Emergence of High Peaks in the Axial Velocity for an Ideal Magnetohydrodynamics Disk Configuration
aa r X i v : . [ a s t r o - ph . H E ] S e p EMERGENCE OF HIGH PEAKS IN THE AXIAL VELOCITY FORAN IDEAL MAGNETOHYDRODYNAMICS DISK CONFIGURATION
Giovanni Montani a, b, c and Nakia Carlevaro a a Department of Physics - “Sapienza” University of Romec/o Dip. Fisica - “Sapienza” Universit`a di Roma, P.le A. Moro, 5 (00185), Roma (Italia). b ENEA C.R. Frascati (Rome), UTFU-MAG. c ICRANet – International Center for Relativistic Astrophysics Network. [email protected] [email protected]
Abstract:
We study the profile of a thin disk configuration as described by anaxisymmetric ideal magnetohydrodynamics steady equilibrium. We consider thedisk like a differentially rotating system dominated by the Keplerian term, butallowing for a non-zero radial and vertical matter flux. As a result, the steadystate allows for the existence of local peaks for the vertical velocity of the plasmaparticles, though it prevents the radial matter accretion rate. This ideal mag-netohydrodynamics scheme is therefore unable to solve the angular momentum-transport problem, but we suggest that it provides a mechanism for the generationof matter-jet seeds.
PACS : 97.10.Gz; 95.30.Qd; 52.30.Cv
1. Basic Statements -
In this paper, we consider an axisymmetric ideal magnetohydro-dynamics (MHD) model for the disk structure surrounding a compact astrophysical object.Our approach, which differs from the standard astrophysical model for the disk morphology,is justified in view of the microphysical properties of the plasma having negligible value of theviscosity and the resistivity coefficients. Indeed, the standard model [1, 2] accounts for theangular momentum transport by postulating a non-zero viscous stress tensor appearing inthe equilibrium as an effect of the internal turbulence. Despite this model is rather successfulin explaining the observations, it remains affected by the non trivial shortcoming that themicroscopic estimations for the viscosity coefficient give much smaller values than those onesrequired to fit data. This is particularly true in accreting objects like X-ray stars, where theobserved rate of accretion is very high. An analogous difficulty concerns the proper balanceof the force acting on electrons, when a significant infalling radial velocity occurs. The solu-tion to this puzzle is offered by the presence of an effective resistivity of the plasma, whichcompensate the azimuthal Lorentz force acting on the electrons. Also such values of theresistivity coefficient are in contradiction with the microscopical estimations related to theastrophysical scales. The turbulent viscosity and resistivity coefficients can be maintainedboth relevant for the equilibrium by postulating a
Prandtl number of order unity for theaccreting plasma. This shortcoming of the standard model is still open to the scientific de-bate and leads to speak of an anomalous visco-resistivity of the disk plasma, rising from theso-called magnetorotational instability described by E. Velikhov [3, 4]. However, the labora-tory experience on turbulent plasmas does not confirm the correlation between the internal1nstabilities and the applicability of a viscoresistive MHD model with Prandtl numbers oforder unity.In our analysis, we address a different scenario as closely related to the transport profile ofmagnetically confined laboratory plasmas [5]. Indeed, in [6], has been shown how the idealMHD profile of a thin disk can have a significant deviation from the Keplerian behavior for astrong vertical Lorentz force. In [7, 8] it has been demonstrated that the local configuration ofa thin disk can be reduced to the profile of a ring sequence, when the electromagnetic backre-action is sufficiently intense and the vertical Lorentz force is able to confine the disk (see also[9]). Despite a solution of the angular momentum transport ( i.e. , a non-zero radial accretionrate) is not jet available in this ideal scheme, it suggests the idea that (instead of a turbulentbehavior) the disk plasma could be characterized by the formation of microstructures whichmight drive the typical ballooning-mode instabilities observed in laboratory systems. Theaccretion mechanism would then rely on the porosity of the plasmas near the x-points of themagnetic field configuration.We here follow the research line traced in [6, 7, 8], but including vertical and radial matterfluxes in the ideal MHD analysis (see also [10, 11, 12]). The resulting equilibrium configura-tion is characterized by a vanishing net mass accretion rate, as in the previous approaches,since contributions due to the radial velocity over and below the equatorial plane cancelout. However, our local configuration permits the existence of specific disk layers where thevertical velocity can acquire a very large amplitude. This scenario suggests the possibility torealize matter jets from the disk equilibrium profile, especially in view of the implementationof this same morphology in presence of the expected plasma instabilities and this issue consti-tutes the main achievement of our analysis. Indeed, the link between highly magnetized diskplasmas and the jet formation (observed in the different electromagnetic bands) constitutes,in the standard model, one of the most relevant motivations for studying the angular mo-mentum transport across these axisymmetric systems (for a proposal on the jet formation see[13]). Our result appears particularly appealing because the viscoresistive model has somedifficulties to create the conditions for the jet formation, because the viscoresistive MHDscenario predicts diffusive magnetic configurations [14, 15].
2. Vertical and Radial Equilibria -
To adapt the MHD equations to the axial symmetry, weintroduce cylindrical coordinates { r, φ, z } and the fundamental variables have no dependenceon φ . The gravitational potential of the central object writes χ ( r, z ) = GM S / p r + z , (1)where M S is the star mass and G the Newton constant. The continuity equation, in thesteady state of the disk, reads as the vanishing divergence of the matter flux ǫ~v ( ǫ being themass density and ~v the velocity field) and it admits the following solution: ǫ~v = − ~e r ( ∂ z Θ) /r + ~e φ ǫr ω ( r, z ) + ~e z ( ∂ r Θ) /r , (2)where, ω ( r, z ) is the disk differential angular velocity (in view of the co-rotation theorem[16], it can be written as ω = ω ( ψ ), ψ ( r, z ) being the magnetic flux surface). Here, thegeneric function Θ( r, z ) must be odd in z to deal with a non-zero mass accretion rate of thedisk ( ˙ M d ): ˙ M d = − πr Z + H − H dz ǫv r = 4 π Θ( r, H ) > , (3)where H is the half-depth of the disk and v r < ~B = − ~e r ( ∂ z ψ ) /r + ~e φ I ( ψ, z ) /r + ~e z ( ∂ r ψ ) /r , (4)with the function I ( ψ, z ) allowing for the existence of poloidal currents in the disk plasma.2n order to investigate the effects induced on the disk profile by the electromagnetic reactionof the plasma, we consider a local model of the equilibrium around a radius value r = r .Thus, we split the mass density ǫ and the thermostatic pressure p into the background (barredterms) and perturbation (hatted terms) components, as ǫ = ¯ ǫ ( r , z ) + ˆ ǫ and, of course, p = ¯ p ( r , z ) + ˆ p , respectively. The same way, we express the magnetic surface function inthe form ψ = ψ ( r )+ ψ ( r , r − r , z ), with ψ ≪ ψ . In particular, the co-rotation theoremallows the representation ω = ω K + ˜ ω ψ , where ω K is the Keplerian term and ˜ ω = const .This form for ω holds locally, as far as ( r − r ) remains sufficiently small so that the dominantdeviation from the Keplerian contribution is due to ψ .When we consider the “drift ordering” for the behavior of the gradient amplitude ( i.e. ,the first-order gradients of the perturbations are of zeroth-order while the second-order onesdominate) we are able to cast the equilibrium configuration by a hierarchical ordering ofthe different contributions. Accordingly, the profile of the disk toroidal currents J φ and theazimuthal component of the Lorentz force F φ have the following form: J φ ≃ − c ( ∂ r ψ + ∂ z ψ ) / πr , (5a) F φ ≃ ( ∂ z I ∂ r ψ − ∂ r I∂ z ψ ) / πr . (5b)We now fix the equations governing the vertical and the radial equilibria of the disk byseparating the fluid components from the electromagnetic backreaction (as discussed in [7,8]). Indeed, we assume that the contribution of the toroidal magnetic field does not affectthese equilibria and therefore the resulting configuration equations overlap that ones derivedoriginally by B. Coppi in 2005. The quantity I ( ψ, z ) plays a crucial role in the azimuthalequation only, allowing for a non-zero radial velocity even in the ideal MHD scenario.The splitting of the MHD equations gives, for the vertical force balance, the followingsystem of equations D ≡ ¯ ǫ/ǫ = exp[ − z /H ] , H ≡ K B T / ( mω K ) , (6a) ∂ z ˆ p + ω K z ˆ ǫ − ∂ z ψ (cid:0) ∂ z ψ + ∂ r ψ (cid:1) / πr = 0 , (6b)where ǫ = ǫ ( r ) ≡ ǫ ( r ,
0) and the temperature T admits the form ( K B being the Boltzmannconstant): 2 K B T ≡ m ( p/ǫ ) = m (¯ p + ˆ p ) / (¯ ǫ + ˆ ǫ ) ≡ K B ( ¯ T + ˆ T ) , (7)The radial equilibrium splits into the decomposition for ω ( i.e. , ω ≃ ω K + δω ≃ ω ( ψ )+ ˜ ω ψ )and the equation 2 ω K r (¯ ǫ + ˆ ǫ )˜ ω ψ + ∂ r ψ ( ∂ z ψ + ∂ r ψ ) / (4 πr ) == ∂ r [ˆ p + ( ∂ r ψ ) / (8 πr )] + ∂ r ψ ∂ z ψ / (4 πr ) . (8)Let us now define the dimensionless functions Y , ˆ D and ˆ P , in place of ψ , ˆ ǫ and ˆ p , i.e. , Y ≡ k ψ / ( ∂ r ψ ) , ˆ D ≡ β ˆ ǫ/ǫ , ˆ P ≡ β ˆ p/p , (9)where p ≡ K B ¯ T ǫ ( ¯ T being ¯ T ( r, β ≡ πp /B z = 1 / (3 ǫ z ) ≡ k H / . (10)Here, we introduced the fundamental wave-number k of the radial equilibrium, defined as k ≡ ω K /v , with v ≡ B z / (4 πǫ ), recalling that B z = ( ∂ r ψ ) /r . Thus, we introducethe dimensionless radial variable defined as x ≡ k ( r − r ), while the fundamental lengthin the vertical direction is taken as ∆ ≡ √ ǫ z H , leading to define u ≡ z/ ∆. This way, therelation D = exp[ − u ǫ z ] holds. 3hese definitions allow us to rewrite the vertical and radial equilibria as follows ∂ u ˆ P + ǫ z ˆ D − (cid:0) ∂ x Y + ǫ z ∂ u Y (cid:1) ∂ u Y = 0 , (11a)( D + ˆ D/β ) Y + ∂ x Y + ǫ z ∂ u Y ++ ∂ x ˆ P + (cid:0) ∂ x Y + ǫ z ∂ u Y (cid:1) ∂ x Y = 0 . (11b)The system above fixes two of the three unknowns Y , ˆ P and ˆ D and has to be completed bythe azimuthal equilibrium and the electron force balance. We stress that, in setting eqs.(11),we have neglect the inertial contributions coming from the radial and the vertical componentsof the matter velocity, because of their small amplitude.
3. Additional Configuration Equations -
The equilibrium along the toroidal symmetry ofthe disk writes ǫv r ∂ r ( ωr ) + ǫv z ∂ z ( ωr ) + ǫωv r = F φ . (12)Using the co-rotation theorem, i.e. , ω = ω ( ψ ), and eqs.(4), (5b), the equation above restatesas ǫr ( v r B z − v z B r ) + 2 ǫv r ω K / ˜ ω == [ ∂ z I ( ∂ r ψ + ∂ r ψ ) − ∂ r I∂ z ψ ] / (4 πr ˜ ω ) . (13)In this scheme, we link the radial and vertical velocity fields to the poloidal currents. Indeed,we aim to set up a picture in which the radial matter flux is induced by the poloidal currents, i.e. , establishing a direct relation between v r and F φ . To this end, as well as for the consistenceof the model, we have to analyze the electron force balance equation: ~E + ( ~v/c ) ∧ ~B = 0. Thebalance of the Lorentz force has radial and vertical components which express the electricfield in the form predicted by the co-rotation theorem [16, 7] and, since the axial symmetryrequires E φ ≡
0, the φ -component of the this equation retains the form v z B r − v r B z = 0 . (14)We now observe that, substituting eq.(14) in eq.(13), we get the searched relation between v r and F φ , i.e. , 2 ǫv r ω K = [ ∂ z I ( ∂ r ψ + ∂ r ψ ) − ∂ r I∂ z ψ ] / (4 πr ) . (15)This relation, together with eq.(14), provides a system for the two unknown functions Θ and I , once ψ and ǫ are given by the vertical and the radial equilibria, i.e , eq.(11a) and eq.(11b),respectively.Substituting v r expressed by eq.(15) into the electron force balance (14), we get the fol-lowing expression for v z : v z = − ( ∂ r ψ + ∂ r ψ ) F φ ω K ǫ ∂ z ψ ( D + ˆ D/β ) . (16)We stress that, addressing the splitting ψ = ψ + ψ , the azimuthal Lorentz force stands as F φ ≃ [ ∂ z I ( ∂ r ψ + ∂ r ψ ) − ∂ r I∂ z ψ ] / (4 πr ) . (17)Expressing eq.(15) for v r and eq.(16) for v z via the function Θ, we get the following integro-differential compatibility condition ∂ z ψ [ R dz ∂ r F φ ] = F φ ( ∂ r ψ + ∂ r ψ ) , (18)from which, provided ψ , i.e. , Y ( x, u ), from eqs.(11), one can determine the toroidal magneticfield I ( ψ, z ). 4 . Non-Linear Limit - Let us now analyze the case Y ≫ i.e. , when the electromagneticreaction within the disk plasma is strong enough to significantly deform the backgroundmagnetic field. In this context, the vertical and radial equilibria are now described by eq.(11a)and ( D + ˆ D/β ) Y + ∂ x ˆ P + ( ∂ x Y + ǫ z ∂ u Y ) ∂ x Y = 0 , (19)respectively. Clearly, we got a new configuration system for which linear terms in the per-turbed magnetic surface Y have been neglected in favor of the quadratic ones. In the samelimit, the azimuthal equation (15) rewrites2 ǫv r ω K = F φ = [ ∂ z I∂ r ψ − ∂ r I∂ z ψ ] / (4 πr ) , (20)and eqs.(16), (18) rewritten in terms of Y as follows: v z = − [ v r ∂ x Y ] / [ √ ǫ z ∂ u Y ] , (21a) ∂ u Y [ R du ∂ x F φ ] = F φ ∂ x Y , (21b)respectively. We now look for a separable solution of the configurational system of the form Y ( u , x ) = F ( u ) sin( αx ) , (22a)ˆ D ( u , x ) = d ( u ) cos( αx ) , (22b)ˆ P ( u , x ) = ξ ( u ) sin ( αx ) + π ( u ) cos( αx ) , (22c) F φ ( u , x ) = G ( u ) sin b ( αx ) . (22d)Substituting eqs.(22) in the set of equations derived above, we get ordinary differential rela-tions in the variable u : from eqs.(11a), (19), we get the following system ∂ u π + ǫ z d = 0 , (23a) π − DF/α = 0 , (23b) ∂ u ξ + 2 (cid:0) − α F + ǫ z F ′′ (cid:1) ∂ u F = 0 , (23c)3 ǫ z F d + αξ + αF ( − α F + ǫ z F ′′ ) = 0 . (23d)Using ǫ = ¯ ǫ + ˆ ǫ , eq.(20) and eq.(21a) give v r = [ G sin b ( αx )] / [2 ω K ǫ ( D + d cos( αx ) /β )] , (24a) v z = − [ αF cotg( αx ) v r ] / [ F ′ √ ǫ z ] , (24b)respectively. Finally, the compatibility equation (21b) can be recast to give the integro-differential relation: bF ′ [ R du G ] = F G ⇒ [ R du G ] = gF b , (25)where we have used ( ... ) ′ = d ( ... ) /du and g = const .
5. Analytical Solution -
To derive an exact solution of the system fixed above, we nowconsider the case of large β -values for the disk plasma, i.e. , ǫ z ≪
1. Such analytical solutionwrites D = 1 − ǫ z u , F = A exp[ − u / , A ≫ , (26a) α = √ , d = F/ ( ǫ z √ , ξ = (3 − ǫ z u ) F , (26b) π = 2(1 − ǫ z u ) F/ √ , G = − bguF b . (26c)5ence the radial velocity (24a) takes the explicit form v r = − ω K ǫ bgA b ue − bu / sin b ( √ x )1 + √ ǫ z Ae − u / cos( √ x ) , (27)where, here and in the following, we neglect the term ǫ z u with respect to terms proportionalto A . It is remarkable that the total mass density is expressed by the relation ǫ = ǫ ( D + ˆ D/β ), i.e. , ǫ = ǫ (cid:2) √ ǫ z Ae − u / cos( √ x ) (cid:3) , (28)and ǫ > √ ǫ z A
1. This feature of the mass density profile (Figure 1, Plot (a))ensures that the ring sequence, predicted in [8], is present in this regime too. Since v r isFigure 1: Behavior of ǫ/ǫ (Plot (a)) and ˜ v z = v z /v ∗ (Plot (b)) ( v ∗ = g/ω K ǫ ), setting A = 100 and ǫ z = 0 . u ( i.e. , Θ is even in z ) the accretion rate of the disk vanishes identically. Despitea local non-zero radial matter flux is present in the equilibrium configuration, however thecompensation between the contributions over and below the equatorial plane, respectively,gives a net zero accretion of the disk.Finally, the vertical velocity component reads as v z = − p /ǫ z ω K ǫ bgA b e − bu / sin b − ( √ x ) cos( √ x )1 + √ ǫ z Ae − u / cos( √ x ) , (29)Let now consider the peculiar case b = 1 of eq.(29), near the equatorial plane. For x √ ∼ π ,we deal with a positive or negative vertical velocity (depending on the sign of Ag ) havinga very large amplitude, | v z |≫ ǫ z √ A ∼ i.e. , in correspondence of the mass-density nodes). Such a condition is satisfied onlyin specific layers of the disk configuration since our model is localized at r but A , g and ǫ z vary with it.Finally, the total term describing the magnetic surface, once we account for the first-ordercorrections around r , reads Y t ≡ x + Y = x + A exp[ − u /
2] sin( √ x ) . (30)The ratios of the magnetic field components to the background magnetic field due to thecentral object, i.e. , ~B = B z ~e z , read as B z /B z = ∂ x Y t ≃ √ A exp[ − u /
2] cos( √ x ) ,B r /B z = −√ ǫ z ∂ u Y t = √ ǫ z Au exp[ − u /
2] sin( √ x ) .
6e stress that the condition to neglect the inertial terms due to the radial and verticalvelocity (despite of the presence of v z -peaks) in the system (11) takes the following form: Ag/ (1 − √ ǫ z A ) ∼ O (1). Moreover, in the regime ǫ z ≪ i.e. , √ ǫ z A
1, ensures the positivity ofthe total pressure too.
6. Astrophysical Characterization -
The configuration scheme we adopted above is appro-priate to describe a local radial layer (a small circular crown) of a thin accretion disk sinceall the morphological prescriptions of this astrophysical system are included. The choice ofthe background magnetic surface as a function of r only, models a dipole magnetic-field ofthe central object, as far as the disk is sufficiently thin and we take ω K ∝ ψ . The grav-itational field is correctly represented in the thin disk approximation, for the limit of anhigh Keplerian angular velocity to have a gravostatic confinement of the vertical profile ofthe disk, i.e. , ω K z ∼ O (1). The idea that the poloidal (radial and vertical) matter fluxesare taken small with respect to the toroidal Keplerian motion is a natural approximation,valid in the standard model too. The vertical velocity must be smaller than the azimuthalone in a thin disk, because of the background vertical equilibrium, while the radial motionis required perturbative since its presence must preserve the steady character of the matterdistribution. In the standard model the amplitude of v r is determined by the turbulent vis-cosity coefficient, while here it is fixed via the self-consistency of the electron force balanceequation. Indeed, we deal with a vanishing net accretion, but, from a physical point of view,we could speak of a weak constant rate of matter infall, since this fact do not affect our maingoal about the vertical matter jets. In this ideal plasma scenario, we can reach very highself-consistent magnetic fields and we show how they can induce intense axial matter fluxes.This picture is commonly believed the basic mechanism of the emission from high energyastrophysical sources (such as Gamma Ray Bursts or Active Galactic Nuclei), but its settingis still rather questioned. We offer here an alternative explanation, surprisingly relying onlyon the morphology of ideal and highly non-linear MHD structures.
7. Conclusions -
We have shown how the steady state of an accretion disk, described byideal MHD, is compatible with the presence of significant radial and vertical matter fluxes.Such matter currents are unable to account for a non-zero accretion rate, but they are relevantin view of the instabilities that can be driven by their existence. In particular, we outlined,as the main issue of our study, that there are conditions under which strong vertical matterfluxes are locally allowed, especially near the equatorial plane. This feature deserves attentionbecause dealing with matter jets along the symmetry axis is one of the expected features ofthe disk configurations, but it is hard to be recovered in the standard viscoresistive model. ∗∗ This work was developed within the framework of the
CGW Collaboration ∗∗ Acknowledgment : NC gratefully acknowledges the CPT - Universit´e de la Mediterran´ee Aix-Marseille 2 and the financial support from “Sapienza” University of Rome. eferences [1] N.I. Shakura, Sov. Astron. , 756 (1973).[2] G.S. Bisnovatyi-Kogan, R.V.E Lovelace, New. Astron. Rev. , 663 (2001).[3] E. Velikhov, Sov. Phys. JETP , 995 (1959).[4] S.A. Balbus, J.F. Hawley, Rev. Mod. Phys. , 1 (1998).[5] B. Coppi, Plasma Phys. Contrl. Fus. , B107 (1994).[6] G.I. Ogivile, Mon. Not. RAS , 63 (1997).[7] B. Coppi,
Phys. Plasmas , 057302 (2005).[8] B. Coppi, F. Rousseau, Astrophys. J. , 458 (2006).[9] B. Coppi,
EuroPhys. L. , 19001 (2008).[10] B. Coppi, F. Rousseau, in the proceedings of The 34 th EPS Conference on Plasma Physics , Warsaw(July 2007), Paper O4.034.[11] G. Montani, R. Benini,
Mod. Phys. L. A , 2667 (2009).[12] M. Lattanzi, G. Montani, EuroPhys. L. , 39001 (2010).[13] D. Lynden-Bell, Mon. Not. RAS , 389 (1996).[14] H.C. Spruit,
Lect. Notes Phys. , 233 (2010).[15] R. Stehle, H.C. Spruit,
Mon. Not. RAS , 587 (2001).[16] V.C.A. Ferraro,
Mon. Not. RAS , 458 (1937)., 458 (1937).