PLASMA RELAXATION

The Super-Comic Edition

Loren SteinhauerUniversity of Washington

Plasma Physics Summer SchoolLos Alamos, 10 August 2006

Sleepy Lumpy Baby

Acknowledgments

With permission from…

RoadmapThe players

Behavioral psychology“Good” behavior

The secret of good behaviorSome essential math

What happy plasmas look likeWhy aren’t all plasmas happy?

Relaxed states (and more)

Part 1: The PlayersPlasma($)

Our friendly plasma

Definition: an “exotic” fluid because…

there is a lot of extra weird stuff going on

How weird is weird?Reference point: a simple fluid:

only one player: the fluid itself

KERPLUNK

time

Stuff that happens:Waves: only sound waves

Friction: viscosity($)

= a fluid dragging against itself

Plasmas($) are weird

Three players: ion fluid (+) electron fluid () electromagnetic (EM) field

ghost

All three occupy

the same space

at the same time

Weird…All three players carry their own kinds of energy

Ion and electron fluids: (a) motion energy (kinetic): a high-grade energy form (b) thermal energy (heat): low-grade energy

low-grade = highly disorganized; high entropy($)

Re second law of thermodynamics

EM field: (a) electric field energy: high-grade energy (b) magnetic field energy: high-grade energy

Fusion: the ions are an isotope of hydrogen

How do the three actors get along?Ions talk to electrons; both talk to the EM field

Blah blah

Blah blahBlah blah

ion fluid to electron fluid: drag against each other = electrical resistance($)

ion fluid to EM field: ions push against the EM field, and vice versa = Lorentz force ($)

electron fluid to EM field: ditto

Waves in a plasmaSimple fluids have sound waves,

but a plasma has many other kinds

Fluids talking to EM field gives rise to…

sound waves,

Alfven waves($)

and many many more

Sibling rivalryIons and electrons get along, sort of…

Get along: they must stay close together,

otherwise gigantic electric fields jerk them back together: quasi-neutrality($)

Fight: they drag against

each other:

= electrical resistance ($)

Oof!

Ugh!

Biff!

Sock!

Energy crisisEnergy implications:

Electrical resistance burns up EM field energy (high-grade energy)

and dumps it into ion and electron

thermal($) energy (low-grade energy)

Electrical resistance is

an entropy($)-builder

Fusion and space plasmasVery hot!

In hot plasma the drag between ions and electrons

is very weak,

i.e. low electrical resistance ($)

experimental plasmas ~ 100 eV 1 eV($) = 11,600 oK

fusion plasmas ~ 10,000 eV = 10 keV

Snobs

In high-temperature plasmas the ions and electrons hardly even speak to each other

Hmmph!

Pah!

Swiss Embassy

EM field: intermediary between ions and electrons.

Tell her “Blah blah

blah!”

Tell him “Blah blah

blah!”

“Blah blah blah!”

“Blah blah blah!”

Important exceptionIons and electrons can have a modest amount of

direct communication, if the EM field is tough

My toys

My toys

Peace! Peace!

This can happen with plasma relaxation($)break

Part II: Behavioral Psychology

Two fundamental classes of behaviorWaves and friction

(1) Waves: very complicated in a plasma because there are three players, not just one. sound waves($): energy of ions and electronsAlfven waves($): energy of magnetic field

Energy: waves move energy around from one place to another.

How fast?

How fast do things happen with waves?

Timescale($) for wave dynamics: wave ~ L/V

L = size scale of plasma (~ 20 cm)

V = (say) speed of sound:

Our friendly plasma

100 eV hydrogen plasma: V ~ 1800 km/s

Timescale for wave dynamics is wave ~ 1 s

Classes of behavior…

(2) Friction: electrons drag against ions

electrical resistance($)

Energy: electrical resistance burns up EM field energy and turns it into low-grade “heat” energy

ZAP!

He He!

magic pencil

YOW!

Dissipation

Electrical resistance burns up EM field energy (high grade) and converts it into plasma heat energy (low grade)

Dissipation moves energy around too It always takes things downhill

Thermal conduction moves heat energy from one place to another; this also is an entropy builder too, making the temperature uniform (less organized)

Viscous friction burns up the flow energy (high-grade form of plasma energy) and turns it into heat (low-grade)

How fast do things happen with dissipation?

Timescale for burning up EM field energy:

resistivity ~ L2/D

L = length scale

D = resistive “diffusivity($)”

In our friendly plasma D ~ 1.2 m2/s;

Timescale for burning up EM field energy resistivity ~ 30 ms

Much slower than the timescale for waves

A fancy name: Lundquist number($)

Lundquist number:

Lu resistivity/wave

Waves move energy around 30,000 times faster than resistance burns up

EM field energy.

Lu = 30,000

The swingWave: oscillating motion of the swing

Dissipation: friction in the air and rope

Push me again Papa If Lu = 30,000, Baby

would swing back and forth ~30,000 times before the swing more-or-less stops.

Key question

If Lu >> 1 (waves much faster than dissipation) can we ignore dissipation?

?

Answer: partly… but not entirely:

In Plasma Relaxation, Lu is large

but dissipation still plays a key role

More later… break

Part III “Good” Behavior

Four types of behavior

(1) Totally good behavior: the “ideal case”

Our friendly plasma as a compliant little angel

This hardly ever happens

Type-2 BehaviorLow-level scrapping that

the baby-sitter tolerates

“A dull roar”

Poke Poke

Keep it

down.

Type-3 Behavior

Oof!

Ugh!

3) Violent scrap

The baby-sitter is a casualty;the kids end up wasted.

Type-4 Behavior4) Violent scrap but the result is different…

Oof!

Ugh!

I’m happy

I’m happy I survived!

Babysitter manages to make some adjustments, after which things are more or less peaceful

Things like this can happen in a plasma

#1) Totally good behavior.

Good, but it hardly ever happens!

Say Lu = 30,000. Waves equalize everything out in about 1 s and then hardly anything happens for the next 30,000 microseconds.

Type-2 Behavior2) Low-level scrappingNo major violence, but continuous, small-scale

scrapping between ions and electrons

= plasma turbulence($).

This leads to a plasma that leaks energy faster than you would like.

Type-3 Behavior

The EM field intervenes, but gets knocked out:

Death ensues = disruption($)

Plasma starts off more or less quiet: quiescent($)

Suddenly, a violent instability (BIG WAVE) appears

Plasma goes crazy;

A lot of dissipation($)

takes place.

Type-4 Behavior

The EM field intervenes, and calms things down:

The plasma survives in a relaxed state($)

Plasma starts off more or less quiet: quiescent($)

Suddenly, a violent instability (BIG WAVE) appears, but the result is different

Some dissipation($)

takes place.

Energy

I need a vacation

The cost of relaxation($)

Type-4 behavior, but at a price

Getting toward …the topic of this lecture

It wears down the EM field

But but but… How can all this violence happen at high–Lu?

Timescale formula dissipation ~ L2/D

~ 30 ms for L = 25 cm (plasma size).

Nasty little

gremlin L

What if a wave process has a much smaller scale,

say L ~ 1/2 cm?

Then dissipation ~ 20 s

…almost as fast

as the wave time.

Example: reconnection($)

Small is big (or beautiful?)

BRRRR!Exampleinsulating your house

Install R100 insulation

keep single-pane windows

Even though Lwindow << Lhouse

most of the heat loss is

through the windows.

The house remains cold and drafty

Can small stuff affect the whole?

(The mouse that roared)

break

Part IV The secret of

“Good” Behavior

The mystery of the relaxed stateWhat’s the difference between type-3 behavior

(passed out) and type-4 (relaxed state)Om

The secret - EM field:

a hidden personality trait of the babysitter

The esoteric mathematics of topology.

Simple illustration #1: Möbius strip “squashed donut”

no twist

one twist

two twists

Consider a particle moving along an edge.In making one complete loop (long way), how many

times does a particle circle around the short way?

Magnetic fields can get twisted around themselves like this.

Magnetic fields can get

tied up in knots too.

Gmmp!

Magnetic helicity($100) Km

measures the knottedness.

Mobius: untwisted = zero

one twist = some

two twists = more

How to express this mathematically?

(2) Knottedness

mini-break

Part VI: The math of relaxed states

Magnetic helicity: the magic personality trait

Km = ABdV

B = magnetic field($) (vector)

A = “vector potential($)”: B = A

A BThanks B! You’re

welcome.

A exists because B = 0

One of Maxwell’s equations($)

What is a “domain”“DOMAIN”: volume occupied by the plasma inside

some well-specified boundary

Example of an idealized domain boundary: a perfectly-electrically conducting metal wall.

domain boundary S

domain volume V

Magnetic energyAnother important volume integral:

total magnetic energy in the domain

Wm is a high-grade

form of energy

02 2BdVWm

magnetic field (magnitude of

B)Natural constant:

permeability of free space

Properties of the two integrals

In the perfectly ideal case

(only waves, no dissipation)

Wm = const Km = const

I cannot tell a lie

i.e. they are

constants of motion($)

or integrals of motion($)

Caveat: not quite true for Wm,

(see later “postscript”)

Taylor theoryNamed after

Brian TaylorTa da!

I’m not Brian Taylor; he’s not nearly as handsome.

Truth telling:

Taylor’s Conjecture

Key elements of Taylor theory

Principle #1: SELECTIVE DECAY($100)

(a) It isn’t a perfect world, but some things play in our favor

i.e. you must take dissipation into account even if Lu is large

(b) Wm is less rugged than Km

i.e., when dissipation goes to work,

Wm burns down a lot faster than Km

All animals are equal, but

some are more equal than

othersGo

elephants!

Same big bad wolfhouse of

strawhouse of bricks

Taylor: principle #2: MINIMUM ENERGY($100)

How far down does Wm burn? As low as possible while keeping fixed Km

Minimize Wm subject to fixed Km relaxed state($) (Taylor state)

Wow, they never taught that in preschool

Math: constrained minimization problem($) variational calculus($) method of Lagrange multipliers($)

Finding Taylor states($): do the math

(Wm Km) = 0

You may not get all this

today.

022 0

2

dVdV

BBA

variation of (…)

Lagrange multiplier (constant)

Substitute

The math (cont.)Remember B = A

0)(22 0

2

dVdV AA

A

Apply the variations

0)(()()(

0

dVdV AA

AA

The math (cont.)

Integrations by parts

Whew!

0ˆ))(

0

dSdV nABB

BA

volume integral

area integral

used B = A and Gauss’ theorem

break

Part VI (cont.) the mathApply principles of variational calculus:

(1) A = 0 on the boundary: gets rid of area integral

00

dVB

BA

(2) The only way for this volume integral to be zero for any A (any “variation” of A) is if […] = 0.

current density

Thus j = B

(used Ampere’s law B = 0j)

Final step: Taylor statesBack substitute to find Wm = Km

What is equal to? Wow, that’s pretty simple

Variational principle finds extrema of Wm:

maxima, minima, relative minima…

Want the absolute minimum

L = domain size

Lowest possible gives the answer.

determined by size of domain:

~ 1/L

What do Taylor states look like?

An example: in a cylindrical geometry

Bessel function model($)

radius coordinate , r

wall =domain boundary

Bz

B

Summary• Plasmas have both waves and friction• Even in hot plasma where the friction is small,

you can’t ignore it• Waves can lead to a violent restructuring (or

death) of the plasma• Friction plays a role too• Plasma relaxation is one of the good outcomes of

the violence• Relaxation burns up some of the EM energy• It leads to “relaxed states”• Taylor states are an example of this

Five brief postscripts

Hey, that’s all I know!

You take over Papa

#1:Wm is not the real constant of motion (Taylor’s

reductionist view)• No ideal constant of motion for the “high grade”

energy form unless… Assume constant density

• In the constant density case the high-grade energy includes both magnetic and flow energy

Wmf (B2/20 + minu2/2)dV• Proper ideal constant of motion measuring “quality

of energy” is the global entropy: S p/n dV• Proper minimization problem: S = 0, not Wmf = 0

#2

• The standard MHD model is “single fluid” It neglects many two-fluid($) effects

• What’s different about a two fluid?

• Two helicity constants of motion instead of one

Ke = ABdV electron helicity

Ki = (A+miu/e)A(B+ miu/e)dV ion helicity

#3But what about the fact that Wmf is not an ideal

constant of motion?

• The minimization problem(Wmf iKi eKe) = 0

• leads to the same result as (S Wmf iKi eKe) = 0

• The results look familiar

je = eB ji = i(B + miu/e)

#4

Is the relaxed state totally relaxed?

• In practice no.

But the core of the plasma may be relaxed and the skin unrelaxed

• The theory of relaxed states is still useful for the core

#5Does a relaxed state always happen?

• If the initial violent phase is too violent then the plasma may disrupt

• You can prevent the possibility of global relaxation by imposing a gigantic toroidal field (Kruskal-Shafranov limit)

The large toroidal field

cripples a tokamak

Heresy!

finis

Banned in Princeton,

Cambridge Mass,

Japan, Culham, San

Diego, Cadarache, etc.