# Presentation: 245203

## Introduction to Fusion andMagnetohydrodynamics 

*Paul Bellan, Caltech*
*Los Alamos Summer School 2006*

## Fusion: If two light nuclei combine, the product has higher binding energy than the 

**Fusion**: If two light nuclei combine, the product has higher binding energy than the
- Ingredients and so energy is released

## Fusion: Explain by example using DT

- DT: the fusion reaction with highest cross-section

## Deuterium-Tritium Fusion

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| --- |

## Deuterium

- 1 proton + 1 neutron

*+*

- Tritium
- 1 proton + 2 neutronS

*+*

*D*

*T*

## In general D and T mutually repel each other

*In general D and T mutually repel each other*
*because both are positive*
*(electrostatic repulsion)*

## For fusion, hurl the D and T towards each other. 

*If D and T get *_*close enough*_* the short range*
*nuclear attractive force takes over and they fuse.*

## You didn’t hurl hard enough!

*You didn’t hurl hard enough!*

## Slide 9

## 14 MeV

*+*

*+*

*14 MeV*
*neutron*

*plus gamma particle (photon)*
*3.5 MeV*

*alpha particle *
*(helium nucleus)*

## Slide 11

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| --- |

## Fusion reaction cross-sections of main interest

***DT is easiest, ***
***can be done at 10 keV, i.e., at only 100 million K !***

## Lawson criterion for fusion

- Reaction rate proportional to *n*2 where *n* is density
- Energy produced scales as *n*2 where is confinement time
- Energy invested scales as *n*
*Energy profit when n*2 > *n*
* *
*When appropriate units inserted, find energy profit when n*cm-3-sec

## Main Issues

- Physics: *satisfy Lawson criterion*
- heating to 10 keV
- keeping heat in (confinement)
- Plasma must be thermal (discuss next slide)
- Engineering: *do this practically*
- neutrons
- tritium handling and recycling
- wall loading and lifetime
- energy extraction

## Why fusion plasma must be thermal

*Cross-*
*Section*
*(log scale)*

*Temperature *
* **(log scale)*

*Scattering by 90**0*

*fusion*

*-Because scattering cross-section exceeds fusion cross-section by at least*
*  **two orders of magnitude, a particle will be scattered over 100 times*
*  **before it undergoes fusion*
*-Hence, particle velocity will be completely randomized, i.e., thermal*

*~10**2*
*ratio*

## Fusion compared to fission

- Fusion ash is just stable helium, much easier to deal with than 100’s of different kinds of radioactive fission products
- Fusion reactor does not contain significant stored energy (reactor contains only what is being currently burned), cannot go prompt critical
- DT fusion does produce a lot of neutrons which can activate structure
- Fusion is difficult:
- high temperature and good confinement not easy
- A fusion reactor has yet to be made
- fusion is at a research stage
- much science still has to be learned

## Fusion and plasma physics

- Binding energy of electrons in atoms is of order of a few eV
- E.g., it takes 13.6 eV to detach the electron from a hydrogen atom
- At temperatures above a few eV all matter becomes ionized
- Plasma is name of this ionized gas
- Fusion invariably involves plasmas
- (unless one believes in cold fusion!)

## plasma

*Room temperature=1/40 eV*

*0.6 eV*

*5 eV*

*1.5 keV*

*10 keV for DT fusion*

*plasma*

## How plasmas are characterized

- Density: particles per cubic meter (or per cubic cm)
- Symbol: n
- Temperature: in eV,  1 eV=11,604 K
- Symbol: T
- Magnetic field (if exists):in Tesla (or Gauss, 1 T=104 G)
- Symbol: B
- Species: e.g., hydrogen, deuterium, neon
- There are many different kinds of plasmas
- Enormous range of parameters, for example
- Magnetosphere tail lobe: n=1 cm-3
- Laser fusion: n=1020 cm-3

## Slide 20

## Fusion plasma physics: two types with very different physics

- Magnetic confinement
- Use magnetic fields to confine hot plasma
- Tokamaks, stellarators, reversed field pinches, spheromaks, field reversed configurations, Z-pinches
- Inertial confinement
- Do fusion fast,
- before plasma has a chance to move
- Laser fusion, particle beam fusion

## Magnetic confinement & space physics

- Magnetic confinement physics has many similarities to plasmas in space
- This is because magnetic fields confine many space plasmas
- Many ideas flow back and forth between magnetic fusion physics and space physics

## Plasmas in astrophysics

*Photos of astrophysical jets from Hubble Space Telescope*

*Plasmas in *_*astrophysics*_

## Photo of solar coronal loop by TRACE spacecraft

*Photo of solar coronal loop by TRACE spacecraft*
*Plasma trapped in arched magnetic fields*

## Plasmas in

- SOHO
- EIT
- 9-14-99

*Plasmas in*
_*Solar physics*_

## Aurora

*Aurora*
*(U. of Alaska)*

## Plasma properties

- plasma properties often _very different_ from ordinary materials
- Plasmas can be trapped in magnetic fields
- Plasmas are excellent electrical conductors and  electrical conductivity improves as the plasma gets hotter, conductivity independent of density!
- Mean free path in plasmas can be very large (e.g. kilometers in fusion plasmas)
- Plasmas can have many internal waves, instabilities, turbulence

## Two approaches to Lawson ncm-3-sec

- Low density for a long time
- Magnetic confinement fusion
- typically * *cm-3 for 1 second
- High density for a short time
- Inertial confinement fusion
- typically * *cm-3 for  second

## Ways of modeling a plasma

- In order of detail (most detailed first)
- Calculate trajectory of every single particle and put in Maxwell’s equations (this is good for thought experiments, but generally impractical)
- Follow trajectories of classes of particles (e.g. all electrons going 60 mph, Vlasov theory)
- Consider plasma as composed of electron gas interspersed with ion gas (2-fluid thery)
- Consider plasma as an electrically conducting fluid (Magnetohydrodynamics, “MHD”)

## Magnetic confinement

- Electrons and ions can be confined by magnetic fields

*Magnetic field*

*Charged particle motion*

## Make  toroidal magnetic field

## Theoretical Models

## Start with the “ideal magnetohydrodyamic equations”:

**Eq. of motion**

**Induction equation**

**Ampere’s law**

-  Mass conservation equation

- Adiabatic relation

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| --- |

*A Reference *
*for the*
*Rest of Us!*

- Magnetohydrodynamics
- FOR
- DUMMIES

## What the ideal MHD

-  equations mean...

## Parallel currents attract each other

*Drawing from http://www.phy6.org/earthmag/oersted.htm*

## Slide 37

## Calculation of force on a current due to a magnetic field

_*Calculation of force on a current due to a magnetic field*_

## Slide 39

## Pinch force

*Pinch force*
*A bundle of parallel wires will all attract each other*
*They will squeeze together tightly*

*end view*

## Pinch force provides confinement

- The pinch force squeezes parallel wires together
- If wires are replaced by current carrying plasma, then the pinch force squeezes the plasma together
- If plasma is hot, then pressure of plasma will push outward
- Can get an _equilibrium_ where pressure pushing out is balanced by pinch force pushing in

## Drawing from http://www.phy6.org/earthmag/oersted.htm

*Drawing from http://www.phy6.org/earthmag/oersted.htm*

## Toroidal direction

*Drawing from ORNL web site*

*Toroidal direction*

*Poloidal*
*direction*

*Source of fields in a tokamak:*
*Toroidal magnetic field produced by *_*external coils*_
*Poloidal magnetic field produced by *_*toroidal current in plasma*_

*Typical magnetic field line*

*Tokamak configuration*

## Slide 44

## Red arrows= outwards force due to pressure

*Red arrows= outwards force due to pressure*
*Black arrows= inward magnetic pinch force*

## Slide 46

## Slide 47

## Poloidal magnetic field provides confinement

*Drawing from ORNL web site*

*Toroidal direction*

*Poloidal*
*direction*

*Poloidal magnetic field provides *_*confinement*_
*Toroidal magnetic field provides *_*stability*_

## “q”is defined as the number

*“**q”is defined as the number*
*of times a field line goes*
*around toroidally for each time*
*it goes around poloidally*
*Drawing from ORNL web site*

*Toroidal direction*

*Poloidal*
*direction*

*Toroidal magnetic field produced by poloidal currents*
*Poloidal magnetic field produced by toroidal currents*

## Equilibrium

- Balance outward pressure of plasma with inward pressure of magnetic force
- Magnetic field lines are helical
- Field lines trace out magnetic flux surfaces
- Ideally, particles can move anywhere on a flux surface, but cannot move off a flux surface
- Magnetic flux surfaces are like surfaces in a smoke ring-vortex

## If field line goes around toroidally many times and if q is not a rational number,

*Smoke ring vortex from Magarvey and MacLatchey 1964*

*If field line goes around toroidally many times and if q is not a rational number,*
*then field line fills up a “magnetic surface” , equilibrium consists of nested toroidal*
*magnetic surfaces like this smoke ring vortex*

## Equilibrium equation

- Predicts that magnetic flux surfaces are surfaces of constant pressure
- Hence can have pressure gradient perpendicular to flux surfaces, confinement
- outward force due to plasma pressure is balanced by inward pinch force due to poloidal field created by toroidal current

## Hand-waving way of thinking about magnetic fields

- Magnetic field behaves as if it produces
-    a pressure in direction perpendicular to the magnetic field
-   a tension in direction parallel to magnetic field
- From this point of view, pinch force is like an elastic band wrapped around a bundle of parallel wires
- Tension in elastic band squeezes wires together
- Proper mathematical description requires vector calculus

## magnetic axis

*Density and temperature are *_*peaked*_* on magnetic axis*

## Magnetic confinement issues

- Stability – MHD, turbulence
- Confinement – leakage across flux surfaces
- Heating –Ohmic, wave, neutral beam
- Beta (efficiency of use of magnetic field)
- Refueling – getting fuel to magnetic axis
- Diagnostics – details inside 10 keV plasma

## Categorization of 

***Categorization of ***
***toroidal confinement***
***configurations***
*Top: least complicated*
*Bottom: most complicated*

*Fields most prescribed,*
*device more expensive,*
*(least magnetic turbulence,*
*instability)*

*Fields least prescribed,*
*Device less expensive,*
*(more magnetic turbulence,*
*Instability)*

## MHD:Frozen-in fields

- Plasma is a very good conductor
- In fact, plasma acts like a superconductor
- This means magnetic field is *frozen into* plasma
- The plasma and magnetic field move together

## Vacuum magnetic fields  

- Vacuum field is lowest energy state for given boundary conditions
- Any perturbation from vacuum configuration requires energy
- Vacuum shape is like shape of a molded radiator hose
- Field wants to snap back to vacuum state, hence concept of “_*field line tension*_”

## Any deformation from initial shape requires work,

*Any deformation from initial shape requires work,*
*configuration wants to snap back to initial shape*

*Radiator hose*

*Vacuum magnetic field*

## Magnetic force between currents

- Anti-parallel currents repel each other
- leads to “_*hoop force*_”
- pressure perpendicular to field

## Physics Concepts (summary)

- Magnetic fields are frozen into plasma
- Magnetic field can be categorized into:
- vacuum (potential) fields which are lowest energy state for given boundary conditions
- non-vacuum fields, due to currents in plasmas

## Kink instabilities: another issue

*Magnetic energy of a circuit can be expressed as inductive energy*

## Because plasma is an excellent electrical conductor, approximate it as a

*Because plasma is an excellent electrical conductor, approximate it as a*
*perfect or ideal conductor*
*This means that the electric field must be zero inside the plasma*
*But Faraday’s law states that a changing magnetic field produces an electric field*
*Since electric field is not allowed in frame of plasma, the magnetic field cannot*
*change in frame of plasma*

## Hence, magnetic flux in the plasma is fixed, i.e., invariant

*Hence, magnetic flux in the plasma is fixed, i.e., invariant*
*Thus, magnetic energy will be lowered if inductance increases*
*Plasma is unstable to motions which increase inductance*
*Plasma would like to coil up, since coils have more inductance*
*Called kink instability *

*Magnetic flux*

*inductance*

*Magnetic energy*

## Experiment at Caltech demonstrating some of these ideas

- (Solar prominence simulation experiment)

## Break down gas

- Form plasma
- Drive field-aligned current

- plasma

## Self field of 

- Self field of
- current causes
- field lines to
- (i)  pinch
- (ii) twist
- (iii) and then...

## Current flow

- Outward force
- due to mutually
- repelling
- currents
- Recall that
- vacuum fields
- were lowest
- energy state

- ....bulge out

## Slide 69

## Slide 70

## Billowing due to hoop force, kinking due to instability of large current

*Billowing due to hoop force, kinking due to instability of large current*

## Particle point of view

- MHD is a fluid picture
- Can also consider motion of individual particles
- Find that particles can move across field lines via drifts
- But particles cannot move across flux surfaces
- Particles are confined to flux surfaces,
- 	provided there is toroidal symmetry

## E cross B Drifts

## Confinement from particle drift point of view

- Because of particle vertical drifts in curved field, a simple toroidal field does not provide confinement
- However, if there is a _toroidal current_, the field due to the current provides confinement
- Net field is helical

## Tokamak plasmas

- Hot plasma is nearly collisionless
- Mean free path is 10’s of km
- Particle motion governed by Lorentz equation
- Many body problem
- Symmetry very important, can show that particles confined to nested toroidal flux surfaces _only if_ configuration is toroidally symmetric

## Large toroidal current in tokamak pinches plasma together and

*Large toroidal current in tokamak pinches plasma together and*
*Balances outward pressure of high temperature coil*
*Toroidal current is unstable with respect to kinking.*
*Kinking prevented by strong toroidal magnetic field produced by toroidal field coil*
* **-toroidal field is a vacuum field, think of toroidal field as a radiator hose that does not*
*  **want to be bent out of shape*

## How to drive a toroidal current?Answer: use transformer action

## Air core transformer also possible

## Balancing the hoop force

- Pinch balanced the thermal pressure
- Toroidal field coils prevent kinking
- But still have hoop force due to current on one side of torus repelling current on opposite side
- Need to add an additional magnetic field to buck the hoop force
- This field, often called “vertical field”, provided by  set of “poloidal field” coils
- These carry a toroidal current flowing in direction opposite to that of plasma current and balance the hoop force

## Complete tokamak configuration

*Human scale*
*for reference*

## Reactor scenario

***Nominal Plasma parameters***
*Density n=10**14** cm**-3*
*Temperature= 10 keV*
*Toroidal current = 10’s of MA*
*Toroidal field =5 Tesla*
*Confinement time =1 second*

## Heating

- Ohmic heating (as in resistor) from dissipation of toroidal current heats to about 2 keV
- At higher temperature Ohmic heating ineffective because plasma is too good a conductor
- Plasma resistivity scales as *T**-*3/2 ,
- Resistivity of 700 eV plasma comparable to copper

## Auxiliary heating

- Inject energetic neutral beams that can cross magnetic field and directly heat particles
- Inject high power radio waves or microwaves (many kinds of waves possible)
- Both methods work

## Status of tokamaks

- Tokamaks are the front-runner
- Break-even has almost been achieved (64% of breakeven on JET)
- Multi-billion dollar machines
- Plans are underway to build a tokamak with net power output (ITER)
- This will be a prototype which produces significant amounts of power (500 MW)

## Slide 85

## Person

*Person*

*ITER*

## ITER

*ITER*
*Cross-section*

## Inside D-III-D tokamak at General Atomics in San Diego

*Inside D-III-D tokamak at General Atomics in San Diego*

## Tritium cycle

- Plan is to recycle tritium within the reactor
- Use lithium reactions
- 		6Li + n       4He + T + 4.8 Mev
- 		7Li +n        4He + T + n -  2.5 Mev
- -Thus convert neutrons partially back into tritium
- -Lithium becomes the consumable,
-  so fuel is effectively deuterium and lithium
- -clad reactor with lithium blankets,
-  use heat exchangers to extract heat from blankets,
-  make steam for turbines

## Caltech

*Room temperature=1/40 eV*

*0.6 eV*

*5 eV*

*1.5 keV*

*10 keV for DT fusion*

*Caltech*
*Experiment*
*regime*

##  Another experiment at Caltechillustrating MHD concepts:Astrophysical Jets/ Spheromak Formation

## Copper annulus 

- Copper annulus
- 50.8 cm diam

- Copper disk
- 20.3 cm diam

- Gas nozzles

- _Coaxial, co-planar electrodes_

## Slide 93

## Slide 94

## Bias field coil

- makes linked
- magnetic flux

## Gas puff valve

- (typical)

## Slide 97

## Gas nozzle

- 50 cm

- disk

- annulus

**Notes:**
Gas injection at 8 toroidal locations

## Side View

-  disk

- annulus

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Installation

## Slide 101

## Sequence

**high**
**vacuum**

- Bias field

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

##   Puff in neutral gas

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Astrophysical jet simulation steps:

- Distinct, arched filamentary plasma loops
- Loops merge to form central column
- Jet-like axial expansion of central column
- Kink instability of central column

## Breakdown, “spider leg formation”

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Spider leg

**Spider leg**

## MHD physics

- Anti-parallel currents repel

## Spider legs get bigger

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Slide 110

## Astrophysical jet simulation steps:

- Creation of distinct, arched filamentary plasma loops
- Filaments merge to form central column
- Jet-like axial expansion of central column
- Kink instability of central column

## MHD physics

- Parallel currents attract

## Spider legs merge to form central column

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Slide 114

## Astrophysical jet simulation steps:

- Creation of distinct, arched filamentary plasma loops
- Filaments merge to form central column
- Jet-like axial expansion of central column
- Kink instability of central column

## Central column lengthens

- I ~ 100 kA

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Slide 117

## Slide 118

## Doppler Shift Measurement Velocities ~ 15-60 km/s  observed

- Blue shift
- seen when
- viewing from
- this end

- Red shift
- seen
- when
- viewing
- from
- this end

## Doppler Shift Measurement Velocities ~ 15-60 km/s  observed

- Blue shift
- seen when
- viewing from
- this end

- Red shift
- seen
- when
- viewing
- from
- this end

## Example of Doppler shift data

- hydrogen, deuterium
- calibration lines
- from reference lamp

- D

- H

## Astrophysical jet simulation steps:

- Creation of distinct, arched filamentary plasma loops
- Filaments merge to form central column
- Jet-like axial expansion of central column
- Kink instability of central column

## Kink Instability of Central Column

**Notes:**
Gun design features
Cylindrical coordinate system
Experimental operation sequence (1) bias field, (2) gas puff, (3) gun discharge
Describe I-V traces

## Slide 125

## Kink occurs when central column becomes sufficiently long to satisfy instability condition

*      *
- S. C. Hsu & P. M. Bellan
- MNRAS 334, 257 (2002)

- L