Post on 26-Dec-2015
PLASMA RELAXATION
The Super-Comic Edition
Loren SteinhauerUniversity of Washington
Plasma Physics Summer SchoolLos Alamos, 10 August 2006
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
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
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-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
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($100) Km
measures the knottedness.
Mobius: untwisted = zero
one twist = some
two twists = more
How to express this mathematically?
(2) Knottedness
mini-break
Magnetic helicity: the magic personality trait
Km = ABdV
B = magnetic field($) (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.)
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!