The MRI in a Collisionless Plasma

Eliot Quataert (UC Berkeley)

Collaborators: Prateek Sharma, Greg Hammett, Jim Stone

Modes of Accretion

thin disk: energy radiated away(relevant to star & planet formation, galaxies, & luminous compact objects)

thick disk (~ torus): energy stored as heat(relevant to BHs & NSs at very low/high accretion rates)

prad >> pgas

ionization state & Hall effects

last stable orbit &radiative efficiency?

jet productionTi/Te ?

efficiency?super-Eddington?

Radiatively Inefficient Accretion Flows• At low densities (accretion rates), cooling is inefficient

• grav. energy ⇒ turbulence (MRI) ⇒ thermal energy: not radiated

• kT ~ GMmp/R (virial): Tp ~ 1011-12 K > Te ~ 1010-11 K near BH

• collisionless plasma: e-p equil. time > inflow time for

Haw

ley

& B

albu

s 20

02

Disk Structure Accretion Rate

(Sto

ne &

Prin

gle

2001

; Haw

ley

& B

albu

s 20

02; I

gum

ensh

chev

et a

l. 20

03)

Relevant Physics in the Kinetic Limit

• Collisionless Damping

• operates at all k, not just high k

• Alfven waves also damped (linearly coupled to slow mode via rotation)

• Free Streaming along B

• Anisotropic Heat & Momentum Transport

• Anisotropic Pressure Tensor

• MFP set by wave-particle interactions, not collisions

Collisionless Damping of Fast & Slow MHD Modes at β = 1

Note: must use anisotropic viscosity (not scalar viscosity)

The Linear MRI in Kinetic Theory

Quataert, Dorland, Hammett 2002; also Sharma et al. 2003; Balbus 2004

fastest growth at low k where tension is weak

linear instability unchanged for Bz, kz only

in general, angular momentum transport

via anisotropic pressure (viscosity!) in addition to magnetic stresses

anisotropic viscosity is destabilizing, unlike

isotropic viscosity

Nonlinear Evolution Simulated using “Kinetic-MHD”

• Large-scale dynamics of collisionless plasmas: expand Vlasov eqn using “slow timescale” and “large lengthscale” assumptions of MHD

• more general than the Braginskii eqns

• Particles efficiently transport heat and momentum along B-field lines

(Kulsrud 1983)

Evolution of the Pressure Tensor

adiabatic invarianceof μ ~ v⊥2/B ~ T⊥/B

closure model in simsk|| ~ L-1 equiv to κ ~ vthL

Shearing Box Sims of the Kinetic MRI

Saturatesat Small

Amplitudes!

w/ δB/B ~ 1/β

Box filled w/ stable

anisotropicAlfven wavespr

essu

rean

isot

ropy

stre

ssm

agne

ticen

ergy

Pressure Anisotropy

• a background pressure anisotropy (p⊥ > p||) can stabilize the MRI

• But .... T⊥ ≠ T|| unstable to small-scale (Larmor radius) instabilities that

act to isotropize the pressure tensor (velocity space instabilities)

• e.g., mirror, firehose, ion cyclotron, electron whistler

• Fluctuations have ω ~ cyclotron: violate μ invariance & pitch angle scatter

• Use subgrid model to account for this physics in Shearing Box Sims

€

µ∝T⊥ /B = constant ⇒ T⊥ > T|| as B ↑

€

∂p⊥∂t

= ... − ν (p⊥ , p||,β) p⊥ − p||[ ]

∂p||

∂t= ... − ν (p⊥, p||,β) p|| − p⊥[ ]

Velocity-Space Instabilities in the Solar Wind

proton || anisotropylimited by firehoseKasper et al. 2002

Te,⊥/

Te,

||

βe,||

whistler

firehose

Electron Anisotropy

Stuart Bale

Shearing Box Sims of the Kinetic MRI(w/ pitch angle scattering & mean Bz)

magnetic energy volume averaged pressure anisotropy

Anisotropic Stress~ Maxwell Stress

Δp/p ~ few %

The Approach to MHD

magneticstress

anisotropicstress

Nonlinear Evolution for Difft. Scattering Rates ν

MHD is reached when ν/Ω > 10 ~ β1/2 collision frequency

Energetics in the (Kinetic) Shearing Box

Shear

Turbulence (MRI)

Grid-scale Loss ofMagnetic & Kinetic

Energy (~ 1/2 of GPE)

AnisotropicPressure

Direct “Viscous”

Heating onResolved

Scales(~ 1/2 of Grav.Pot. Energy)

Collisionless Damping of Fluctuations

(small)

• Heating ~ Shear*Stress

• Collisional Plasma

• Coulomb Collisions set Δp ➝ q+ ~ m1/2 T5/2

• Primarily Ion Heating

• Collisionless Plasma

• Microinstabilities Regulate Δp

• Significant Electron Heating

Ti/Te in Two-Species Kinetic MRI Sims

Ti/Te ~ 10 at late times

Astrophysical Implications

Use q+ ~ T1/2 in 1D modelsto determine Te(r) & radiation

Line of fixed L = 1036 erg/s

for Sgr A*

Sgr A*: Predicted consistent w/ Faraday Rotation & measured Te from VLBI

Lines w/ smaller T at a given R correspond

to larger accretion rate

‘best

gues

s’

Summary• MRI growth is enhanced in a collisionless plasma

• anisotropic viscosity is destabilizing so Pm effects fundamentally difft.

• Nonlinear amplification & saturation requires isotropization of the plasma pressure by small-scale kinetic instabilities

• Anisotropic Stress ~ Maxwell Stress

• ~ 1/2 of the grav. pot. energy thermalized on large scales

• ⇒ significant electron heating

• ⇒