fusion energy: mit’s pathway to unlimited clean energy

Fusion Energy: MIT’s Pathway toUnlimited Clean Energy

AJ Cavallaro

MIT HSSP’20

Contents

1 Week 01 - Setting the Scene, Evaluating Fusion Energy 11.1 Why do we care? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 What is fusion? (Energy Perspective) . . . . . . . . . . . . . . . . . . . . . . 31.3 Q1: What are the fuels? Why? . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Q2: What are the conditions required to achieve net energy? . . . . . . . . . 101.5 Q3: What fusion approaches exist, and how have they done? . . . . . . . . . 13

2 Week 02 - Building a Tokamak Power Plant 222.1 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 History of the Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Components of a Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Week 03 - 25

4 Week 04 - Problems in Plasma Shaping 25

5 Week 05 - Turbulence, Stability, Disruptions 325.1 Stability Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Other Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Week 06 - 36

§1 Week 01 - Setting the Scene, Evaluating Fusion Energy

§1.1 Why do we care?

The world is rapidly approaching massive climate catastrophe, and the very collapse ofour society as well as the majority of life on this planet. This cannot be overstated, andsolutions must be found urgently. As a quick recap of why,

� Energy use is correlated strongly with all measures of quality of life.

� Humans strive to improve (at least their own) quality of life.

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AJ Cavallaro (MIT HSSP’20) Fusion Energy: MIT’s Pathway to Unlimited Clean Energy

� Plentiful energy sources are needed, and use is accelerating every year.

� An easy and efficient way to gain meaningful energy is to spin a turbine to generatea current.

� Steam generators are the most efficient way to convert heat into electricity, byspinning a turbine.

� The easiest way to generate heat is to burn something.

� Combustion, generally, produces CO2 and H2O.

� High levels of CO2 and other industrial waste products in the atmosphere lead tomore light from the sun being allowed to hit the Earth’s surface than can radiateaway (the CO2 ‘blocks’ the light, and therefore heat, in).

� More energy accumulates on Earth, resulting in higher temperatures.

� Lots of things die.

Fusion energy, like other renewables, is a means to generate electricity without producingCO2. Here are the main methods for carbon neutral electricity generation.

� Solar photovoltaic - directly generates a current when photons from the Sun impactthe panels. Intermittent, unable to ‘load-follow.’ Uses a lot of land, not very powerdense.

� Concentrated solar - same thing, but uses mirrors to concentrate it. Also good forheat generation, which is currently done by fossil fuels. Intermittent, unable to‘load-follow.’

� Wind - use the wind to spin a turbine. Intermittent, unable to ‘load-follow.’

� Hydro - use water falling to spin a turbine. Quite bad for the local environment,but the lowest CO2 option existing.

� Nuclear fission - split heavy atoms, generate heat, steam turbine. Scary, but canrun constantly and load-follow. Issues of proliferation and waste can be persuasive.

All of these sources have problems. Load following is when the power generation is ableto scale its power output to the needs of the grid. This allows for much cheaper electricity,as you only have to produce as much at any given time as is needed. Intermittencyis a property of renewable options; when the sun don’t shine and the wind don’t blow,you need an alternative. Right now they burn natural gas to make up for it. The mainoption is to produce way more energy than is needed when the sun shines/wind blows,and use massive battery installations, increasing costs and impact on the climate.

To get to 0 carbon electricity generation, we would need something like the followingenergy mix

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The level of scaling from today is massive, and requires a major international effort.Several of these are optimistic; only so many rivers are suitable for dams, and the politicalwill for nuclear fission makes a 12x increase... unlikely. Fusion represents a complementto these options, to take the strain off of each respective field.

§1.2 What is fusion? (Energy Perspective)

Fusion is when two atoms of light elements are fused to form one atom of a heavier atom.

This reaction, like fission, generates heat, which can spin a turbine or be used in industrialheating applications. Essentially, in all the existing coal, natural gas, and other fossilfuel power plants, you could ‘slot in’ a fusion reactor, and reuse all the normal steamturbines to convert that heat to electricity.

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What are the properties that make fusion exciting?

� Efficient - the most efficient energy source ever achieved (20 gallons of fuel producesas much energy as over 150 million gallons of oil).

� Sustainable - there exists enough fusion fuel on Earth to power humanity forhundreds of millions of years, and accessing it is not harmful to the environment.Fuel cost essentially 0.

� Carbon-free - the reaction produces no CO2, simply Helium (which we are lackinganyways).

� Waste-free - the reaction produces no nuclear waste; just Helium. Technically thereactor itself will become radioactive, but depending on the materials used it couldreturn to safe levels within a few days. Not inherent to the reaction; engineering.

� Proliferation-free - you can’t use the supplies to make a bomb.

� No meltdowns - the reaction just stops under any interruption; lightly breathinginto the reactor would cease it.

� Power dense - a reactor the size of a large room could power an entire large city.

� On when you want - can follow loads, works no matter if the Sun shines.

� Site where needed - easier than building solar farms in the desert and using longtransmission lines.

� Equal access to all - the fuel is available everywhere, so all parts of the worldhave equal access to fusion for energy.

The downside: it’s really hard! We do tons of fusion, trillions of reactions in ourmachines, but the actual power-plant bits are difficult.

However, once it gets off the ground (and we are confident it will), there is reason toexpect it to lift-off, quickly.

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When new energy methods reach a critical point of being acceptable for grid-scale powergeneration, they experience rapid expansion. On this log-log plot (i.e. every line is a 10xincrease), you can see the rapid growth of fission, wind, and solar after their mainstreamintroductions, until they hit fundamental limits on how fast they could grow. Fusion,once it is working and cost-effective, has no such limits, so this plot shows a speculativegrowth rate for it.

MIT specifically is on a pathway for fusion energy, fast, but I must say: most fusionscientists do not believe fusion will happen fast enough to help fight climatechange. MIT’s pathway is encouraging, and our driving goal is to combat climate change,but others are more pessimistic. We do know one thing:

The first demonstration must happen by the mid-2030s!

§1.3 Q1: What are the fuels? Why?

There are four fundamental forces in nature

� Gravity

� Electromagnetic

� Weak nuclear

� Strong nuclear

With the latter two having very short ranges. All chemical reactions (like combustion)are a result of electromagnetism (so is electricity, of course). Nuclear power results fromnuclear interactions; splitting or fusion atoms to convert some mass to energy. This is anexpression of the strong nuclear force. We’ll expand on that further.

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Definition 1.1 (Strong Nuclear Force). The strong nuclear force is about 100 timesstronger than electromagnetism, and 1038 times stronger than gravity. It is the strongestfundamental force, and attracts nucleons (protons, neutrons) of all types to each other,over short distances. This counteracts the natural Coulomb Repulsion of like-chargedparticles, like protons in the nucleus.

Definition 1.2 (Coulomb Repulsion). The repulsive force between two like-chargedparticles, like protons or electrons.

You may be familiar with the model of an atom, as a nucleus consisting of Protonsand Neutrons tightly bound with each other, with electrons orbiting farther away. Ifunfamiliar with the nuclear force, it is quite odd to see so many positively-charged thingslike Protons practically contacting each other! This is because the nuclear force overcomesthis repulsion, like so

As nuclei grow, their orientation becomes more complicated, and it is possible fordifferent configurations to appear. We call these isotopes

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Definition 1.3 (Isotope). An isotope is a particular configuration of an element. Anelement is uniquely defined by the number of protons in its nucleus, however the numberof neutrons can vary. 1H (Hydrogen-1) is the normal type, and is just a single proton(the number given is the count of protons and neutrons). In this class we’ll also discuss2H (Deuterium, hydrogen with an extra neutrons) and 3H (Tritium, hydrogen with twoextra neutrons).

Different atoms have more or less favourable configurations of nucleons. This is bestsummarized in the below graph of Binding Energies

Remark 1.4 (Minimizing Energy). In general, the universe wants to achieve a state ofminimal energy. More precisely, lower energy states are thermodynamically favourable.

From the plot, we see that 56Fe (Iron-56) is the most stable element, and that lightelements like Hydrogen are quite unstable, and heavy isotopes like 238U (Uranium-238)are also less stable, if only slightly slow. Transmuting an atom into a lower-energyone, either by fusing light elements or fissioning heavy ones, moves you down the curve,releasing the excess binding energy of the nucleus.

Remark 1.5 (Mass → Binding Energy). When this binding energy is released, what formdoes it take? This may be surprising, but the mass of the nucleus actually decreases, andthe resulting element is faster. This is a direct conversion of mass into energy, the amountof which depending on the reaction.

As an example with fake numbers, consider the following initial condition

� 2 atoms of Hydrogen, each with mass 1.0 and speed of 100

When fused, producing something like

� 1 atom of helium, with total mass 1.9, and speed 10000

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Isn’t that wild? Another strange takeaway, is that a Proton weighs less in a Helium nucleusthan in Hydrogen. Wild.

So, when does fusion happen? How do you get the nuclei to touch, even though theyboth have positive charges and therefore repel? Get them moving real fast andcross your fingers!.

Remark 1.6 (Conditions for Fusion). Fundamentally, you need to cross this barrier

(Remember, lower energy states are ideal). To do this, some input energy is needed toovercome the hump. If the particles are moving fast enough, randomly, some amount ofthem will overcome the barrier and get close enough for the strong nuclear force to kick in.

This sounds binary right; either they get close enough or they don’t, right? Well...quantum effects are weird. Sometimes the particles can ‘teleport’ over the steepest partof the distribution, making fusion actually more likely. The cross section for fusion isshown below, with the actual probability shown in red and the theoretical one (if the‘getting over the hump’ model were 100% true with 0 quantum effects) shown in red

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Definition 1.7 (Cross Section). A cross section is essentially the likelihood of aninteraction ocurring at a certain energy level; think of it as probability.

Remark 1.8 (Ideal Fusion Fuel). The ideal fusion fuel will have

� 1. Low input energy to induce a fusion reaction

Technologically easier to achieve

Economically requires less input energy

� 2. A high probability of fusion at that energy

� 3. Produce a lot of energy when it does fuse

So... let’s look at some candidates!

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From this plot, we see that heavy elements have way too much repulsion due to theircharge, while many lighter elements simply have low probabilities of fusing and/or producelittle energy when they do fuse. We see we have three candidates (with a fourth pickedfor engineering reasons)

We see D + T (Deuterium and Tritium, heavy hydrogens) peaks at the lowest temperature(still 175M degrees!), has the highest reactivity as its peak, and produces quite highenergy. This is the most common reaction chosen for power plant concepts. D + D isused for science experiments, as the conditions are nearly identical except without all theexcess power generation.

§1.4 Q2: What are the conditions required to achieve net energy?

How do you get and keep a campfire going? You need three things

� 1. Density of wood.

� 2. High temperature in the wood.

� 3. Energy confinement in the wood.

If you have insufficient wood, the fire won’t last long. If the wood is wet, the heat will gointo the water first to convert it to steam, and the wood itself will not get hot. If youhave a bunch of logs strewn about, as they burn they will not heat each other, and willgo out. As a result, certain configurations like the one shown above are used to keep asmuch heat on your fuel as possible, to keep it going. Plasma is analogous.

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Definition 1.9 (Triple Product). The Triple Product, defined as the product ofplasma density, plasma temperature, and energy confinement time (how long it takes forenergy in the plasma to escape it, on average)

n× T × τE

Is a measure of how good of a fusion reaction you have. The higher the better. Theoriginal term was defined by John D. Lawson in 1955, often rewritten as the LawsonCriterion.

Definition 1.10 (Lawson Criterion). The Lawson Criterion was an earlier formulationof the triple product, that was just

n× τEand expressed a minimum value required for net energy output. A particular value ofthe Lawson Criterion, at a particular temperature, defines a value called Q

Definition 1.11 (Q (Physics)).

Q =Fusion energy output

Energy input

Q is the ratio of the amount of power your reactor produces, vs. how much energy youput in to either initiate or maintain it. In a fossil fuel example, low Q would be using ablowtorch to light a match (high energy to initiate, low energy produced). High Q wouldbe using a single lit match to initiate a burn of thousands of gallons of oil. Low initiationenergy, high energy produced.

In fusion, we want to initiate a reaction that will then ‘burn up’ on its own. The higherQ, the better.

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Decades of research are striving to move up this plot, by increasing temperature (to apoint, along the X-axis), and the Lawson Criterion (density times energy confinementtime) along the Y-axis. We summarize as

Remark 1.12 (Validating fusion). Proximity to burning plasma conditions is the ultimatearbiter of the viability of any fusion energy approach.

� T and n × τE giving Q ≥ 0.1 is ready for fusion energy/reactor design. Baselineviability.

� T and n × τE giving Q < 0.1 is still just a physics experiment. Be careful ofextrapolating further in that space.

Remark 1.13 (Why confinement?). The bogeyman of fusion is the following:

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I.e. the probability of two particles just... ‘bouncing’ off of each other (repelling due to likecharge) is always much higher than the probability that they fuse. Period. When this wasdiscovered, people thought fusion was dead. Fortunately, there’s another framework we canuse: if two particles repel, does all of their energy have to be lost?

Think of it as a pool table; if the walls didn’t exist, balls would simply fly off the edge andpocketing them would be super difficult. If you add walls (of some sort, i.e. confinement),particles that repel won’t lose all of their energy. So even if it takes 100 tries for elements tofuse, if the majority of the energy stays confined, no harm done!

§1.5 Q3: What fusion approaches exist, and how have they done?

We broadly have two categories of fusion approach, in terms of performance

� Tokamaks (stellerators as well, a bit)

� Everything else.

Here is a plot of non-tokamak performance (stellerators shown as squares)

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And here is the same plot with tokamaks added

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Even the very first tokamak, in the bottom left, is far and away better than the bestrepresentative of the vast majority of methods at that point or since. The best performer,JET (Joint European Torus) in the UK produced Q = 0.65 in the mid-90s, tantalizinglyclose to fusion conditions.

In fact, another tokamak achieved nTτE values that would have produced Q = 1.25,however it was never cleared to use D + T fuel, only D + D, so the actual power producedwas little. ITER, the world’s largest and most expensive science project, costing 10s ofbillions of dollars and involving dozens of leading scientific nations

Is finishing up construction and is aiming for Q = 10. So, have we done it? Have wecompleted fusion? Well... not quite.

First, let’s look at some methods!

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Remark 1.14 (Cold Fusion / Low Energy Nuclear Reactions (LENR)).

Well... you can’t put 0 on a log-log plot. Claims to use some process to create fusionenergy conditions at room temperature.

� First ‘claimed’ by Fleishmann and Pons in 1989.

Debunked at MIT.

� Proposed processes cannot be rectified with any known model of physics.

� Used as an example of a pathological science: research that continues long after it hasbeen established as false.

Remark 1.15 (Gravitational Confinement).

� Works! Just look at the universe.

� Stars initially fuse hydrogen, but progress to heavier elements.

� Energy release from fusion reactions generates tiny power densities over a massivevolume

0.27 W/m3 average power density (about same as typical compost heap)

1027m3 volume

� Balance plasma pressure with gravity

impossible to replicate on Earth

Remark 1.16 (Hydrogen Bombs).

� Inertia with implosion driven by fusion bomb confines the energy.

� Only succesful net-fusion gain on Earth. Not great for energy.

� Fission bomb is ignited next to fusion fuel

Resulting X-rays rapidly heat and compress fuel to fusion conditions prior todestruction

Fusion boosts the fission explosion energy by 1000x

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� Note: it requires fission to explode (fusion isn’t a bomb, seriously)

� Not a good power source!

Remark 1.17 (Inertial Confinement (ICF)).

� Similar to bombs, but implosion driven by lasers

� Gives insight into how bombs work, primary focus of the research and development inICF.

� Performance is impressive (Q = 0.1), but scaling to a reactor seems unlikely

Maintanence: significant machine componenets are destroyed each implosion

Rep rate: currently get about 1/day at maximum, would need about 1/s for apower plant.

Efficiency: less than 1% of the electricity used to generate those laser beamsactually makes it to the fuel target, so you can think of the ‘relevant’ Q as being afactor of 100 smaller.

Remark 1.18 (Particle Accelerators).

� Confines plasma by accelerating it with electric fields.

� Fires beam of high energy particles into other particles

Easy to build compact 100 keV (way ‘hotter’ than needed for fusion) beam.

Can fuse standard DT, but also heavy ions with a powerful enough beam. Onlyway to do that outside of stars.

� But the particles are still way more likely to simply repel than to fuse.

When particles repel, their energy is entirely lost as there is no proper confinement

Beam always requires more energy than it makes from fusion

� Good neutron source, good science, bad fusion gain

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Remark 1.19 (Electrostatic Potential Wells (Fusors)).

� Confines plasma by trapping it with electric fields.

� A spherical ion accelerator with a potential well to collide ions against each other inthe center.

� Multiple mechanisms slow or eject the ions before enough fusion happens for net gain.

Coulomb collisions

Particle losses

Conduction losses

Bremsstrahlung

� Can be an effective simple neutron source.

� Fun and easy to build for hobbyists

Remark 1.20 (Magnetic Mirrors).

� Uses magnetic fields to confine plasma in 2 directions, tries to ‘crimp’ at the ends toconfine along that axis, unsuccessful.

� Charged particles spiral around magnetic field lines

But confinement only in 2D

Some particles always leak out the ends

� Many different configs tried to plug the ends of the mirror

Large $1B-class experiments

Losses always domination fusion unless the mirror is very long

� Conclusion: a net energy device is unreasonably long ( km), still a good fusion neutronsource.

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Remark 1.21 (Pinches, Magnetized Targets).

� Confines particles with ‘squeezed’ magnetic fields.

� Rapidly compress the plasma and heat it by changing magnetic field quickly.

� Many configs tried.

Requires large pulsed power systems

Often with ‘sacrificial conductors’ surrounding plasma (i.e. they break)

� Large instabilities and plasma cooling occur before net-energy conditions are reached

Useful as a high-power X-ray or neutron source or particle accelerator.

Remark 1.22 (Torus of mirrors or cusps).

� Confines particles with ‘bumpy’ magnetic fields.

� Instead of plugging the mirror end losses, feed them into another mirror

ad infinitum ⇒ a torus

� Tried with many geometric configs in the ‘70s and ‘80s in large programs.

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Oak Ride National Lab: ELMO bump torus

UC California: TORMAC

NASA: Bumpy torus

� Breaking symmetry created additional instabilities in the plasma.

Limited temperatures, ruined confinement

Interesting physics, though!

Remark 1.23 (Field-reverse Configurations, Spheromaks).

� Confines particles with ‘self-twisted’ magnetic fields.

� Instead of torus of many mirrors, make a torus with the magnetic field spiraling in ahelix

Increases the confinement at cost of stability

� Plasma can create these field shapes through ‘self-organization’

Transient effects, limited to milliseconds.

Sudied widely over a long period of time.

� Very rich plasma physics, but very difficult to control and confinement still lacking.

Have not yet reached energy-relevant confinement or temperatures.

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Remark 1.24 (Stellerators).

� Confines particles with magnetic fields twisted by external coils.

� Use many external magnetic coils to create precisely the desired magnetic field shape

Stable, steady-state

� Requires highly-optimized field shapes and magnets to obtain best performance

One of the original fusion concepts.

Ongoing work world-wide

� Higher performanmce but with complex engineering to create the exact 3D shapesneeded.

Makes an expensive reactor.

Remark 1.25 (Tokamaks).

� Confines particles with magnetic fields twisted by (simpler) external coils and currentin the plasma.

� Simplify the magnets by carrying a toroidal current in the plasma to create a slightlyhelical field

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Good stability and can be made steady-state.

Symmetry provides good confinement.

� High initial performance led to lots of research for the past 50 years,

170 devices built (6 at MIT).

Extensive physics understanding.

Technologies well-developed

Only deviced to make significant fusion energy (17 MW, Q 0.65)

� Consensus among world plasma physics community is that the tokamak will be ableto generate net energy.

§2 Week 02 - Building a Tokamak Power Plant

§2.1 Recap

The three factors we looked at last week for evaluating approaches to generating fusionenergy were

� Rule 1: Fuel choice fundamentally sets the difficulty of any approach to fusionenergy.

� Rule 2: Proximity to burning plasma conditions is the ultimate arbiter of theviability of any fusion energy approach.

� Rule 3: Many approaches to fusion energy have been or are being tried; only thetokamak has demonstrated energy-ready performance.

To elaborate further, ‘proximity to burning plasma conditions’ is set by the tripleproduct, of plasma density (n), temperature (T ), and energy confinement time (τE). Toget to net energy generation (Q > 1, where Q is the ratio of energy in:energy produced),we want to increase this triple product, summarized below, with tokamaks shown as bluedots and all other methods shown as various green dots.

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We increase the energy confinement time using a host of confinement methods derivedin the last lecture (electrostatic, inertial, etc), but our chief interest is in magneticconfinement as it can be run in steady-state, which is desirable for a power plant. Thistakes advantage of the fact that a plasma, defined as a mass of ionized particles (i.e.particles whose nuclei and electrons are separated), is affected by magnetic fields, and itsparticles will tend to follow the magnetic field lines. This keeps the energy in the mainplasma body, preventing plasma from losing its heat to the walls and cooling down tosub-fusion temperatures (and destroying your machine). Some complicated configurationneeds aside, the tokamak is the most effective magnetic configuration observed.

§2.2 History of the Tokamak

For decades the tokamak’s performance marched steadily up the plot of the triple product,increasing at a faster rate than the fabled Moore’s Law for computing, but somethinghappened in the mid ‘90’s: it stopped

Why is this? We’ll get to it over the course of the class.

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Needless to say, the tokamak is a well-understood technology

� Over 170 tokamaks have been built, running for over 1800 tokamak-years of study.

� Design and construction are well understood.

� Tokamak-specific technologies are well established.

� Have run D-T safely under various regulatory bodies, globally.

� Knowledgeable existing workforce.

� Built in many sizes and configurations.

Here is a website containing info on all the world’s tokamaks.

§2.3 Components of a Tokamak

A fusion power plant uses the following mechanisms to generate its own fuel and produceelectricity

� Deuterium (D) - Tritium (T), heavier isotopes of hydrogen, are fused and producea helium atom and a neutron.

� The helium is confined by the plasma, and gives its excess heat to the plasma,keeping the plasma hot. The helium is essentially ‘ash,’ and hinders the reaction onceit thermalizes (achieves thermal equilibrium with its surroundings). We deconfineit, and when it is exhausted we use its remaining heat to generate electricity.

� The neutron streams out of the plasma. We allow it to bounce around in a blanket,which will warm up as a result. We use its heat to generate electricity. The blanketcontains Lithium, which when interacting with a neutron will produce Tritium, ourmain fuel, along with some extra heat.

Essentially, it’s a normal power plant, except the heat comes from fusion power. Let’slook more closely at this fusion ‘core.’

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tokamak.info


This cross-sectional view is a common one that will appear in this class again and again,so get familiar with it. We’ll break these things down one-by-one

§3 Week 03 -

§4 Week 04 - Problems in Plasma Shaping

Baseline tokamak design:

� Toroidal field is primary confinement

� Poloidal field gives helical magnetic field lines - comes predominantly from internalplasma current induced by cenbtral solenoid. Handles most obvious instabilities.

� Circular cross section.

This is where we were at in the 1960s. What were the issues with this?

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Remark 4.1 (Toroidal field scaling, structural issues). Toroidal field strength scales like

BT ∝1

R

i.e. the toroidal field strength decreases as you move out, like so (from SPARC)

This means that the coils themselves experience imbalanced stresses as you move out. Ifyou use a circular cross section, the inside edge will be pulled more strongly than theoutside, leading to stretching. There will also be forces that flip the coils over. This meantthat increasing BT was being limited by the structure of the machine. The solution? ThePrinceton D-Shaped coil.

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This minimizes the magnetic tension.

Remark 4.2 (Increasing Ip). Ip (the plasma current) is an important factor for confinement,so why don’t we make it arbitrarily large? We get a disruptive mode. What does thismean? Well, the plasma breaks. If you don’t have enough helicity, we get the drifts wediscussed. Too much, the plasma can ‘kink;’ imagine holding a fluid with rubber bands.If the bands aren’t twisted, the fluid will leak out. If the bands are twisted enough, thefluid will be confined in an air-tight way. Twist them too much, you could imagine them‘cinching’ shut, and flipping over themselves. So, how do we prevent this, while increasingIp?

Remark 4.3 (Plasma Shaping, Elongation, Triangularity). This is where the externalpoloidal field coils come in. The D-shaped coils give you some wiggle room to stretch theplasma, and it just so happens this is good for confinement - selecting optimal values forelongation and triangularity.

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Recall that R,Z are our X,Y in toroidal geometries. We have triangularity

δupper =(Rgeo −Rupper)

a

Where the overall triangularity is the average of the upper and lower triangularity.

Let’s unpack this; the further horizontally the center of the plasma (Rgeo - geometriccenter) is from the upper and lower extends Rupper, on the R-axis, the more triangularity.This is somewhat intuitive; imagine keeping the upper and lower extents at the same points(vertically in-line with each other), and bringing the center of the plasma further and furtherout, we’d see a triangular shape.

What is the point of triangularity? Well, making it bigger tends to lead to betterconfinement, but making it too large tends to lead to disruptions (classic). This is why theoperating regimes of tokamaks tend to be a bit stretched out. Why? Poorly understoodphysics, magnetohydrodynamics - MHD. Negative triangularity has recently showninteresting results, but is not really compatible with the Princeton D-shaped coil design.

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We also have elongation

κ =Zmax − Zmin

2a

This one’s simple; the taller the plasma, with width held constant, gives greater elongation.It’s basically a measure of how stretched the plasma is. Why do we care? Well, it allows usto increase Ip and β (defined later) without getting disruptions! More elongation is better,although it can lead to disruptions when increased too much.

Conclusion: increasing these two shape parameters is key, but once they are set, thereisn’t much more room; treat them as roughly constants for a power-plant relevant design.

Remark 4.4 (β). We defined beta as

β =〈p〉B2

2µ0

i.e. the average plasma pressure divided by the magnetic pressure. Why do we care? Higherβ means, for a given set of magnets, you have more pressure. More pressure, more fusion.Why don’t we make this as high as possible? Things tend to break!

The Troyon factor and the Greenwald limit define an operating regime where youwon’t experience this type of collapse.

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Recalling our primary scalling factor,

Pfusion ∝ Rβ2B4

operating near the limit of β is desirable. This value is very dependent on configuration;Spherical Tokamak designs (more like a cored apple than a donut) can achieve 10x theβ! Unfortunately, their design makes shielding essentially impossible, but the U.K. is veryinvested in this tech.

Remark 4.5 (Operating Regimes). There exist different regimes for confinement. Thedefault one until the 80s is now known as L-mode, while a new regime discovered inGermany is now called H-mode. Let’s look at a plot of plasma pressure along themidplane.

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So, what does this show? H-mode demonstrates much higher plasma pressure (as well asbetter energy confinement time), and is noted by this ‘pedastal’ thing. What is this?

Profiles of pressure and other parameters, shown like above, demonstrate this slopedpattern because near the edge of the plasma, there are more interactions with the wall,giving impurities. These impurities keep the temperature of the plasma lower near the edge.Consequently, the vast majority of your fusion is done near the very center of the core.The pedestal refers to this steep gradient; the slope is quite stark at the edge, up to thepedestal, then the normal scaling is shown. Why is this? No clue, above my pay-grade. Itessentially comes from different sources of heating controlling turbulence.

H-mode gets in excess of twice the performance of L-mode, but it’s not all rosy - it comeswith instabilities, of course, called ELMs.

Remark 4.6 (Edge-Localized Modes (ELMs)). H-mode is slightly unstable; the pedestaltends to move. The steepness of the gradient increases as more energy is confined in theplasma, until it gets too steep and abruptly collapses, releasing 10-20% of the energy in theplasma at once, similar to a solar flare. This is no issue, as they are predictable, but theyare the biggest problem for exhaust; releasing heat all at once tends to melt things, ratherthan a slow burn.

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§5 Week 05 - Turbulence, Stability, Disruptions

Last week we discussed some of the basic concepts in plasma shaping, and a briefintroduction of turbulence. Today we’ll continue the latter half, and introduce somestability limits in standard operating regimes for Tokamaks in H-Mode.

§5.1 Stability Limits

Remark 5.1 (Greenwald Limit). Limits the number density of particles in tokamaks.Empirically determined

nG =Ipπa2

Where nG is the density in units of 1020m−3, Ip is the plasma current in MA, and a is theplasma’s minor radius in m. Since density is the most direct driver of fusion power, thislimit is a bummer!

Since this is an edge-behaviour, peaked pressure profiles like we’ve seen before are typicallyused. But what is the proposed mechanism of this limit? What happens if you cross it?

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Remark 5.2 (Resistive Tearing Modes).

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The formation of these magnetic islands allow for instabilities to become trapped, constantlyradiating heat away. As the islands grow, more energy is escaping the reaction faster andfaster, from the very core of your plasma, creating a runaway effect called a NeoclassicalTearing Mode. Proposed solutions?

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The one ITER is using is to just shoot ‘em.

§5.2 Other Disruptions

Remark 5.3 (Sawteeth). Caused by magnetic reconnection in the core of the tokamak,leading to abrupt relaxations of temperature and pressure profiles. Since they’re periodicand abrupt, often compared to a sawtooth plot

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Definition 5.4 (Magnetic Reconnection). Magnetic reconnection refers to the breakingand reconnecting of oppositely directed magnetic field lines in a plasma. In the process,magnetic field energy is converted to plasma kinetic and thermal energy.

§6 Week 06 -

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fusion energy: mit’s pathway to unlimited clean energy

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