# Presentation: 056151

## PLASMA RELAXATIONThe Super-Comic Edition

- Loren Steinhauer
- University of Washington
- Plasma Physics Summer School
- Los Alamos, 10 August 2006

## Acknowledgments

***Acknowledgments***

## Roadmap

- The players
- Behavioral psychology
- “Good” behavior
- The secret of good behavior
- Some essential math
- What happy plasmas look like
- Why aren’t all plasmas happy?
- Relaxed states (and more)

## Part 1: The Players

***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

- 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

- All three occupy
- the same space
- at the same time

## Weird...

- All three players carry their own kinds of energy

**Ion**** ****and**** electron ****fluid****s**:
- 	     (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

- 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 plasma

- Simple 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 rivalry

- Ions 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* ($)

## Energy crisis

- Energy 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 plasmas

***Very 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

## Swiss Embassy

- EM field: intermediary between ions and electrons.

## Important exception

- Ions and electrons can have a modest amount of direct communication, if the EM field is tough

- This can happen with plasma relaxation($)

## Part II: Behavioral Psychology

## Two fundamental classes of behavior

*Waves and friction*

**(1) Waves**: very complicated in a plasma because there are three players, not just one.
- 	*sound waves*($):  energy of ions and electrons
- 	*Alfven waves*($): 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

## 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* ~ *L*2/*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.

## The swing

*Wave*: oscillating motion of the swing
*Dissipation*: friction in the air and rope

- 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...

## 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 Behavior

- Low-level scrapping that
-        the baby-sitter tolerates

- “A dull roar”

## Type-3 Behavior

- 3) **Violent scrap**

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

## Type-4 Behavior

- 4) Violent scrap but the result is different...

- 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 Behavior

**2) Low-level scrapping**

- No 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

- 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* ~ *L*2/*D *
*	**	**~ *30 ms for *L* = 25 cm (plasma size).

- 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?) 

- Exampleinsulating your house
- 	Install R100 insulation
- 	keep single-pane windows

- Even though *L**window* << *L**house*
- 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)

## Part IV The secret of “Good” Behavior

## The mystery of the relaxed state

- What’s the difference between type-3 behavior 						(passed out) and 						type-4 (relaxed state)

- The secret - EM field:
- 	a hidden personality trait of the babysitter

- The esoteric mathematics of topology.

## Simple illustration #1: Möbius strip 

-  “*squashed    donut”*

- 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.

-    tied up in knots too.

*Magnetic helicity***($100)** *K**m*
-    measures the knottedness.

**Mobius**: untwisted = zero
- 	     one twist = some
- 	     two twists = more

- How to express this mathematically?

**(2) Knottedness**

## Part VI: The math of relaxed states

## Magnetic helicity: the magic personality trait

*K**m* = **A********B***dV*
**     ****B** = *magnetic field*($) (vector)
**     ****A** = “vector potential($)”:     **B** = ******A**

**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.

## Magnetic energy

- Another important volume integral: total magnetic                         energy in                                      the domain

*W**m* is a *high-grade*
- form of energy

## Properties of the two integrals 

- In the perfectly ideal case
- (only waves, no dissipation)
*W**m* = *const*   *K**m* = *const*

- i.e. they are
- 	*constants of motion*($)
- 	or *integrals of motion*($)

*Caveat*: not quite true for *W**m*,
- (see later “postscript”)

## Taylor theory

- Named after       Brian Taylor

- 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) *W**m* is less rugged than *K**m*
- i.e., when dissipation goes to work,
-  *W**m* burns down a lot faster than *K**m*

- All animals are equal, but some are more equal than others

*Go elephants!*

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

- How far down does *W**m* burn?      As low as possible while   keeping fixed *K**m*

- Minimize *W**m* subject to fixed *K**m  * relaxed state($) (*Taylor state*)

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

## Finding Taylor states($): do the math

** (*W**m*  ***K**m*) = 0

- variation of (...)

- Lagrange multiplier (constant)

- Substitute

## The math (cont.)

- Remember **B** = ******A**

- Apply the variations

## The math (cont.)

- Integrations by parts

- volume integral

- area integral

- used **B** = ******A** and Gauss’ theorem

## Part VI (cont.) the math

**Apply principles of variational calculus:**
**    **(1) ****A** = 0 on the boundary: gets rid of area integral

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

## Final step: Taylor states

- Back substitute to find  *W**m*   =   ***K**m*
- 	What is *** *equal to?

- Variational principle finds extrema of *W**m*:
- 	maxima, minima, relative minima...
- Want the *absolute *minimum

## What do Taylor states look like?

- An example: in a cylindrical geometry
*Bessel function model*($)

## 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 

**#1:**

*W**m* 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
-  			*W**mf*      (*B*2/2**0 + *m**i**nu*2/2)*dV*
- Proper ideal constant of motion measuring “quality of energy” is the global entropy:  *S*      *p/n*** *dV*
- Proper minimization problem: ***S* = 0, not ***W**mf* = 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
*	**	**K**e*  =  **A********B***dV  *electron helicity
*	**	**K**i*  =  (**A**+*m**i***u**/*e*)**A******(**B+*** m**i*******u**/*e*)*dV  *ion helicity

## #3

- But what about the fact that *W**mf* is not an ideal constant of motion?

- The minimization problem
*	**	***(*W**mf*  ***i**K**i** *** ***e**K**e*) = 0
- leads to the same result as
*	**	***(*S*  ***W**mf*  ***i**K**i** *** ***e**K**e*) = 0
- The results look familiar
- 		**j***e*  =  ***e***B**		**j***i*  =  ***i*(**B** + *m**i*******u**/*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

## #5

- Does 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

## Banned in Princeton, Cambridge Mass,  Japan, Culham, San Diego, Cadarache, etc.