Linear Instabilities Driven by Differential Rotation in Very Weakly Magnetized Plasmas
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed September 14, 2018 (MN L A TEX style file v2.2)
Linear Instabilities Driven by Differential Rotation in VeryWeakly Magnetized Plasmas
E. Quataert ? , T. Heinemann , & A. Spitkovsky Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley Kavli Institute for Theoretical Physics, University of California, Santa Barbara Department of Astrophysical Sciences, Princeton University
September 14, 2018
ABSTRACT
We study the linear stability of weakly magnetized differentially rotating plasmasin both collisionless kinetic theory and Braginskii’s theory of collisional, magnetizedplasmas. We focus on the very weakly magnetized limit in which β (cid:38) ω c / Ω, where β isthe ratio of thermal to magnetic energy and ω c / Ω is the ratio of the cyclotron frequencyto rotation frequency. This regime is important for understanding how astrophysicalmagnetic fields originate and are amplified at high redshift. We show that the singleinstability of fluid theory - the magnetorotational instability mediated by magnetictension - is replaced by two distinct instabilities, one associated with ions and onewith electrons. Each of these has a different way of tapping into the free energy ofdifferential rotation. The ion instability is driven by viscous transport of momentumacross magnetic field lines due to a finite ion cyclotron frequency (gyroviscosity);the fastest growing modes have wavelengths significantly longer than MHD and HallMHD predictions. The electron instability is a whistler mode driven unstable by thetemperature anisotropy generated by differential rotation; the growth time can beorders of magnitude shorter than the rotation period. The electron instability is anexample of a broader class of instabilities that tap into the free energy of differentialrotation or shear via the temperature anisotropy they generate. We briefly discuss theapplication of our results to the stability of planar shear flows and show that such flowsare linearly overstable in the presence of fluid gyroviscosity. We also briefly describethe implications of our results for magnetic field amplification in the virialized halos ofhigh redshift galaxies.
Key words: instabilities – plasmas – accretion, accretion disks – galactic halos
Balbus & Hawley (1991) demonstrated that even energet-ically weak magnetic fields can be dynamically important:when the magnetic energy in a plasma is very small comparedto the thermal or rotational energies, magnetic tension isnonetheless important for small wavelength fluctuations. Forthe specific case of the magnetorotational instability (MRI)in magnetohydrodynamics (MHD), the fastest growing modehas a growth rate of ∼ Ω (independent of the field strength)and a wavelength ∼ v a / Ω, where Ω is the rotation rate and v a is the Alfvén speed.For sufficiently weak magnetic fields the fastest growingmode predicted by the MHD theory of the MRI can havea wavelength sufficiently small that the single fluid MHD ? E-mail: [email protected] approximation breaks down. Unless there are large primordialmagnetic fields generated in the early Universe, weak fieldsof this magnitude will inevitably be the ‘initial condition’during the formation of the first stars and galaxies. A naturalquestion is how such magnetic fields get amplified to thepoint where MHD becomes a plausible model of the plasmadynamics? And are magnetic stresses dynamically importantfor the formation of even the first astrophysical objects? Inthis paper we address aspects of this problem by consideringthe linear stability of differentially rotating plasmas with veryweak magnetic fields in both collisionless kinetic theory andthe collisional magnetized theory of Braginskii (1965). Thisallows us to begin to address how very weak initial magnetic For brevity we shorten ‘collisionless kinetic theory’ to ‘kinetictheory’ in most places in this paper. And by Braginskii’s collisionalmagnetized theory we specifically mean the anisotropic viscousc (cid:13) a r X i v : . [ a s t r o - ph . GA ] D ec E. Quataert, T. Heinemann, & A. Spitkovsky fields can be amplified even when ions and electrons are onlypartially magnetized (in the sense of having Larmor radiicomparable to the size of the system under study).There is a significant literature studying extensions ofthe MRI beyond the ideal MHD approximation (e.g., Blaes &Balbus 1994; Wardle 1999; Quataert et al. 2002; Ferraro 2007).One approach to studying the weak field limit is based on HallMHD, which takes into account the difference between theion and electron dynamics when fluctuations have timescalescomparable to or shorter than the ion cyclotron period. TheMRI persists even in this limit (as a destabilized whistlerwave), with the same maximum growth rate as in MHD(Wardle 1999; Balbus & Terquem 2001). The Hall MHDtheory of the MRI is motivated primarily by the applicationto protostellar disks where the plasmas are collisional butdeviations from MHD are due to the very low density ofcharge carriers (e.g., Lesur et al. 2014). By contrast, HallMHD does not provide a good description of low-collisionalityweakly magnetized plasmas with β (cid:29)
1, as Ferraro (2007)emphasized in the context of the MRI (see, e.g., Howes 2009for a more general discussion of some of the limitations ofHall MHD). Our approach in this paper is to carry outa kinetic linear stability calculation, valid so long as thefluctuations of interest have wavelengths smaller than theelectron and proton mean free paths. Our work draws heavilyon that of Heinemann & Quataert (2014) (hereafter HQ),who studied the linear kinetic theory of local instabilities indifferentially rotating plasmas. We also show that the kineticion instability described in this paper has a fluid analoguein which both planar shear flows and differentially rotatingplasmas are destabilized by gyroviscosity.In §2 we summarize the aspects of HQ’s formalism im-portant for our analysis. We then present numerical solutionsfor linear instabilities of differentially rotating plasmas forthe case of weak magnetic fields aligned or anti-aligned withthe rotation axis of the system (§3.1); §3.2 & 3.3 presentanalytical approximations to these numerical instability cal-culations and elucidate the physics. In §4 we show how theresults derived in §3 can be applied to the problem of pla-nar shear flows in addition to differentially rotating plasmas.In §5 we briefly describe the application of our results tomagnetic field amplification in the virialized halos of highredshift galaxies. Finally, in §6 we summarize and discussour results.
HQ derived the linear theory of the shearing sheet for acollisionless plasma. We review here some of their resultsthat are important for our analysis but we largely defer totheir paper for details. For consistency, we utilize the samenotation as HQ throughout (including their use of SI unitsfor electromagnetism). We use the subscript s to represent aparticular particle species (e.g., electron, ion) but drop thesubscript for clarity when it is not required.We describe the dynamics of a local patch of a differ-entially rotating flow in a rotating reference frame. The transport present in a collisional plasma when the cyclotron fre-quency is larger than the collision frequency (see §3.2.1). coordinate system is locally cartesian and we neglect verticalstratification. The coordinate system is such that the rotationaxis is in the e z direction so that the background rotationalvelocity is along e y . The equilibrium magnetic field is givenby B = B y e y + B z e z . The equilibrium thus corresponds toa uniform density n in x, y, and z , with the bulk motion ofeach plasma species as viewed in the rotating reference framegiven by u = − q Ω x e y where q ≡ − d Ω d ln r . (1)The Maxwellian distribution function with uniform densityand a background velocity given by equation 43 is f ( K ) = n exp( −K /v t )(2 π ) / v t √ − ∆ , (2)where v t = const is the thermal velocity, K = 12 (cid:20) v x + ( v y + q Ω x ) − ∆ + v z (cid:21) (3)is the gyration energy (the difference between a particle’senergy and the energy of a hypothetical particle with thesame angular momentum but on a circular orbit),∆ = q Ω ω c b z + 2Ω , (4)is the tidal anisotropy and ω c = eB/m is the cyclotronfrequency. The sign conventions used here are that e can bepositive or negative while B > b z = B z /B denotes thecomponent of the total field along the rotation axis. Notethat b z = ± b z = 1) or anti-aligned ( b z = −
1) with respect tothe rotation axis.Equation 3 shows that ∆ in equation 4 is the species-dependent temperature anisotropy imposed by the back-ground differential rotation. This is the level of temperatureanisotropy inevitably created by differential rotation in acollisionless plasma and is distinct from the more familiartemperature anisotropy with respect to a mean magneticfield. The temperature anisotropy implied by equation 3 is T x = T y − ∆ . (5)Depending on the sign of ∆, T y can be either less than orlarger than T x . For the case of a vertical magnetic field, thetidal anisotropy is entirely in the plane perpendicular to themean magnetic field, i.e., it is distinct from the typical tem-perature anisotropy considered in homogeneous magnetizedplasmas. By contrast, for the more general case of B z (cid:44) B y (cid:44)
0, the tidal anisotropy includes both an anisotropy inthe plane perpendicular to the magnetic field and anisotropywith respect to the mean magnetic field. Finally, we note thatthe dynamics of differential rotation only imposes an x − y ( r − φ ) temperature anisotropy. In equation 2 we have forsimplicity taken T z = T x although this in general need notbe true. Relaxing the restriction to T z = T x would generatean even larger class of instabilities driven by temperatureanisotropy than those that we present in this paper.For an unmagnetized plasma, ∆ = q/
2, which is alsothe standard anisotropy of stellar dynamics (Shu 1969); thiscorresponds to T y = T x / q =3 /
2. In the opposite guiding center limit in which ω c b z / Ω → c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas ∞ , ∆ →
0. In this paper, we are interested in the weak fieldlimit, which corresponds to finite ∆ (cid:44) k y = 0) perturbations about theequilibrium state described above, with perturbations ∝ exp( − iωt + k · x ). Note that Im ω > D · (cid:16) − q Ω iω e x e y (cid:17) · δ ˜ E = 0 (6)where ˜ E = E − q Ω x e y × B is the electric field as seen by anobserver that is locally at rest with respect to the backgroundshear flow. Note that det( − q Ω /iω e x e y ) = 1 so that thedispersion relation is determined solely by the dispersiontensor, which is given by D = (cid:0) k − kk − iωµ σ (cid:1) v a (7) − q Ω X s n s m s ρ ω cs b z ω cs b z + 2Ω e y e y where the sum is over each species s , ρ = P s n s m s is thetotal mass density, v a = B / ( µ ρ ) is the square of the Alfvénspeed, and σ is the linear shearing sheet conductivity tensor(summed over species) calculated by HQ. The latter is relatedto the linear response tensor of each species Λ s using σ = − iω X s e s n s m s Q s · Λ s · Q s . (8)where Q s = e x e x + e y e y p − ∆ s + e z e z . (9)The full plasma response tensor Λ s for each species for thedistribution function in equation 2 is given in HQ’s eqs 49-53.This in turn depends on the plasma response function W ( ξ )(Ichimaru 1973), which is related to the standard plasmadispersion function Z via W ( ζ ) = 1 + ξZ ( ξ ) with ζ = √ ξ . k k Ω k B HQ numerically solved for the linear theory of the MRI for k k Ω k B (including the case of B and Ω anti-parallel) forthe case of kinetic ions and cold, massless electrons. Here wegeneralize their results and show that in the weak field limitaccounting for kinetic electrons substantially changes thephysics and growth rates. We also clarify some of the physicsof modes driven by kinetic ions with finite cyclotron frequency,a case considered by Ferraro (2007) & HQ. We take q = 3 / Figure 1 shows growth rates as a function of k z v a / Ω forseveral different values of ω cp / Ω and β p = β e . Figures 2 &3 then show the maximum growth rate and the wavelengthof the fastest growing mode as a function of ω cp / Ω and β ,respectively. In all three Figures, we take b z = 1, i.e., Ω and B are parallel. The anti-parallel case is shown in Figure 4discussed below. Figures 1-3 also show solutions for threedifferent approximations to the physics: Hall MHD, i.e., thecold ion and cold, massless electron limit of kinetic theory(black solid lines), kinetic ions and cold, massless electrons(dashed colored lines), and kinetic ions and electrons (solidcolored lines). In all of our calculations with kinetic electrons,we take T p = T e (and hence β p = β e ) and m p = 1836 m e .The Hall MHD results shown in Figures 1-3 repro-duce the well-known MRI results in the literature, withthe maximum growth rate of q Ω / k z v a / Ω ∼ min(1 , [ ω cp / Ω] / ) (Wardle 1999; Balbus & Terquem 2001).Figures 1-3 show, however, that the physics is very dif-ferent for the case of kinetic ions and cold, massless elec-trons. Most notably, the maximum growth rate can reach( − d Ω /d ln r ) / = √
3Ω (for q = 3 /
2) for β (cid:29) ω cp / Ω (cid:38) kv a / Ω ∼ β − / ( ω cp / Ω) / , which is equiv-alent to kv tp / Ω ∼ ( ω cp / Ω) / . These significant differencesbetween the kinetic ion and Hall MHD results are particu-larly striking in the low k peak in the dispersion relation inFigure 1 and the ω cp / Ω (cid:38)
10 solution for the fastest growingmode in Figure 2. Note, moreover, that although this kineticion mode is not the fastest growing mode in the presence ofkinetic electrons, the low k peak in the dispersion relation isnot significantly modified by the inclusion of kinetic electrons.It is an essentially ion driven mode. We elucidate the physicsof this ion-driven instability in §3.2.Finally, we turn to the case of kinetic electrons. Figures1-3 show that the inclusion of kinetic electrons introduces afundamentally new unstable mode. Remarkably, althoughthe wavelength of the fastest growing mode is similar tothe case of Hall MHD, the growth rate is far faster, with γ (cid:29) Ω for high β e and finite ω cp / Ω. Moreover the growth rateincreases ∝ β / e at fixed ω cp / Ω (Fig. 3). This can exceed byorders of magnitude the previously known fastest growth ratefor modes driven by the free energy in differential rotation( − d Ω /d ln r ; Quataert et al. 2002). We explain these resultsanalytically in §3.3.Figure 4 shows growth rates as a function of k z v a / Ω forcase of b z = −
1, i.e., Ω and B anti-parallel. As before, we con-sider several different values of ω cp / Ω and β p = β e and showthe dispersion relation curves for same three physics casesconsidered in Figures 1-3, namely Hall MHD, kinetic ionsand cold, massless electrons, and kinetic ions and electrons.A comparison of Figures 1 and 4 shows that the ion-driveninstability at k z v a / Ω (cid:28) b z , withrapid growth only for b z = 1. By contrast, the kinetic electroninstability at k z v a ∼ Ω is insensitive to the sign of b z . Weexplain these results analytically in the following subsections.For the antiparallel case shown in Figure 4 we were c (cid:13)000
1, i.e., Ω and B anti-parallel. As before, we con-sider several different values of ω cp / Ω and β p = β e and showthe dispersion relation curves for same three physics casesconsidered in Figures 1-3, namely Hall MHD, kinetic ionsand cold, massless electrons, and kinetic ions and electrons.A comparison of Figures 1 and 4 shows that the ion-driveninstability at k z v a / Ω (cid:28) b z , withrapid growth only for b z = 1. By contrast, the kinetic electroninstability at k z v a ∼ Ω is insensitive to the sign of b z . Weexplain these results analytically in the following subsections.For the antiparallel case shown in Figure 4 we were c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky − − − − − − γ / Ω ω cp b z / Ω = 0 . β p ω cp b z / Ω = 1 10 − − − − − k z v a / Ω10 − − − − − − γ / Ω ω cp b z / Ω = 10 10 − − − − − k z v a / Ω ω cp b z / Ω = 100 10 Figure 1.
Numerical growth rates γ predicted by the kinetic ion and electron dispersion relation (solid colored lines) for k k Ω k B fordifferent values of β p = β e and ω cp / Ω (taking T e = T p and m p = 1836 m e ). Also shown is the dispersion relation for kinetic ions butcold, massless electrons (dashed lines) and the Hall MHD (cold ion, cold massless electron) dispersion relation (black solid lines, whichcorrespond to β p = 0). For weak magnetic fields (high β ) kinetic ion physics generates an instability at much longer wavelengths relativeto that predicted by MHD or Hall MHD. This produces the distinctive peaks in the growth rate at low k z v a / Ω (see §3.2 and eq. 20 forthe interpretation). Moreover, at high β kinetic electrons lead to substantially enhanced growth rates (cid:29) Ω at k z v a / Ω ∼
1. This is anelectron whistler instability driven by the temperature anisotropy in a differentially rotating plasma (see §3.3 and eqs. 32 & 37). unable to find growing modes for ω cp (cid:46) Ω, unlike for b z = 1,where there can be rapid growth associated with kineticelectrons (Fig. 2) even when the ions become effectivelyunmagnetized at low cyclotron frequency. As we discussin §3.3.1, this lack of growth at ω cp (cid:46) Ω for Ω and B anti-parallel is likely an artifact of restricting our numericalsolutions to k = k z e z only.Taken together, Figures 1-4 demonstrate that the physicsof kinetic ions and electrons dramatically change the proper-ties of instabilities driven by differential rotation in high β low collisionality plasmas relative to that predicted by theMHD or Hall MHD theory of the MRI. Ion-driven instabili-ties grow on large scales k z v a (cid:28) Ω where magnetic tensionis irrelevant. In addition, there is a new kinetic electron in-stability with growth rates (cid:29)
Ω, far exceeding previouslyknown instabilities driven by differential rotation.
In this section we analytically derive an approximate disper-sion relation for the large-scale ( kv a (cid:28) Ω) unstable modeshown in Figures 1-4, and provide a physical interpretationof the results. On these scales magnetic tension is negligible so the physics of the instability is quite different from thatof the more familiar ideal MHD theory of the MRI.As is evident from Figures 1-4, the large scale modesare not influenced significantly by electron dynamics. Wemay thus treat the electrons as a cold, massless, charge-neutralizing fluid (the Vlasov-fluid approximation, see Frei-dberg 1972). For the sake of brevity we shall also simplifynotation in this section so that that species dependent quan-tities without a subscript are proton quantities.For parallel modes with b z = ± k = k z e z , thelinear response tensor Λ required to calculate the dispersionrelation (eq. 7) is given by equations HQ71 and HQ72. Thearguments of the plasma dispersion function are ζ ± = ω ± ω g kv t , (10)where k = | k z | , ω g = (1 − ∆) S z S z = ω c b z + 2Ω (11)and ω g > k peakin the dispersion relation in Figure 1 due to kinetic ions canbe derived analytically by assuming the ordering β (cid:29) ω c / Ω (cid:29) c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas − γ / Ω β p = 10 β p = 10 β p = 010 − − ω cp b z / Ω10 − − − k z v a / Ω Figure 2.
Growth rate and wavevector of the fastest growing modefor the kinetic ion and electron dispersion relation (solid coloredlines) as a function of proton cyclotron frequency ω cp / Ω for k k Ω k B , T e = T p , and m p = 1836 m e . Dashed lines show solutions forkinetic ions and cold, massless electrons and black solid lines showthe Hall MHD solution (the cold ion and cold, massless electronlimit of kinetic theory, i.e., β p = 0). At high β p (cid:38) ω cp / Ω (cid:38) ω cp / Ω, i.e.,for very weakly magnetized plasmas, the solutions with kineticelectrons differ significantly from the Hall MHD or kinetic ionsolutions, with much faster growth rates. This is the electronwhistler instability driven by the temperature anisotropy in adifferentially rotating plasma (see eq. 35-37 in §3.3). and looking for modes with ω/ Ω ∼ kv t / ( ω c Ω) / ∼ . (13)The first inequality in equation 12 will always be satisfied forsufficiently weak magnetic fields. The second equality saysthat we are roughly in the guiding center limit (though inpractice the analytic theory is a reasonable approximationeven for ω c ∼ Ω). In the following we will refer to equations12 and 13 together as the gyroviscous ordering . Note thataccording to this ordering we have v a / Ω (cid:28) k − (cid:28) v t / Ω . (14)The modes under consideration are thus large scale comparedto the MRI scale v a / Ω but small scale compared to thethermal scale height v t / Ω of the disk.Physically, the ratio kv t / ( ω c Ω) / is the ratio of the rateof angular momentum redistribution by viscous stresses (off-diagonal components of the pressure tensor) to the rotationrate. This is most easily seen in the fluid theory of Braginskii(1965), in which the anisotropic momentum transport in acollisional, magnetized plasma contains a contribution alongthe magnetic field (that depends on the collision frequencybut is independent of the cyclotron frequency) and a cross-field component (the gyroviscous stress) that depends on the γ / Ω ω cp / Ω10 − β p − − − k z v a / Ω Figure 3.
Growth rate and wavevector of the fastest growingmode for the kinetic ion and electron dispersion relation (solidcolored lines) as a function of β p = β e (taking k k Ω k B , T e = T p ,and m p = 1836 m e ). Dashed lines show solutions for kinetic ionsand cold, massless electrons while the black solid lines show theHall MHD solution, which corresponds to β p = 0. At low β p ,Hall MHD provides a reasonable approximation to the full kineticsolution, as shown by the convergence of the kinetic and HallMHD solutions at low β p . For weak fields, however, i.e., high β p ,the fastest growing modes have growth rates much larger thanpredicted by either Hall MHD or a kinetic ion, fluid electron theory(and with γ ∝ β / ). This rapid growth is an electron whistlerinstability driven by the temperature anisotropy in a differentiallyrotating low collisionality plasma (see eq. 35-37 in §3.3). cyclotron frequency but is independent of the collision fre-quency (see §3.2.1 below for more details). Thus the orderingin equation 13 implies that gyroviscous transport of angularmomentum is dynamically important. Magnetic tension onthe other hand is negligible. Indeed, the gyroviscous orderingimplies kv a Ω ∼ (cid:18) ω c β Ω (cid:19) / (cid:28) . (15)Magnetic tension thus has no effect on the dynamics of themodes under consideration (which have ω ∼ Ω).Given the gyroviscous ordering, it also follows that thearguments of the plasma dispersion function ζ ± ∼ (cid:16) ω c Ω (cid:17) / (cid:29) . (16)We may thus employ the large argument expansion of theplasma dispersion function: W ( ζ ) ’ − ζ − .For parallel modes with b z = ± k = k z e z , theunstable branch of the dispersion relation is obtained fromthe perpendicular (with respect to e z ) dispersion tensor D ⊥ in equation 7. We wish to find the leading order incarnationof this tensor in the gyroviscous ordering. In order to doso, we introducing the ordering parameter (cid:15) (cid:28) β ∼ /(cid:15) and ω c / Ω ∼ /(cid:15) . It is not difficult to verify that c (cid:13)000
Growth rate and wavevector of the fastest growingmode for the kinetic ion and electron dispersion relation (solidcolored lines) as a function of β p = β e (taking k k Ω k B , T e = T p ,and m p = 1836 m e ). Dashed lines show solutions for kinetic ionsand cold, massless electrons while the black solid lines show theHall MHD solution, which corresponds to β p = 0. At low β p ,Hall MHD provides a reasonable approximation to the full kineticsolution, as shown by the convergence of the kinetic and HallMHD solutions at low β p . For weak fields, however, i.e., high β p ,the fastest growing modes have growth rates much larger thanpredicted by either Hall MHD or a kinetic ion, fluid electron theory(and with γ ∝ β / ). This rapid growth is an electron whistlerinstability driven by the temperature anisotropy in a differentiallyrotating low collisionality plasma (see eq. 35-37 in §3.3). cyclotron frequency but is independent of the collision fre-quency (see §3.2.1 below for more details). Thus the orderingin equation 13 implies that gyroviscous transport of angularmomentum is dynamically important. Magnetic tension onthe other hand is negligible. Indeed, the gyroviscous orderingimplies kv a Ω ∼ (cid:18) ω c β Ω (cid:19) / (cid:28) . (15)Magnetic tension thus has no effect on the dynamics of themodes under consideration (which have ω ∼ Ω).Given the gyroviscous ordering, it also follows that thearguments of the plasma dispersion function ζ ± ∼ (cid:16) ω c Ω (cid:17) / (cid:29) . (16)We may thus employ the large argument expansion of theplasma dispersion function: W ( ζ ) ’ − ζ − .For parallel modes with b z = ± k = k z e z , theunstable branch of the dispersion relation is obtained fromthe perpendicular (with respect to e z ) dispersion tensor D ⊥ in equation 7. We wish to find the leading order incarnationof this tensor in the gyroviscous ordering. In order to doso, we introducing the ordering parameter (cid:15) (cid:28) β ∼ /(cid:15) and ω c / Ω ∼ /(cid:15) . It is not difficult to verify that c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky − − − − − − γ / Ω ω cp b z / Ω = − β p − − − − − k z v a / Ω10 − − − − − − γ / Ω ω cp b z / Ω = −
100 10 Figure 4.
Numerical growth rates γ predicted by the kinetic ionand electron dispersion relation as a function of k z v a / Ω (solidcolored lines) for b z = −
1, i.e., Ω antiparallel to B , taking T e = T p and m p = 1836 m e . Dashed lines show solutions for kinetic ionsand cold, massless electrons while black solid lines show the HallMHD solution. Comparison to Figure 1 shows that the kinetic ioninstability at long wavelengths ( k z v a (cid:28) Ω) is sensitive to the signof Ω · B while the kinetic electron mode at k z v a ∼ Ω is not. Forthe antiparallel case considered here, there are no growing modesfor ω cp / Ω (cid:46)
1, unlike for the case of Ω k B shown in Figure 1where we show solutions for ω cp / Ω = 0 . kv a / Ω ∼ (cid:15) / (cid:28) ζ ± ∼ /(cid:15) / (cid:29)
1, consistent withequations 15 and 16.The matrix representation of D ⊥ in the basis ( e x , e y )is given in equation HQ73. Letting the gyroviscous orderingparameter (cid:15) → D ⊥ = − ω iω (cid:18) − k v t ω c b z (cid:19) − iω (cid:18) − k v t ω c b z (cid:19) − ω − q Ω . (17)In deriving this expression we have used the “circular com-ponents” of the response tensor (see HQ71 & HQ72), whichare approximately given byΛ ± = ± ωω c − ω ω c ∓ ωω c h(cid:16) − q (cid:17) Ω ω c b z − k v t i + O ( (cid:15) ) (18)and ω g /S z = b z − q Ω / (2 ω c ) + O ( (cid:15) ) . (19) Substituting equations 18 and 19 into equations HQ73 andHQ74 and letting (cid:15) → ω (cid:2) ω − κ + α Ω (4 − α ) (cid:3) = 0 , (20)where α = k v t Ω ω c b z (21)and κ = 2(2 − q )Ω is the square of the epicyclic frequency.We stress again that this dispersion relation contains no traceof magnetic tension.It is straightforward to show from equation 20 that for b z > γ max = (cid:18) − d Ω d ln r (cid:19) / = p q Ω (22)occurs at k max v t = (2Ω ω c b z ) / . (23)The maximum growth rate and wavenumber predicted byequations 22 & 23 are in very good agreement with thenumerical solutions show in Figures 1-3 (see, in particular,the dependence on ω c in Fig. 2). One can also readily showthat growth in the gyroviscous limit only occurs for2 − p q < k v t Ω ω c < p q. (24)For q = 3 /
2, i.e., κ = Ω, this corresponds to growth for kv t / √ Ω ω c ∈ [0 . , . k for the ion-scale gyroviscous mode explains the unusual shape of thedispersion relation curve for ω c / Ω = 100 in Figure 1 (seealso Fig. 5 discussed below).Equation 23 implies that the wavelength of the fastestgrowing mode in the very weak field limit is smaller than thethermal scale height of the plasma ∼ v t / Ω by only a factorof ∼ (Ω /ω c ) / . This is contrary to the intuition from theMHD theory of the MRI, where tension requires that growthis restricted to very small scales ∼ v a / Ω when the magneticfield is weak.The maximum growth rate derived here (eq. 22) is iden-tical to that derived by Quataert et al. (2002) and Balbus(2004) in the case of very different physics: B y (cid:44) b z = −
1, i.e., Ω and B anti-aligned, eq. 20 predictsthat there are no unstable modes. This is consistent withthe significant difference between the b z = 1 and b z = − γ max = (cid:0) − d Ω /d ln r (cid:1) / occurs only for b z = 1, consistent with theanalytic dispersion relation. In the numerical solutions, thereis growth at long wavelengths for b z = − ω c (cid:29) Ω and is not well described by the ordering used inderiving our analytic approximations (eq. 12 & 13). c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas In a magnetized collisional plasma with collision frequency ν i (cid:28) ω c , there are three conceptually distinct contributionsto the momentum transport (Braginskii 1965): (1) transportof momentum along magnetic field lines, which is equivalentto the field-free transport and thus depends on ν i but not ω c , (2) cross-field transport which is smaller than the field-aligned transport by a factor of ∼ ( ω c /ν i ) (cid:29)
1, and (3)cross-field transport which is independent of the collisionfrequency and is thus suppressed relative to the field-alignedtransport by a factor of ∼ ( ω c /ν i ). The latter is an exampleof the gyroviscous stress, which is in general the componentof the stress in a magnetized plasma that is perpendicular tothe magnetic field and independent of the collision frequency(Ramos 2005). Physically, this momentum transport arisesbecause of spatial variations in drifts across the Larmororbits of particles (Kaufman 1960). For example, in thepresence of a background shear, the mean E × B velocity ina plasma varies spatially across a Larmor orbit, generatinga net momentum flux. The resulting cross-field gyroviscousstress is the leading order finite Larmor radius contributionto the momentum transport.Ferraro (2007) derived the dispersion relation for theMRI in a magnetized, collisional plasma accounting for finiteLarmor radius effects via the inclusion of gyroviscosity. Hecorrectly pointed out that at high β , the dominant correc-tion to the ideal MHD theory of the MRI is not the Halleffect but rather gyroviscosity. Ferraro also emphasized thatgyroviscosity can stabilize short wavelength modes for whichmagnetic tension is destabilizing in MHD. He did not, how-ever, explain the physics of the instabilities that remain inthe presence of gyroviscosity, nor identify the fact that theyare physically quite distinct from the MRI. Given this, andthe close connection between our kinetic theory instabilitycalculation and the analogous fluid calculation with Bragin-skii gyroviscosity, we find it useful to briefly summarize thefluid theory of the MRI with gyroviscosity.In the collisional, magnetized limit, kv t (cid:28) ν i (cid:28) ω c , theion gyroviscous stress is given by (Braginskii 1965) P gv = ρv t ω c h b × W · (cid:16) + 3 bb (cid:17) − (cid:16) + 3 bb (cid:17) · W × b i (25)where is the unit tensor and W ij = ∂ i u j + ∂ j u i − δ ij ∇ · u is the standard fluid strain tensor.It is straightforward to solve for the MHD linear disper-sion relation of the shearing sheet including equation 25 asthe only non-ideal MHD term in the equations. We reiteratethat neglecting the Hall effect is self-consistent in the limit β (cid:29)
1. In addition, parallel viscosity along field lines is unim-portant for the case considered here with k k Ω k B . Forsimplicity of presentation we also neglect magnetic tensionto be consistent with the analytic approximations to our ki-netic theory results derived in §3.2. The resulting dispersionrelation in the Boussinesq approximation is ω − ω Ω ( α − (cid:18) α − κ (cid:19) + q Ω (cid:18) α − α (cid:19) = 0 , (26)where again α = ( kv t ) / (Ω ω c b z ). Equation 26 is equivalentto Ferraro (2007)’s eq. 10 with his K →
0. Equation 26 isalso the fluid analog of our analytic kinetic theory dispersionrelation in the gyroviscous ordering (eq. 20). Equation 26 can be solved directly to yield the growingmode predicted by MHD with Braginskii gyroviscosity. Theunstable root of the dispersion relation corresponds to thenegative branch, i.e. ω = 12 Ω (cid:20) ( α − α − q ) − (cid:2) ( α − ( α − q ) + α q (4 − α ) (cid:3) / (cid:21) . (27)Equation 27 can be solved for the wavevector and growth rateof the fastest growing mode but the solution is sufficientlyunwieldy as to not provide much insight (this is because theresulting equation is a quartic in k ). For q = 3 / b z >
0, the result is k max v t ’ . √ Ω ω c with a correspondinggrowth rate of γ max / Ω ’ .
31. Note that the growth rate ofthe fastest growing mode in Braginskii theory is somewhatless than the corresponding kinetic theory result in equation22. The instability criterion with Braginskii gyroviscosity canbe readily determined from the dispersion relation in equation26 evaluated for ω = 0. This yields that the condition forinstability is (Ferraro 2007) 0 < α <
4, i.e. b z > kv t < ω c ) / . This implies that growth is restricted tolong wavelengths where kv A (cid:28) Ω, i.e., tension is negligible(so long as β (cid:29) ω cp / Ω). The solutions of the fluid dispersionrelation thus have some similarity to the kinetic dispersionrelation, with rapid growth requiring b z > kv t (cid:46) (Ω ω c ) / . However, the fluid andkinetic theory results differ in that in fluid theory there isgrowth even for k →
0, while this is not true in kinetic theory(see eq. 24).Figure 5 compares our full kinetic theory numericaldispersion relation (for cold, massless electrons, b z = 1, ω c / Ω = 100, and β p = 10 ) with the kinetic theory analyticapproximation (eq. 20) and the dispersion relation in MHDwith Braginskii gyroviscosity (eq. 27). The kinetic theoryanalytic approximation is in excellent agreement with thefull numerical solution. The Braginskii gyroviscous modelpredicts growth over a broader range of wavelengths andwith a somewhat smaller peak growth rate. As we shall nowdiscuss, the difference between the fluid and kinetic resultslies in the different form of the viscous stress in kinetic theoryand collisional Braginskii theory. In MHD, the MRI is driven unstable by the redistributionof angular momentum by magnetic tension. By contrast,the numerical and analytic solutions described in §3.2 showthat magnetic tension plays no role in the ion-driven modespresent for β (cid:29) ω c / Ω (cid:38)
1. This ion instability is thus physi-cally quite distinct from the MHD theory of the MRI.A closer analog of the ion instabilities described here isthe guiding center kinetic theory instability of differentiallyrotating plasmas studied by Quataert et al. (2002) (and itsfluid analog studied by Balbus 2004). In this ‘kinetic MRI’(or ‘magneto-viscous’ instability) momentum transport byviscosity (i.e., the off-diagonal components of the pressuretensor in a collisionless plasma) is the key to destabilizingthe mode. In the guiding center limit ( ω c → ∞ ), transport ofangular momentum requires an initial B y (cid:44)
0, which is whyQuataert et al. (2002) and Balbus (2004) found significant c (cid:13)000
0, which is whyQuataert et al. (2002) and Balbus (2004) found significant c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky . . . . . k z v t / (Ω ω cp ) / . . . . . . . . . . γ / Ω ω cp b z / Ω = 100 β p = 10 Kinetic (numerical)Kinetic (analytic)Braginskii
Figure 5.
Growth rates for ion-driven instabilities in the limit β (cid:29) ω cp / Ω (cid:29)
1. We compare the full kinetic theory solution forcold, massless electrons, β p = 10 , and ω cp / Ω = 100 (black line)with our analytic kinetic theory dispersion relation (blue line; eq.20) and the analogous collisional fluid theory dispersion relationincluding the Braginskii gyroviscous stress (green line; eq. 26;Ferraro 2007). The modest quantitative differences between thekinetic and Braginskii results are due to different models for thecross-field viscous transport in these two regimes (§3.2.2). Notethat the analytic solutions are independent of β and ω cp / Ω solong as β (cid:29) ω cp / Ω (cid:29) k z is normalized as on the x-axis. deviations from the MHD theory of the MRI only for B y (cid:44) B y , a small perturbation to the initial magneticfield enables viscous stresses to remove angular momentumfrom the plasma, allowing it to fall inwards. This distorts themagnetic field in such a way as to promote further viscousredistribution of angular momentum, leading to a runaway.In guiding center theory, the physics of the MRI inkinetic theory is identical to that in MHD for the case k = k z e z and B = B z e z . This is because if momentum transportis solely along magnetic field lines, there is no destabilizingviscous redistribution of angular momentum for k = k z e z and B = B z e z . Here, however, we have shown that finitecyclotron frequency effects produce significant differencesrelative to MHD even for k = k z e z and B = B z e z (solong as β (cid:29) ω cp / Ω). The physical interpretation is closelyrelated to the kinetic MRI, but with the viscous stress thattransports angular momentum due to cross-field transportassociated with a finite ion cyclotron frequency. To see this,it is helpful to explicitly write out the equations of motionfor the Lagrangian displacement. In the presence of a finitepressure tensor but neglecting magnetic tension, these takethe form: ∂ ξ ∂t + 2 Ω × ∂ ξ ∂t − q Ω ξ x e x + ∇ · δ P ρ = 0 . (28)We show in the Appendix that the linearly perturbed pressureforce in kinetic theory in the shearing sheet is ∇ · δ P ρ = k z v t ω c ∂ ξ ∂t × b (29)where we have continued to take k = k z e z . Note that whenit is expressed in terms of the Lagrangian displacement, theviscous force in equation 29 contains no explicit dependenceon the rotation rate (only implicitly, through the fact thatthe Lagrangian displacement itself evolves differently in arotating medium). By contrast, the linearly perturbed gyro-viscous stress (eq. 25) in the magnetized collisional regime is ∇ · δ P gv ρ = k z v t ω c (cid:16) ∂ ξ ∂t × b + q Ω2 ξ ⊥ b z (cid:17) . (30)The dispersion relation accounting for Braginskii’s gyrovis-cous stress (eq. 26) can be derived by combining the equationsof motion in a differentially rotating plasma (eq. 28) with thelinearly perturbed gyroviscous stress in equation 30. Like-wise, the kinetic theory dispersion relation in the gyroviscousordering (eq. 20) can be derived by combining equations 28and 29.The two expressions above for the viscous stress (eqs 29& 30) agree in the non-rotating limit q Ω = 0 but not in therotating case. In general, the exact form of the gyroviscousstress depends on the plasma conditions and is different forcollisionless plasmas vs. highly collisional plasmas (Ramos2005). Thus it is not surprising that the kinetic theory andBraginskii models give qualitatively similar but quantitativelydifferent results (Fig. 5).The above analysis of the cross-field momentum trans-port leads to a simple interpretation of the kinetic theorynumerical and analytical results. In particular, for linearperturbations with k = k max given by equation 23 the az-imuthal viscous force in equation 29 exactly cancels theCoriolis force. The solution in this case is pure radial motionwith ∂ ξ x /∂t = 2 q Ω ξ x , which yields the maximum growthrate in equation 22. This basic physics - pure radial motiondue to efficient viscous redistribution of angular momentum- is similar to that identified by Quataert et al. (2002) andBalbus (2004) in the guiding center limit. In guiding centertheory, viscous redistribution of angular momentum yieldsmodes with growth rates ∼ √ q Ω over a wide range of k ,provided that there is an azimuthal component of the back-ground magnetic field, that magnetic tension is negligible,and that the timescale for viscous stresses to redistributeangular momentum is short compared to the rotation period.The existence of rapid growth over a wide range of k is aconsequence of the fact that in guiding center theory mo-mentum transport is only along field lines and there is nostabilizing radial viscous force for k = k z e z . By contrast, inthe gyroviscous limit β (cid:29) ω c / Ω (cid:38) k where the viscous and Coriolis forces are comparable inmagnitude (Fig. 5). The numerical solutions in Figures 1-4 show that in theweak magnetic field limit kinetic electrons produce a newunstable mode that can have a growth rate (cid:29)
Ω. This doesnot have any analog in the previous literature on the MRI.We now show analytically that this instability is producedby electrons tapping into the free energy of the temperatureanisotropy present in a differentially rotating plasma.Taking ω g ∼ ω c and kv a ∼ Ω, the electron-drivenmodes in Figures 1-4 correspond to the argument ofthe plasma dispersion function (eq. 10) being ζ ± ∼ ( | ω ce | / Ω) β − / ( m e /m p ) / for the electrons. We thus see thatfor any finite value of | ω ce | / Ω, the approximation ζ ± (cid:28) β . We make this approximationin what follows and confirm its domain of validity after the c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas fact (see eq. 38). In addition, we utilize the fact that for aproton-electron plasma with m p (cid:29) m e , the proton contribu-tion to the total conductivity tensor (eq. 8) is negligible forthe modes of interest, as can also be checked after the fact.As in §3.2, we use equations HQ71 and HQ72 for thelinear response tensor Λ appropriate for parallel modes with b z = ± k = k z e z . For ζ ± (cid:28)
1, the plasma responsefunction can be expanded as W ( ζ ) = 1 + i p π/ ζ . In thiscase, the linear response tensor simplifies greatly and becomesdiagonal. The components transverse to the magnetic fieldare given by[ Q · Λ · Q ] ⊥ = − iω p π/ kv t h e x e x + (1 − ∆) e y e y i . (31)With equation 31, it follows directly from equation 7 that thedispersion tensor itself is diagonal. As a result, the dispersionrelation factors into three simple contributions, one for eachdiagonal component of the dispersion tensor. The unstablebranch is the yy component, which yields the dispersionrelation: ω = − ikv te p π/ − ∆ e ) (cid:20) k v a ω cp m e m p + 2∆ p ∆ e q (cid:21) (32)= − ikv te p π/ (cid:20) k v a ω cp m e m p − ∆ e + 2 q Ω ( ω cp b z + 2Ω)( ω ce b z + κ / (cid:21) . The first term in [ ] in the top line of equation 32 can alsobe written as k ‘ e where ‘ e is the electron inertial length.This will be useful below.In equation 32, we have written the dispersion relationboth in terms of the electron and proton tidal anisotropies ∆ e and ∆ p , respectively, and explicitly in terms of the cyclotronand orbital frequencies using equation 4. Note that we restrictourselves to ∆ < ω ) <
0, which is stabilizing, i.e., damping. Thedestabilizing term is the second term in [ ]. The conditionfor instability is thus that ∆ p ∆ e /q <
0, i.e., that2 q Ω ( ω cp b z + 2Ω)( ω ce b z + 2Ω) < q >
0, which is appropriate for a typi-cal astrophysical rotation law. The condition for instabilitydepends on the ratio of the disk rotation frequency to thecyclotron frequencies of the particles: | ω ce | (cid:46) Ω: there is no instability for q >
0. This suggeststhat an unmagnetized plasma ( | ω cs | / Ω →
0) is linearly stable.As we discuss in §6, however, the unmagnetized shearingsheet is in fact linearly unstable but to non-axisymmetricmodes not considered in equations 32 and 33. ω cp (cid:38) Ω: because ω ce and ω cp have opposite signs, equation33 is always satisfied and there are unstable modes for both b z = ±
1, i.e., for Ω and B parallel and anti-parallel. ω cp (cid:46) Ω (cid:46) | ω ce | : there are only unstable axisymmetricmodes for b z > Ω · B >
0) given our sign conventionthat ω ce <
0. As we discuss in §3.3.1, however, there arevery likely unstable modes for b z <
0, but only for non-axisymmetric wavevectors not considered in eqs. 32 and 33. Equation 32 predicts growth for kv a Ω < p q ( ω cp (cid:38) Ω) kv a Ω < (cid:16) q ω cp b z Ω (cid:17) / ( ω cp (cid:46) Ω (cid:46) | ω ce | ) (34)though the numerical solutions in Figure 1 show that theinstability ceases to be driven primarily by the electrons atsufficiently low kv a / Ω. The fastest growing electron modeassociated with equation 32 has k max v a Ω = (cid:18) − e ∆ p q m p m e ω cp Ω (cid:19) / (35) ’ (cid:16) q (cid:17) / ( ω cp (cid:38) Ω) ’ (cid:16) q ω cp b z Ω (cid:17) / ( ω cp (cid:46) Ω (cid:46) | ω ce | )This can also be written more compactly in terms of theelectron intertial length as k max ‘ e = (cid:18) − e ∆ p q (cid:19) / . (36)The results for the wavelength of the fastest growing modesin equation 35 are very similar to (though not identical to)the Hall MHD results for the fastest growing MRI modes(Wardle 1999). The maximum growth rates are, however,very different. In particular, the fastest growing modes have γ max ’ . β / e q / (cid:18) Ω ω cp (cid:19) (cid:18) m e m p (cid:19) / ( ω cp (cid:38) Ω) (37) ’ . β / e q / (cid:18) Ω | ω ce | b z (cid:19) / ( ω cp (cid:46) Ω (cid:46) | ω ce | )Equation 37 shows analytically that in the weak field limitof high β and finite ω c / Ω, the growth rate can be (cid:29)
Ω, incontrast to all existing known limits of the MRI. Moreover,this rapid growth requires kinetic electrons.It is important to reiterate that the analytic results ofthis section (and, in particular, eq. 32) were derived assumingthat ζ ± (cid:28) β e (cid:29) m e m p (cid:16) ω ce Ω (cid:17) ( ω cp (cid:38) Ω) β e (cid:29) | ω ce | Ω ( ω cp (cid:46) Ω (cid:46) | ω ce | ) . (38)Note that since the growth rates increase ∝ β / e and theanalytic solutions are only valid for sufficiently large β e , thisimplies that the analytic results are valid precisely when thegrowth rates are (cid:29) Ω, i.e., in the regime of most interest.The analytic results in equations 32-37 are in good agree-ment with the numerical solutions of the kinetic dispersionrelation discussed in §3.1. We compare the two directlyin Figure 6. At sufficiently low k z v A / Ω, the low ζ expan-sion of the plasma dispersion function used in the analyticderivation breaks down, but the analytics are an excellentapproximation for the fastest growing modes. More generally,compared to the numerical solutions in Figures 1-4, the ana-lytic results correctly capture that the fastest growing modefor sufficiently high β e has (1) a growth rate (cid:29) Ω and ∝ β / e c (cid:13)000
Ω, incontrast to all existing known limits of the MRI. Moreover,this rapid growth requires kinetic electrons.It is important to reiterate that the analytic results ofthis section (and, in particular, eq. 32) were derived assumingthat ζ ± (cid:28) β e (cid:29) m e m p (cid:16) ω ce Ω (cid:17) ( ω cp (cid:38) Ω) β e (cid:29) | ω ce | Ω ( ω cp (cid:46) Ω (cid:46) | ω ce | ) . (38)Note that since the growth rates increase ∝ β / e and theanalytic solutions are only valid for sufficiently large β e , thisimplies that the analytic results are valid precisely when thegrowth rates are (cid:29) Ω, i.e., in the regime of most interest.The analytic results in equations 32-37 are in good agree-ment with the numerical solutions of the kinetic dispersionrelation discussed in §3.1. We compare the two directlyin Figure 6. At sufficiently low k z v A / Ω, the low ζ expan-sion of the plasma dispersion function used in the analyticderivation breaks down, but the analytics are an excellentapproximation for the fastest growing modes. More generally,compared to the numerical solutions in Figures 1-4, the ana-lytic results correctly capture that the fastest growing modefor sufficiently high β e has (1) a growth rate (cid:29) Ω and ∝ β / e c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky − − − − − k z v a / Ω10 − − − − − γ / Ω ω cp b z / Ω = 0 . β e = 10 NumericalAnalytical
Figure 6.
Numerical growth rates for the electron-driven instabil-ity (solid black line) compared to the analytic dispersion relationin equation 32 (dashed red line). The agreement is excellent for thefastest growing modes. At low k z v a / Ω, the expansion of the plasmadispersion function used in the analytic derivation is inapplicable. (2) a growth rate ∝ (Ω /ω cp ) / for ω cp (cid:28) Ω and b z = 1, and(3) a wavelength of kv a / Ω ∼ min[1 , ( ω cp / Ω) / ]. In addition,the analytics confirm that for ω cp (cid:38) Ω, the kinetic electroninstability is independent of the sign of b z (compare Figs. 1and 4). We find analytically that there are no instabilitiesassociated with kinetic electrons for b z = − ω cp (cid:46) Ω(eq. 33). This is consistent with our inability to find anygrowing modes numerically in this regime.
It is straightforward to combine equations 4, 5, & 32 toshow that the dispersion relation can be written explicitly interms of the temperature anisotropy imposed by differentialrotation: ω = − ikv te p π/ (cid:18) q (cid:20) T xe − T ye T ye (cid:21)h T xi − T yi T xi i + k ‘ e T xe T ye (cid:19) ’ − ikv te p π/ (cid:18) T xe − T ye T ye + k ‘ e (cid:19) ( ω cp (cid:46) Ω (cid:46) | ω ce | ) (39)where ‘ e is the electron inertial length. Equation 39 is eas-iest to interpret when ω cp (cid:46) Ω (cid:46) | ω ce | , in which case thedispersion relation reduces to the approximate equality onthe second line.Because the unstable root here is associated with the yy component of the dispersion tensor, it is also straightforwardto show that the polarization of the unstable mode is δ ˜ E z = 0 δ ˜ E x = q Ω iω δ ˜ E y δ B = − k z δ ˜ E y ω e x δ J = − ik z δ ˜ E y ωµ e y (40)Note that equation 40 implies that the linearly unstable modehas no Maxwell or Reynolds stress, in contrast to the MRI.In addition, substituting eqs. 31 and 40 in equation A9 showsthat the linearly perturbed pressure force also vanishes.Equations 39 and 40 demonstrate that the unstable modeis a circularly polarized electron whistler driven unstable by the background temperature anisotropy (which is itselfcreated by the differential rotation). The resulting instabilityis also closely related to the electron Weibel instability ofan unmagnetized plasma (Weibel 1959): the second line ofequation 39 is identical to the electron Weibel instabilitydispersion relation in the limit of a small fractional electrontemperature anisotropy. This result is not particular to theshearing sheet: in a homogeneous magnetized plasma witha gyrotropic temperature anisotropy ∆ T e (cid:46) T e , the electronWeibel instability persists as an electron whistler instabilityso long as | ω ce | (cid:28) kv te , which corresponds to v a /v te (cid:28) ( m e /m p )(∆ T e /T e ) / for the fastest growing mode.Equation 39 shows that for ω cp (cid:46) Ω (cid:46) | ω ce | instabilityis present only if T x e < T y e . Given equations 4 & 5, this isequivalent to the constraint b z >
0, i.e., Ω · B >
0, notedin §3.3. In the velocity space instability interpretation pro-vided here, this constraint arises for the following reason: theWeibel instability in a homogeneous plasma requires thatthe wavevector have a component along the low temperaturedirection. For our equilibrium shearing sheet model with ω cp (cid:46) Ω (cid:46) | ω ce | and Ω · B > T x = T z < T y while for Ω · B < T x = T z > T y . Thus modes with k k Ω willonly be unstable for Ω · B >
0, as is indeed the case. Thisanalysis demonstrates, however, that the case Ω · B < ω cp (cid:46) Ω (cid:46) | ω ce | , but probably only tonon-axisymmetric modes with k y (cid:44) The results derived in the previous sections can also beapplied to study the stability of non-rotating planar shearflows. Although a planar shear flow is linearly stable inideal hydrodynamics or MHD, the inclusion of non-idealphysics such as ambipolar diffusion or the Hall effect cangenerate linear instability (Kunz 2008). The importance ofthese linear instabilities is uncertain given the well-knownnon-linear hydrodynamic instabilities afflicting planar shearflows. Nonetheless, the presence of magnetically-mediatedlinear instabilities in planar shear flows might in some casesalter the resulting turbulence, transport properties, and/ormagnetic field amplification relative to that predicted bynon-linear hydrodynamic turbulence.A shear flow with an equilibrium velocity v = Sx e y satisfies the identical dispersion relation to that derived in§2 (eq. 32) with Ω → q Ω → − S (and thus finite).Using this transformation, it is straightforward to assess thestability of a planar shear flow to the instabilities of rotatingplasmas highlighted in this paper. Here we briefly summarizethe conclusions drawn from making this transformation, butwe defer a detailed study of the stability of planar shear flowsin the shearing sheet to future work. As in the bulk of thispaper, we restrict our analysis to B = B z e z and k = k z e z .A planar shear flow has an equilibrium temperatureanisotropy set by ∆ s = − S/ω cs b z (see eq. 5). We restrictourselves to ∆ s < ω g < > c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas a planar shear flow is stable to the electron temperatureanisotropy instability described in §3.3 (at least for B = B z e z and k = k z e z ). This follows from equation 32 bysetting Ω → q Ω → − S . Physically, the reason is thatthe perturbed ion current associated with the temperatureanisotropy exactly cancels the analogous perturbed currentdue to the electrons. Mathematically, this corresponds to thefact that the nominally destabilizing term in the dispersionrelation (eq. 32) is the last term ∝ q Ω , which vanishes for ashear flow. Since Weibel instabilities are driven by currentbunching, the fact that the ion current shields the electroncurrent for a planar shear flow leads to linear stability.A planar shear flow is also linearly stable to the kinetictheory version of the ion gyroviscous instability describedin §3.2. This follows from equation 20 by setting Ω → q Ω → − S . However, a planar shear flow is linearlyunstable in the presence of Braginskii gyroviscosity, i.e., inthe magnetized, collisional limit. Indeed, setting Ω → q Ω → − S , equation 26 becomes ω − ω ( f − Sf ) + S f , (41)where f = k v t /ω c b z and equation 41 is derived assumingmagnetic tension is negligible, which requires β (cid:29) ω c /S . Thesolution to equation 41 is ω = 12 h f ( f − S ) ± (cid:0) f ( f − Sf ) (cid:1) / i (42)For f (cid:29) S , equation 42 corresponds to dispersive waves with ω = k v t /ω c . It is straightforward to show from equation42, that a linear shear flow with β (cid:29) ω c /S is subject tolinear over-stabilities (i.e., solutions that both oscillate andexponentiate in time) provided that b z > f < S . Thefastest growing mode has k max v t = ( Sω c b z ) / and ω max = ± S i S . (43)The growth rate of the fastest growing mode for the planarshear flow is thus comparable to that for a differentiallyrotating flow shown in Figure 5. The physical interpretationof this ion-driven instability of planar shear flows in themagnetized, collisional (Braginskii) limit is also similar tothat described for rotating flows in §3.2.2: a perturbation tothe magnetic field generates a viscous force that displaces theplasma further from its initial equilibrium position, enhancingthe initial perturbation to the magnetic field and the resultingviscous force. Here we briefly describe the application of our work to theorigin of magnetic fields at high redshift. This is a complexproblem whose full solution is well beyond the scope ofthis paper. Here we focus on providing simple estimatesof magnetic field amplification in the virialized plasma inthe outskirts of high redshift dark matter halos, becausethese plasma conditions are reasonably well understood and This conclusion only applies to modes with ω ∼ S and kv t ∼√ Sω c , assuming β (cid:29) ω c /S (cid:29)
1, which is the shear flow analogof the ordering used in §3.2. because this plasma is the least likely to be magnetized byother processes. We assume that a seed magnetic field isalready present, generated by, e.g., the Biermann battery(e.g., Naoz & Narayan 2013) or Weibel-like instabilities (e.g.,Lazar et al. 2009; Spitkovsky & Quataert, in prep).We first show that the halo plasma conditions of interestare relatively collisional. As a result, we argue that the mostimportant instability is likely to be the collisional versionof the ion-driven instability described in §3.2 & §4. Wescale typical estimates to dark matter halos with masses of10 M M (cid:12) at redshift ∼ −
20. In particular, 2 − σ densityfluctuations at z ∼
10 lead to collapsed halos with masses ∼ − M (cid:12) (e.g., Barkana & Loeb 2001). The virialtemperature of plasma in such halos is T vir ’ × M / (cid:16) z (cid:17) K (44)while the characteristic plasma density in the outer parts ofthe halo is roughly 200 times the mean baryonic density ofthe Universe (independent of halo mass): n ∼ . (cid:16) z (cid:17) cm − . (45)These estimates compare well to the temperatures and densi-ties in numerical simulations of the formation of high redshiftproto-galaxies (e.g., Wise & Abel 2007). Given these plasmaparameters, the corresponding proton collision frequency andmean free path are ν p ∼ − M − (cid:16) z (cid:17) / s − (46)and ‘ p ∼ M / (cid:16) z (cid:17) − cm . (47)The electron mean free path is comparable to the protonmean free path while the electron collision rate is p m p /m e times larger.The characteristic shear rate in the plasma is set by thedynamical time, which is of order a tenth of the Hubble time,i.e. S ∼ t − ∼ − (cid:16) z (cid:17) − / s − (48)The typical rotation rate for inflowing matter in the outskirtsof dark matter halos is Ω ∼ . S (e.g., Barkana & Loeb 2001)but rotation of course becomes more important as matterflows in to smaller radii.A comparison of equations 46 and 48 shows that on therotation/shear timescale the plasma is reasonably collisional.This implies that the temperature anisotropy cannot reachthe full value in equation 5 (which is valid only in a colli-sionless plasma) but will instead be limited to ∆ T /T ∼ S/ν (Schekochihin et al. 2005). As a result, the electron temper-ature anisotropy driven instability derived here cannot bedirectly applied to the collisional halo plasma. We defer to fu-ture work an investigation of this instability under collisionalconditions. We stress, however, that there will be regions ofmuch higher temperature and much lower collisionality inhigh redshift galaxies; e.g., supernova remnants and blackhole and/or neutron star accretion flows will occur soon afterthe formation of the first stars. The conditions in such re-gions are not as well understood but it is very plausible that c (cid:13)000
10 lead to collapsed halos with masses ∼ − M (cid:12) (e.g., Barkana & Loeb 2001). The virialtemperature of plasma in such halos is T vir ’ × M / (cid:16) z (cid:17) K (44)while the characteristic plasma density in the outer parts ofthe halo is roughly 200 times the mean baryonic density ofthe Universe (independent of halo mass): n ∼ . (cid:16) z (cid:17) cm − . (45)These estimates compare well to the temperatures and densi-ties in numerical simulations of the formation of high redshiftproto-galaxies (e.g., Wise & Abel 2007). Given these plasmaparameters, the corresponding proton collision frequency andmean free path are ν p ∼ − M − (cid:16) z (cid:17) / s − (46)and ‘ p ∼ M / (cid:16) z (cid:17) − cm . (47)The electron mean free path is comparable to the protonmean free path while the electron collision rate is p m p /m e times larger.The characteristic shear rate in the plasma is set by thedynamical time, which is of order a tenth of the Hubble time,i.e. S ∼ t − ∼ − (cid:16) z (cid:17) − / s − (48)The typical rotation rate for inflowing matter in the outskirtsof dark matter halos is Ω ∼ . S (e.g., Barkana & Loeb 2001)but rotation of course becomes more important as matterflows in to smaller radii.A comparison of equations 46 and 48 shows that on therotation/shear timescale the plasma is reasonably collisional.This implies that the temperature anisotropy cannot reachthe full value in equation 5 (which is valid only in a colli-sionless plasma) but will instead be limited to ∆ T /T ∼ S/ν (Schekochihin et al. 2005). As a result, the electron temper-ature anisotropy driven instability derived here cannot bedirectly applied to the collisional halo plasma. We defer to fu-ture work an investigation of this instability under collisionalconditions. We stress, however, that there will be regions ofmuch higher temperature and much lower collisionality inhigh redshift galaxies; e.g., supernova remnants and blackhole and/or neutron star accretion flows will occur soon afterthe formation of the first stars. The conditions in such re-gions are not as well understood but it is very plausible that c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky these regions are critical sites of magnetogenesis because thelow collisionality conditions enable a wider range of plasmainstabilities to be important. Indeed, we have shown thatunder low collisionality conditions, electron instabilities canamplify the magnetic field on a timescale much less than therotation period by tapping into the temperature anisotropygenerated by differential rotation.The fact that ν p (cid:29) S in the halo plasma suggests thatthe most important instability is likely to be the collisionalversion of the gyroviscous instability described in §3.2, alongwith the magneto-viscous instability generated by collisionaltransport of momentum along magnetic field lines (Balbus2004). Both of these instabilities require ω c (cid:38) ν p , whichcorresponds to B (cid:38) − G M − ([1 + z ] / / under theconditions of interest. For our fiducial parameters and a10 − G field, β ∼ (cid:29) ω c /S ∼ , so that the gyrovis-cous stress is indeed dynamically important. The results ofthis paper demonstrate that both rotation and planar shearflows in the halos of high redshift galaxies will be unstable togyroviscosity-mediated instabilities that will exponentiallyamplify the magnetic field on a timescale comparable to,or somewhat shorter than, the rotation/shear time. As themagnetic field grows in strength, eventually β (cid:46) ω c /S andgyroviscosity will cease to be dynamically dominant. At thatpoint, the unstable mode of interest will become the stan-dard MRI. The ion-driven instabilities described in this paperthus provide a way of amplifying the magnetic field from aninitially small value to the point where the magnetohydrody-namic MRI can take over. The key stage that we have notaddressed is how the magnetic field is amplified to the pointwhere ω c (cid:38) ν p , so that Braginskii’s collisional, magnetizedtheory applies. We have studied the linear stability of weakly magnetizeddifferentially rotating plasmas in both collisionless kinetictheory and Braginskii’s theory of collisional, magnetized plas-mas. We have focused in particular on the limit of very weakmagnetic fields, for which the ion and/or electron cyclotronfrequencies are not much larger than the rotation frequencyof the plasma. Our motivation is primarily to understandhow magnetic fields can get created and/or amplified fromvery small initial values. Astrophysically, this is particularlyimportant in the context of understanding at what stage dur-ing structure formation at high redshift do magnetic fieldsbecome dynamically significant and need to be included intheoretical and numerical models of star formation, galaxyformation, and massive black hole growth.For very weak magnetic fields, ideal MHD predicts thatthe most unstable MRI mode driven by differential rotationhas a short wavelength ∼ v a / Ω that gets smaller for weakerfields. Hall MHD represents an extension of this theory to ω cp (cid:46) Ω, i.e, to conditions in which the proton cyclotronfrequency can be small compared to the disk rotation fre-quency. In Hall MHD the MRI persists as a whistler modedestabilized by magnetic tension (Wardle 1999). The HallMHD approximation corresponds, however, to the cold ion,cold massless electron limit of kinetic theory. Thus it is nota good approximation for high β low-collisionality plasmas,nor can it capture any kinetic electron physics. In particular, momentum transport along and across magnetic field lines isa more important non-ideal MHD effect than Hall currentsin the high β dilute plasmas of interest in this paper.These considerations motivate the linear kinetic theorycalculation described in this paper, which self-consistentlyincorporates finite electron and ion cyclotron frequencies(and Larmor radii). Our analysis draws heavily on the for-malism developed in Heinemann & Quataert (2014) (HQ),who carried out a general linear stability calculation in ki-netic theory for local instabilities in differentially rotatingplasmas. Their primary assumptions were charge neutralityand axisymmetry. In this paper we have further restrictedour analysis to the simplest non-trivial problem, in which B k Ω k k (including both parallel and anti-parallel fields).The case of B k Ω k k captures the key physics of theMRI in MHD. Moreover, in guiding center kinetic theory,in which one averages over the Larmor orbits of ions andelectrons, the linear theory of the MRI for B k Ω k k isidentical to that in MHD (Quataert et al. 2002). We haveshown, however, that the case of kinetic ions and electronswith finite cyclotron frequencies is far more interesting. Thesingle instability of ideal MHD that is mediated by magnetictension is replaced by two distinct instabilities, one associatedwith kinetic ions and one with kinetic electrons. Each of theseinstabilities has a different way of tapping into the free energyof differential rotation.In kinetic theory, if β (cid:38) ω cp / Ω (cid:38)
1, there is an instabilityat long wavelengths kv tp / Ω ∼ ( ω cp / Ω) / which has a max-imum growth rate of γ max = ( − d Ω /d ln r ) / (Figs. 1 & 2and eqs. 22 & 23). This instability is associated with kineticions and is indifferent to the electron physics, being presentfor both fluid and kinetic electron models. Note that for ω cp ∼ Ω the wavelength of the fastest growing mode is com-parable to the thermal scale height of the plasma ∼ v tp / Ω,i.e., the fastest growth is for the largest scale modes. This iscontrary to the predictions of MHD and Hall MHD, in whichtension requires that growth is restricted to very small scaleswhen the magnetic field is weak.The maximum growth rate found here for the ion insta-bility in the limit of B k Ω k k is identical to that derivedby Quataert et al. (2002) for the ‘kinetic MRI’ (and its fluidanalog, the magnetoviscous instability; Balbus 2004), whichrequired B φ (cid:44) β low-collisionality plasmas. The angular momentum re-distribution is in turn coupled to the magnetic field geometry(because of the Larmor motion of particles), which is whatleads to an instability: perturbations to the initial magneticfield structure enhance the viscous redistribution of angularmomentum, allowing plasma to fall inwards, which drags thefield with it, further enhancing the redistribution of angularmomentum. A runaway ensues.In guiding center theory, the transport of angular mo-mentum is only along magnetic field lines so that a finite B φ is needed to generate viscous transport in linear theory.By contrast, in our present analysis the momentum trans-port is due to cross-field terms associated with a finite ioncyclotron frequency. This cross-field momentum transport isgenerically known as the gyroviscous stress (Ramos 2005). c (cid:13) , 000–000 eak Field Instabilities in Differential Rotating Plasmas We have shown that this produces an instability driven byviscous transport of angular momentum even for the caseof B k Ω k k . Previously, Ferraro (2007) highlighted theimportance of gyroviscosity for the MRI using Braginskii(1965)’s result for the gyroviscous stress in collisional, mag-netized plasmas (see §3.2.1). He did not, however, identifythe fact that the instabilities that remain in the presence ofgyroviscosity are physically quite distinct from the MRI inideal MHD.We have shown that the collisionless kinetic theory andBraginskii models of cross-field gyroviscous transport bothproduce an instability driven by viscous transport (not mag-netic tension) when β (cid:29) ω c / Ω. Growth is on similar spatialscales and with similar growth rates in the fluid and collision-less cases, though the kinetic theory growth rates exceed thefluid growth rates by a modest amount (Fig. 5). It is impor-tant to reiterate that these relatively large scale ion-driveninstabilities exist in both the collisionless and magnetized,collisional (Braginskii 1965) regimes (see Fig. 5). This is sig-nificant because it implies that they are likely to be relativelyrobust and present under a wide range of plasma conditions.In addition to the long-wavelength instability driven byion momentum transport, we also find a shorter wavelengthinstability with kv a ∼ Ω that is present only for the caseof kinetic electrons. This instability is in many ways moreremarkable than that due to kinetic ions because the growthrate is ∝ β / e and can exceed the rotation rate by manyorders of magnitude, particular for modest values of ω cp / Ω(Figs. 1-4). This is despite the fact that the free energy sourcefor this electron instability is still differential rotation, in thesense that the growth rate is ∝ d Ω /dr .We have shown that this kinetic electron instability isthe whistler mode driven unstable by the background tem-perature anisotropy present in the kinetic equilibrium of adifferentially rotating plasma (see §3.3). This temperatureanisotropy (the ‘tidal anisotropy’; eqs. 4 & 5) is requiredto satisfy the Vlasov equation in the equilibrium state. Itis distinct from the more familiar temperature anisotropyrelative to a magnetic field typically considered in homoge-neous magnetized plasmas. The kinetic electron instabilitywith growth rates (cid:29) Ω is thus a consequence of a uniquefeature of the kinetic equilibrium of a differentially rotatingplasma. This temperature anisotropy is in fact well-knownin the theory of collisionless stellar disks, which exhibit ananalogous anisotropy (e.g., Shu 1969).The whistler instability found here is also closely relatedto the Weibel (1959) instability of unmagnetized plasmaswith a temperature anisotropy. In particular, in the limit | ω ce | (cid:38) Ω (cid:38) ω cp , the growth rate of the whistler instabil-ity that we have derived (eq. 39) is the same as that ofthe electron Weibel instability, provided one uses the elec-tron temperature anisotropy implied by the shearing sheetequilibrium (eqs. 4 & 5) in the Weibel dispersion relation.This connection is further highlighted by the fact thatan initially unmagnetized differentially rotating plasma itselfhas a temperature anisotropy - with T φ = T r / (cid:28) Ω − ,with the initial length scale of the growing modes of orderthe electron skin depth. In a future paper, we will studythe saturation of these instabilities using PIC simulations(Spitkovsky & Quataert, in prep).The instabilities described in this paper driven by thetemperature anisotropy in differentially rotating plasmasare examples of a broader class of instabilities in which thefree energy in shear or differential rotation can be tappedvia the temperature anisotropy it induces. For example, ina system nominally described by MHD, the existence of avelocity shear and a finite collisionality implies that thereis a temperature anisotropy ∆ T /T ∼ S/ν where S is theshear rate in the plasma and ν is the Coulomb collisionrate (e.g., Schekochihin et al. 2005). If | ∆ T /T | (cid:38) β − , thenthe system nominally described by MHD will in fact beunstable to a set of velocity-space instabilities including thefirehose and mirror instabilities (and, in some cases, theelectron whistler and ion cyclotron instabilities). The impactof these instabilities on the dynamics of astrophysical plasmasremains an area of active investigation (e.g., Kunz et al. 2014;Riquelme et al. 2014).For the Ω k B case that we have focused on in thispaper, the temperature anisotropy induced by differentialrotation is entirely in the plane perpendicular to the localmagnetic field. For the more general case of B φ (cid:44)
0, the tidalanisotropy will include an anisotropy with respect to thebackground magnetic field. Because the tidal temperatureanisotropy is ∆
T /T ∼ Ω /ω c for ω c (cid:38) Ω (see eq. 4) we expectthat for B φ (cid:44) β (cid:38) ω c / Ω, differentially rotating plasmaswill be unstable to the firehose and mirror instabilities inaddition to the whistler instability highlighted in this paper.This remains to be explicitly demonstrated in future work.The derivations in this paper can be readily applied tothe stability of planar shear flows in addition to differentiallyrotating plasmas (§4). Utilizing this fact, we have foundthat planar shear flows are subject to linear overstabilitiesin the presence of the Braginskii gyroviscosity, i.e., in themagnetized, collisional limit. The resulting overstabilitieshave growth rates comparable to the shear rate (eq. 43).This suggests that turbulence with a weak magnetic fieldin the magnetized, collisional regime is likely to be a farricher physics problem than suggested by standard kinematicdynamo models.The critical question not addressed by our analysis isthe ultimate saturation of the instabilities described here andtheir impact on astrophysical plasmas. We suspect that theseinstabilities are important for the amplification of magneticfields at high redshift. In particular, we have demonstratedthat the virialized plasma in the halos of high redshift galax-ies is unstable to instabilities mediated by the collisional,magnetized gyroviscous stress (§5). This is true for bothrotating flows and planar shear flows. These instabilities pro-vide a way of amplifying magnetic fields to the point wherethe canonical MRI of ideal MHD takes over.Several important questions remain to be addressedin future work. In particular, the equilibrium temperatureanisotropy in a collisional plasma is significantly less thanin a collisionless plasma. This will decrease the growth rate c (cid:13)000
T /T ∼ Ω /ω c for ω c (cid:38) Ω (see eq. 4) we expectthat for B φ (cid:44) β (cid:38) ω c / Ω, differentially rotating plasmaswill be unstable to the firehose and mirror instabilities inaddition to the whistler instability highlighted in this paper.This remains to be explicitly demonstrated in future work.The derivations in this paper can be readily applied tothe stability of planar shear flows in addition to differentiallyrotating plasmas (§4). Utilizing this fact, we have foundthat planar shear flows are subject to linear overstabilitiesin the presence of the Braginskii gyroviscosity, i.e., in themagnetized, collisional limit. The resulting overstabilitieshave growth rates comparable to the shear rate (eq. 43).This suggests that turbulence with a weak magnetic fieldin the magnetized, collisional regime is likely to be a farricher physics problem than suggested by standard kinematicdynamo models.The critical question not addressed by our analysis isthe ultimate saturation of the instabilities described here andtheir impact on astrophysical plasmas. We suspect that theseinstabilities are important for the amplification of magneticfields at high redshift. In particular, we have demonstratedthat the virialized plasma in the halos of high redshift galax-ies is unstable to instabilities mediated by the collisional,magnetized gyroviscous stress (§5). This is true for bothrotating flows and planar shear flows. These instabilities pro-vide a way of amplifying magnetic fields to the point wherethe canonical MRI of ideal MHD takes over.Several important questions remain to be addressedin future work. In particular, the equilibrium temperatureanisotropy in a collisional plasma is significantly less thanin a collisionless plasma. This will decrease the growth rate c (cid:13)000 , 000–000 E. Quataert, T. Heinemann, & A. Spitkovsky of the kinetic electron instability found here under manyastrophysical conditions (but the instability is so strongthat it may nonetheless remain important). In future work,it would also be valuable to study the interplay betweeninstabilities driven by cross-field viscous transport of angularmomentum (such as the ion instability studied here) andinstabilities driven by field-aligned viscous transport (suchas those studied by Quataert et al. 2002 and Balbus 2004).Our focus on k k Ω k B in this paper precludes the latterfrom being important (see §3.2.2). ACKNOWLEDGEMENTS
We thank Steve Balbus, Alex Schekochihin, Sean Ressler,and Greg Hammett for useful conversations and the refereefor a particularly thoughtful and constructive report thatimproved the paper. This work was supported in part by NSFgrants AST-1333682 and PHY11-25915, Simons Investigatorawards from the Simons Foundation (to EQ and AS), theDavid and Lucile Packard Foundation, and the ThomasAlison Schneider Chair in Physics at UC Berkeley.
References
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Let f be the one-particle distribution function. The number density n and fluid velocity u are then defined as the zeroth andfirst moment of the distribution function: n = Z d v f and n u = Z d v f v . (A1)Taking the second moment with respect to the peculiar velocity v − u yields the pressure tensor P = Z d v f ( v − u )( v − u ) (A2)The Vlasov equation in the shearing sheet is ∂f∂t + v · ∇ f + em ( E + v × B ) · ∂f∂ v − (2 Ω × v + ∇ ψ ) · ∂f∂ v = 0 , (A3)where the tidal potential ψ = − q Ω x . (A4)Taking the first moment of eq. (A3) yields the momentum equation nm (cid:16) ∂ u ∂t + u · ∇ u + 2 Ω × u + ∇ ψ (cid:17) + ∇ · P = en ( E + u × B ) . (A5)Written in terms of the relative velocity and electric field, defined by˜ u = u + q Ω x e y and ˜ E = E − q Ω x e y × B , (A6)the momentum equation is given by nm (cid:16) ∂ ˜ u ∂t + ˜ u · ∇ ˜ u + 2 Ω × ˜ u − q Ω˜ u x e y (cid:17) + ∇ · P = en ( ˜ E + ˜ u × B ) . (A7)In linear theory, the relative fluid velocity is related to the electric field via − iωδ ˜ u = ( Q · Λ · Q + ∆ e y e y ) · (cid:16) − q Ω iω e x e y (cid:17) · em δ ˜ E (A8)Linearizing eq. (A7) and solving for the ∇ · δ P , the result may be written as ∇ · δ P = T · enδ ˜ E , (A9)where the tensor T = + 1 iω h − iω + (cid:16) Ω + em B (cid:17) × − q Ω e y e x i · ( Q · Λ · Q + ∆ e y e y ) · (cid:16) − q Ω iω e x e y (cid:17) . (A10) A1 Gyroviscous Ordering
Up until now our analysis has been general and can be applied to determine the perturbed pressure tensor in the shearingsheet for any linear wave or instability. We now specialize to the case of the gyroviscous ordering discussed in §3.2 (includingtaking k = k z e z ), namely 1 /β ∼ (cid:15) , Ω /ω c ∼ (cid:15) (cid:28) , ω ∼ Ω , k v t ∼ ω c Ω . (A11)In this limit the tensor T reduces to lim (cid:15) → ω c T = − k v t ω c ( − bb ) · (cid:16) − q Ω iω e x e y (cid:17) (A12)so that lim (cid:15) → ∇ · δ P ρ = − k v t ω c ( − bb ) · (cid:16) − q Ω iω e x e y (cid:17) · δ ˜E B . (A13)Using the fact that the Hall effect is negligible in the gyroviscous ordering, we can rewrite ∇ · δ P in terms of the perturbedion velocity using δ ˜ E = − δ ˜ u × B . Rewriting the perturbed velocity δ ˜ u in terms of the Lagrangian displacement using ∂ ξ /∂t = δ ˜ u − ξ x q Ω e y yields equation 29 of the main text. c (cid:13)000