Importance of magnetic fields in highly eccentric discs with applications to tidal disruption events
MMNRAS , 1–20 (2019) Preprint 6 January 2021 Compiled using MNRAS L A TEX style file v3.0
Importance of magnetic fields in highly eccentric discs withapplications to tidal disruption events
Elliot M. Lynch (cid:63) and Gordon I. Ogilvie
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences,Wilberforce Road, Cambridge CB3 0WA, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Whether tidal disruption events (TDEs) circularise or accrete directly as a highlyeccentric disc is the subject of current research and appears to depend sensitively onthe disc thermodynamics. In a previous paper we applied the theory of eccentric discsto TDE discs using an α − prescription for the disc stress, which leads to solutions thatexhibit extreme, potentially unphysical, behaviour. In this paper we further explore thedynamical vertical structure of highly eccentric discs using alternative stress modelsthat are better motivated by the behaviour of magnetic fields in eccentric discs. We findthat the presence of a coherent magnetic field has a stabilising effect on the dynamicsand can significantly alter the behaviour of highly eccentric radiation dominated discs.We conclude that magnetic fields are important for the evolution of TDE discs. Key words: accretion, accretion discs – hydrodynamics – black hole physics – MHD
Tidal disruption events (TDEs) are transient phenomenawhere an object on a nearly parabolic orbit passes withinthe tidal radius and is disrupted by the tidal forces, typ-ically a star being disrupted by a supermassive black hole(SMBH). Bound material from the disruption forms a highlyeccentric disc, which in the classic TDE model of Rees (1988)are rapidly circularised as the material returns to pericentre.It has, however, been proposed that circularisation in TDEsmay be inefficient resulting in the disc remaining highly ec-centric (Guillochon et al. 2014; Piran et al. 2015; Kroliket al. 2016; Svirski et al. 2017). In Lynch & Ogilvie (2020)(henceforth Paper I) we presented a hydrodynamical modelof these highly eccentric discs applied to TDEs where circu-larisation is inefficient.Two issues were highlighted in Paper I. One was con-firming that radiation pressure dominated discs are ther-mally unstable when the viscous stress scales with total pres-sure in highly eccentric discs, a result that has long beenknown for circular discs (Shakura & Sunyaev 1976; Pringle1976; Piran 1978). Circular radiation pressure dominateddiscs can be stabilised by assuming stress scales with gaspressure (Meier 1979; Sakimoto & Coroniti 1981). For highlyeccentric discs it appears that the thermal instability is stillpresent when stress scales with gas pressure; however thereexists a stable radiation pressure dominated branch which (cid:63)
E-mail: [email protected] is the outcome of the thermal instability. For typical TDEparameters, this branch is very hot and often violates thethin-disc assumptions.The second issue was the extreme behaviour that canoccur during pericentre passage. For the radiation pressuredominated disc where stress scales with gas pressure the so-lution is nearly adiabatic and undergoes extreme compres-sion near pericentre. In models where the viscous stressescontribute to the dynamics we typically found that the ver-tical viscous stress is comparable to or exceeds the (total)pressure, which is possibly problematic for the α -model as itwould indicate transonic turbulence. In some of the solutionsthe vertical viscous stress can exceed the total pressure byan order of magnitude, strongly violating the assumptionsof the α -model.In this paper we focus on the second of the two issuesby considering alternative turbulent stress models which arebetter motivated by the physics of the underlying magneticfield to see if this resolves some of the extreme behaviourseen in the α -models. We will also see if alternative stressmodels are more thermally stable than the α -model, al-though it’s possible the solution to this issue is outside thescope of a thin disc model .Two additional physical effects, not present in an α -model, may be important for regulating the extreme be-haviour at pericentre. One is the finite response time of themagnetorotational instability (MRI) (see for instance theviscoelastic model of Ogilvie (2001) and discussion therein)which means the viscous stress cannot respond instantly to © a r X i v : . [ a s t r o - ph . H E ] J a n E. M. Lynch and G. I. Ogilvie the rapid increase in pressure and velocity gradients dur-ing pericentre passage, potentially weakening the viscousstresses so they no longer exceed the pressure. Another isthe relative incompressibility of the magnetic field, com-pared with the radiation or the gas, with the magnetic pres-sure providing additional support during pericentre passagewhich could prevent the extreme compression seen in somemodels.Various attempts have been made to rectify some ofthe deficiencies of the α − prescription using alternative clo-sure models for the turbulent stress. Ogilvie (2000, 2001)proposed a viscoelastic model for the dyadic part of theMaxwell stress (i.e. the contribution from magnetic tension B i B j µ ) to account for the finite response time of the MRI.It was shown in Ogilvie & Proctor (2003) that for incom-pressible fluids there is an exact asymptotic correspondencebetween MHD in the limit of large magnetic Reynolds num-ber and viscoelastic fluids (specifically an Oldroyd-B fluid)in the limit of large relaxation time. Ogilvie (2002) improvedupon the compressible viscoelastic model of Ogilvie (2000,2001) by including an isotropic part to the stress to modelthe effects of magnetic pressure and correcting the heatingrate so that total energy is conserved. Ogilvie (2003) pro-posed solving for both the Maxwell and Reynolds stressesand suggested a nonlinear closure model based on requiringthe turbulent stresses to exhibit certain properties (such aspositive definiteness, and relaxation towards equipartitionand isotropy) known from simulations and experiments.Simulations of MRI in circular discs typically find thatthe magnetic pressure tends to saturate at about 10% of thegas pressure. However in the local linear stability analysisof Pessah & Psaltis (2005) the toroidal magnetic field onlystabilises the MRI when it is highly suprathermal (specifi-cally when the Alfv´en speed is greater than the geometricmean of the sound speed and the Keplerian speed). Daset al. (2018) confirmed this result persists in a global lineareigenmode calculation. In light of this, Begelman & Pringle(2007) have suggested that, for a strongly magnetised disc,the viscous stress may scale with the magnetic pressure andshowed that such a disc would be thermally stable even whenradiation pressure dominates over gas pressure. Such a discwas simulated by S (cid:44) adowski (2016), who indeed found ther-mal stability.Throughout this paper we will make use of certain con-ventions from tensor calculus, such as the Einstein summa-tion convention and the distinction between covariant andcontravariant indices, along with the notation for symmetris-ing indices, X ( ij ) := 12 (cid:16) X ij + X ji (cid:17) . (1)This paper is structured as follows. In Section 2 we dis-cuss the geometry of eccentric discs and restate the coor-dinate system of Ogilvie & Lynch (2019). In Section 3 wederive the equations for the dynamical vertical structure, in-cluding the effects of a Maxwell stress, in this coordinate sys-tem. In Section 4 we consider a model with an α − viscosityand a coherent magnetic field which obeys the ideal induc-tion equation. In Section 5 we consider a nonlinear consti-tutive model for the magnetic field. In our discussion wediscuss the stability of our solutions (Section 6.1) and the possibility of dynamo action in the disc (6.2). We presentour conclusions in Section 7 and additional mathematicaldetails are in the appendices. Similar to Paper I we assume the dominant motion in aTDE disc consists of elliptical Keplerian orbits, subject torelatively weak perturbations from relativistic precessionaleffects, pressure and Maxwell stresses. This model is unlikelyto be applicable to TDEs where the tidal radius is compara-ble to the gravitational radius owing to the strong relativisticprecession.Let ( r, φ ) be polar coordinates in the disc plane. Thepolar equation for an elliptical Keplerian orbit of semimajoraxis a , eccentricity e and longitude of periapsis (cid:36) is r = a (1 − e )1 + e cos f , (2)where f = φ − (cid:36) is the true anomaly. A planar eccentricdisc involves a continuous set of nested elliptical orbits. Theshape of the disc can be described by considering e and (cid:36) to be functions of a . The derivatives of these functionsare written as e a and (cid:36) a , which can be thought of as theeccentricity gradient and the twist, respectively. The discevolution is then described by the slow variation in timeof the orbital elements e and (cid:36) due to secular forces suchas pressure gradients in the disc and departures from thegravitational field of a Newtonian point mass.In this work we adopt the (semimajor axis a , eccentricanomaly E ) orbital coordinate system described in Ogilvie& Lynch (2019). The eccentric anomaly is related to the trueanomaly bycos f = cos E − e − e cos E , sin f = √ − e sin E − e cos E (3)and the radius can be written as r = a (1 − e cos E ) . (4)The area element in the orbital plane is given by dA =( an/ J da dE where J is given by J = 2 n (cid:20) − e ( e + ae a ) √ − e − ae a √ − e cos E − ae(cid:36) a sin E (cid:21) , (5)which corresponds to the Jacobian of the (Λ , λ ) coordinatesystem of Ogilvie & Lynch (2019). Here n = (cid:113) GM • a is themean motion with M • the mass of the black hole. The Ja-cobian can be written in terms of the orbital intersectionparameter q of Ogilvie & Lynch (2019): J = (2 /n ) 1 − e ( e + ae a ) √ − e (1 − q cos( E − E )) (6)where q is given by q = ( ae a ) + (1 − e )( ae(cid:36) a ) [1 − e ( e + ae a )] , (7) MNRAS000
E-mail: [email protected] is the outcome of the thermal instability. For typical TDEparameters, this branch is very hot and often violates thethin-disc assumptions.The second issue was the extreme behaviour that canoccur during pericentre passage. For the radiation pressuredominated disc where stress scales with gas pressure the so-lution is nearly adiabatic and undergoes extreme compres-sion near pericentre. In models where the viscous stressescontribute to the dynamics we typically found that the ver-tical viscous stress is comparable to or exceeds the (total)pressure, which is possibly problematic for the α -model as itwould indicate transonic turbulence. In some of the solutionsthe vertical viscous stress can exceed the total pressure byan order of magnitude, strongly violating the assumptionsof the α -model.In this paper we focus on the second of the two issuesby considering alternative turbulent stress models which arebetter motivated by the physics of the underlying magneticfield to see if this resolves some of the extreme behaviourseen in the α -models. We will also see if alternative stressmodels are more thermally stable than the α -model, al-though it’s possible the solution to this issue is outside thescope of a thin disc model .Two additional physical effects, not present in an α -model, may be important for regulating the extreme be-haviour at pericentre. One is the finite response time of themagnetorotational instability (MRI) (see for instance theviscoelastic model of Ogilvie (2001) and discussion therein)which means the viscous stress cannot respond instantly to © a r X i v : . [ a s t r o - ph . H E ] J a n E. M. Lynch and G. I. Ogilvie the rapid increase in pressure and velocity gradients dur-ing pericentre passage, potentially weakening the viscousstresses so they no longer exceed the pressure. Another isthe relative incompressibility of the magnetic field, com-pared with the radiation or the gas, with the magnetic pres-sure providing additional support during pericentre passagewhich could prevent the extreme compression seen in somemodels.Various attempts have been made to rectify some ofthe deficiencies of the α − prescription using alternative clo-sure models for the turbulent stress. Ogilvie (2000, 2001)proposed a viscoelastic model for the dyadic part of theMaxwell stress (i.e. the contribution from magnetic tension B i B j µ ) to account for the finite response time of the MRI.It was shown in Ogilvie & Proctor (2003) that for incom-pressible fluids there is an exact asymptotic correspondencebetween MHD in the limit of large magnetic Reynolds num-ber and viscoelastic fluids (specifically an Oldroyd-B fluid)in the limit of large relaxation time. Ogilvie (2002) improvedupon the compressible viscoelastic model of Ogilvie (2000,2001) by including an isotropic part to the stress to modelthe effects of magnetic pressure and correcting the heatingrate so that total energy is conserved. Ogilvie (2003) pro-posed solving for both the Maxwell and Reynolds stressesand suggested a nonlinear closure model based on requiringthe turbulent stresses to exhibit certain properties (such aspositive definiteness, and relaxation towards equipartitionand isotropy) known from simulations and experiments.Simulations of MRI in circular discs typically find thatthe magnetic pressure tends to saturate at about 10% of thegas pressure. However in the local linear stability analysisof Pessah & Psaltis (2005) the toroidal magnetic field onlystabilises the MRI when it is highly suprathermal (specifi-cally when the Alfv´en speed is greater than the geometricmean of the sound speed and the Keplerian speed). Daset al. (2018) confirmed this result persists in a global lineareigenmode calculation. In light of this, Begelman & Pringle(2007) have suggested that, for a strongly magnetised disc,the viscous stress may scale with the magnetic pressure andshowed that such a disc would be thermally stable even whenradiation pressure dominates over gas pressure. Such a discwas simulated by S (cid:44) adowski (2016), who indeed found ther-mal stability.Throughout this paper we will make use of certain con-ventions from tensor calculus, such as the Einstein summa-tion convention and the distinction between covariant andcontravariant indices, along with the notation for symmetris-ing indices, X ( ij ) := 12 (cid:16) X ij + X ji (cid:17) . (1)This paper is structured as follows. In Section 2 we dis-cuss the geometry of eccentric discs and restate the coor-dinate system of Ogilvie & Lynch (2019). In Section 3 wederive the equations for the dynamical vertical structure, in-cluding the effects of a Maxwell stress, in this coordinate sys-tem. In Section 4 we consider a model with an α − viscosityand a coherent magnetic field which obeys the ideal induc-tion equation. In Section 5 we consider a nonlinear consti-tutive model for the magnetic field. In our discussion wediscuss the stability of our solutions (Section 6.1) and the possibility of dynamo action in the disc (6.2). We presentour conclusions in Section 7 and additional mathematicaldetails are in the appendices. Similar to Paper I we assume the dominant motion in aTDE disc consists of elliptical Keplerian orbits, subject torelatively weak perturbations from relativistic precessionaleffects, pressure and Maxwell stresses. This model is unlikelyto be applicable to TDEs where the tidal radius is compara-ble to the gravitational radius owing to the strong relativisticprecession.Let ( r, φ ) be polar coordinates in the disc plane. Thepolar equation for an elliptical Keplerian orbit of semimajoraxis a , eccentricity e and longitude of periapsis (cid:36) is r = a (1 − e )1 + e cos f , (2)where f = φ − (cid:36) is the true anomaly. A planar eccentricdisc involves a continuous set of nested elliptical orbits. Theshape of the disc can be described by considering e and (cid:36) to be functions of a . The derivatives of these functionsare written as e a and (cid:36) a , which can be thought of as theeccentricity gradient and the twist, respectively. The discevolution is then described by the slow variation in timeof the orbital elements e and (cid:36) due to secular forces suchas pressure gradients in the disc and departures from thegravitational field of a Newtonian point mass.In this work we adopt the (semimajor axis a , eccentricanomaly E ) orbital coordinate system described in Ogilvie& Lynch (2019). The eccentric anomaly is related to the trueanomaly bycos f = cos E − e − e cos E , sin f = √ − e sin E − e cos E (3)and the radius can be written as r = a (1 − e cos E ) . (4)The area element in the orbital plane is given by dA =( an/ J da dE where J is given by J = 2 n (cid:20) − e ( e + ae a ) √ − e − ae a √ − e cos E − ae(cid:36) a sin E (cid:21) , (5)which corresponds to the Jacobian of the (Λ , λ ) coordinatesystem of Ogilvie & Lynch (2019). Here n = (cid:113) GM • a is themean motion with M • the mass of the black hole. The Ja-cobian can be written in terms of the orbital intersectionparameter q of Ogilvie & Lynch (2019): J = (2 /n ) 1 − e ( e + ae a ) √ − e (1 − q cos( E − E )) (6)where q is given by q = ( ae a ) + (1 − e )( ae(cid:36) a ) [1 − e ( e + ae a )] , (7) MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs and we require | q | < E , which determinesthe location of maximum horizontal compression around theorbit, is determined by the relative contribution of the ec-centricity gradient and twist to q : ae a − e ( e + ae a ) = q cos E . (8)Additionally it can be useful to rewrite time deriva-tives, following the orbital motion, in terms of the eccentricanomaly: ∂∂t = n (1 − e cos E ) ∂∂E . (9) A local model of a thin, Keplerian eccentric disc was devel-oped in Ogilvie & Barker (2014). In Paper I we developed apurely hydrodynamic model, which included an α -viscosityprescription along with radiative cooling, allowing for contri-butions to the pressure from both radiation and the gas, inthe ( a, E ) coordinate system of Ogilvie & Lynch (2019). Ina similar vein we here develop a local model that allows fora more general treatment of the turbulent/magnetic stress.The equations, formulated in a frame of reference thatfollows the elliptical orbital motion, are the vertical equationof motion, Dv z Dt = − GM • zr − ρ ∂∂z (cid:18) p + 12 M − M zz (cid:19) , (10)the continuity equation, DρDt = − ρ (cid:18) ∆ + ∂v z ∂z (cid:19) , (11)and the thermal energy equation, DpDt = − Γ p (cid:18) ∆ + ∂v z ∂z (cid:19) + (Γ − (cid:18) H − ∂F∂z (cid:19) , (12)where, for horizontally invariant “laminar flows”, DDt = ∂∂t + v z ∂∂z (13)is the Lagrangian time-derivative,∆ = 1 J dJdt (14)is the divergence of the orbital velocity field, which is aknown function of E that depends of e , q and E . F = F rad + F ext is the total vertical heat flux with F rad = − σT κρ ∂T∂z (15)being the vertical radiative heat flux and F ext containing any additional contributions to the heat flux (such as fromconvection or turbulent heat transport). The tensor M ij := B i B j µ (16)is the part of the Maxwell stress tensor arising from magnetictension. This can include contributions from a large scalemean field and from the disc turbulence. Its trace is denoted M = M ii , which corresponds to twice the magnetic pressure.In this paper we shall explore two different closure modelsfor the time-evolution of M ij .Following Paper I, we write the heating rate per unitvolume, resulting from the dissipation of magnetic/turbulentenergy, as H = f H np v , (17)where f H is a dimensionless expression that depends on theclosure model and p v is a pressure to be specified in theMaxwell stress closure model.In addition to the magnetic pressure, which is includedthrough the M term in equation (10), the pressure includescontributions from radiation and a perfect gas with a ratioof specific heats γ . We define the hydrodynamic pressure tobe the sum of the gas and radiation pressure, p = p r + p g = 4 σ c T + R ρTµ , (18)and β r to be the ratio of radiation to gas pressure: β r := p r p g = 4 σµ c R T ρ . (19)We assume a constant opacity law, applicable to theelectron-scattering opacity expected in a TDE, with theopacity denoted by κ .We consider a radiation+gas mixture where F ext is as-sumed to be from convective or turbulent mixing and thefirst and third adiabatic exponents are given by (Chan-drasekhar 1967)Γ = 1 + 12( γ − β r + (1 + 4 β r ) ( γ − β r )(1 + 12( γ − β r ) , (20)Γ = 1 + (1 + 4 β r )( γ − γ − β r . (21)As in Paper I, we propose a separable solution of theform ρ = ˆ ρ ( t )˜ ρ (˜ z ) ,p = ˆ p ( t )˜ p (˜ z ) ,M = ˆ M ij ( t ) ˜ M (˜ z ) ,F = ˆ F ( t ) ˜ F (˜ z ) ,v z = dHdt ˜ z, (22)where˜ z = zH ( t ) (23) MNRAS , 1–20 (2019)
E. M. Lynch and G. I. Ogilvie is a Lagrangian variable that follows the vertical expansionof the disc, H ( t ) is the dynamical vertical scaleheight of thedisc, and the quantities with tildes are normalized variablesthat satisfy a standard dimensionless form of the equationsof vertical structure.In order to preserve separability the (modified) Maxwellstress M ij must have the same vertical structure as the pres-sure ( ˜ M = ˜ p ) . This assumption has a couple of importantconsequences. It corresponds to a plasma- β , defined as theratio of hydrodynamic to magnetic pressure β m := p/p m ,independent of height. Additionally it has implications forthe realisability of M ij : for a large scale field we require M zz = 0 in order that the underlying magnetic field obeysthe solenoidal condition. For small scale/turbulent fields thesolenoidal condition instead implies the mean of B z is inde-pendent of height; however M zz has the same vertical struc-ture as pressure.The separated solution works provided that d Hdt = − GM • r H + ˆ p ˆ ρH (cid:32) M p − ˆ M zz ˆ p (cid:33) , (24) d ˆ ρdt = − ˆ ρ (cid:18) ∆ + 1 H dHdt (cid:19) , (25) d ˆ pdt = − Γ ˆ p (cid:18) ∆ + 1 H dHdt (cid:19) +(Γ − (cid:32) f H n ˆ p v − λ ˆ FH (cid:33) , (26)ˆ F = 16 σ ˆ T κ ˆ ρH , (27)ˆ p = (1 + β r ) R ˆ ρ ˆ Tµ , (28)where the positive constant λ is a dimensionless cooling ratethat depends on the equations of vertical structure (furtherdetails can be found in Paper I) and β r = 4 σµ c R ˆ T ˆ ρ . (29)We must supplement these equations with a closure modelfor M ij and f H .Note that the surface density and vertically integratedpressures are (owing to the definitions of the scaleheight andthe dimensionless variables)Σ = ˆ ρH, P = ˆ pH, P v = ˆ p v H. (30)The vertically integrated heating and cooling rates are f H nP v , λ ˆ F . (31)The cooling rate can also be written as λ ˆ F = 2 σ ˆ T s (32) We can have an additional height independent contribution to M ij (e.g. coming from a height-independent vertical magneticfield), but this has no effect on the dynamics. where ˆ T s ( t ) is a representative surface temperature definedbyˆ T s = 8 λ T ˆ τ (33)andˆ τ = κ Σ (34)is a representative optical thickness.We then have1
H d Hdt = − GM • r + P Σ H (cid:32) M p − ˆ M zz p (cid:33) , (35) J Σ = constant , (36) (cid:18) − (cid:19) dPdt = − Γ P Γ − (cid:18) ∆ + 1 H dHdt (cid:19) + f H nP v − λ ˆ F , (37)with λ ˆ FP n = λ σ ( µ/ R ) κn P Σ − (1 + β r ) − . (38)We assume for a given β ◦ m , β ◦ r and n there exists anequilibrium solution for a circular disc and use this solutionto nondimensionalise the equations. As in the hydrodynam-ical models considered in Paper I, we use ◦ to denote theequilibrium values in the reference circular disc (e.g. H ◦ , T ◦ etc). Depending on the closure model there can be multipleequilibrium solutions, some of which can be unstable (par-ticularly in the radiation dominated limit). Our choices ofsolution branch for our two closure models are specified inAppendices B and D.Scaling H by H ◦ , ˆ T by T ◦ , M ij by p ◦ , t by 1 /n and J by 2 /n we obtain the dimensionless version¨ HH = − (1 − e cos E ) − + TH β r β ◦ r (cid:32) Mp − M zz p (cid:33)(cid:104) M ◦ p ◦ − ( M zz ) ◦ p ◦ (cid:105) , (39)˙ T = − (Γ − T (cid:18) ˙ JJ + ˙ HH (cid:19) + (Γ −
1) 1 + β r β r T (cid:18) f H P v P − C ◦ β ◦ r β r J T (cid:19) , (40)where a dot over a letter indicates a derivative with respectto rescaled time. We have written the thermal energy equa-tion in terms of the temperature. The factor Γ − β ∝ c V where c V is the specific heat capacity at constant volume. β r can be obtained through β r = β ◦ r JHT , (41) MNRAS000
1) 1 + β r β r T (cid:18) f H P v P − C ◦ β ◦ r β r J T (cid:19) , (40)where a dot over a letter indicates a derivative with respectto rescaled time. We have written the thermal energy equa-tion in terms of the temperature. The factor Γ − β ∝ c V where c V is the specific heat capacity at constant volume. β r can be obtained through β r = β ◦ r JHT , (41) MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs where we have introduced β ◦ r , which is the β r of the refer-ence circular disc. The equilibrium values of the referencecircular disc H ◦ , ˆ T ◦ , etc., are determined by β ◦ r and n . Thereference cooling rate can be obtained by setting it equal tothe reference heating rate: C ◦ = f ◦H P ◦ v P ◦ .Additionally we introduce the (nondimensional) en-tropy, s := 4 β r + ln( JHT / ( γ − ) , (42)which has contributions from the radiation and the gas. In Paper I we found that (when p v = p g ) our radiationdominated solutions exhibit extreme compression at peri-centre, similar to the extreme behaviour of the adiabaticsolutions of Ogilvie & Barker (2014). Many of our solutionswith more moderate behaviour have strong viscous stressesat pericentre which call into question the validity of the α − prescription.What additional physical processes could reverse thecollapse of the fluid column and prevent the extreme com-pression seen in the radiation dominated model? Can the col-lapse be reversed without encountering unphysically strongviscous stresses? An obvious possibility is the presence of alarge scale horizontal magnetic field within the disc whichwill resist vertical compression. Such a field could be weakfor the majority of the orbit but, owing to the relative in-compressibility of magnetic fields, become dynamically im-portant during the maximum compression at pericentre. InAppendix E we show that in an eccentric disc, a solution tothe steady ideal induction equation in an inertial frame is B a = 0 , B E = Ω B E ( a, ˜ z ) nJH , B z = B z ( a ) J . (43)Here B E is the component parallel to the orbital motion(quasi-toroidal) and B z is the vertical component. We usequasi-poloidal to indicate the components B a and B z .The magnetic field of a star undergoing tidal disrup-tion has been studied by Guillochon & McCourt (2017) andBonnerot et al. (2017). In these papers it was found thatthe stretching of the fields during the disruption causes anincrease in the magnetic pressure from the field aligned withthe orbital direction. Meanwhile the gas pressure and mag-netic pressure from the field perpendicular to the orbit drop.Guillochon & McCourt (2017) found that this tends to re-sult in the magnetic pressure from the parallel field becomingcomparable to the gas pressure. Similar results were foundin Bonnerot et al. (2017), although with a dependence onthe initial field direction. This supports our adopted fieldconfiguration, with the vertical field set to zero. As the ver-tical field does not contribute to the dynamics of the verticaloscillator we can do so without loss of generality.In addition to the large scale magnetic field, we assumethat the effects of the small-scale/turbulent magnetic fieldcan be modelled by an α -viscosity, µ s,b = α s,b p v ω orb , (44) where µ s,b are the dynamic shear and bulk viscosities, α s,b are dimensionless coefficients, ω orb is some characteristic fre-quency of the orbital motion (here taken to be n ) and p v issome choice of pressure. As in Paper I we set the bulk vis-cosity to zero ( α b = 0).As discussed in Section 3, in order to preserve separa-bility of the equations we require B E ( a, ˜ z ) to depend on ˜ z in such a way as to make β m independent of height. The di-mensionless equations for the variation of the dimensionlessscale height H and temperature T around the orbit (derivedin Appendix E) are then¨ HH = − (1 − e cos E ) − + TH β r β ◦ r (cid:18) β ◦ m (cid:19) − × (cid:34) β m − α s P v P ˙ HH − (cid:18) α b − α s (cid:19) P v P (cid:18) ˙ JJ + ˙ HH (cid:19)(cid:35) , (45)˙ T = − (Γ − T (cid:18) ˙ JJ + ˙ HH (cid:19) + (Γ −
1) 1 + β r β r T (cid:18) f H P v P − α s P ◦ v P ◦ β ◦ r β r J T (cid:19) , (46)and the plasma- β is given by β m = β ◦ m JHT β r β ◦ r − e cos E e cos E (47)where β ◦ m is the plasma beta in the reference circular disc.These equations can be solved using the same relaxationmethod used to solve the purely hydrodynamic equations inPaper I. However caution must be taken when solving theequations with low β ◦ m (i.e. strong magnetic fields through-out the disc) as the method does not always converge to aperiodic solution (or at least takes an excessively long timeto do so). This is most likely due to the absence of dissipativeeffects acting on the magnetic field, so any free oscillationsin the magnetic field are not easily damped out. We believethat the quasiperiodic solutions we find for low β ◦ m are thesuperposition of the forced solution and a free fast magne-tosonic mode. For now we only consider values of β ◦ m whichsuccessfully converge to a periodic solution. p v = p g ) Figures 1-3 show the variations of the scale height, β r and β m around the orbit for a disc with α s = 0 . α b = 0, e = q = 0 . E = 0. The magnetic field has a weak effect on the gaspressure dominated ( β ◦ r = 10 − ) solutions. For the radiationpressure dominated ( β ◦ r = 10 − ) case, a strong enough mag-netic field stabilises the solution against the thermal instabil-ity and, instead of the nearly adiabatic radiation dominatedsolutions seen in the hydrodynamic case, the solution is onlymoderately radiation pressure dominated and maintains sig-nificant entropy variation around the orbit. This solution issimilar to the moderately radiation pressure dominated hy-drodynamic solutions. If the field is too weak (e.g. β ◦ m = 100 MNRAS , 1–20 (2019)
E. M. Lynch and G. I. Ogilvie
Log r L o g H r =0.0001 m =100.0 r =0.0001 m =10.0 r =0.001 m =100.0 r =0.001 m =10.0 Figure 1.
Variation of the scale height of the disc with radiation+ gas pressure with different β ◦ r and magnetic fields. Disc param-eters are p v = p g , α s = 0 . α b = 0, e = q = 0 . E = 0. Redline has β ◦ r = 10 − and β ◦ m = 100, blue line has β ◦ r = 10 − and β ◦ m = 10, grey line has β ◦ r = 10 − and β ◦ m = 100, green line has β ◦ r = 10 − and β ◦ m = 10. The discs with β ◦ m = 100 are nearlyindistinguishable from an unmagnetised disc. considered here) the magnetic field isn’t capable of stabilis-ing against the thermal instability and the solution tends tothe nearly adiabatic radiation dominated solution.Most of the solutions in Figures 1-3 are not sufficientlyradiation pressure dominated to represent most TDEs. Fig-ures 4-6 show solutions which attain much higher β r . We seeit is possible to attain significantly radiation pressure dom-inated solutions which do not possess the extreme variationof the scale height around the orbit present in the hydro-dynamic case. In particular, consider the green curve with β ◦ r = 1, β ◦ m = 0 . β ◦ r = 1, β ◦ m = 0 . β ◦ r =1, β ◦ m = 1) in Figures 4-6 has not converged. The magneticfield is unimportant for this solution. Based on the radiationdominated hydrodynamic models of Paper I, the disc with β ◦ r = 1 will converge on a solution with β r much largerthan the β r ∼ − which were the most radiationdominated, converged, solutions obtained in Paper I. As theentropy gained per orbit is small compared to the entropyin the disc, this will take a large number of orbits ( > p v = p g , β ◦ r = 1, β ◦ m = 0 . α s = 0 . α b = 0, e = q = 0 . E = 0(i.e. the green solution from Figures 4-6). This shows thatthe dominant balance in this solution is between the verti-cal acceleration, gravity and the magnetic force. This sug- Log r L o g r r = 0.0001 m =100.0 r = 0.0001 m =10.0 r = 0.001 m =100.0 r = 0.001 m =10.0 Figure 2.
Variation of the ratio of radiation to gas pressure ( β r )around around the orbit for each model in Figure 1. Log r L o g m r = 0.0001 m =100.0 r = 0.0001 m =10.0 r = 0.001 m =100.0 r = 0.001 m =10.0 Figure 3.
Variation of the plasma- β around the orbit for eachmodel in Figure 1. Log r L o g H r =0.001 m =10.0 r =0.01 m =0.2 r =1.0 m =1.0 r =1.0 m =0.005 Figure 4.
Same as Figure 1 but attaining larger β r . Red line has β ◦ r = 10 − and β ◦ m = 10, blue line has β ◦ r = 10 − and β ◦ m = 0 . β ◦ r = 1 and β ◦ m = 1, green line has β ◦ r = 1 and β ◦ m = 0 .000
Same as Figure 1 but attaining larger β r . Red line has β ◦ r = 10 − and β ◦ m = 10, blue line has β ◦ r = 10 − and β ◦ m = 0 . β ◦ r = 1 and β ◦ m = 1, green line has β ◦ r = 1 and β ◦ m = 0 .000 , 1–20 (2019) agnetic fields in eccentric TDE Discs Log r L o g r r = 0.001 m =10.0 r = 0.01 m =0.2 r = 1.0 m =1.0 r = 1.0 m =0.005 Figure 5.
Variation of β r around the orbit for each model in Figure4. Log r L o g m r = 0.001 m =10.0 r = 0.01 m =0.2 r = 1.0 m =1.0 r = 1.0 m =0.005 Figure 6.
Variation of the plasma- β around the orbit for eachmodel in Figure 4. gests that dynamics of radiation pressure dominated TDEsmay be controlled by the magnetic field. Being the leastcompressible pressure term, the magnetic pressure tends todominate at pericentre, even if it is fairly weak throughoutthe rest of the disc. However for radiation dominated TDEspressure is only important near pericentre so even a weakmagnetic field will have a disproportionate contribution tothe dynamics. This suggests that ignoring even subdomi-nant magnetic fields in TDE discs can lead to fundamentalchanges to the TDE dynamics.While it is possible to find a combination of β ◦ r and β ◦ m which yields a solution with the desired β r exhibiting “rea-sonable” behaviour, it is not clear that the magnetic fieldin the disc will always be strong enough to produce the de-sired behaviour. It is possible that this represents a tuningproblem for β ◦ m .To explore this we look at what happens if β ◦ m is ini-tially too weak to stabilise against the thermal instabilitybut we gradually raise it over several thermal times. Figure8 shows what happens when the magnetic field is increased Eccentric Anomaly a / r ) Log TH r r Logp m Log Viscous StressLog | HH | Figure 7.
Magnitude of the different terms in the vertical momen-tum equation for a magnetised radiation-gas mixture; the blackline is the disc gravity, the blue is the hydrodynamic pressure, thecyan is the magnetic pressure, green is the viscous stress and redis the vertical momentum. Disc parameters are p v = p g , β ◦ r = 1, β ◦ m = 0 . α s = 0 . α b = 0, e = q = 0 . E = 0. Thebalance at pericentre is now between the gravity, the magneticpressure and the vertical acceleration. gradually from β ◦ m = 100 to β ◦ m = 10 for a disc with p v = p g , β ◦ r = 10 − , α s = 0 . α b = 0, e = q = 0 . E = 0. Thiscorresponds to moving from the grey to the green solutionin Figures 1-3. This is done by periodically stopping the cal-culation and restarting with a larger β ◦ m The resulting β r in fact increases with time and remains close to that of thegrey solution in Figures 1-3 even as we increase the magneticfield strength, and does not transition to a value consistentwith the green solution. This suggests that the solution issensitive to the path taken and that a magnetic field whichgrows (from an initially weak seed field), in a nearly adia-batic radiation pressure dominated disc, may not cause thedisc to collapse to the gas pressure dominated branch. Thisis likely because the disc is very expanded meaning the mag-netic field is still quite weak and incapable of influencing thedynamics.We carried out a similar calculation for a disc with p v = p g , β ◦ r = 1, α s = 0 . α b = 0, e = q = 0 . E = 0 moving from β ◦ m = 1 to β ◦ m = 5 × − (correspond-ing to the grey and green solutions of Figures 4-6). In thiscase β r steadily increases with time (apart from a small vari-ation over the orbital period) with the magnetic field havingno appreciable influence on the solution. Owing to the rel-atively large β ◦ r this solution never reached steady state,as discussed previously. The implication of these two testsis that radiation pressure dominated, magnetised, discs canhave two stable solution branches, with the choice of branchdetermined by the magnetic field history.Figure 9 shows the pericentre passage for a magnetiseddisc with p v = p g , β ◦ r = 10 − , β ◦ m = 10, α s = 0 . α b = 0, e = q = 0 . E = 0. The magnetic pressure is extremelyconcentrated within the nozzle and near to the midplane.Like the hydrodynamic nozzle structure considered in PaperI, the nozzle is asymmetric and located prior to pericentre,which is appears to be characteristic of dissipative highlyeccentric discs. MNRAS , 1–20 (2019)
E. M. Lynch and G. I. Ogilvie time L o g r Figure 8. β r for a disc where the magnetic field strength is grad-ually increased from β ◦ m = 100 to β ◦ m = 10. Disc parametersare p v = p g , β ◦ r = 10 − , α s = 0 . α b = 0, e = q = 0 . E = 0. Colours indicate where we have stopped the calculationand restarted with a different magnetic field strength. The result-ing solution remains in the nearly adiabatic radiation pressuredominated state and doesn’t converge on the green solution ofFigures 1-3. The solution with the final magnetic field strength( β ◦ m = 10) was run for longer to allow it to relax to a steady state. Figure 9.
Pericentre passage for a magnetised disc with p v = p g , β ◦ r = 10 − , β ◦ m = 10, α s = 0 . α b = 0, e = q = 0 . E = 0 showing the magnetic pressure scaled by the maximumhydrodynamic pressure. The magnetic field is highly concentratedin the nozzle. An unmagnetised disc, with the same parameters,is thermally unstable and would be considerably thicker. Begelman & Pringle (2007) have suggested that discs withstrong toroidal fields may be stable to the thermal instabil-ity if the stress depends on the magnetic pressure. In thissubsection we explore this possibility for a highly eccentricdisc.Figures 10-12 show variation of the scale height, β r andplasma beta for a disc with p v = p m , α s = 0 . α b = 0, e = q = 0 . E = 0. These have essentially the same be-haviour as the nearly adiabatic radiation pressure dominateddiscs for the hydrodynamic case. This is not surprising as inthis limit the gas and magnetic pressures are essentially neg- Log r L o g H r =0.001 m =1.0 r =0.01 m =0.1 r =1.0 m =1.0 r =1.0 m =0.1 Figure 10.
Variation of the scale height of the disc when the viscousstress is proportional to the magnetic pressure. Disc parametersare p v = p m , α s = 0 . α b =, e = q = 0 . E = 0. ligible, which also results in negligible viscous stress/heatingwhen it scales with either of these pressures. Increasing themagnetic field strength stabilises the “gas pressure domi-nated” branch, where the magnetic field and viscous dissi-pation become important. This branch can have p r (cid:29) p g around the entire orbit; this is similar to the behaviour ofthe radiation pressure dominated hydrodynamic discs con-sidered in Paper I with large α s .Figures 13-15 show the variation of the scale height, β r and β m for a disc with p v = p + p m , α s = 0 . α b = 0, e = q = 0 . E = 0. Here we find that, with a strongenough magnetic field, we can obtain thermally stable solu-tions despite the dependence of the stress on the radiationpressure. Generally for thermal stability the magnetic fieldneeds to dominate (over radiation pressure) over part of theorbit. Having such a strong horizontal magnetic field overa sizable fraction of the orbit may lead to flux expulsionthrough magnetic buoyancy, an effect we do not treat here.If the magnetic field is too weak, however, we encounter thethermal instability similar to the hydrodynamic radiationpressure dominated discs when p v = p .Part of the motivation for introducing the magnetic fieldwas to regularise some of the extreme behaviour encoun-tered at pericentre. Unfortunately, while the prescriptions p v = p m and p v = p + p m are promising as a way of tamingthe thermal instability they exhibit the same extreme be-haviour that the hydrodynamic models possess. In particu-lar when p v = p m the solutions exhibit extreme compressionat pericentre, while for the more magnetised discs (with ei-ther p v = p m or p v = p + p m ) we again encounter the issueof the viscous stresses being comparable to or exceeding thepressure (including the magnetic pressure). See, for exam-ple, Figure 16 which shows that the viscous stresses exceedthe magnetic, gas and radiation pressures during pericentrepassage. MNRAS000
Variation of the scale height of the disc when the viscousstress is proportional to the magnetic pressure. Disc parametersare p v = p m , α s = 0 . α b =, e = q = 0 . E = 0. ligible, which also results in negligible viscous stress/heatingwhen it scales with either of these pressures. Increasing themagnetic field strength stabilises the “gas pressure domi-nated” branch, where the magnetic field and viscous dissi-pation become important. This branch can have p r (cid:29) p g around the entire orbit; this is similar to the behaviour ofthe radiation pressure dominated hydrodynamic discs con-sidered in Paper I with large α s .Figures 13-15 show the variation of the scale height, β r and β m for a disc with p v = p + p m , α s = 0 . α b = 0, e = q = 0 . E = 0. Here we find that, with a strongenough magnetic field, we can obtain thermally stable solu-tions despite the dependence of the stress on the radiationpressure. Generally for thermal stability the magnetic fieldneeds to dominate (over radiation pressure) over part of theorbit. Having such a strong horizontal magnetic field overa sizable fraction of the orbit may lead to flux expulsionthrough magnetic buoyancy, an effect we do not treat here.If the magnetic field is too weak, however, we encounter thethermal instability similar to the hydrodynamic radiationpressure dominated discs when p v = p .Part of the motivation for introducing the magnetic fieldwas to regularise some of the extreme behaviour encoun-tered at pericentre. Unfortunately, while the prescriptions p v = p m and p v = p + p m are promising as a way of tamingthe thermal instability they exhibit the same extreme be-haviour that the hydrodynamic models possess. In particu-lar when p v = p m the solutions exhibit extreme compressionat pericentre, while for the more magnetised discs (with ei-ther p v = p m or p v = p + p m ) we again encounter the issueof the viscous stresses being comparable to or exceeding thepressure (including the magnetic pressure). See, for exam-ple, Figure 16 which shows that the viscous stresses exceedthe magnetic, gas and radiation pressures during pericentrepassage. MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs Log r L o g r r = 0.001 m =1.0 r = 0.01 m =0.1 r = 1.0 m =1.0 r = 1.0 m =0.1 Figure 11.
Variation of β r around the orbit for each model inFigure 10. Log r L o g m r = 0.001 m = 1.0 r = 0.01 m = 0.1 r = 1.0 m = 1.0 r = 1.0 m = 0.1 Figure 12.
Variation of the plasma- β around the orbit for eachmodel in Figure 10. The model considered in Section 4 has a number of draw-backs. The first is that the viscous stress and the coherentmagnetic field are treated as separate physical effects whenthey are in fact intrinsically linked (although subsection 4.3partially addresses this issue). Secondly, the turbulent mag-netic field, responsible for the effective viscosity, cannot storeenergy. Lastly the model neglects resistive effects and, whilenonideal MHD effects would be weak if the flow were strictlylaminar, the turbulent cascade should always move magneticenergy to scales on which nonideal effects become important.Thus the coherent magnetic field should be affected by somedissipative process.To address these issues we consider a model of the (mod-ified) Maxwell stress where the magnetic field is forced by aturbulent emf and relaxes to a isotropic field proportional tosome pressure p v on a timescale τ . While the “turbulence”in this model acts to isotropise the magnetic field, the pres- Log r L o g H r =0.001 m =0.1 r =0.01 m =0.01 r =1.0 m =0.0001 r =1.0 m =1e-05 Figure 13.
Variation of the scale height of the disc when the vis-cous stress is proportional to the total gas+radiation+magneticpressure. Disc parameters are p v = p + p m , α s = 0 . α b = 0, e = q = 0 . E = 0. This confirms that a strong magneticfield can stabilise the thermal instability in an eccentric disc if p v includes the magnetic pressure. Log r L o g r r = 0.001 m =0.1 r = 0.01 m =0.01 r = 1.0 m =0.0001 r = 1.0 m =1e-05 Figure 14.
Variation of β r around the orbit for each model inFigure 13. Log r L o g m r = 0.001 m =0.1 r = 0.01 m =0.01 r = 1.0 m =0.0001 r = 1.0 m =1e-05 Figure 15.
Variation of the plasma- β around the orbit for eachmodel in Figure 13.MNRAS , 1–20 (2019) E. M. Lynch and G. I. Ogilvie
Eccentric Anomaly a / r ) Log TH r r Logp m Log Viscous StressLog | HH | Figure 16.
Magnitude of terms for a disc with p v = p + p m , β ◦ r =10 − , β ◦ m = 0 . α s = 0 . α b = 0, e = q = 0 . E = 0. Theviscous stress exceeds the magnetic, gas and radiation pressuresin the nozzle, during pericentre passage. ence of the background shear flow feeds off the quasi-radialfield component and produces a highly anisotropic field thatis predominantly quasi-toroidal. A possible justification forthis model based on a stochastically forced induction equa-tion is given in Appendix E. This model has much in com-mon with Ogilvie (2003), but does not solve for the Reynoldsstress explicitly.The Maxwell stress in this model evolves according to D M ij = − ( M ij − B p v g ij ) /τ , (48)where g ij is the metric tensor and B is a nondimensionalparameter controlling the strength of forcing relative to p v . B can can be taken to be constant by absorbing any varia-tion into the definition of p v . D is the operator from Ogilvie(2001) (a type of weighted Lie derivative) which acts on arank (2,0) tensor by D M ij = DM ij − M k ( i ∇ k u j ) + 2 M ij ∇ k u k . (49)As noted in Ogilvie (2001), D M ij = 0 is the equation for theevolution of the (modified) Maxwell stress for a magneticfield which satisfies the ideal induction equation; it statesthat the magnetic stress is frozen into the fluid.We adopt the following prescription for the relaxationtime: τ = De z (cid:114) p v M , (50)where Ω z = (cid:112) GM • /r is the vertical oscillation frequencyand De is a dimensionless constant; this matches the func-tional form for the relaxation time τ given in the compress-ible version of Ogilvie (2003). In subsequent equations it willbe useful to express this relaxation time as a Deborah num-ber De = nτ , a dimensionless number used in viscoelasticfluids that is the ratio of the relaxation time to some char-acteristic timescale of the flow. When τ is given by Equa-tion 50 then Equation 48 corresponds to the equation for the (modified)-Maxwell stress given in Ogilvie (2003) if theReynolds stress is isotropic and proportional to some pres-sure p v . One emergent property of such a stress model isthat the stress will naturally scale with magnetic pressure,as the latter is the trace of the former (see Appendix A).From this stress model, we have a nondimensional heat-ing rate, f H = 12De (cid:18) Mp v − B (cid:19) , (51)which ensures that magnetic energy loss/gained via the re-laxation terms in Equation 48 is converted to/from the ther-mal energy (this is shown in Appendix B). Thus energy isconserved within the disc, although it can be lost radiativelyfrom the disc surface.In Appendix B we obtain the hydrostatic solutions for acircular disc. If p v is independent of M the vertical equationof motion, rescaled by this reference circular disc, is¨ HH = − (1 − e cos E ) − + TH β r β ◦ r (cid:32) Mp − M zz p (cid:33)(cid:104) B P ◦ v P ◦ (cid:16) + De P ◦ v P ◦ (cid:17)(cid:105) , (52)while the thermal energy equation is˙ T = − (Γ − T (cid:18) ˙ JJ + ˙ HH (cid:19) + (Γ −
1) 1 + β r β r T (cid:20) (cid:18) Mp − B p v p (cid:19) − C ◦ β ◦ r β r J T (cid:21) , (53)where we have introduced a reference cooling rate, C ◦ = 94 B De ◦ P ◦ v P ◦ = (cid:18) (cid:19) / B / De (cid:32) (cid:114) B (cid:33) − / P ◦ v P ◦ . (54)Here De ◦ is the equilibrium Deborah number in the referencecircular disc, which is in general different from De .We solve these equations along with the equations forthe evolution of the stress components,˙ M λλ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M λλ = − ( M λλ − B p v g λλ ) / De , (55)˙ M λφ − M λλ Ω λ − M λφ Ω φ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M λφ = − ( M λφ − B p v g λφ ) / De , (56)˙ M φφ − M λφ Ω λ − M φφ Ω φ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M φφ = − ( M φφ − B p v g φφ ) / De , (57)˙ M zz + 2 ˙ JJ M zz = − ( M zz − B p v g zz ) / De . (58)We solve for these stress components in the ( λ, φ ) coordinatesystem of Ogilvie (2001) as this simplifies the metric tensor. MNRAS000
1) 1 + β r β r T (cid:20) (cid:18) Mp − B p v p (cid:19) − C ◦ β ◦ r β r J T (cid:21) , (53)where we have introduced a reference cooling rate, C ◦ = 94 B De ◦ P ◦ v P ◦ = (cid:18) (cid:19) / B / De (cid:32) (cid:114) B (cid:33) − / P ◦ v P ◦ . (54)Here De ◦ is the equilibrium Deborah number in the referencecircular disc, which is in general different from De .We solve these equations along with the equations forthe evolution of the stress components,˙ M λλ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M λλ = − ( M λλ − B p v g λλ ) / De , (55)˙ M λφ − M λλ Ω λ − M λφ Ω φ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M λφ = − ( M λφ − B p v g λφ ) / De , (56)˙ M φφ − M λφ Ω λ − M φφ Ω φ + 2 (cid:18) ˙ JJ + ˙ HH (cid:19) M φφ = − ( M φφ − B p v g φφ ) / De , (57)˙ M zz + 2 ˙ JJ M zz = − ( M zz − B p v g zz ) / De . (58)We solve for these stress components in the ( λ, φ ) coordinatesystem of Ogilvie (2001) as this simplifies the metric tensor. MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs Log r L o g H r =0.0001, hydro r =0.0001 r =0.001, hydro r =0.001 r =0.1 Figure 17.
Variation of the scale height of the disc with radiation +gas pressure with different β ◦ r using our modified Maxwell stressprescription. Disc parameters are e = q = 0 . E = 0 and p v = p g ; α -discs have α s = 0 . α b = 0 while the Maxwell stressprescription has De = 0 . B = 0 .
1. Black line is an α -disc with β ◦ r = 10 − ; red line has β r = 10 − with the Maxwell stressprescription, magenta line is an α -disc with β ◦ t = 10 − , grey linehas β r = 10 − with the Maxwell stress prescription, green linehas β r = 10 − with the Maxwell stress prescription. Log r L o g r r = 0.0001, hydro r = 0.0001 r = 0.001, hydro r = 0.001 r = 0.1 Figure 18.
Variation of β r around the orbit for each model inFigure 17. We can do this as, apart from M zz (which is the same inboth coordinate systems), our equations only depend on M ij through scalar quantities.Figures 17-19 show the variations of the scale height, β r and plasma beta (defined as β m = pM ) around the orbit fora disc with p v = p g , De = 0 . B = 0 . e = q = 0 . E = 0. Like the ideal induction equation model of Section3, the coherent magnetic field has a stabilising effect on thedynamics. The effect is not as strong as that seen in the idealinduction equation model as, in that model, we could choose β ◦ m so as to achieve a much stronger field than achieved bythe constitutive model here. Compared with the ideal induc-tion equation model the plasma- β is more uniform aroundthe orbit; there is still an abrupt decrease in the plasma- β near pericentre, which highlights the importance of themagnetic field during pericentre passage. Log r L o g m r = 0.0001 r = 0.001 r = 0.1 Figure 19.
Variation of the plasma- β around the orbit for eachmodel in Figure 17. Figure 20.
Pericentre passage for a disc using our modifiedMaxwell stress prescription, with p v = p g , β ◦ r = 10 − , De = 0 . B = 0 . e = q = 0 . E = 0 showing the magnetic pressure( M/ Figure 20 shows the pericentre passage for a disc with p v = p g , β ◦ r = 10 − , De = 0 . B = 0 . e = q = 0 . E = 0. As with the ideal induction equation model,the magnetic pressure is extremely concentrated within thenozzle and near to the midplane. The nozzle is far moresymmetric compared to the ideal induction equation modelas the weaker field means that the disc is in a modified formof the nearly adiabatic radiation pressure dominated state.In addition to considering the situation where the fluc-tuation pressure scales with the gas pressure p v = p g , wealso considered p v = p + p m . As in the ideal induction equa-tion model we found it is possible to stabilise the thermalinstability with a strong enough magnetic field; however wefound that this requires fine tuning of De and B , for whichthere is no obvious justification. However, instead of stabil-ising the thermal instability, it is possible to delay its onsetby choosing a small enough De , so that the thermal run-away occurs on a timescale much longer than the orbitaltime (occurring after ∼ MNRAS , 1–20 (2019) E. M. Lynch and G. I. Ogilvie solution and instead have a long phase of quasi-periodic evo-lution, where the mean scale height remains close to its ini-tial value, before eventually experiencing thermal runaway.A quasi-periodic solution of our model is not self consistent,so the possibility that the thermal instability is delayed inthe nonlinear-constitutive MRI model needs to be exploredusing an alternative method.The possibility that the thermal instability stalls or isdelayed has some support from simulations looking at thethermal stability of MRI active discs (Jiang et al. 2013;Ross et al. 2017). In both these papers it was found thatthe disc was quasi-stable, with thermal instability occur-ring when a particularly large turbulent fluctuation causeda strong enough perturbation away from the equilibrium.The (modified) Maxwell stress considered here is equal tothe expectation value of a modified Maxwell stress which isstochastically forced by fluctuation with amplitude propor-tional to p v (see Equation E5 of Appendix E and discussiontherein), so it is possible that our stress model captures thethermal quasi-stability seen in Jiang et al. (2013) and Rosset al. (2017) in some averaged sense. The possibility that thethermal instability is delayed or slowed is particularly rele-vant for TDEs which are inherently transient phenomena –if the timescale for thermal runaway is made long enoughthen eccentric TDE discs maybe thermally stable over thelifetime of the disc. As discussed in Paper I, one advantage of our solutionmethod is that the solutions it finds are typically nonlin-ear attractors (or at least long lived transients) and so arestable against (nonlinear) perturbations to the solution vari-ables ( H , ˙ H , T and M ij when present). Generally we expectsuch perturbations to damp on the thermal time or faster.Instabilities such as the thermal instability manifest as afailure to converge to a 2 π − periodic solution.For the ideal induction equation model, our methodcannot tell us about the stability of the solution to pertur-bations to the horizontal magnetic field. Showing this wouldrequire a separate linear stability analysis. Perturbations tothe vertical field typically have no influence on the dynamicsof the disc vertical structure.However, for the constitutive model, perturbations tothe magnetic field are encapsulated in perturbations to M ij so these solutions are stable against (large scale) pertur-bations to the magnetic field. This is most likely because,unlike the ideal induction equation, dissipation acts on themagnetic field.Our solution method doesn’t tell us about the stabilityof our solutions to short wavelength (comparable or less thanthe scale height) perturbations to our system. So our discstructure could be unstable to such perturbations. Like thehydrodynamic solutions in Paper I, it is likely our discs areunstable to the parametric instability (Papaloizou 2005a,b;Wienkers & Ogilvie 2018; Barker & Ogilvie 2014; Pierenset al. 2020). Additionally if, as assumed, turbulence in highlyeccentric discs is caused by the MRI then there must beperturbations to the magnetic field in the ideal inductionequation model which are unstable. Interestingly the simulations of S (cid:44) adowski (2016) foundthat the strength of turbulence in magnetised and unmag-netised TDE discs is broadly comparable, something thatwould not be expected in a circular disc. S (cid:44) adowski (2016)suggested that a hydrodynamic instability might be respon-sible for the disc turbulence. The discs considered by S (cid:44) a-dowski (2016) still have appreciable eccentricity at the endof their simulation (with e ≈ .
2) so an obvious contenderwould be the parametric instability feeding off the disc ec-centricity and breathing mode.
Even when the magnetic field in our models does a goodjob of resisting the collapse of the disc, the stream will stillbe highly compressed at pericentre. The highly compressedflow combined with a very strong field (with β m (cid:28)
1) makesthe nozzle a prime site for magnetic reconnection. This willrequire that magnetic field lines on neighbouring orbits canhave opposite polarities. Our solutions are agnostic to themagnetic field polarity and, in principle, support this possi-bility.The simulations of Guillochon & McCourt (2017) sug-gest that the initial magnetic field in the disc will be (quasi-)toroidal with periodic reversals in direction. When sucha field is compressed both horizontally and vertically dur-ing pericentre passage, neighbouring toroidal magnetic fieldlines of opposite polarity can undergo reconnection, gener-ating a quasi-poloidal field. We thus have a basis for aneccentric α − Ω dynamo, where strong reconnection in thenozzle creates quasi-poloidal field, which in turn creates asource term in the quasi-toroidal induction equation fromthe shearing of this quasi-radial field (see Appendix C). Inhighly eccentric TDE discs, with their large vertical and hor-izontal compression, this dynamo could potentially be quitestrong.The constitutive model we considered in Section 5 im-plicitly possesses a dynamo through the source term propor-tional to p v . This is a small scale dynamo which arises fromthe action of the turbulent velocity field and will be limitedby the kinetic energy in the turbulence. A dynamo closed byreconnection in the nozzle would be a large scale dynamoand the coherent field produced could potentially greatlyexceed equipartition with the turbulent velocity field.Lastly reconnection sites can accelerate charged parti-cles. We would therefore expect that TDEs in which there isa line of sight to the pericentre should include a flux of high-energy particles. From most look angles the line of sight tothe pericentre“bright point”will be blocked (see also Zanazzi& Ogilvie (2020)). This will both block the X-ray flux fromthe disc and any particles accelerated by reconnection in thenozzle. Hence these high energy particles should only be seenin X-ray bright TDEs. In this paper we considered alternative models of theMaxwell stress from the standard α − prescription appliedto highly eccentric TDE discs. In particular we focus on theeffects of the coherent magnetic field on the dynamics. We MNRAS000
1) makesthe nozzle a prime site for magnetic reconnection. This willrequire that magnetic field lines on neighbouring orbits canhave opposite polarities. Our solutions are agnostic to themagnetic field polarity and, in principle, support this possi-bility.The simulations of Guillochon & McCourt (2017) sug-gest that the initial magnetic field in the disc will be (quasi-)toroidal with periodic reversals in direction. When sucha field is compressed both horizontally and vertically dur-ing pericentre passage, neighbouring toroidal magnetic fieldlines of opposite polarity can undergo reconnection, gener-ating a quasi-poloidal field. We thus have a basis for aneccentric α − Ω dynamo, where strong reconnection in thenozzle creates quasi-poloidal field, which in turn creates asource term in the quasi-toroidal induction equation fromthe shearing of this quasi-radial field (see Appendix C). Inhighly eccentric TDE discs, with their large vertical and hor-izontal compression, this dynamo could potentially be quitestrong.The constitutive model we considered in Section 5 im-plicitly possesses a dynamo through the source term propor-tional to p v . This is a small scale dynamo which arises fromthe action of the turbulent velocity field and will be limitedby the kinetic energy in the turbulence. A dynamo closed byreconnection in the nozzle would be a large scale dynamoand the coherent field produced could potentially greatlyexceed equipartition with the turbulent velocity field.Lastly reconnection sites can accelerate charged parti-cles. We would therefore expect that TDEs in which there isa line of sight to the pericentre should include a flux of high-energy particles. From most look angles the line of sight tothe pericentre“bright point”will be blocked (see also Zanazzi& Ogilvie (2020)). This will both block the X-ray flux fromthe disc and any particles accelerated by reconnection in thenozzle. Hence these high energy particles should only be seenin X-ray bright TDEs. In this paper we considered alternative models of theMaxwell stress from the standard α − prescription appliedto highly eccentric TDE discs. In particular we focus on theeffects of the coherent magnetic field on the dynamics. We MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs consider two separate stress models: an α − disc with an ad-ditional coherent magnetic field obeying the ideal inductionequation and a nonlinear constitutive (viscoelastic) modelof the magnetic field. In summery our results are1) The coherent magnetic field in both models has a sta-bilising effect on the dynamics, making the gas pressuredominated branch stable at larger radiation pressures andreducing or removing the extreme variation in scale heightaround the orbit for radiation pressure dominated solutions.2) The coherent magnetic field is capable of reversing thecollapse at pericentre without the presumably unphysicallystrong viscous stresses seen in some of the hydrodynamicmodels.3) For the radiation pressure dominated ideal inductionequation model with a moderate magnetic field the dynamicsof the scale height is set by the magnetic field (along withgravity and vertical motion) and doesn’t feel the effects ofgas or radiation pressure. This is because magnetic pressuredominates during pericentre passage which is the only partof the orbit where pressure is important.At present the behaviour of magnetic fields in eccen-tric discs is an understudied area. Our investigation suggeststhat magnetic fields can play an important role in TDE discsand that further work in this area is needed. ACKNOWLEDGEMENTS
We thank J. J. Zanazzi, L. E. Held and H. N. Latter for help-ful discussions. E. Lynch would like to thank the Science andTechnologies Facilities Council (STFC) for funding this workthrough a STFC studentship. This research was supportedby STFC through the grant ST/P000673/1.
The data underlying this article will be shared on reasonablerequest to the corresponding author.
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APPENDIX A: PROPERTIES OF OUR STRESS MODEL
In this appendix we show some important properties of our constitutive model for the Maxwell stress. Restated here for clarity,our model for the (modified) Maxwell stress is M ij + τ D M ij = B p v g ij , (A1)where p v is some pressure which controls the magnitude of the magnetic fluctuations. The Deborah number is given byDe = τ n = De n Ω z (cid:114) p v M , (A2)where Ω z = (cid:112) GM • /r and De is a dimensionless constant. This matches the decay term in the compressible version ofOgilvie (2003). A1 Viscoelastic behaviour
This model is part of a large class of possible viscoelastic models for M ij . The elastic limit τ → ∞ of this equation isfairly obvious, corresponding to a magnetic field which obeys the ideal induction equation through the “freezing in” of M ij ( D M ij = 0). However, to obtain the viscous behaviour responsible for the effective viscosity of circular accretion discs requiresmore work. The viscous limit is obtained when De (cid:28)
1. For simplicity, in what follows, we shall assume τ and p v areindependent of the magnetic field ( M ij ).We propose a series expansion in τ , M ij = ∞ (cid:88) k =0 τ k M ijk . (A3)This expansion is only likely to be valid in the short τ limit and may break down for material variations on timescales shorterthan τ . With that caveat we can find a series solution for equation A1, M ij = B ∞ (cid:88) k =0 ( − τ D ) k p v g ij . (A4)Keeping the lowest order terms in the expansion we have M ij = B p v g ij − B τ D ( p v g ij ) + O ( τ ) . (A5)The lowest order term is an isotropic stress and evidently a form of magnetic pressure. The operator D = D when acting ona scalar and when acting on the metric tensor D g ij = − S ij + 2 ∇ k u k g ij . Using the product rule, M ij ≈ B p v g ij + 2 B τ p v S ij − B τ ( p v ∇ k u k + Dp v ) g ij = B p v g ij + 2 B τ p v S ij + 2 B τ (cid:18)(cid:18) ∂p v ∂ ln ρ (cid:19) s − p v (cid:19) ∇ k u k g ij − B τ (cid:18) ∂p v ∂s (cid:19) ρ ( Ds ) g ij . (A6)So the first O (1) term gives rise to a magnetic pressure, the first O ( τ ) term is a shear viscous stress, the second is a bulkviscous stress and the final term is an additional nonadiabatic correction which has no obvious analogue in the standardviscous or magnetic models. Higher order terms contain time derivatives of the pressure p v and the velocity gradients ∇ i u j .This dependence produces a memory effect in the fluid causing the dynamics of the fluid to depend on its history. The fluidhas a finite memory and becomes insensitive to (“forgets about”) its state at times (cid:38) τ in the past.If τ and p v depend on M ij then the terms in equation A6 are modified. However the equation still decomposes into anisotropic magnetic pressure term, a shear stress term ∝ S ij , a bulk stress term ∝ ∇ k u k g ij and a non-adiabatic term ∝ ( Ds ) g ij (as in equation A6). A2 Realisability
In addition to its behaviour in the viscous and elastic limits another necessary property of a model of a Maxwell stress is itsrealisability from actual magnetic fields. As M ij = B i B j µ , M ij must be positive semi-definite. Thus for all positive semi-definiteinitial conditions M ij (0) our constitutive model equation A1 must conserve the positive semi-definite character of M ij . Thisis equivalent to requiring that the quadratic form Q = M ij Y i Y j satisfy Q ≥ Y i , at all points in the fluid.We will show by contradiction that an initially positive semi-definite M ij cannot evolve into one that is not positivesemi-definite. Suppose, to the contrary, that at some point in the flow Q < X i at some time after the initial MNRAS000
In addition to its behaviour in the viscous and elastic limits another necessary property of a model of a Maxwell stress is itsrealisability from actual magnetic fields. As M ij = B i B j µ , M ij must be positive semi-definite. Thus for all positive semi-definiteinitial conditions M ij (0) our constitutive model equation A1 must conserve the positive semi-definite character of M ij . Thisis equivalent to requiring that the quadratic form Q = M ij Y i Y j satisfy Q ≥ Y i , at all points in the fluid.We will show by contradiction that an initially positive semi-definite M ij cannot evolve into one that is not positivesemi-definite. Suppose, to the contrary, that at some point in the flow Q < X i at some time after the initial MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs state. Then let us consider a smooth, evolving vector field Y i that matches the vector X i at the given point and time. Thecorresponding quadratic form Q is then a scalar field that evolves according to D Q = Y i Y j D M ij + M ij D ( Y i Y j )= (cid:0) B p v Y − Q (cid:1) /τ + M ij D Y i Y j . (A7)By assumption, Q is initially positive and evolves continuously to a negative value at the given later time. Therefore Q must pass through zero at some intermediate time, which we denote by t = 0 without loss of generality. We can also assume,without loss of generality, that the vector field evolves according to D Y i = 0, which means that it is advected by the flow.The equation for Q then becomes DQ = (cid:0) B p v Y − Q (cid:1) /τ , (A8)where we have made use of the fact that D = D when acting on a scalar. Within the disc we expect p v >
0, additionally as M ij is positive semi-definite M ≥
0. Thus at t = 0, Q = 0 and the time derivative of Q is given by DQ | t =0 = B p v Y /τ ≥ . (A9)This contradicts the assumption that Q passes through zero from positive to negative at t = 0. We conclude that M ij remainspositive semi-definite if it is initially so. A3 Energy Conservation
In order that the interior of our disc conserve total energy we need to derive the appropriate magnetic heating/cooling rateso that energy lost/gained by the magnetic field is transferred to/from the thermal energy. The MHD total energy equationwith radiative flux is ∂ t (cid:20) ρ (cid:18) u e (cid:19) + B µ (cid:21) + ∇ · (cid:20) ρ u (cid:18) u h (cid:19) + u B µ − µ ( u · B ) B + F (cid:21) = 0 , (A10)where e is the specific internal energy and h = e + p/ρ is the specific enthalpy. In terms of the modified Maxwell stress, ∂ t (cid:20) ρ (cid:18) u e (cid:19) + M (cid:21) + ∇ i (cid:20) ρu i (cid:18) u h (cid:19) + u i M − u j M ij + F i (cid:21) = 0 . (A11)From this we deduce the thermal energy equation, ρT Ds = M ij S ij −
12 ( DM + 2 M ∇ i u i ) − ∇ i F i . (A12)Using the constitutive relation, Equation A1, we obtain ρT Ds = 12 τ ( M − B p v ) − ∇ i F i , (A13)so we have the nondimensional heating rate, f H = 12De (cid:18) Mp v − B (cid:19) . (A14)Substituting Equation A6 into Equation A12 we recover, in the viscous limit, terms proportional to S ij S ij and ( ∇ i u i ) which act like a viscous heating rate. APPENDIX B: STRESS MODEL BEHAVIOUR IN A CIRCULAR DISC
In this appendix we consider the behaviour of our nonlinear constitutive model in a circular disc. We derive the referencecircular disc, with respect to which our models are scaled. For a circular disc, the fixed point of equation 48 is M RR = M zz = B p v (B1) RM Rφ = −
32 De B p v (B2) R M φφ = (cid:18) (cid:19) B p v , (B3) MNRAS , 1–20 (2019) E. M. Lynch and G. I. Ogilvie which results in a magnetic pressure of p m = 12 M = (cid:18)
32 + 94 De (cid:19) B p v . (B4)As De and p v can depend on M this equation needs to be solved to determine the equilibrium M . When p v is independent of M this yields M = 32 (cid:32) (cid:114) B (cid:33) B p v , (B5)while for p v = p + p m we obtain a quadratic equation for p m , (cid:20) − B (cid:18) (cid:19)(cid:21) p m = B p (cid:18) (cid:19) p m + 94 De B p . (B6)For physical solutions ( p, p m >
0) the right hand side is positive. Therefore we require 2 > B (cid:0) De (cid:1) in order that p m isreal. This equation has a singular point at 2 = B (cid:0) De (cid:1) , which results in a negative magnetic pressure. Solving for themagnetic pressure, p m = 32 B p De ± (cid:113) De B − B (cid:0) De (cid:1) . (B7)Given the requirement that 2 > B (cid:0) De (cid:1) , the negative root always results in a negative magnetic pressure and is thusunphysical.In addition to the equilibrium values of M ij , the circular reference disc obeys hydrostatic and thermal balance. Theequation for hydrostatic equilibrium in a circular disc is P Σ H (cid:18) M p − M zz p (cid:19) = n , (B8)while the equation for thermal balance is C ◦ = f H P ◦ v P ◦ = 94 B De ◦ P ◦ v P ◦ , (B9)whereDe ◦ = De (cid:114) p ◦ v M ◦ . (B10)Taking the solution to equations B8-B9 which has β r = β ◦ r and substituting this into equations 39 and 40 as the circularreference state we obtain¨ HH = − (1 − e cos E ) − + TH β r β ◦ r (cid:32) Mp − M zz p (cid:33)(cid:104) B P ◦ v P ◦ (cid:0) + (De ◦ ) (cid:1)(cid:105) , (B11)˙ T = − (Γ − T (cid:18) ˙ JJ + ˙ HH (cid:19) + (Γ −
1) 1 + β r β r T (cid:20) (cid:18) Mp − p v p (cid:19) − C ◦ β ◦ r β r J T (cid:21) , (B12)where the reference cooling rate is given by equation B9. APPENDIX C: SOLUTION TO THE INDUCTION EQUATION
In this appendix we derive the the structure of a steady magnetic field in an eccentric disc. The equations for a horizontallyinvariant laminar flow in a magnetised disc in an eccentric shearing box were derived in Ogilvie & Barker (2014). In theircoordinate system the induction equation is DB ξ = − B ξ (∆ + ∂ ς v ς ) , (C1) MNRAS000
In this appendix we derive the the structure of a steady magnetic field in an eccentric disc. The equations for a horizontallyinvariant laminar flow in a magnetised disc in an eccentric shearing box were derived in Ogilvie & Barker (2014). In theircoordinate system the induction equation is DB ξ = − B ξ (∆ + ∂ ς v ς ) , (C1) MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs DB η = Ω λ B ξ + Ω φ B η − B η (∆ + ∂ ς v ς ) , (C2) DB ζ = − B ζ ∆ , (C3)where Ω λ = ∂ Ω ∂λ and Ω φ = ∂ Ω ∂φ .In order to rewrite the terms involving derivatives of Ω we introduce functions ℵ and (cid:105) defined by˙ ℵℵ = − Ω φ , ˙ (cid:105) ℵ = − Ω λ , (C4)which in Ogilvie & Barker (2014) were denoted α and β . Noting that ∆ = ˙ JJ and ∂ ς v ς = ˙ HH the ξ and ζ components of theinduction equation become˙ B ξ B ξ + ˙ JJ + ˙ HH = 0 , (C5)˙ B ζ B ζ + ˙ JJ = 0 , (C6)which have solutions B ξ = B ξ ( λ, ˜ z ) JH , B ζ = B ζ ( λ ) J , (C7)where we have additionally made use of the solenoidal condition to show B ζ is independent of ˜ z . Substituting the solution for B ξ into the η component of the induction equation and rearranging we get ℵ JH ˙ B η + ℵ J ˙ HB η + ℵ ˙ JHB η + ˙ ℵ JHB η + ˙ (cid:105) B ξ = 0 , (C8)which has the solution B η = Ω B η nJH − Ω (cid:105) B ξ nJH , (C9)where we have used ℵ ∝ Ω − from Ogilvie & Barker (2014). Thus the large scale magnetic field solution in an eccentricshearing box is given by B ξ = B ξ ( λ, ˜ z ) JH , B η = Ω B η nJH − Ω (cid:105) B ξ nJH , B ζ = B ζ ( λ ) J . (C10)The equation for (cid:105) is given in Ogilvie & Barker (2014) as (cid:105) = 32 (cid:18) eλe λ − e (cid:19) (cid:18) GMλ (cid:19) / t − λe λ (2 + e cos θ ) sin θ (1 − e )(1 + e cos θ ) − λω λ (1 + e cos θ ) + constant , (C11)which in the ( a, E ) coordinate system is given by (cid:105) = 3 nt − e ( e + 2 ae a )] √ − e − ae a (1 − e ) / − e cos E − e − e ( e + 2 ae a ) sin E − a(cid:36) a (1 − e cos E ) (1 − e )[1 − e ( e + 2 ae a )] + constant , (C12)this contains a term that grows linearly in time. This means that B η can be expected to grow linearly in the presence of a“quasiradial” field.The ideal induction equation can be written in tensorial form using the operator D as D ( B i B j ) = 0 (Ogilvie 2001). Thesolution to this equation for a horizontally invariant laminar flow (along with the solenoidal condition) in an eccentric shearingbox is given by Equation C10. In the limit τ → ∞ the Maxwell stress obeys D M ij = 0 and the corresponding magnetic fieldobeys the induction equation.The solenoidal condition is not automatically satisfied by the solutions to D M ij = 0 (although it can be imposed). In J here being the Jacobian of the (Λ , λ ) coordinate system, as used throughout; as opposed to the Jacobian of the coordinate systemof Ogilvie & Barker (2014) which shares the same symbol. In fact our J is closer to J of Ogilvie & Barker (2014).MNRAS , 1–20 (2019) E. M. Lynch and G. I. Ogilvie particular the assumption that the stress has the same vertical dependence as the pressure breaks the solenoidal condition, if M zz (cid:54) = 0.Finally, when B ξ = 0 it is convenient to write the magnetic field in a form which is independent of the horizontalcoordinate system used, B = B h (˜ z ) nJH v orbital + B v J ˆ e z , (C13)where B v is a constant, B h (˜ z ) is a function of ˜ z only and v orbital is the orbital velocity vector. APPENDIX D: DERIVATION OF THE IDEAL INDUCTION EQUATION MODEL
We here derive the full set of equations for a horizontally invariant laminar flow in an eccentric disc with a magnetic field.Assume the magnetic field can be split into mean and fluctuating parts: B = ¯ B + b . (D1)In order that we have a steady field we require B ξ = 0, otherwise there is a source term in the η component of the inductionequation from the winding up of the “quasiradial” ( B ξ ) field. This trivially satisfies the ξ component of the induction equation.We assume the fluctuating field b is caused by the MRI and its effect on the dynamics is captured by the turbulent stressprescription. Thus keeping the mean field only and dropping the overbar the equations for a horizontally invariant laminarflow in a magnetised disc in the eccentric shearing coordinates of Ogilvie & Barker (2014) are the η and ζ components of theinduction equation DB η = Ω λ B ξ + Ω φ B η − B η (∆ + ∂ ς v ς ) , (D2) DB ζ = − B ζ ∆ , (D3)the momentum equation Dv ζ = − φ ζ − ρ ∂ ζ (cid:18) p + B µ − T zz (cid:19) + Tension , (D4)where Tension are the magnetic tension terms. The solenoidal condition gives ∂ ζ B ζ = 0, thus B ζ is independent of ζ and themagnetic tension terms in the vertical momentum equation are zero. Finally the thermal energy equation is Dp = − Γ p (∆ + ∂ ζ v ζ ) + (Γ − H − ∂ ζ F ζ ) , (D5)and we must specify the equation of state. Making use of the solutions to the induction equation (Equation C13), we obtainan expression for the magnetic pressure, p M = B h (˜ z )2 µ ( nJH ) v + B v µ J , (D6)with v = | v orbital | is the square of the magnitude of the orbital velocity.The contribution of the vertical component of the magnetic field to the magnetic pressure is independent of the height inthe disc (in order to satisfy the solenoidal condition) and makes no contribution to the dynamics of the vertical structure. Assuch we neglect the vertical component of the magnetic field from this point on. The magnetic pressure simplifies to p M = B h (˜ z )2 µ ( nJH ) v . (D7)On a circular orbit v = ( na ) so that p m ∝ ( JH ) − ∝ ρ and the magnetic pressure behaves like perfect gas with γ = 2.Thus, for a magnetised radiation-gas mixture, the magnetic field is the least compressible constituent of the plasma and willbe the dominant source of pressure when the plasma is sufficiently compressed. On an eccentric orbit there is an additionalsource of variability owing to the stretching and compressing of the field by the periodic variation of the velocity tangent tothe field lines.The vertical component of the momentum equation becomes¨ HH = − φ − ρ ˆ H ˜ z ∂ ˜ z (cid:18) p + B h (˜ z )2 µ ( nJH ) v − T zz (cid:19) , (D8) MNRAS000
We here derive the full set of equations for a horizontally invariant laminar flow in an eccentric disc with a magnetic field.Assume the magnetic field can be split into mean and fluctuating parts: B = ¯ B + b . (D1)In order that we have a steady field we require B ξ = 0, otherwise there is a source term in the η component of the inductionequation from the winding up of the “quasiradial” ( B ξ ) field. This trivially satisfies the ξ component of the induction equation.We assume the fluctuating field b is caused by the MRI and its effect on the dynamics is captured by the turbulent stressprescription. Thus keeping the mean field only and dropping the overbar the equations for a horizontally invariant laminarflow in a magnetised disc in the eccentric shearing coordinates of Ogilvie & Barker (2014) are the η and ζ components of theinduction equation DB η = Ω λ B ξ + Ω φ B η − B η (∆ + ∂ ς v ς ) , (D2) DB ζ = − B ζ ∆ , (D3)the momentum equation Dv ζ = − φ ζ − ρ ∂ ζ (cid:18) p + B µ − T zz (cid:19) + Tension , (D4)where Tension are the magnetic tension terms. The solenoidal condition gives ∂ ζ B ζ = 0, thus B ζ is independent of ζ and themagnetic tension terms in the vertical momentum equation are zero. Finally the thermal energy equation is Dp = − Γ p (∆ + ∂ ζ v ζ ) + (Γ − H − ∂ ζ F ζ ) , (D5)and we must specify the equation of state. Making use of the solutions to the induction equation (Equation C13), we obtainan expression for the magnetic pressure, p M = B h (˜ z )2 µ ( nJH ) v + B v µ J , (D6)with v = | v orbital | is the square of the magnitude of the orbital velocity.The contribution of the vertical component of the magnetic field to the magnetic pressure is independent of the height inthe disc (in order to satisfy the solenoidal condition) and makes no contribution to the dynamics of the vertical structure. Assuch we neglect the vertical component of the magnetic field from this point on. The magnetic pressure simplifies to p M = B h (˜ z )2 µ ( nJH ) v . (D7)On a circular orbit v = ( na ) so that p m ∝ ( JH ) − ∝ ρ and the magnetic pressure behaves like perfect gas with γ = 2.Thus, for a magnetised radiation-gas mixture, the magnetic field is the least compressible constituent of the plasma and willbe the dominant source of pressure when the plasma is sufficiently compressed. On an eccentric orbit there is an additionalsource of variability owing to the stretching and compressing of the field by the periodic variation of the velocity tangent tothe field lines.The vertical component of the momentum equation becomes¨ HH = − φ − ρ ˆ H ˜ z ∂ ˜ z (cid:18) p + B h (˜ z )2 µ ( nJH ) v − T zz (cid:19) , (D8) MNRAS000 , 1–20 (2019) agnetic fields in eccentric TDE Discs where we have used ˆ H to denote the dimensionful scale height, to distinguish it from the dimensionless scale height H .We propose separable solutions with p = ˆ p ( τ )˜ p (˜ z ) , T zz = ˆ T zz ( τ )˜ p (˜ z ) , ρ = ˆ ρ ( τ )˜ ρ (˜ z ) . (D9)The dimensionless functions obey the generalised hydrostatic equilibrium which means the pressure obeys d ˜ pd ˜ z = − ˜ ρ ˜ z . (D10)To maintain separability we require the reference plasma beta to be independent of height, β ◦ m = 2 µ ˜ p (˜ z ) p ◦ a B h (˜ z ) . (D11)From this we obtain the equation for variation of the scale height around the orbit,¨ HH = − φ + ˆ p ˆ ρ ˆ H (cid:32) β ◦ m J H P ◦ ˆ P v ( an ) − ˆ T zz ˆ p (cid:33) , (D12)where square of the velocity is v = ( an ) e cos E − e cos E . (D13)The reference circular disc has f H = α s as in the hydrodynamic models considered in Paper I. In the reference circulardisc, hydrostatic balance is given by P ◦ Σ ◦ H ◦ H ◦ (cid:18) β ◦ m (cid:19) = n . (D14)Rescaling Equation D12 by this reference circular disc we obtain¨ HH = − (1 − e cos E ) − + TH β r β ◦ r (cid:16) β ◦ r β r β ◦ m JHT e cos E − e cos E − ˆ T zz p (cid:17)(cid:16) β ◦ m (cid:17) , (D15)with the rest of the equations proceeding as in the hydrodynamic laminar flow model considered in Paper I. APPENDIX E: MICROPHYSICAL BASIS OF THE NONLINEAR CONSTITUTIVE MODEL
Several authors have looked at the possibility of using stochastic calculus as a model of the MRI (Janiuk & Misra 2012; Rosset al. 2017). Here we assume the magnetic field satisfies a Langevin equation: d B + ( B · ∇ u − B ∇ · u ) dt = − λdt + F d X , (E1)where X is a Wiener process in the sense of Ito calculus. The left hand side of this equation is the ideal terms in theinduction equation, the − λdt term models damping from resistivity, while the F d X represents stochastic forcing by a turbulentelectromotive force, where F is some scale factor controlling the strength of the forcing. In the absence of a mean velocityfield u , the magnetic field would evolve like a damped Brownian motion.Introducing (cid:104)·(cid:105) to denote the expectation value, we have the standard result for the Wiener process X , (cid:104) X i X j (cid:105) = g ij t , (E2)where g ij is the inverse metric tensor. So X is a statistically isotropic vector field. Physically, in this model, turbulentfluctuations act to isotropise the magnetic field. The orbital shear can feed on these fluctuations and induce a highly anisotropicmagnetic field that is predominantly aligned/antialigned with the orbital motion. The change in the Maxwell stress can beobtained from Ito’s formula, µ dM ij = (cid:88) n ∂M ij ∂B n dB n + 12 (cid:88) nm ∂ M ij ∂B n ∂B m d (cid:104) B n B m (cid:105) , (E3) MNRAS , 1–20 (2019) E. M. Lynch and G. I. Ogilvie with the partial derivatives given by ∂M ij ∂B n = 2 B ( i δ j ) n , ∂ M ij ∂B n ∂B m = 2 δ ( in δ j ) m . (E4)After substituting in these and the equation for dB Equation E3 becomes µ dM ij = − B k B ( i ∇ k u j ) − B ( i B j ) ∇ k u k ) dt − B ( i λ j ) dt + 2 F B ( i dX j ) + (cid:88) nm δ ( in δ j ) m F d (cid:104) X n X m (cid:105) = − B k B ( i ∇ k u j ) − B ( i B j ) ∇ k u k ) dt − B ( i λ j ) dt + 2 F B ( i dX j ) + F g ij dt . (E5)Making use of the definition of D , and the fact the Ito integral preserves the martingale property, we can take the expectationof Equation E5 to obtain an equation for the expected Maxwell stress, D(cid:104) M ij (cid:105) = − µ (cid:104) B ( i λ j ) (cid:105) + 1 µ F g ij . (E6)We should caution that this procedure may not be valid if F depends on B . Henceforth we shall drop the angle brackets onthe modified Maxwell stress and use M ij to denote the expected modified Maxwell stress.What’s left now is to determine appropriate forms for λ i and F . A priori there is no obvious way of directly obtainingthese using the underlying physics. However, in a similar vein to Ogilvie (2003), we can place certain constraints on thepossible forms of λ i and F . In particular on dimensional grounds they both have dimensions of magnetic field over time. λ i transforms as a vector and F transforms as a scalar. Other than this we assume:1) Following Ogilvie (2003) neither λ i or F directly know about the mean velocity field, although they could know aboutthe various orbital frequencies.2) F is non-negative.3) There is no preferred direction.This leaves us to construct a vector and a scalar which have the same dimensions as magnetic field over time from B i , p g , p r , ρ , µ along with the vertical and horizontal epicyclic frequencies and mean motion Ω z , κ , n . Immediately it is apparentthat the the only vectorial quantity we have available is the magnetic field B i . As such λ i must have the form λ i = B i τ , (E7)where τ is some relaxation time which can depend on the mean field quantities. Next on dimensional grounds ρ cannot appearin either F or τ and the other terms must only appear in the combination | B | , µ p g and µ p r along with the various orbitalfrequencies. Without loss of generality we can write F = (cid:114) µ B p v τ , (E8)where τ is the relaxation time, B is a dimensionless constant and p v is some reference pressure of the fluctuations. This meansour equation for M ij becomes D M ij = − τ (cid:16) M ij − B p v g ij (cid:17) , (E9)where we must specify how the relaxation time and fluctuation pressure depend on M , p g , p r and the orbital frequencies inorder to close the model. This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000