# Presentation: 816664

## Kinetic effects in astrophysical plasmas

- Prateek Sharma (Princeton)
- (thanks to G. Hammett, E. Quataert, A. Schekochihin, J. Stone,...)

- Chandra  image of Hydra A cluster

**Notes:**
First of all I’d like to Thank you for having me here. It’s a pleasure to be here.
I’ll be talking about “kinetic effects in astrophysical plasmas”. The work has been done with my thesis advisor Greg Hammett and  Jim Stone from Princeton; Eliot Quataert from Berkeley and Alex Schekochihin from Cambridge.

I’ll discuss two examples of collisionless astrophysical plasmas: Low luminosity accretion flows, and hot plasmas in x-ray clusters. Image on the left is IR observations from  Keck telescope by the UCLA group of the stellar orbits around the SMBH in the Galactic center taken over a period of 10 years. The red spots indicate the location of stars at a particular time. Similar observations have been carried out by Genzel’s group at Max Planck. Using the orbital parameters for the ellipses described by the trajectory of the stars, the mass of the BH is estimated to be 4 million solar masses. The accretion disk around the SMBH is fed by the outflows from these massive stars.

On the right is the Chandra image of Hydra A cluster showing a few tens of million Kelvin hot plasma emitting x-rays.

## Motivation and Outline

- Accretion important for formation of planets, stars, BHs, galaxies,...
- How does matter loose angular momentum and fall in? How does it radiate?
- MRI turbulence responsible for transport [Balbus & Hawley ’91]
- MRI studied in the collisional (MHD) limit, but low luminosity accretion flows collisionless
- MRI in collisionless limit: linear characteristics, nonlinear shearing box simulations
- Pressure anisotropy as a result of amplification of B
- Anisotropy driven instabilities, isotropization
- Applications to disks and clusters

**Notes:**
Accretion is important in astrophysics. It’s important for the formation of planets, stars, black holes and galaxies. Many of the luminous objects that we observe in the sky are powered by accretion.
Key question is, how does matter lose its angular momentum and fall into the potential well? Molecular viscosity is terribly small to cause accretion. In 1991 Balbus and Hawley proposed the MRI as the source of turbulence in the disk. Since then MHD simulations of accretion disks have been very successful in elucidating the physics of accretion.
Low luminosity accretion flows like the one in our galactic center are believed to be collisionless with mean free path comparable to the system size. So we have studied the MRI in the collisionless limit. We have carried out linear analysis and nonlinear simulations in a local shearing box.
Pressure anisotropy is a natural consequence of magnetic field amplification in collisionless plasmas because of conservation of the adiabatic invariant. Pressure anisotropy cannot be arbitrarily large. It drives small scale instabilities like mirror, ion-cyclotron, firehose instabilities which isotropizes the pressure. I’ll discuss the application of these microinstabilities to MRI turbulence in collisionless disks and to clusters where collisionless effects can be important.

## Accretion disk

- Local Simulations in
- Shearing-Box

**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.

## Transport in disks

- molecular viscosity not enough => turbulent ‘viscosity’
- vth H, Wu [Shakura & Sunyaev 1973]
-    Source of turbulence is MRI
- [Balbus & Hawley ‘91, Hawley, Gammie & Balbus ‘95]

- Spring force :  B.B, -.(p║-p┴)bb, anisotropic force causes instability even if B=0

- Anisotropic stress arises in collisionless plasma where pressure is anisotropic

- V~1/R1/2

**Notes:**
Molecular viscosity between differentially rotating disk is too small to account for accretion. Ideas of turbulent viscosity were introduced.
Shakura and Sunyaev introduced the -prescription. In this model coefficient of turbulent viscosity corresponds to a turbulent eddy of disk height scale. It was not until 1991 that the source of disk turbulence was identified. Balbus and Hawley identified MRI as the source of turbulence. MHD turbulence results in correlations of magnetic fields and a non-zero stress. Stress consists of Maxwell and Reynolds stress.  
Its easy to understand MRI in terms of a simple model by Balbus. Consider two fluid elements connected by a spring. Since the flow is Keplerian, the inner fluid element is moving faster than the outer one, the spring is stretched and pulls back on the inner fluid element. The inner fluid element slows down and loses the rotational support. This enhances the effect of the spring and there is a run-away.
Magnetic fields act like a spring. In magnetized collisionless plasmas pressure is anisotropic, pressure anisotropy also gives rise to a spring force which is non-zero even when B->0. The destabilizing force due to pressure anisotropy can be important for magnetic amplification from small magnetic field strengths. As I shall show later,  anisotropic stress is important in transport in collisionless disks. 

## A 3-D global MHD simulation

- [Movies by John Hawley]

- MHD simulations immensely successful. Need to study this important
- instability in different physical regimes: e.g., collisionless accretion flows

**Notes:**
These are John Hawley’s global MHD disk simulations. Contours of density are shown. A constant angular momentum torus is initialized in equilibrium. The magnetic field lines are aligned with the density contours. This equilibrium configuration  is unstable to the MRI. The initial torus is turned into a turbulent Keplerian disk by the action of the MRI. The movie on the left is the 3-D volume rendering of density. The movie on the right shows azimuthally averaged inner accretion disk. The simulations shows a turbulent disk with a net mass inflow. There are indications of  weak outflows. 
Ideal MHD simulations have been immensely successful. Since the MRI is very rubust instability, we need to study its importance in different physical conditions like the kinetic regime. 

## RIAFs: Sgr A*

- Strikingly low L~ 10-6LBondi ~ 10-8LEdd
- Non-radiating => hot plasma=> thick
- disk, unlike luminous thin disks
- Massive stars feed the disk around
- the Black Hole at 10-5 M****** **per year
- Observed plasma: (RBondi~106 Rs)
- T ~ 107 K, n~10 cm-3
-  mfp ~ 1016 cm >>
-            RS (horizon)~1012 cm
- Plasma collisionless at few 1000 Rs (H)
- We study MRI in the collisionless
- limit

- Chandra image

**Notes:**
This is a Chandra image of the hot plasma around the SMBH  in the Galactic center. Bright spots are the massive stars and the accretion disk around the black hole. Massive stars feed the accretion disk at 10-5M0 per year. The plasma parameters in the outer regions of the disk are well measured.
As the plasma does not emit efficiently, the particles are believed to be hot and form a pressure supported thick disk, unlike the cold luminous thin disks. Sgr A* is the best studied of the low luminosity accretion flows that are believed to be present in the nuclei of quiescent galaxies.
Observed plasma density at the Bondi Radius (~1pc) is 10 particles/cc and temperature is 107K. This give a mfp of 1016cm, which is much larger than a few thousands of the Schwarzchild radius. Plasma is collisionless at the disk length scales. So we study the MRI in the collisionless regime.

OMIT:
At Bondi radius: Vi = 1.5e-10 /s, ve = 9.2e-9 /s, tdyn=3e-7, mfp=1.9e15 cm
Estimated conditions near BH:
Tp~1012 K, Te~1010 -1011 K
n~106 cm-3
B~30G (b=10)
ri ~re ~104 cm, R~H~ a few RS
r/L << 1 everywhere
ni~10-12 /s, ne~10-8 /s, cs/L~10-2/s

## Kinetic MHD

-  Vlasov equation for  << i and kri << 1=> DKE
- Moments of DKE => KMHD, for collisionless effects on MHD scales
- But particles free stream (rather than diffuse) along field-lines

- [Kulsrud 1983, *Handbook of Plasma Phys.*]

- Drift kinetic equation
- f(x,v║,,t) in 5-D

- KMHD

**Notes:**
Vlasov equation is the complete description of a collisionless plasmas. We obtain the drift kinetic equation on an asymptotic expansion of the Vlasov equation in the  , klimit. This approximation retains the slow timescales and large length scales aprox. of MHD but provides a correct description of collisionless plasma unlike MHD. Particles free stream along the field lines, rather than diffuse as in collisional regime.

DKE evolves the distribution function f(x,vpar,mu,t) in a 5-D phase space. Moments of the DKE gives the continuity eq. and the eq. of motion. The difference from MHD is that plasma pressure is anisotropic w.r.t. local magnetic field direction unlike in MHD. We need equations for the evolution of ppar and pperp which is obtained by the 3rd order moments of the DKE.

## Closure approx. for heat fluxes

- q=0 CGL/double adiabatic approximation

- adiabatic invariance
- of  ~ mv┴2/B ~T┴/B

- p┴  pII  as B 

- Closure Models for heat flux (temp. gradients wiped out on ~ a crossing time) => multipole approx. to plasma Z function, recovers Landau damping
- dispersion relation for all waves (and instabilities) consistent with the DKE, correct threshold for mirror instability
- non-local q in real space we use a local expression with a typical kL
- recovers Braginskii/Chapman-Enskog in large collision frequency limit

- [Snyder et al.* *1997, *Hammett & Perkins 1990,*
*similar to Chang & Callen 1992*]

**Notes:**
3rd order moments of the DKE gives rise to eqs. for ppar, pperp in terms of qpar, qperp. We run into the usual closure problem and we need equation of q’s in terms of lower order moments. From the equation for ppar and pperp its easy to see that as B increases, pperp increases and ppar decreases. This is a direct consequence of adiabatic invariance of mu. 

For the heat fluxes, we use the results from Snyder et al., who work out closure approximations for the case of the kinetic MHD equations, based on earlier work on fluid models of kinetic effects.

These closure models for the heat flux basically cause temperature gradients to be
wiped out in the time it takes for particles to cross a perturbation, and can be shown
to be equivalent to a multipole approximation to the plasma dispersion Z-function and
thus provide a model of kinetic effects like Landau damping.

It is thus able to reproduce the dispersion relation for all waves and instabilities
consistent with the Drift-Kinetic equation.  In particular, it gives the correct
instability criterion for mirror instability, which is not possible if the heat fluxes
are dropped.  Kulsrud showed that the CGL equations give a large error in the threshold for mirror instability.
{heat fluxes give a factor of 6 error in the threshold for the mirror instability}

Because of the k-dependence of this closure model, the heat flux would be non-local in
real space.  For these simulations, we instead replace k_par with a typical constant k_L
in order to get a simple local heat flux expression, and do some tests with different
values of k_L to investigate the sensitivity of the results to this parameter.  This is
somewhat like flux-limited diffusion models of radiation transport.

## Microinstabilities

- Change in B gives rise to pressure anisotropy due to  conservation
- Anisotropy results in:
- 					ion-cyclotron
- 					mirror              (p ┴/ p║-1) > (a few)/
- 
-                  		firehose (p ║/ p┴-1) > (a few)/
- 
- For MRI turbulence B increases
-                                             and   p ┴ > p║  => IC/mirror

- Even with microinstabilities  can
- be conserved unless  i, kri  
- Pressure anisotropy can be larger
- than marginal. Pitch angle
- scattering at large anisotropies

- [Gary et al 1997, *JGR* **102**, 27159]

- 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.  

## KMHD to Braginskii if  k║vth

- Pressure equation reduces to Braginskii’s anisotropic stress tensor, P<<P
- Correct result for anisotropic heat flux
- Kinetic MHD more general, transitions smoothly from collisionless to Braginskii, to MHD  as increases
- Mirror, firehose, ion-cyclotron instabilities can occur in Braginskii, growing at smallest scales, also needs isotropization due to microinstabilities

**Notes:**
In the limit when time between collisions is smaller than the time it takes for the ions to free stream along a perturbation, KMHD reduces to the Braginskii equations. The difference of the two equations for ppar and pperp, on making the assumption that dP<P, and balancing the velocity gradient terms and the collision terms give the result for Braginskii’s anisotropic stress tensor. Similarly Braginskii’s result for local anisotropic conduction is obtained in the high collisionality limit.

Kinetic MHD is a more general description, transitions smoothly from fully kinetic to Braginakii to MHD regime as collision frequency is increased. 

Mirror, firehose and cyclotron instabilities can also exist in the Braginskii limit resulting in fastest growth at smallest scales. Need to include anomalous scattering due to  microinstabilities.

## Collisionless MRI: linear

- Fastest mode ~ 2 larger than MHD
- Fastest growth at much larger
- scales: magnetic fields in
- protogalaxies
- Transition to MHD: 

- [Sharma et al.* *2003, *ApJ ***596**, 1121
- Balbus 2004, *ApJ* **616**, 857
- Islam & Balbus 2005, *ApJ* **633**, 328]

- Magnetoviscous instability

**Notes:**
Graph shows the growth of the MRI normalized to the disk rotation frequency as a function of kVa/W, where Va is the Alfven velocity based on the vertical magnetic field. Omega is the disk frequency. The instability in KMHD is twice faster and occurs at larger length-scales as compared to MHD. This can be important on amplifying fields in protogalaxies where field strength is very weak. 
Top dashed line shows the growth rate in the collisionless limit, and as we increase the collision frequency we transition to MHD. Transition to MHD occurs roughly when nu>Omega root(beta). Balbus has studied the same instability in the Braginskii formalism and shown that it exists even when magnetic field strength is vanishingly small, he called it the magnetoviscous instability and emphasized the fact that it can amplify fields at large scales beginning with very weak fields. 

## MHD wave families in

- a plasma in Keplerian
- rotation
- Presence of damped modes
- can influence energetics
- preferential heating of
- resonant particles
- Parallel eq. of motion:
- mdv║/dt = -║B + e E║

- Barnes damping
- 	(TTMP)    Landau damping

- [Sharma et al 2003, ApJ
- Quataert* *1998, *ApJ ***500**, 978]

**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.

## KMHD code (based on ZEUS)

- From 8 variables in MHD to 15 in KMHD
- Subgrid model for microinstabilities
- Nontrivial implementation of anisotropic
- transport

**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.

## Mirror instability

- 1D test for mirror instability in kinetic MHD
- Magnetic field 750 to the box
- ║=1
- CGL approx. gets the wrong marginal stability
- Parallel heat conduction important for mirror
- instability
- Correct linear growth rate
- Nonlinearly: magnetic field strength and
- density anticorrelated at small scales
- because  particles are trapped in magnetic
- mirrors
- Qualitatively similar to PIC simulations

- p┴

- p║

- density

- B

- time

- X, saturated state

**Notes:**
We have tested the KMHD code with several test problems. This is a 1-D test, the magnetic field is at 75 degrees to the box. We begin with an initial pressure anisotropy which is unstable to mirror instability. Linear mirror instability fastest at smallest scales. We get correct linear growth rate. As mentioned earlier parallel heat conduction is crucial for mirror instability and this anisotropy would have been stable if we use the CGL approx. of q=0.

Nonlinear saturation of the mirror qualitatively similar to fully kinetic PIC simulations. As I showed earlier in the physical picture for mirror instability, we see the formation of anticorrelated density and magnetic perturbations indicating presence of magnetic mirrors.   

## Firehose Instability

- 2-D 100X100, initially isotropic
- shear  decreases B, p┴, p║ firehose instability
- Bx=By initially, 
- <<<< k║ vth
- =200

- p║

- p┴

- (p║-p┴)/B2

- Contours of Bx, By, vx, vy

- time

- time

**Notes:**
This is a 2-D test of firehose instability caused by pressure anisotropy which is driven by the shear in plasma.
We start with an isotropic pressure and initial magnetic field at 45 degrees to the box. We impose a shear in the box which unfolds the field lines and the field lines move further apart. As B is reduced pressure anisotropy is created with ppar becoming larger than pperp, plasma becomes unstable to the firehose instability. Pressure anisotropy increases and settles down to the marginally stable value. Notice the emergence of firehose mode structure along the magnetic field lines at small scales in the 2D plot.

## Collisionless Damping

- LFC: correct , 
- CGL: no damping

- Initialized a fast wave eigenmode along
- the field line
- LFC correctly gives linear damping rate
- No damping in CGL as expected.
- anisotropic conduction required for Landau
- damping

- time

- u^2

- Initial eigenmode damped
- as expected

- x

- initial

- final

- time

**Notes:**
In this problem we show that kinetic MHD reproduces correct linear damping of the linear MHD modes. We initialize a fast wave  eigenmode and see that it is damped according to the dampnig rate given by the kinetic theory. Figure shows the kinetic energy which decays with time as a result of collisionless damping. The eigenmode has both a real and imaginary part and that’s borne by the simulation as shown in the figure on the right. The initial perturbation is damped and also has a phase shift. The CGL simulation does not give any damping as expected. We require parallel heat conduction to capture collisionless damping.

## SB sims.: no limit on Δp, CGL

- As B amplified by MRI, p┴
- increases, field lines become
- stiff, stabilizing all the resolved
- modes, low amplitude anisotropic
- Alfven waves
- B.B  [1+(p┴-p║)/B^2]B.B
- pressure anisotropy Δp/B^2~
- few 10s
- Mirror, cyclotron instabilities
- expected to arise 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.  

## Collisionless MRI simulations

- Pitch angle scattering due to microinstabilities (mirror, IC, firehose) has been
- included
- Similar turbulent  saturation of the instability as in MHD. Stronger channel
- solution, variability

- MHD

- KMHD

- =400

**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.

## Physical limit on pressure anisotropy:

- p┴/p║-1<7/┴ (mirror instability violates 
-                        conservation, dominates at   5)
- p┴/p║-1<S/║1/2 (ion-cyclotron waves)

- [Gary and coworkers,
- Hasegawa 1969 *Phys. Fluids* **12**, 2642]

- Anisotropic stress ~ Maxwell stress
- Consequences on disk heating: resistive vs.
- collisionless damping

- MHD

- kinetic

**Notes:**
These are some more results from our standard kinetic MHD run shown in the previous slide. The first figure shows pressure anisotropy as a function of time in the dark line. The dot-dashed line which follows the pressure anisotropy very closely is the pressure anisotropy limit due to the ion-cyclotron hard-wall. The dashed line in the bottom is due to the mirror hard-wall. For this choice of pitch angle scattering models, ion-cyclotron instability dominates the  isotropization of the pressure tensor. Since the physics of isotropization because of microinstabilities is not entirely clear, we have used a range of plausible models for pitch angle scattering. 
The second figure shows the comparison of Maxwell stress to the anisotropic stress. The two are well correlated and the anisotropic stress is almost as important as the Maxwell stress in transporting angular momentum. Anisotropic stress also important for disk heating, and heating by anisotropic stress is comparable to resistive heating in KMHD.  
We have studied the transition from collisionless to collisional regime using an explicit collision frequency in the KMHD eqs. The image on top right shows the anisotropic and Maxwell stress for different collision frequencies. As collision frequency becomes larger than a few tens, anisotropic stress becomes smaller and smaller whereas Maxwell stress reaches its MHD value, as we do not expect anisotropy in MHD limit.

## A fiducial cluster

- Hot plasma in the ICM
- n~10-3 /cc,  T~108 K, vthi~103 km/s
- U (rms turbulent velocity) ~ 300 km/s
- L~200 kpc (outer scale)
- mfp ~1kpc
- Viscosity ~ mfp vthi  => Re=UL/~50
-    		        Rm=UL1029
- L/U~109 yr. (outer timescale)
- Re-1/2L/U~108 yr (viscous scale)
- L~LRe-3/4~10kpc (viscous cutoff)
- Beq = (8nT)1/2 ≈ 20 G
- Bturb = Beq U/vthi ≈ 6 G
- i~104 km << mfp , plasma magnetized
- Braginskii regime

- Coma cluster

**Notes:**
The next application of collisionless plasma physics I discuss is for clusters of galaxies. This is a Chandra image of a few tens of million degrees hot plasma in the intracluster medium (ICM) of Coma cluster. The mass of the cluster is dominated by the dark matter but the mass of gas in the ICM is larger than that of stars in the galaxies.

Cluster plasma is hot and dilute.The typical densities are ..., temperatures in few tens of million Kelvin. This gives an ion thermal speed of a thousand km/s, The turbulent speeds are estimated to be ~ 300 km/s. The outer scale of the cluster is 200 kpc. The mean free path is 1 kpc. Because of large mean free path, plasma viscosity is large and Reynolds number is small 50s and the Magnetic Reynolds number of 10 to the 29! 

 Equipartion field is around a few microGauss. Larmor radius is much smaller than mean free path, plasma is magnetized.

## Magnetic fields in Clusters

- No rotation
- subsonic (galactic mergers, AGN feedback)
- Tangled magnetic fields at small scales

- [Schekochihin et al* *2004, *ApJ ***612**, 276]

- l

**Hydra A**
- [Vogt & Enßlin* *2005, *A&A ***434**, 67]

**Notes:**
Clusters are simple MHD systems as we do not have to worry about rotation and can do subsonic, isotropic, homogeneous turbulence in a box.

If the Prandtl numer is much larger than one, magnetic field can be concentrated at scales much smaller than the turbulent velocities. Consider a simple picture of a sheared eddy with a magnetic field. The shear grows the magnetic field strengths at small scales until it reaches the magnetic diffusion scales. This has been borne out by the simulation. This image shows the velocity and magnetic field strengths in 3-D simulations of MHD turbulence in large Prandtl number limit be Schekochihin and coworkers. Magnetic fields arrange in form of folded field structure.

The graph is an observational result from the Faraday rotation measurements of Hydra A  cluster by Vogt and Ensslin. It also shows an indication for growth of magnetic energies at subviscous scales.

## Magnetic fields in Clusters

- Large Prandtl number (viscosity >> resistivity) in galactic clusters gives wide range of scales where collisionless effects are important Re~100>>Rm
-  Pm~/=2.6 X10-5 T4/n (1029 for clusters!!)
- Pm >>1 for hot, dilute plasmas, Pm << 1for earth’s core

**Notes:**
This plot illustrates different scales in the cluster turbulence. The largest cluster scale is few 1 Mpc and a Kolmogorov viscous cut-off at ~ 10 kpc., mean free path of 1-10 kpc. Below this scale we expect the cluster plasma to be collisionless. The next scale is the proton Larmor radius at 10to5 km followed by the resistive scale at 1000 km! There is a big range of important scales. Need to identify important effects at each scale to know the small scale structure of magnetic fields. 

## Folded fields & microinstabilities

- Magnetic field amplified
- Susceptible to mirror, ion-cyclotron

- Magnetic field reduced, firehose

- Microinstabilities prevent the field to accumulate at small scales (l),
- Saturate at a few 10’s i
- Particle scattering by microinstabilities, increases effective collisionality,
- Increases Re
- (1/B)dB/dt = bb:u ~ Re1/2U/L
- Can give rise to fast growth of magnetic fields at large scales, reduces
- effective Pm

- [Schekochihin et al* *2005, *ApJ ***629**, 139,
- In progress]

- =(p┴ - p║)/p┴

**Notes:**
Shown is the folded field structure that arises in the Pm>>1 dynamos. The longer section of the field line structure compresses as the magnetic field is increased in those regions. The field lines in the folded region is expanding and magnetic field strength is reduced. Mirror instability is expected to arise in regions of increasing B and firehose in the regions of reducing B. The dispersion reln. For mirror and firehose instability are shown and firehose occurs when ppar>pperp and mirror occurs in the opposite limit.
These instabilities cause mode structure at Larmor radius scale which is large compared to lnu. Provides a length scale for the cut-off of magnetic energy.

Anisotropic instabilities cause pitch angle scattering that increases the effective collision frequency and hence the large Reynolds number => fast growth of magnetic fields. Alex has been looking into this scenario of growing magnetic field at fast rate.

## Future Work

- Previous calculations with vertical B. Future nonlinear sims. w/ initial z, fastest growth at much larger scales
- Global 2-D calculations with T0, effect of MTI (convective instability driven by T)
- Direct kinetic simulation of the DKE (evolving f(x,v║,,t)) like gyrokinetic codes in fusion

**Notes:**
We want to investigate the case with Bphi=Bz in the collisionless limit. That case does show a factor of two faster growth rate and at large scales. We are still investigating the nonlinear saturation of this case.
We want to investigate the global stability of MRI turbulence in global disk simulations. The convective instability driven by temperature gradient unlike the entropy gradients in a collisional plasma can affect the disk turbulence.
To investigate the kinetic effects in detail we want to go beyond a fluid approximation of anisotropic conduction. With the available computational resources we can hope to solve the Eulerian full drift kinetic equation to study collisionless plasmas. These will be analogous to the gyrokinetic codes used in the fusion simulations.

## Conclusions

- -Plasma kinetic effects important for RIAFs, clusters
- -mirror, firehose, ion-cyclotron instabilities
- -Subgrid models for microinstabilities to model flows at large (L>>ri) scales in kinetic MHD
- -Anisotropic stress, collisionless damping, energetics: Landau damping, resistive, viscous heating
- -Convective instability, MTI, effect on global disk simulations
- -Identify important scales and processes for collisionless astrophysical plasmas

- Thank you for your attention!

**Notes:**
To conclude,
I have described our linear and nonlinear studies of the MRI in the collisionless regime. We discovered that plasma kinetic effects can be important for dynamics and energetics of the collisionless disks. Small scale microinstabilites arise as a result of pressure anisotropy caused by fluctuating magnetic fields. We use sub-grid models to impose a limit on pressure anisotropy due to mirror, ion-cyclotron and firehose instabilities. Anisotropic stress and collisionless damping will have important effects on heating of different species in the plasma. Need to study relative importance of small scale viscous and resistive heating and large scale collisionless damping.

It will be important to study the convective stability of MRI turbulent disk in global MHD simulations. Since there are so many important scales in collisionless plasmas it will be important to identify important physics at each scale and its effect on the global dynamics and energetics in the collisionless systems.