# Presentation: 474765

## Kinetic Effects on MRI Turbulencein Black Hole Accretion

- Kinetic Effects on MRI Turbulencein Black Hole Accretion

- Prateek Sharma, 24th Aug.’06
- FPOE

## Accretion disks 
star & planet formation
Energetic sources, XRBs, AGN
gravitational energy         radiation
Consider matter falling from r1 to r2<r1
Decrease in gravitational energy        radiation
Luminosity (energy per time) M(dm/dt)
How does matter accrete?
Stable Keplerian orbit, R-3/2
molecular viscosity (friction between layers) insufficient
turbulent “viscosity”
What sustains nonlinear motions? MRI  [Balbus & Hawley’91]
Ionized, magnetized disk,                      MRI

- star & planet formation
- Energetic sources, XRBs, AGN
- gravitational energy         radiation
- Consider matter falling from r1 to r2<r1
- Decrease in gravitational energy        radiation
- Luminosity (energy per time) M(dm/dt)
- How does matter accrete?
- Stable Keplerian orbit, R-3/2
- molecular viscosity (friction between layers) insufficient
- turbulent “viscosity”
- What sustains nonlinear motions? MRI  [Balbus & Hawley’91]
- Ionized, magnetized disk,                      MRI

- How does matter fall in?

- r1

- r2

## Spring picture of the MRI

- Keplerian rotation
- Magnetic fields like a rubber-band

- Inner fluid element loses angular momentum to outer one
- Stable Keplerian          unstable MRI        nonlinear motions & turbulence

- Spring slow down mi, loses rotational support,
- falls in, spring stretches further. Run-away!
- Stabilized if B very strong—stable Alfven oscillations

- mo

- mi

## Standard model

- Gravitational energy radiated       cold, thin disk
- Dense collisional disk
- MHD valid        MRI, turbulence, and transport
- Good for luminous objects: quasars & binaries
- Are all disks thin & luminous? No. low luminosity accretion flows, energy retained as heat; hot, dilute, collisionless, thick disk.

## Thesis results

- Motivation is to understand physics of low-luminosity thick disks
- Transition from kinetic to collisional MRI
- Kinetic              Braginskii             MHD as 
- Presence of damped modes
- Local nonlinear shearing box sims. of collisionless MRI
- Pressure anisotropy natural as B changes, invariance
- Intermittent pitch angle scattering due to mirror/IC
- Scattering imposes an MHD like behavior
- Anisotropic stress~Maxwell stress
- Monotonic anisotropic diffusion
- Simple test problems to illustrate
- Limited differencing of transverse gradient

## Radiatively inefficient accretion flows (RIAFs)

- Low luminosity        grav. energy retained as heat, not radiated
- hot, dilute, collisionless, thick disk
- Radiative efficiency,             ,
- Accretion on BH/NS more efficient than fusion!
- ambient density
- But           << observed luminosity, Why?
-  accreted                     , or

## Sgr A*, a RIAF

- Sphere  of influence of SMBH barely resolved

## RIAFs in galactic centers

- Self-similar models for density profiles for
- -ADAF/Bondi, p=0,
- -CDAF/ADIOS, p=0-1,

- n

- r

## RIAF parameters: Sgr A*

- Plasma is collisionless for
- Plasma is magnetized with

- Realistic plasma model
- MHD not enough

## Plasma description of RIAFs

- Vlasov eq. for                 &			    DKE
- Moments of the DKE: Kinetic MHD eqs.

- PDF in a 5-D phase space

## Closure approx. for heat fluxes

- Next order moment
- Closure problem
- Adiabatic invariance
-                     = const.
- q=0, CGL approx., free streaming particles
- Closure Models for heat flux (temp. gradients wiped out on ~ a crossing time)      multi-pole approx. to plasma Z function, recovers Landau damping
- Use k║=kL=const. for sims.

## Braginskii (intermediate collisionality) limit,  k║vth

- Kinetic MHD more general, transitions smoothly from collisionless, to Braginskii, to MHD  as  increases

## Linear analysis of collisionless MRI

- Also occurs in collisionless regime
- Collisionless (pressure anisotropy) effects enhanced at larger 
- Fastest mode ~ twice faster than MHD
- Fastest growth at much larger scales
- Similar nonlinear saturation though

- Keplerian rotation frequency

## Kinetic      Braginskii      MHD 

- Terms in KMHD eq. of motion
- Transitions as  increased
- for             :
- Kinetic to Braginskii
- Braginskii to MHD

- Kinetic      Braginskii      MHD

## MHD modes in plasma 

- MHD modes in plasma
- w. Keplerian rotation
- MRI, the destabilized
- slow mode
- Damped modes can
- preferentially heat
- resonant particles
- Parallel eq. of motion:
- dv║/dt = -║B + e E║/m

- Barnes     Landau
- damping  damping
- (TTMP)    electrons
- ions

**Notes:**
This is a busy slide. Shown are the MHD wave families in a Keplerian disk in MHD and collisionless plasmas. The important point to note is that MRI is present in the collisionless limit as well,  and in addition there are damped modes present in KMHD. These can heat the resonant particles if the energy released from accretion can be tapped. Presence of collisionless damping at large scales affect important questions like:  How is the disk plasma heated? Relative importance of resistive, viscous, and wave heating? Wave-particle interactions occur through both magnetic mirrors and parallel electric field. W-p interactions in magnetic perturbations is TTMP or Barnes damping a magnetic perturbations analogue of Landau damping due to parallel electric field.

To reiterate,
Key difference between MHD and collisonless plasma is that only way the turbulent energy is dissipated in MHD is at small scales whereas collisionless damping at large scales due to wave particle interactions can be quite important for kinetic plasmas.

Epar = Sum(  (e/m)b.grad.P) /Sum(ne^2/m) 

## Nonlinear Shearing box sims.

- Local shearing-box sims. analogous
- to flux tube sims. in fusion; shearing
- periodic BCs

**Notes:**
This is an artist’s impression of a binary accretion system. Consider a binary system of a star and a compact object like a BH or NS or a WD. The plasma in the stellar companion fills the Roche lobe and a stream of matter is pulled by the compact object. The tidally stripped plasma has angular momentum and forms an accretion disk around the compact object.
The green box is a local box cut from the accretion disk. In the shearing box approx. widely used for numerical simulations in the local limit, equations are written in the frame rotating with the box. This gives rise to centrifugal and coriolis forces. The simulations that I describe later have been done in the local shearing box limit. Periodic boundary conditions are used in vertical & azimuthal directions. Shearing periodic BC is used in the radial direction, there is a jump in velocity vphi corresponding to a Keplerian shear in the box. This shear in the box drives the MRI.

## KMHD code (based on ZEUS)

- Anisotropic pressure unlike MHD
- Thermal conduction along field lines, local
- Landau fluid closure instead of more accurate
- non-local expressions

**Notes:**
Use operator splitting as used by ZEUS, into transport and source steps. The advection terms are treated conservatively using upwind methods. Rest other terms are treated by using centered differences in space. Number of dependent variables increased from 8 in MHD to 15 in KMHD. Instead of internal energy energy eqn. we have separate eqns. For the evolution of  ppar, pperp. Eqs. for ppar & pperp uses parallel heat fluxes based on Snyder’s result.
We include subgrid model for microinstabilites that limit pressure anisotropy within a given pressure anisotropy threshold. Isotropization by microinstabilities act like effective collision frequency. The subgrid models are based on hard wall limits on pressure anisotropy due to firehose, mirror and ion-cyclotron instabilities. 

Density, diagonal pressure terms are at zone centers, components of the vectors, q, v, B are at face centers. Anisotropic pressure terms and electric fields at the zone edge.

## CGL SB sims., =400, vertical B

- As B amplified by MRI, p┴ , as
-                          = const.
- Field lines become stiff, stabilizing
- all the resolved modes, linear
- anisotropic Alfven waves
- B.B  [1+(p┴-p║)/B2]B.B
- pressure anisotropy 4Δp/B2~
- few 10s
- Mirror, ion-cyclotron instabilities
- expected to at these pressure
- anisotropies

**Notes:**
This and next few slides show the simulations of the collisionless MRI with an initial beta of 400. This simulation is in the CGL limit with no limit imposed on pressure anisotropy by the subgrid models. As magnetic field is amplified by the MRI, pperp increases by mu conservation and the field lines become more stiff, resulting in the stabilization of all the MRI modes in the box. The steady state consists of small amplitude Alfven waves in the box. Notice that the total stress is very small and oscillatory in time. The pressure anisotropy relative to magnetic energy is rather large and we expect mirror and ion-cyclotron instabilities to isotropize the pressure tensor. We include the effect of microinstabilities in the next set of simulations that I’ll show.  

## Microinstabilities

- Anisotropy results in instabilities:

- ion-cyclotron
- mirror
- Firehose
- For MRI turbulence B sustained      p ┴ > p║         IC/mirror
- Even with microinstabilities  conserved unless
- Pressure anisotropy slightly larger than marginal allowed

- firehose

- mirror

**Notes:**
Pressure anisotropy cannot become arbitrarily large. Plasma microinstabilities are driven by pressure anisotropy. Ion-cyclotron and mirror instability occur when pperp>ppar and firehose when ppar>pperp. In MRI turbulence since magnetic field is increasing, pperp is larger than ppar over most of the flow. This results in mirror & cyclotron instabilities. The presence of microinstabilities does not imply that magnetic moment is not conserved. To violate mu conservation the scale of the microinstabilities should be ~ Larmor radius or the growth rate ~ gyrofrequency. Thus, pressure anisotropy can become more than the marginal criterion. We explore different models for pitch angle scattering by microinstabilities based on the work by Gary et al.

The firehose instability is similar to instability of a garden hose that is unstable to fast flow through it. The centrifugal force of free-streaming particles exceed magnetic restoring force and makes the initial perturbation grow.

Mirror instability causes magnetic compressions and rarefactions. Particles are trapped in regions of small magnetic field due to mirror force. This causes density and magnetic field strength to be anticorrelated.  

## MRI simulations w. subgrid models

- Similar turbulent  saturation, variability as in MHD
- Stronger channels

- =400, initially vertical field

- Much smaller magnetic
- energy when  pressure
- anisotropy not restricted
- Factor of ~105
- Pitch angle scattering
- makes a qualitative
- difference; imposes MHD
- like behavior
- That’s why MHD so good
- for collisionless plasmas!

**Notes:**
The results I present now are the runs with parallel heat conduction the pressure is isotropized by microinstabilities. We include a pitch angle scattering model due to mirror and ion-cyclotron instabilities (based on the work by Gary and coworkers) in the shearing box simulations. This is a comparison of the magnetic energy in the saturated state for MHD and kinetic MHD. The subgrid model imposes an upper limit on the pressure anisotropy.  

The magnetic energy show a similar saturation level as in MHD but the initial transient phase, the channel solution is much stronger in kinetic MHD. Kinetic MHD also shows slightly different time variability as compared to MHD. The final beta in the turbulent state is of the order of a few. Vertical field MRI causes near equipartition fields from initial beta of 100s.

## Pitch angle scattering

- Pitch angle scattering makes
- a qual. difference
- Mirror dominates over IC for 
- Scattering only in small fraction
- of box
- Scattering centers intermittent
- Effective scatteringat Sc. Ctrs.
- But Sc. Ctrs. are rare
- So, m.f.p. can be large even after pitch angle scattering

- Volume fraction where pitch angle
- scattering occurs

## Anisotropic (viscous) stress

- Viscous heating, collisionless damping in anisotropic stress
- Important for e & ion heating & hence radiation
- Aniso. stress  as  , transition to MHD

## Global MHD sims.

- ang. mom. torus surrounded
- by a low density corona.
- Density/temp. jump at bdry.
- Black hole at the origin
- Flow settles into a turbulent
- thick disk, dilute corona, &
- weak magnetized outflows

- Q. mfp~disk height, what is the effect of anisotropic conduction?
- Implementation of anisotropic diffusion non-trivial

## Anisotropic thermal conduction

- Conduction along field lines,
-  Parallel temp. gradient

## Centered Differencing

- Asymmetric
- Symmetric

- ni,j

- (i+1/2,j)

- (i+1/2,j+1/2)

- [Gunter et al. 2005]

## Negative temp. w. asymm. method

- Reflective B.C.
- Heat flux out of
- temp. minimum!
- T<0

## Negative temp. w. symm. method

- Same temp. as before,
-  different (i,j) labels

- qy=0, dT/dx non-zero only at (i-1/2,j+1/2)
- Only non-vanishing heat flux is

## Why negative temperature?

- For asymm. method, qxx always from
- higher to lower temp., but qxy can have
- any sign.
- Need to carefully interpolate qxy
- responsible for heat flowing in wrong
- direction
- What is the best interpolation?
- Arithmetic average for dT/dy?No.
- Centered differencing not always the
- best,e.g., gas dynamics
- Limiters for interpolating will solve it.

## Limiting transverse gradient

- We limit transverse temperature gradient to calculate qx
- L is a limiter like: minmod, van Leer, monotonized central (MC)
- Limiters return zero if arguments of opposite sign
- Only normal term remain at extrema
- Temperature extrema not amplified

- At extremum
- dT/dx = 0
- dT/dy = 0

## Perpendicular numerical diffusion

- MC limiter just twice more
- diffusive than asymm.
- MC is the least diffusive of
- Limiters
- (for 100 grid pts. In each
- dirn.) Sufft. to study
- qualitative effects,e.g., MTI
- Symmetric method least
- diffusive, but T<0 problem!

## Thesis contributions

- Transition from kinetic to collisional MRI
- Kinetic              Braginskii             MHD as 
- Presence of damped modes
- Local shearing box sims.
- Pressure anisotropy natural as B changes
- Intermittent pitch angle scattering due to mirror/IC
- Scattering imposed an MHD like behavior
- Anisotropic stress~Maxwell stress
- Monotonic anisotropic diffusion
- Simple test problems to illustrate
- Limited differencing of transverse gradient

## Future Work

- Two fluid sims., electron vs. ion heating
- Effect of anisotropic stress on electrons
- Global sims. w. anisotropic conduction
- Effect on
- Outflows, convection, role of MTI
- Directly solving DKE in 5-D phase space
- Role of intermittent pitch angle scattering
- On heat conduction
- MHD vs. free streaming, collisionless damping
- Pitch angle sc. common in collisionless plasmas, e.g, X-ray clusters, solar wind

- Thank you for your attention!