Study of MRI in stratified viscous plasma configuration
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Study of MRI in Stratified Viscous Plasma Configuration
Nakia Carlevaro , , and Giovanni Montani , Fabrizio Renzi ENEA, Fusion and Nuclear Safety Department, C.R. Frascati, Via E. Fermi, 45 (00044) Frascati (RM), Italy; L.T. Calcoli, Via Bergamo, 60 (23807) Merate (LC), Italy; Department of Physics, “Sapienza” University of Rome, P.le Aldo Moro, 5 (00185) Roma, Italy.
PACS – Accretion and accretion disks
PACS – Plasma and MHD instabilities
PACS – Electric and magnetic fields
Abstract –We analyze the morphology of the Magneto-rotational Instability (MRI) for a strat-ified viscous plasma disk configuration in differential rotation, taking into account the so-calledcorotation theorem for the background profile. In order to select the intrinsic Alfv´enic natureof MRI, we deal with an incompressible plasma and we adopt a formulation of the local per-turbation analysis based on the use of the magnetic flux function as a dynamical variable. Ourstudy outlines, as consequence of the corotation condition, a marked asymmetry of the MRI withrespect to the equatorial plane, particularly evident in a complete damping of the instability overa positive critical height on the equatorial plane. We also emphasize how such a feature is alreadypresent (although less pronounced) even in the ideal case, restoring a dependence of the MRI onthe stratified morphology of the gravitational field.
Introduction. –
In 1959, E.P. Velikhov discovered anew type of magneto-hydrodynamics (MHD) instability,associated to the coupling of the plasma differential rota-tion with the Alfv´enic modes [1]. In this original work, itis investigated the behavior of a magnetized plasma, lyingbetween two rotating cylinders at different angular veloc-ity and the corresponding unstable mode spectrum wasnamed Magneto-Rotational Instability (MRI). In 1960, S.Chandrasekar proposed the implementation of MRI to theastrophysical context, with reference to a stellar accre-tion disk rotating mainly due to the gravity of the centralbody [2]. Then, in 1991, MRI was re-analyzed [3] withthe awareness of the role it can play in rising turbulencein accretion disks and, hence, the required effective viscos-ity to account for the Shakura idea of accretion [4, 5] (fora detailed discussion of MRI in accreting structures andits contribution to the angular momentum transport, seeRef.[6] and references therein).Since MRI is an Alfv´enic mode, it is not associated tomatter transport and it survives also in incompressibleplasmas, where its real nature is indeed well-traced. Fora study of MRI in the case of a liquid magnetized metal,see Ref.[7], and a similar astrophysical study can be foundin Ref.[8]. In the present paper, we concentrate on sucha restricted case of an incompressible plasma in order toselect a pure Alfv´enic instability and evaluate the role the so-called corotation theorem [9] plays in the MRI features.Such a theorem states that the differential rotation profileof a steady axisymmetric disk (actually the backgroundconfiguration on which our perturbation analysis is per-formed) must depend on the magnetic field morphologyvia the magnetic flux function only. In Refs.[10–12], thesame question has been addressed in the context of idealMHD mainly focusing on parallel propagating perturba-tions, while here we address the behavior of a viscous in-compressible rotating plasma. A valuable discussion ofMRI in the case of a viscous-resistive plasma disk (butwithout retaining the corotation theorem and the strati-fied nature of the disk) has been provided in Ref.[13]. Inwhat follows, we will neglect (differently from Ref. [13])the effect of dissipation on the background profile, but wedeal with a stratified disk for which the angular velocitydepends on the vertical coordinate too [14].The main merit of the present analysis is to outline thedependence of the MRI growth rate on the vertical pro-file of the background and, in particular, we show a clearasymmetry with respect to the equatorial plane. In fact,MRI is suppressed over a critical height on the equatorialplane (in the region of positive vertical cylindrical coordi-nate z , when the magnetic field has a natural dipole likeconfiguration), by a mechanism very similar to the one in-vestigated in Ref.[13] as an effect of the increasing valuep-1. Carlevaro, G. Montani, F. Renziof the viscosity coefficient. Here, the same dimensionlessparameter adopted in the discussion of that article varieswith z as a consequence of the stratified structure of thedisk, i.e., the magnetic field morphology depends on theheight on the equatorial plane. Since such dimensionlessviscosity parameter results to increase with z , we observea significant damping of the MRI growth rate. However,we emphasize how such a damping is the combined effectof what clarified above, together with an ideal propertyof the MRI emerging only when the corotation theorem isconsidered. In fact, a critical z value, over which MRI issuppressed, exists already in the ideal case and it remainsthe same for the viscous plasma, but the vertical dampingis much more marked for that case.Also the critical wavenumber at which MRI is removedappears to be the same both in the inviscid and viscouscase, but in the latter case, the suppression is evidentlywell marked before the critical wave number, since thegrowth rate rapidly decays. We also show how the idealdispersion relation can be consistently derived from theanalysis in Ref.[14], as soon as the corotation constraintis taken into account. However, also in the ideal case,we are able to trace the dependence of the MRI growthrate on the magnetic field profile, a feature due just to thecorotation theorem and, indeed, absent in Ref.[14].With respect to the study in Ref.[13], in addition to theco-rotation theorem, we also generalize the perturbationscheme to a stratified configuration. For a purely verti-cal magnetic field and a disk angular frequency dependingon the radial variable only (as assumed in Ref.[13]) ourdispersion relation overlaps the one derived in Ref. [13].This is due to the fact that, under such hypotheses, thecorotation theorem is automatically fulfilled.For what concerns studies which demonstrate the rele-vance of the vertical matter distribution in the disk towardthe efficiency of MRI in generating turbulence, see Ref.[15],where the relevance of the boundary condition is outlined,and Ref.[16,17]. Moreover, an interesting analysis concern-ing an effect of saturation for MRI due to the presence ofmagnetosonic waves, which also influences the emergenceof a turbulent regime, is provided in Ref.[18–21].Summarizing, the present analysis provides a clear pic-ture of how the coupling of the corotation condition forthe background and the stratified nature of the config-uration can alter the expected morphology of the MRI,producing an important asymmetry of the correspondinggrowth rate with respect to the equatorial plane of thebackground configuration. Basic equations. –
In order to set up the fundamen-tal formalism and notation at the ground of the perturba-tion analysis we are going to perform, it is worth writingdown the full set of basic equations governing the dynam-ics of a magnetized viscous fluid. As a first step, let usconsider the system composed by the Faraday equation and the generalized Ohm law: ∂ t B = − c ∇ × E , (1) E + v × B /c = 0 , (2)where E and B denote the electric and magnetic field,respectively, and v is the disk velocity field. In terms ofthe electric potential Φ and the vector potential A , Eq.(2)can be rewritten as follows ∇ Φ + ∂ t A /c = v × B /c , (3)where B = ∇ × A and ∇ · A = 0. In what follows, weconsider a two-dimensional axisymmetric system (usingcylindrical coordinates ( r, φ, z )) in which all the physicalvariables are independent of the azimuthal angle φ . With-out any loss of generality, we express the magnetic fieldvia the magnetic flux surface ψ as B ≡ − e r ∂ z ψ/r + e φ ¯ B φ /r + e z ∂ r ψ/r , (4)(here e r,φ,z denotes the coordinate versors). This expres-sion of the magnetic field is associated to a vector potentialhaving the following form A = A p + e φ r ψ , (5)where the poloidal vector potential A p satisfies, in theCoulomb gauge, the conditions ∇ × A p = e φ r ¯ B φ , ∇ · A p = 0 . (6)In the same way, the velocity field can be split into apoloidal and an azimuthal component defined as v = v p + rω e φ , v p = v r e r + v z e z , (7)and we can thus separate the azimuthal and poloidal partsof Eq.(3) as ∂ t ψ + v p · ∇ ψ = 0 , (8a) c ∇ Φ + ∂ t A p = ω ∇ ψ + v p × ( ∇ × A p ) , (8b)respectively. Taking the curl of Eq.(8b), we can built upa scalar equation which governs the azimuthal magneticfield ¯ B φ : ∂ t ¯ B φ + v p · ∇ ¯ B φ + ¯ B φ ( ∂ r v r + ∂ z v z ) − B φ v r /r == r ( ∂ z ω∂ r ψ − ∂ r ω∂ z ψ ) . (9)This equation coincides with the azimuthal component ofthe so-called induction equation, and, it is worth stressingthat Eq.(8a) is gauge invariant, since it corresponds to theazimuthal component of the generalized Ohm law whichis intrinsically gauge independent.Let us now face the analysis of the momentum conserva-tion equations. In the case of a viscous fluid, the azimuthalcomponent of the MHD Navier-Stokes equation reads ρr ( ∂ t ω + v p · ∇ ω ) + 2 ρv r ω == 14 πr (cid:0) ∂ r ψ∂ z ¯ B φ − ∂ z ψ∂ r ¯ B φ (cid:1) + η v ∇ ( rω ) , (10)p-2RI in Stratified Plasma Diskswhere ρ is the mass density and η v the dynamical viscosityof the plasma, taken here as a constant quantity. Further-more, the poloidal component assumes the form [22] ρ ( ∂ t v p + v p · ∇ v p ) − ρrω e r == − πr h ∂ r (cid:16) r ∂ r ψ (cid:17) + 1 r ∂ z ψ i ∇ ψ + −∇ P + F p − ∇ ¯ B φ πr + η v ∇ v p , (11)here P is the thermostatic pressure and F p an externalforce acting on the plasma and it coincides with the stargravity, being written as F p = − ρω k r p , (12)where r p = r e r + z e z and ω k = p GM ( r + z ) − / is theKeplerian angular frequency ( M being the central objectmass and G the gravitational constant).The dynamical system composed by Eqs.(8a), (9), (10)and (11) is completed by the continuity equation (massconservation): ∂ t ρ + ∇ · ( ρ v ) = 0 . (13)We do not assign a specific equation of state for a plasmaas a whole (relating the mass density to the pressure) dis-cussing separately the background and perturbation cases. Linear perturbations in a viscous magnetizedfluid: dispersion relation. –
The linear perturbationanalysis is performed by choosing a background config-uration corresponding to a purely differentially rotating(at ω ) plasma disk, i.e., the background poloidal compo-nent of the velocity field vanishes so that v ≡ rω e φ .The plasma is embedded in a poloidal magnetic field,associated to the background magnetic flux function ψ via Eq.(4) (here and in what follows, we denote by thesuffix 0 the background and by the suffix 1 the corre-sponding linear perturbations, i.e., for a generic quantity A → A + A ).The main assumption we adopt in our analysis is theAlfv´enic nature of the perturbations, thus we can neglectthe contribution of the magnetosonic waves and we can as-sume the plasma incompressibility, i.e., ∇· v p = ∇· v p = 0(indeed, Alfv´en waves do not transport matter). In thecase of a purely differentially rotating plasma, Eqs.(9)and (10) are automatically satisfied at zeroth order, whileEq.(11) splits in two background equations ∇ P − ρ ( ω r e r − ω k r p ) = 0 , (14)¯∆ ψ ≡ πr h ∂ r (cid:16) r ∂ r (cid:17) + 1 r ∂ z i ψ = 0 . (15)Let us now focus our attention on this background system.Eq.(15) is the force-free condition for the vacuum magneticfield of the central object. In what follows, we will adopta dipole configuration for ψ which is a natural choice forcompact astrophysical objects. In fact, far enough fromthe center (in the disk), the magnetic field is essentially dipole-like. We moreover stress that, for a thin disk (al-most coinciding with the equatorial plane), the dipole fieldreduces to a pure vertical one. Eq.(14) describes insteadthe gravostatic equilibrium and determines the disk mor-phology: note that the background pressure gradient isnot negligible in our analysis. From this equation, we seethat the presence of vertical shear in the problem ( i.e., the z -dependence of the angular velocity) is due to the verti-cal pressure gradient. In what follows, we develop a localperturbation approach dealing with wavelengths smallerthan the scale of background variation. As far as the ver-tical shear is smooth (in typical accretion disks it is ofthe half-depth order), the local approach almost overcomethe global one [14, 23]. But, when the coupling betweenthe background vertical gradient and the perturbation isrelevant (see for instance the analyses in [24] and [25]),the prediction of the local and global approach can de-viate, the latter depending significantly on the boundaryconditions.Finally, we observe that, a steady background hav-ing B φ = const. (in particular B φ ≡
0) and vanish-ing poloidal velocities is characterized by a vanishing left-hand-side of Eq.(9). Therefore, from the right-hand-side,the proportionality between the angular velocity and sur-face function gradients comes out, leading to ω = ω ( ψ ):this issue corresponds to the so-called corotation theorem[9].Let us now separate, without loss of generality, the an-gular velocity into its corotation and generic parts as fol-lows ω = ¯ ω ( ψ ) + ω ∗ , (16)where, we have ω ∗ = 0 (since ω = ¯ ω ( ψ )) and ω = ¯ ω + ω ∗ = ψ d ¯ ωdψ (cid:12)(cid:12)(cid:12) ψ + ω ∗ ≡ ω ′ ψ + ω ∗ , (17)here, the relation ∇ ω = ω ′ ∇ ψ holds. Introducing thepoloidal plasma shift ξ p = ξ r e r + ξ z e z defined by v p ≡ ∂ t ξ p and perturbing Eq.(8a), one can now write the basicrelation ψ = − ξ p · ∇ ψ ⇔ ∂ t ψ = − v p · ∇ ψ . (18)We now observe that, in the linear perturbation regime,the induced poloidal magnetic field remains much smallerthan the background component, i.e., |∇ ψ | ≪ |∇ ψ | .Moreover, the behavior of the perturbed pressure P willbe determined by preserving the incompressibility alongthe plasma dynamics.We now write the first order quantity as follows (pre-serving the axial symmetry, i.e., assuming no propagationalong the φ direction) A = ˜ Ae − i ( k p · r p − Ω t ) = ˜ Ae − i ( k r r + k z z − Ω t ) , (19)where ˜ A is a small constant amplitude while k p = k r e r + k z e z and Ω are the poloidal wave vector and the frequencyp-3. Carlevaro, G. Montani, F. Renziof the perturbation, respectively. According the local ap-proach to the perturbation dynamics, we require that thecondition k p · r p ≫ k p , we can now rewrite Eqs.(9), (10)and (11) as Ω ¯ B φ − r ω ∗ k p · B = 0 , (20) ir ( i Ω + νk p ) ω ∗ − ω ξ r ++ k p · B πrρ ¯ B φ + iνk p rω ψ = 0 , (21) i Ω( i Ω + νk p ) ξ p − rω ( ω ′ ψ + ω ∗ ) e r + − i k p P ρ − k p ∇ ψ πr ρ ψ = 0 , (22)respectively, where ν ≡ η v /ρ and we have to take into ac-count the incompressibility constraint k p · ξ p = 0. Above,we also used the relation ω ′ ∂ t ψ = − ω ′ v p · ∇ ψ ≡− v p · ∇ ω which is guaranteed by Eq.(18). Analogously,in Eq.(20) the contribution ω ′ ψ naturally cancels in theright-had side.Combining Eqs.(20) and (21), we get r∂ r ψ ( − i Ω( i Ω + νk p ) − ω A ) ω ∗ ++2 ω Ω ∂ r ψ ψ − νk i Ω y r / ω = 0 , (23)where ω A ≡ ( k p · v A ) is the frequency associated to theAlfv´en speed v A = p B / πρ and ϕ ≡ rω ω ′ ∂ r ψ in-cludes information about the angular velocity gradient.Let us now take the scalar product of Eq.(22) with thewave vector k p to obtain the behavior of the perturbedpressure P iP = − h rρ ω ( ω ′ ψ + ω ∗ ) k r k p + k p · ∇ ψ πr ψ i , (24)where we have required that the incompressibility condi-tion is preserved along the dynamics. Substituting theexpression above in Eq.(22), multiplying it by ∇ ψ andnoting that ψ = − ξ p · ∇ ψ , we obtain the following basicequation (cid:2) − i Ω( i Ω + νk p ) − ω A − δϕ (cid:3) ψ = 2 δrω ∂ r ψ ω ∗ , (25)where δ ≡ − k r ( k p · ∇ ψ ) k p ∂ r ψ . (26)In the same way, the radial component of Eq.(22) provides ξ r in terms of ψ and ω ∗ : − i Ω( i Ω + νk p ) ∂ r ψ ξ r == − ( αϕ + δk p v Az ) − αrω ∂ r ψ ω ∗ , (27)with α = 1 − k r /k p , v Az = ∂ r ψ / p πρ r . (28) Combining together Eqs.(23), (25) and (27), leads tothe following dispersion relation q + δ (cid:2) ϕω A − s Ω (cid:3) = 0 , (29)where we have defined q = − i Ω( i Ω + νk p ) − ω A , s = 4 αω /δ + ϕ . (30)In what follows, we extract information from such a dis-persion relation to characterize the MRI validity regions. Physical implications. –
For further analysis, it isconvenient to rewrite Eq.(29) in a dimensionless form.Thus, we introduce the following variables: y = i Ω ω , ¯ k = ω A ω , ¯ s = s ω , ¯ ν = νω χ A , (31)where, χ A = ω A /k p = δ v Az /α denotes an effective Alfv´enspeed and we underline how, for fixed magnetic field, thevariable ¯ k properly represents a normalized wave vector.Using such definitions, the dispersion relation takes theform y + 2¯ ν ¯ k y + (2¯ k + δ ¯ s + ¯ ν ¯ k ) y ++2¯ ν ¯ k y + ¯ k ( δ (¯ s − α/δ ) + ¯ k ) = 0 . (32)Such a quartic equation in the y variable can be analyt-ically studied only in some simplified cases [13, 26, 27].Thus, we numerically integrate Eq.(32) focusing on thesolutions in y having a positive real part, correspondingto unstable modes with Im[Ω] < α = const. , constrainingthe orientation of the wavenumber k p in the ( r, z ) plane.Moreover, the stellar magnetic field can be reliably repre-sented by a background dipole-like configuration (satisfy-ing the force-free condition, i.e., Eq.(15)) as ψ ( r , z ) = µ r ( r + z ) − / , (33)with µ = const. The gravitational field is retained asNewtonian, and we use the functional dependence ofthe angular velocity on the equatorial plane ω | z =0 = ω k | z =0 = GM µ /ψ | z =0 , assuming to extend this expres-sion everywhere in the disk by virtue of the corotationtheorem: ω = GM µ /ψ . (34)This relation allows to predict the behavior of the rotationprofile even far away from the midplane. The stratified ideal case.
Let us now discuss the so-lutions of Eq.(32) in the inviscid limit. For vanishing vis-cosity, the dispersion relation takes the following form y + y ( δ ¯ s + 2¯ k ) + ¯ k ( δ (¯ s − α/δ ) + ¯ k ) = 0 . (35)This equation is the same found in Ref.[14] for a divergentpolytropic index and, in the limit of k r = 0 (or equiva-lently for vanishing radial magnetic field), it reduces top-4RI in Stratified Plasma Disksthe proper dispersion relation for a thin accretion diskwith a Keplerian rotation profile [3, 11]. The main differ-ence, here, is the appearance of the factor δ which containsinformations on the magnetic field behavior in the three-dimensional space through the ratio ∂ z ψ /∂ r ψ (once fixedthe value of α ). The physical content of the inviscid dis-persion relation is then summarized by the instability con-dition: ¯ k < ¯ k c ≡ p δ (4 α/δ − ¯ s ) , (36)while, for ¯ k > ¯ k c , one gets y = 0. Moreover, Eq.(35) is asimple quadratic form in Ω and it is easy to show that amaximum unstable growth rate exists: y M = δr∂ r ω / (2 ω α ) , (37)occurring when the Alfv´en frequency assumes the follow-ing expression ω A = ω A ( M ) ≡ = − rβ∂ r ω ( ω + rβ∂ r ω / α ) . (38)For a dipole-like configuration, which properly describesstellar magnetic fields, we get ∂ z ψ /∂ r ψ = 3 zr/ ( r − z )and three different cases can be distinguished accordingly: k R e @ y D z = r z = rz = r z = r Fig. 1: Effect of the height on the MRI in inviscid disks withdipole-like magnetic field. The curves represent real solutions(unstable) of Eq.(35) as function of the dimensionless wavevector ¯ k and they are obtained fixing the radial coordinate r and the disk background configuration: this leads to unstablemodes which differ only for the value of z (as indicated in theplot). Increasing the height, the MRI interval shifts to lowwave vectors and shrinks. (i) z >
0. In this case, δ decreases for increasing value of z . For δ >
0, a critical height z ∗ exists where ¯ k c = 0and the MRI is completely suppressed by virtue ofthe condition (36). In fact, the standard stabilityconstraint for a magnetized disk [14] is found to bealways satisfied for z > z ∗ . It is worth stressing that,where α < δ <
0, the criterium (36) reads ω A < − δϕ .Clearly, this region becomes more and more stablewhile approaching z ∗ . In Fig. 1, the δ suppression isreported for different heights. (ii) z = 0. On the equatorial plane, ∂ z ψ = 0 and, thus, δ = α . The resulting dispersion relation is a gener-alization of what is found for a Keplerian disk under the assumption k p k B (which, in the flux surfaceformalism, reads k p · ∇ ψ = 0 [11]). In this case,changing the orientation of the wave vector in the( r, z ) plane reproduces a suppression similar to thatdiscussed for the case z >
0. In Fig. 2, the midplanebehavior corresponding to α = 0 . (iii) z <
0. Below the equatorial plane, δ > α , and thesituation is the opposite of the case (i) . The resultingMRI mode is enhanced but the amplification is weakand the midplane behavior is resembled because δ isessentially generally of order unity. Nonetheless, forpeculiar magnetic configurations, it is possible to getsignificant enhancement for the MRI mode. k R e @ y D Fig. 2: Maximum growth rate for inviscid disks with dipole-like magnetic field. The radial coordinate is the same of Fig.1. The curve represents the real solution of Eq.(35) for z = 0as function of the dimensionless wave vector ¯ k . - k R e @ y D Fig. 3: Same as in Fig. 2 for viscous disks with ¯ ν = 1. Thecritical point ¯ k c does not vary with respect to the inviscid case,but the mode amplitude decreases by about 1/3 (cfr. Fig. 2).For ¯ k > ¯ k c , the mode turns into a damping proportional to¯ ν ¯ k . The stratified viscous case.
We are now going to dis-cuss the general solution of Eq.(32). It describes the sta-bility behavior of an incompressible stratified viscous diskand, as discussed above, the system geometry enters thedispersion relation through the factor δ . It is moreovereasy to verify that it reduces to the proper dispersion re-p-5. Carlevaro, G. Montani, F. Renzilation for adiabatic perturbations in a thin viscous mag-netized disk [13, 27], or equivalently to that found for arotating metal annulus [7], for vanishing k r . The effectof viscosity is to make more stable the disk configurationand, consequently, the MRI has a lower growth rate withrespect to the inviscid case.In Fig. 3, we plot the unstable solution on the equatorialplane for ¯ ν = 1 and α = 0 .
7. This is actually the samecase of Fig. 2 and, here, the effect of the viscosity can beeasily recognized comparing the maximum growth rate. - k R e @ y D z = rz = rz = rz = r Fig. 4: Influence of the height on MRI in viscous disks withdipole-like magnetic field. The curves are the unstable solu-tions to Eq.(32), in a narrow range of z near z ⋆ . In this region,viscosity dominates the dynamics and enhances the suppres-sion due to a stratified configuration leading to unstable modeswhose features depend on ¯ ν . The same transition is observedin a thin disk at increasing viscosity [13, 27]. As in Fig. 1,curves differ only in the vertical coordinate value. It is worth stressing that, in a stratified disk, the di-mensionless viscosity parameter ¯ ν is also a function of theheight z (through the effective Alfv´en speed χ A ) and, inthe region where α δ
0, it grows by several orders ofmagnitude. In fact, the magnetic field decreases in am-plitude approaching the critical height z ∗ and the effectof viscosity combines with the height suppression. When¯ ν ≫
1, viscosity dominates the perturbation dynamics andthe morphology of the instability changes. One can alsoverify that the z dependence of the growth rate (in thisregion of the disk) reproduces exactly the same behaviorof the increase the disk viscosity at z = 0. In Fig. 4, theunstable solution is plotted for different heights near z ∗ .As demonstrated in Ref.[27], an analytical solution forthe maximum growth rate and the critical wave number ofthe instability (named here ¯ k νc ) can be derived for visco-resistive disks. For vanishing resistivity, however, whilethe critical wave number becomes that of the ideal case¯ k νc = ¯ k c , i.e., it does not directly depend on viscosity(see Appendix A), the maximum growth rate can not bederived analytically and the solution: y νM = (¯ s δ ) − / p − ¯ s δ/ (2¯ ν ) , (39)has to be intended as the exact solution for very largeviscosity and as an upper limit for ¯ ν >
1. It is worth stressing that even if the critical wavenumber coincideswith that found in the inviscid case, the viscosity dampingmakes the mode amplitude for ¯ k . ¯ k νc always negligible.Clearly, the instability is not only suppressed in amplitudeby the vanishing δ value, but it is additionally damped for¯ k νM < ¯ k < ¯ k c , where ¯ k νM corresponds the wavenumberassociated to the maximum growth rate y νM . We concludethis Section underlining that the situation is reversed inthe region where δ > α . Conclusions. –
We developed a perturbation anal-ysis of the stratified configuration concerning an incom-pressible plasma disk, by using the magnetic-flux func-tion as the basic dynamical variable. We considered thebackground as associated to a purely differentially rotat-ing inviscid stratified profile and we explicitly imposed thecorotation theorem, i.e., the dependence of the disk an-gular velocity on the unperturbed magnetic-flux function.We emphasize how, including viscosity on the backgroundwould not affect the validity of such a theorem, becausethe electron force balance equation is not influenced byviscosity.We first analyzed the case of ideal perturbations to thebackground and then we included viscous effects. Wedemonstrated, in both these cases, the emergence of avertical cut-off on the MRI in the positive z -axis, overa critical height and for a sufficiently large wavenumber(the same in the viscid and inviscid cases). However, thedamping is much more marked in the presence of viscosity,since the growth rate is significantly suppressed alreadybefore the critical height and wavenumber. Such an asym-metry of the MRI can have significant consequences onthe transport features of the plasma disk, especially whenwe recall that MRI is the only reliable mechanism to in-duce the necessary turbulence postulated by the Shakuraidea for the accretion mechanism [22,26]. By other words,when the induced effective viscosity, responsible for a non-zero infalling velocity, is sufficiently relevant to influencethe perturbation dynamics, the vertical asymmetry of theMRI (which we demonstrated to be already present in theideal case) is enhanced so much that the accretion processcan take place efficiently only on one side of the equato-rial plane. On the other side, the MRI generates viscosityand it is, in turn, suppressed by its own product (actuallythe turbulent flow), so that we are led to think that theinfalling velocity should be much weaker there. Clearly,this is just a qualitative statement, which requires furtherinvestigation to be applied to a real disk-like accretionstructure.We conclude by stressing how the present analysis hasa relevant astrophysical interest, since it concerns realplasma configurations accreting around compact objects(for a review on stellar accretion disks see Ref.[22], whilefor a discussion on the plasma stability within such sys-tems, see Ref.[26]). Although the thin disk approximationsucceeds in describing the basic features of many types ofstellar accreting plasma, some structures are significantlyp-6RI in Stratified Plasma Disksthick and require a separate analysis, especially in view oftheir non-Keplerian differential rotation [28]. In this thickplasma configurations, the pressure gradients play an im-portant role in fixing the steady equilibrium configurationand then in determining the behavior of linear perturba-tions. When the vertical pressure and mass density gradi-ents are significantly stiff, the approximation of an angularfrequency depending on the radial distance from the centeronly, appears rather rough. In fact, Eq.(14) directly linksthe pressure and mass density vertical behaviors to thevertical variation of the angular frequency. The analysishere addressed applies just to systems of plasma arrangedin such a way to be both thick and stratified disks. Thechoice of a dipole-like magnetic field is very reliable foraccreting systems, whose background configuration doesnot react to the central object field, which in the disk re-gion, i.e., far enough from the star surface, is essentiallydescribed by its dipole component [22].We observe that a dipole-like magnetic field is essen-tially vertical in a thin disk configuration, almost coincid-ing with the equatorial plane of the accreting structure.Therefore, the most important deviation from the stan-dard thin disk morphology is expected far enough fromthe equatorial plane, where the dipole-like field acquires asufficiently large radial component. This is just what weobserved in fixing a vertical quote for the MRI suppres-sion both in the ideal and viscous cases. In this respect,the present study is reliably applicable to thick structures,having a sufficiently large vertical shear, like AdvectiveDominated Accretion Flows [29], but it can be also in-teresting for transient collapsing configurations, like Cat-aclysmic Variables [30] (see also Ref.[31]).Our study rises interesting questions concerning howthe turbulence and accretion profiles [6] are deformed inview of the vertical shear. In particular, the suppressionof MRI in the upper half plane of the configuration sug-gests that there the accretion mechanism can no longerrely on the effective viscosity due to turbulence and itwould require alternative processes for the angular mo-mentum transport. Appendix A. –
Here, we are going to discuss theexpression of the critical dimensionless wavenumber, ¯ k νc ,as it arises from the dispersion relation (29). In orderto derive an equation for ¯ k νc , we introduce the followingdimensionless variables X = y + ¯ ν Γ , Γ = ¯ k , ¯ ϕ = ϕ/ω . (40)Therefore, Eq.(29) can be rewritten in the equivalent form X − νδX + (cid:0) δ + ¯ ν δ + ¯ s (cid:1) X + − νδ (cid:0) δ + ¯ s (cid:1) X + δ + β ¯ ϕδ + ¯ ν δ ¯ s = 0 . (41)Clearly, at the critical point determined by the condition y (Γ c ) = 0, we have X = ¯ ν Γ c (¯ k νc ). Consequently, it iseasy to verify that the following equation for Γ c holds:(Γ c + ¯ ϕδ )Γ c = 0, and the only non-trivial solution is ¯ k νc = √− δ ¯ ϕ , which is equivalent to Eq.(36). The sameresult can be found in Ref.[13, 27] for vanishing magneticresistivity. REFERENCES[1]
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