Plasmas in Laboratory and AstrophysicsPlasmas in Laboratory and AstrophysicsAlfven Wave Turbulence and Heating.Alfven Wave Turbulence and Heating.

Center for Multi-scale Plasma Dynamics.Center for Multi-scale Plasma Dynamics.Bill Dorland, Maryland.Bill Dorland, Maryland.Alex Schekochihin, Cambridge.Alex Schekochihin, Cambridge.Steve Cowley, UCLA/Imperial College.Steve Cowley, UCLA/Imperial College.Troy Carter, Brian Brugman,Warren Lyberger, Pat Pribyl, UCLA.Troy Carter, Brian Brugman,Warren Lyberger, Pat Pribyl, UCLA.Greg Howes, Eliot Quataert, Berkeley.Greg Howes, Eliot Quataert, Berkeley.Greg Hammett, Princeton.Greg Hammett, Princeton.Thanks to: Stewart Prager, U. Wisconsin and the members of theThanks to: Stewart Prager, U. Wisconsin and the members of theCenter for Magnetic Self Organization Center for Magnetic Self Organization for data.for data.

Accretion and MHD.Accretion and MHD.

Instability Tangled BField.

Momentum transport.

Gravitational energy loss

Turbulent energy

Dissipation, Heat to ??• Electrons -- radiation.• Ions -- swallowed by black hole.Quataert and Gruzinov.

Magnetic self-organization in the lab., MST Magnetic self-organization in the lab., MST Reversed Field Pinch at Wisconsin, Reversed Field Pinch at Wisconsin, Prager Prager et. al.et. al.

–2–1012Time (ms)1.50.50 Q

(MW/m2)

30200 V

(km/s)

100.40.20 Tion

(KeV)

C4+.07.06Φ/πa2

( )T

.04.020 bB(a)

~1.0

€

˜ B

B

toroidal magnetic flux

heat flux(MW/m2)

rotation(km/s)

ion temperature(keV)

dynamo

magnetic fluctuations

energy transport

momentum transport

ion heating

time (ms)

Small Scales.Small Scales.

Kraichnan argued that the small scale motions will see a large quasi-uniform field. The small scale motions will become Alfven/slow waves traveling along a roughly uniform B.

Most of the dissipation happens at small scales -- thesearguments suggest that the dissipation is of small amplitude Alfven/slow waves.

Introduction.Introduction.

• Need to understand the turbulent cascade of Alfven waves.Need to understand the turbulent cascade of Alfven waves.

Many applications: astrophysics, fusion, space.Many applications: astrophysics, fusion, space.

• MHD regime -- Goldreich - Sridhar theory.MHD regime -- Goldreich - Sridhar theory.

• Collisionless regime -- gyrokinetics.Collisionless regime -- gyrokinetics.

• Heating in gyrokineticsHeating in gyrokinetics

• Simulations. Simulations. Bill Dorland, Greg Howes.Bill Dorland, Greg Howes.

• Experiments. Experiments. Troy Carter, Brian Brugman, Warren Troy Carter, Brian Brugman, Warren Lyberger, Pat Pribyl.Lyberger, Pat Pribyl.

MHD Waves. (Review)MHD Waves. (Review)

Slow Waves

kk, B0 and displacementcoplanar.

Damped by transit time damping when k|| mfp~1.

Alfven Waves

k

No damping until k i ~1

Displacement perpendicular toB0 and k.

Collision MeanFree path

Ion Larmor radius

Alfven Wave Interaction.Alfven Wave Interaction.Incompressible MHD, Kraichnan, Goldreich and Sridhar.

B0=B0z

Positive and negative waves traveling along B0.

Only oppositely going waves interact nonlinearly

Two Wave packets - passing through.Two Wave packets - passing through.

Three wave weak interaction for oppositely directedwaves gives.

Thus and there is no cascade in

Strong Alfven Cascade.Strong Alfven Cascade.Goldreich-Sridhar 1995. Komolgorov argument.

Energy flux from scale l to 2l

Cascade time

Constant from scale to scale.

GS argue that the scale l is the perpendicular scale andthat the parallel scale is set by “critical balance.” Nonlinear rate = Linear rate

Assuming strong interaction

Alfven waves stirred at scale l0 with velocity v0

Collisionless

Spectrum.Spectrum.

Slow WaveAlfven Wave

k

Ek

Gyrokinetic regime

MHD simulations Maron & Goldreich confirm scaling. Cyclotron frequency

B0

B

Vion

.Vflow

€

Vflow ≈E × B

B2

L

L||

Parallel scale >> Perpendicular scale, L|| >>L~i

Gyro-kinetic Fluctuations.

Bp

Particle displacement, p ~ Of Order Field line displacement, B ~ O(L)

B0

B

Vion

€

Vflow ≈E × B

B2

L||

Parallel scale >> Perpendicular scale, L|| >>L~i

Gyro-kinetic Fluctuations.

Particle displacement, p ~ Of Order Field line displacement, B ~ O(L)

“Ring” Center=R

Ion position

Ring radius= = v/i

Gyro-kinetic Equation. Gyro-kinetic Equation.

Maxwellian evolving on slow heating timescale

Shift in energyDue to potential

Perturbed DistributionOf “rings”

+ Maxwell’s equations.

Evolution of the Rings

Taylor and Hastie,Frieman and Chen ……Taylor and Hastie,Frieman and Chen ……

5D!

Ion kinetic energy

Collisions

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Box Gyro-kinetic Simulation MHD regime -- Bill Dorland.

Waves launchedby “numericalantenna” drivenby stochastic oscillator.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Gyro-kinetic heating.Gyro-kinetic heating.

Time averaged qE.vWork done on particles

Collisional energy exchange

Numerically it is hard to compute the heating this way because energy sloshes in and out of particles. We canshow that it is equivalent to an expression derived from ENTROPY PRODUCTION.

This term is positive and much easier to average, heating always comes from collisions even in collisionless regime!

Mixing in Phase Space.Mixing in Phase Space.

• f(f(r,vr,v,t) is conserved along particle orbits in ,t) is conserved along particle orbits in collisionless motion. collisionless motion.

• Stochasticity in orbits leads to fine scale mixing - Stochasticity in orbits leads to fine scale mixing - jagged f. Less collisions jagged f. Less collisions more jagged f. more jagged f. Entropy production becomes independent of Entropy production becomes independent of collision rate. collision rate.

f

v

New SimulationsNew Simulations

• Current simulations are doing the kinetic Current simulations are doing the kinetic regime and caculating the ion and electron regime and caculating the ion and electron heating. heating. QQee=Q=Qee(T(Tee/T/Tii, , ), Q), Qii=Q=Qii(T(Tee/T/Tii, , ) ) Bill Bill

Dorland, Greg Howes.Dorland, Greg Howes.

• Need to understand the cascade in this Need to understand the cascade in this regime and examine the dynamics.regime and examine the dynamics.

• Single Pass Design.Single Pass Design.• Alfvén waves are Alfvén waves are

launched into the open launched into the open plasma column.plasma column.

• Only co-propagating Only co-propagating geometry is possible.geometry is possible.

Cavity Launched Wave Experiment Cavity Launched Wave Experiment Design LAPD.Design LAPD.

• Reflecting Cavity DesignReflecting Cavity Design• Alfvén waves are Alfvén waves are

launched into a plasma launched into a plasma column terminated by a column terminated by a reflecting platereflecting plate

• Co- and Counter-Co- and Counter-propagating interactions propagating interactions occuroccur

• Using the Using the Natural MASERNatural MASER as the pump wave and the as the pump wave and the Driven Driven MASERMASER as a sideband the as a sideband the dependence of the instability dependence of the instability may be investigated.may be investigated.

The Spontaneous and Driven cavity waves The Spontaneous and Driven cavity waves in Concert are also Unstablein Concert are also Unstable

Possible Experiment GeometriesPossible Experiment Geometries

• Single Pass, co-propagating and Single Pass, co-propagating and counter-propagating separatelycounter-propagating separately

Multiple pass, co-propagating and Multiple pass, co-propagating and counter-propagating combinedcounter-propagating combined

CONCLUSIONSCONCLUSIONS

• Well defined problem that needs a solution.Well defined problem that needs a solution.

• Excellent use of fusion codes - well adapted Excellent use of fusion codes - well adapted to the problem.to the problem.

• Experimental study on LAPD will yield Experimental study on LAPD will yield further results this year.further results this year.