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Introduction to Fusion and Magnetohydrodynamics Paul Bellan, Caltech Los Alamos Summer School 2006

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

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Fusion: Explain by example using DT DT: the fusion reaction with highest cross-section

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Deuterium-Tritium Fusion

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Deuterium 1 proton + 1 neutron + Tritium 1 proton + 2 neutronS + D T

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

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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.

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+ D + T You didn’t hurl hard enough!

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+ +

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MeV neutron plus gamma particle (photon) 3.5 MeV alpha particle (helium nucleus)

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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 !

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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

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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

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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

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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

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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!)

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Room temperature=1/40 eV 0.6 eV 5 eV 1.5 keV 10 keV for DT fusion plasma

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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=10 4 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=10 20 cm -3

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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

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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

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Photos of astrophysical jets from Hubble Space Telescope Plasmas in astrophysics

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

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SOHO EIT Plasmas in Solar physics

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Aurora (U. of Alaska)

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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

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Two approaches to Lawson n cm -3 -sec 1.Low density for a long time –Magnetic confinement fusion –typically cm -3 for 1 second 2.High density for a short time –Inertial confinement fusion –typically cm -3 for second

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Ways of modeling a plasma In order of detail (most detailed first) 1.Calculate trajectory of every single particle and put in Maxwell’s equations (this is good for thought experiments, but generally impractical) 2.Follow trajectories of classes of particles (e.g. all electrons going 60 mph, Vlasov theory) 3.Consider plasma as composed of electron gas interspersed with ion gas (2-fluid thery) 4.Consider plasma as an electrically conducting fluid (Magnetohydrodynamics, “MHD”)

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Magnetic confinement Electrons and ions can be confined by magnetic fields Magnetic field Charged particle motion

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Make toroidal magnetic field

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Theoretical Models

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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

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What the ideal MHD equations mean…

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Parallel currents attract each other Drawing from

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Calculation of force on a current due to a magnetic field

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

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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

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Drawing from

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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

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

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Drawing from ORNL web site Toroidal direction Poloidal direction Poloidal magnetic field provides confinement Toroidal magnetic field provides stability

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“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

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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

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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

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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

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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

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Density and temperature are peaked on magnetic axis

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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

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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)

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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

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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”

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Any deformation from initial shape requires work, configuration wants to snap back to initial shape Radiator hose Vacuum magnetic field

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Magnetic force between currents Anti-parallel currents repel each other –leads to “hoop force” –pressure perpendicular to field

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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

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Kink instabilities: another issue Magnetic energy of a circuit can be expressed as inductive energy

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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

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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

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Experiment at Caltech demonstrating some of these ideas (Solar prominence simulation experiment)

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Break down gas Form plasma Drive field-aligned current plasma

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

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Current flow Outward force due to mutually repelling currents Recall that vacuum fields were lowest energy state ….bulge out

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

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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

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E cross B Drifts

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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

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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

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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

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

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Air core transformer also possible

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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

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Complete tokamak configuration Human scale for reference

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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

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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

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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

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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)

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Person ITER

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Cross-section

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

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Tritium cycle Plan is to recycle tritium within the reactor Use lithium reactions 6 Li + n 4 He + T Mev 7 Li +n 4 He + T + n 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

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Room temperature=1/40 eV 0.6 eV 5 eV 1.5 keV 10 keV for DT fusion Caltech Experiment regime

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

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Copper annulus 50.8 cm diam Copper disk 20.3 cm diam Gas nozzles Coaxial, co-planar electrodes

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Bias field coil makes linked magnetic flux

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Gas puff valve (typical)

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Gas nozzle 50 cm disk annulus

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Side View disk annulus

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Installation

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Sequence high vacuum Bias field

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Puff in neutral gas

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Astrophysical jet simulation steps: 1.Distinct, arched filamentary plasma loops 2.Loops merge to form central column 3.Jet-like axial expansion of central column 4.Kink instability of central column

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Breakdown, “spider leg formation”

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Spider leg

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MHD physics Anti-parallel currents repel

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Spider legs get bigger

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Astrophysical jet simulation steps: 1.Creation of distinct, arched filamentary plasma loops 2.Filaments merge to form central column 3.Jet-like axial expansion of central column 4.Kink instability of central column

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MHD physics Parallel currents attract

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Spider legs merge to form central column

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Astrophysical jet simulation steps: 1.Creation of distinct, arched filamentary plasma loops 2.Filaments merge to form central column 3.Jet-like axial expansion of central column 4.Kink instability of central column

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Central column lengthens I ~ 100 kA

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Doppler Shift Measurement Velocities ~ km/s observed Blue shift seen when viewing from this end Red shift seen when viewing from this end

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Doppler Shift Measurement Velocities ~ km/s observed Blue shift seen when viewing from this end Red shift seen when viewing from this end

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Example of Doppler shift data hydrogen, deuterium calibration lines from reference lamp DD HH

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Astrophysical jet simulation steps: 1.Creation of distinct, arched filamentary plasma loops 2.Filaments merge to form central column 3.Jet-like axial expansion of central column 4.Kink instability of central column

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Kink Instability of Central Column

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Kink occurs when central column becomes sufficiently long to satisfy instability condition S. C. Hsu & P. M. Bellan MNRAS 334, 257 (2002) L

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