m ! (cid:20) d ln Ω d ln R + X (cid:18) ( χ + − ˜ G ) (cid:19)(cid:21) = . (13)Here, Re M ≡ v A / Ω η is the magnetic Reynolds number.In order to solve Eq. (13) numerically, we need to make as-sumptions on the relative importance of the non-ideal MHDterms. Sano & Stone (2002) show that the PPDs can be sep-arated into three regions which are classified depending onthe relative importance of the Hall e ff ect: (i) the outer regionof the disc where | χ | < Re M ≫ ff ect and ohmic dissipation are unimportant), (ii) the inter-mediate region of the disc where | χ | > Re M > ff ect is important but ohmic dissipation is still canbe neglected), (iii) the inner part of the disc where | χ | ≫ Re M ≪ ff ects are important). We investi-gate the e ff ect of the magnetic field gradient on MRI in threeregions of the disc.Eq. (13) gives two fast, two slow and one standing wavemodes. One of the slow modes is unstable. The right panelof Fig. 3 shows the dispersion relations for Ω · B >
0. To seethe e ff ect of the magnetic field gradient, we plot our graphsfor ˜ G = , . , .
25. Modes with smaller wavenumber than
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S. Do˘gan: Magnetic field gradient e ff ects on MRI s X(a) Re m =100, χ =0.2 G = 0 G = 0.1 G = 0.25 0 0.4 0.8 0 1 2 3 4 5 6 s X(d) Re m =100, χ =-0.2 G = 0 G = 0.1 G = 0.25-0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 s X(b) Re m =2, χ =2 G = 0 G = 0.1 G = 0.25 0 0.2 0.4 0 0.5 1 1.5 2 s X(e) Re m =2, χ =-2 G = 0 G = 0.1 G = 0.25 0 0.2 0.4 0 0.02 0.04 0.06 0.08 0.1 s X(c) Re m =0.02, χ =100 G = 0 G = 0.1 G = 0.25 -0.1 0 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 s X(f) Re m =0.02, χ =-100 G = 0 G = 0.1 G = 0.25 Fig. 3
Growth rates found from Eq. 13. The figures are plotted for ˜ G =
0, 0.1, 0.25. Left panels are for Ω · B >
0; theright panels are for Ω · B < k c ) are unstable, modes exceeding thecritical value are stable for the MRI. Fig. 3a shows that, inthe outer region where we assume that χ = . Re M = χ = Re M = G = , . , .
25. Here, the growth ratesare reduced by the e ff ect of ohmic dissipation. The criti-cal wavenumbers are also found to be smaller than the firstcase. But the larger magnetic field gradient again enhancesthe growth rates and the critical wavenumber. As shown inFig. 3c, decreasing Re M leads to even smaller growth rates and critical wavenumbers. Here, the growth rates are plottedfor χ = Re M = .
02 and ˜ G = , . , .
25. In this case,the magnetic field gradient has very little e ff ect on the maxi-mum growth rate, but increasing the field gradient enhancesthe critical wavenumber of the instability.The left panel shows the dispersion relations for Ω · B <
0. If we compare Fig. 3a and 3d, we see that negative χ pro-duces wider unstable regions. Fig. 3e and 3f show that de-creasing the Re M (i.e. increasing the e ff ect of ohmic dissipa-tion) reduces the growth rates significantly. There is no char-acteristic scale for the instability as the critical wavenumbergoes to infinity when χ = − Re M = Copyright line will be provided by the publisher sna header will be provided by the publisher 7
In conclusion, the net e ff ect of the magnetic field gra-dient is to shift the critical wavenumber towards highervalues. This is an expected result, as the role of the mag-netic field strength is to determine a characteristic scalefor unstable wavelengths, a scale that makes k compara-ble to Ω /v A (Balbus & Hawley 1991, 1998). If the magneticfield is weakened, then the critical wavenumber requiredfor the MRI increases. In Fig. 3, the curves with ˜ G , ff ect. On the other hand, increasing the non-idealMHD e ff ects produces much smaller growth rates and crit-ical wavenumbers. Fields oriented such that Ω · B < Ω · B > Any realistic model on accretion onto a central object with adipole magnetic field should take into account the e ff ects ofmisalignment of the magnetic axis. Previously, it has beenshown that the inclusion of the magnetic field gradient in theradial direction plays an important role in triggering MRI.We now show that the magnetic field gradient in the az-imuthal direction also makes MRI more powerful and pro-duces a new unstable region. Inclusion of the magnetic fieldgradient influences the growth rates and the characteristicwavenumbers significantly. While increasing the non-idealMHD e ff ects suppresses the growth of the instability, in-creasing the magnitude of the magnetic field gradient pro-duces higher growth rates. The behaviour we have discussedin this paper is relevant to T Tauri discs where the stellarmagnetic axis is misaligned with respect to the rotation axis.For example, AA Tau is known to have a 2 - 3 kG magneticdipole tilted at about 20 ◦ to the rotation axis (Bouvier et al.2007a). If we naively assume that the rotation axis of thestar and the disc coincide, we find the variation in the fieldstrength in the azimuthal direction that the disc particles ex-perience during their orbital motion as 14 per cent for thisinclination. This variation gives ˜ G ≈ . ff ects of radialand azimuthal gradients on MRI, we can see that both gra-dients produce a new unstable region. The maximum valuesof the growth rate found from these new unstable regions arevery similar for a weak gradient in the radial and azimuthaldirection. The maximum growth rate is 0.19 for ˜ G = . G = . G is the radial gradient in di-mensionless form, i.e. G = dlnB / dlnR (cf. DP12, Table 1).We should note that the magnitude of the azimuthal gra-dient caused by the misalignment of dipole field may notbe as high as the radial gradient produced by magnetiza-tion. However, the influence of a strong azimuthal gradienton the new unstable region seems to be slightly stronger than that of a radial gradient. Here, we find a wider unsta-ble region and a higher maximum growth rate from UR-IIfor ˜ G = .
25 than we found for G = . G = .
25 and found the maximum growth rateas 0.94 which is slightly lower than we found in this studyfor ˜ G = .
25 (see Table 1). It is a condition for the growthof MRI that the field be weak and both investigations revealthat weakening an already weak magnetic field triggers MRIand enhances the growth rate of the instability. In conclu-sion, the inclusion of magnetic field gradient tends to makethe instability more powerful.Previously, Balbus & Terquem (2001) pointed out theimportance of identifying the Hall parameter precisely.When the Hall term is defined depending on the product Ω · B , this raises the following question: what happens whenthe magnetic field lies in the disc plane, i.e. Ω · B = k · Ω )( k · B ). If the magnetic field has a radial component,then it is always possible to find a wavevector that makesthe Hall parameter negative. The analysis described here isrestricted to axial fields and axial wavenumbers, thereforewe could not address this problem. In this work, we con-sider the vertical component of the magnetic field ( B z ) only,as it is crucial for MRI. However, if the central object hasa misaligned dipole field, then the magnetic field threadingthe disc will always have a radial component. We shall makemore general analysis including the radial field componentand explore this problem in a future publication.In realistic case, the problem is much more complicatedthan described here. Because the magnetic field may not bepurely dipole. The strength and the structure of the mag-netic field is very important as it determines the magne-topause radius where the disc plasma is separated from thestellar magnetosphere, and therefore the inner disc radius(Aly 1980). But if the magnetic axis is misaligned withthe rotation axis of the disc, we always expect a magneticfield gradient to occur in the disc plane whatever the fieldstructure is. We should also note that a non-axisymmetricfield is expected to generate non-axisymmetric perturba-tions. In this paper, we restricted our attention to a magneticfield subject to axisymmetric perturbations. Previously, ithas been demonstrated that non-axisymmetric MRI modescan appear as transient events only (Balbus & Hawley1992; Terquem & Papaloizou 1996). However, exponen-tially growing non-axisymmetric MRI modes have alsobeen obtained by Kitchatinov & R¨udiger (2010) for Kep-lerian discs. For non-axisymmetric field configurations, itmay be argued that non-axisymmetric modes may be moreimportant than the axisymmetric modes, at least transiently.A full stability analysis of a disc in a misaligned dipole fieldwould require the derivation of a dispersion relation for ageneral perturbation and this will be the purpose of our fu-ture work. Copyright line will be provided by the publisher
S. Do˘gan: Magnetic field gradient e ff ects on MRI Acknowledgements.
I would like to thank the referee for his / hervaluable suggestions which helped to improve the manuscript, andProf. Rennan Pek¨unl¨u for his useful comments. I thank the Theo-retical Astrophysics Group at University of Leicester for their hos-pitality. This work was supported by The Scientific and Techno-logical Research Council of Turkey (T ¨UB˙ITAK) through the Post-doctoral Research Fellowship Programme (2219). References