# Presentation: 058246

## Anisotropic conduction with large temperature gradients

- Prateek Sharma (Princeton)
- (thanks to G. Hammett)

- Chandra  image of Hydra A cluster

- Trace image

**Notes:**
I’ll be talking about “anisotropic conduction, specially its numerical implementation in regions with large temperature gradients”. The work has been done in collaboration with my adviser Greg Hammett. Images show two examples where plasma is collisionless, and anisotropic conduction is important. Image on the left is a trace image of coronal loops emerging from the solar surface. Plasma is magnetically dominated and about a million degrees hot. Thermal conduction is maily along the field lines. The image on the right is a more exotic example. It shows a cluster of galaxy in x-rays indicating the presence of hot plasma, which is expected to be collisionless with thermal conduction being anisotropic.

## Motivation and Outline

- Anisotropic transport for hot, dilute plasmas
-     (c»nT-3/2
- Thermal conduction along B
- Finite differencing anisotropic conduction
- Symmetric, Asymmetric methods
- Negative temperature: simple tests
- Basic review of slope limiters in CFD.
- Limiting temperature gradient: slope, entropy limiters
- Tests
- Applications

**Notes:**
Here I give an outline and motivation for the talk. Anisotropic conduction in low collisionality plasmas is pervasive. Conduction becomes anisotropic when the collision frequency is much smaller than the gyrofrequency. In this case particles move along  

## Anisotropic thermal conduction

- T = e/n(for ideal
- gas in 3-D

- e : internal energy density
- q : anisotropic heat flux
- T : temperature
- t : time
- ║conduction coefficients
- Finite difference equation in conservative form in 2-D:

**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.

## Grid

- Staggered grid with scalars
- at zone centers, vectors at
- zone faces.
- Natural location for
- conservative form

## Asymmetric differencing

- Most natural differencing

- Min used so that Courant stability condition is not severe.

## Negative temperature with asymmetric method

- Reflecting BC for temperature

## Symmetric method

- Primary heat fluxes at cell corners
-   [Gunter et al., JCP, 2005]

## Why Symmetric method?

- Numerical cross-field diffusion does not scale with ║/ ,Sovinec’s test
-  Self-adjointness of                        , matrix is symmetric, good for Krylov methods
- Entropy condition satisfied at the cell corners,   -q.T0
- good when temperature gradients are not enormous
- Less sensitive to angle between b and coordinate axes

## Problems with symmetric method

- Small scale overshoots are not damped.
- Unable to diffuse away a chess-board pattern.

- =0

- =0

- , q=0

## Negative temperature with symmetric method

- Heat flows out of (i,j) despite it being a minimum.
- Reflective BC.
- qx, qy at (i-1/2,j-1/2) <0

- symmetric

- entropy-limited

- asymmetric

- slope-limited

## Why negative temperature?

- qxx satisfies the entropy condition,
- with heat flowing from higher to lower
- temp., but qxy can have any sign.
- Need to limit transverse term qxy
- Responsible for heat flowing in wrong
- direction
- What is the best interpolation?
- Arithmetic average for dT/dy?
- Limiters for averaging?

## Basic Eulerian/Continuum Advection Algorithms

- Discrete grid, f(zj) = fj Conservative differencing:

- Std 2nd order centered differencing (okay for smooth regions, phase errors too large for sharp-gradient regions, gives unphysical oscillations):

- 1st order upwind (eliminates unphysical oscillations, but too dissipative):

*thanks to Greg Hammett*
*for introductory slides on *
*limiters.*

## Higher-order upwind Methods withclever monotonicity-preserving slope limiters

- Reconstruct f(z) in each cell, extrapolate to bdys:

- Piecewise constant = 1st order upwind :

- van Leer’s (MC) limiter:
- “Monotonized Central”

- Simplest, minmod limiter:  		minmod(a,b) =	sign(a,b). min(|a|,|b|)

- Higher order extensions, e.g., 2nd order PPM of Colella & Woodward

## Advection tests

- 2cd order Centered
- Algorithm
- okay in smooth regions
- Phase errors large
- for sharp gradients

- 1st Order upwind
- Too dissipative

- From R.J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press (2002).

## Advection tests:   Higher order upwind w/ limiters 

- Advection tests:   Higher order upwind w/ limiters

- 1st Order upwind
- Too dissipative

- 2cd order upwind
- With MC limiter
- Much better

- From R.J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press (2002).

## Just going to higher order doesn’t help near sharp gradient regions (Gibb’s phenomena)

- Lax-Wendroff equivalent to downwind
- Slope. Can lead to overshoots in
- reconstruction

- Top Fig. From  R.J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press (2002).
- 2cd Fig. From C.B. Laney, Computational Gasdynamics, Cambridge Univ. Press (1998).

## MC limiter gives much more robust and accurate result.

- Central differencing to determine
- slopes can lead to overshoots in
- reconstruction, Slope limiter uses
- s=0 at extrema to avoid oscillations

- From R.J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press (2002).

## Limiting transverse gradient

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

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

## Limiting symmetric method

- L2 not symmetric in its arguments
- Need tolimit both normal and transverse gradients.
- Normal derivative limited so that qxx is always from higher to lower temp.
- Chess-board pattern will not diffuse if normal derivative not limited!

## Entropy limiting

- Using face pairs to satisfy entropy condition

- qx

- qy

- If dT/dx=0, then an arbit. qx can give neg.
- temp.
- Not strictly monotonic, but overshoots
- highly damped
- Entropy condition satisfied at some point
-  is not a sufficient condition for heat flowing
- in the right dirn.

## Ring diffusion test

- Initial hot patch 0.5<r<0.7, 11/12<<13/12
- Coefft. ║=0.01,
- =0, tend=200
- Reflective BC
- Circular magnetic
-     field lines

## Small temperature gradient

- Asymmetric
-        MC

- Symmetric
-        MC

- 400 X 400 box
- Asymmetric and symmetric
- methods non-monotonic
- even late times
- Slope limited methods
- monotonic
- Sharp boundaries even with
- limiting
- For lower resln. slope lim.
- methods are more diffusive.

## Ring diffusion with large temp. gradient

- Initially Tmax=10, Tmin=0.1
- Both symmetric and asymmetric
- methods give negative temp.
- at late times
- Slope limited methods are
- strictly monotonic with Tmin=0.1
- at all times
- Entropy limiting damps the
- undershoots.

## Perpendicular numerical diffusion

- Test problem by *Sovinec et al. 2005*
- Solve anisotropic diffusion with source term to get steady state, circular field lines
- Lx=Ly=1, in SS heat diffusion balances Q
- Q =
- An explicit Tanal(0,0)=1/ 
- num = 1/T(0,0)-1, correct defn. is
- num = 1/T(0,0)-1/Tiso(0,0)

## 

- 

- 

- Symmetric method is least diffusive
- (also entropy limited)
- num independent of ||/
- Asymmetric method & MC limiter
- close, num scales with  ||/
- Second order convergence for all
- except minmod
- Correct defn. for num implies even
- tinier diffusion

- num= fewfor N=100

- symm.

- minmod

- van Leer

- MC

- asymm.

- entropy

## Applications

- Problems with large temperature gradients where negative temperature cause numerical problems (spurious instabilities)
- Astrophysical systems e.g., Disk-corona interface, warm-hot phase interface in ISM
- Systems where a huge ||/is notreqd., or where need not be resolved.

## Future Directions

- Methods that are both monotonic and less diffusive, higher order reconstructions
- Faster implicit methods for anisotropic conduction
- Applications to problems with large temperature gradients and anisotropic thermal conduction, e.g., global models of RIAFs

**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

- -Non-monotonic behavior of centered differencing in presence of large temp. gradients
- -simple test problems for negative temp.
- -slope limited methods are monotonic, second order convergence
- -test problem to measure num
- - Astrophysical applications, ISM, disk-corona interface

- 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.