IDL gotchas and misfeatures

IDL is a popular language in the plasma physics & astrophysics world. It has a number of pitfalls and poor design choices that lead to nasty pitfalls and gotchas. Here’s a list of the ones I’ve encountered. Maybe this will save some poor grad student a headache.

https://www.harrisgeospatial.com/docs/automatic_compilation_an.html
“Versions of IDL prior to version 5.0 used parentheses to indicate array subscripts. Because function calls use parentheses as well, the IDL compiler is not able to distinguish between arrays and functions by examining the statement syntax.”
This causes all kind of problems. One nasty one is that a function that IDL can’t find, which has keyword arguments when called, will give a non-descriptive ‘syntax error,’ because it is interpreted to be an attempt to index an array using a keyword argument! This is a very common problem when dealing with a new codebase, because if you don’t set your bash config file up to export the correct directories in the IDL_PATH environment variable so that IDL can find the libraries where utility functions are defined, then you’ll get dozens of errors like this. It took me forever to realize what was happening! 😡

https://www.harrisgeospatial.com/docs/catch.html
There is no ‘try.’ Error catching works by jumping backward in the file to the place where the error-handler is located, executing whatever is in that block, and then trying the failed statement again. There is no way to go around a statement if it doesn’t work.

You can ‘declare’ variables by passing them to a function! They are essentially null variables, and then, in lieu of the procedure being turned into a function (with a return value), the output of the function is placed into the null variable as a ‘side effect.’ This is extremely confusing when reading someone else’s code, if you are coming from another language that makes any kind of sense. Because a function can only produce a single output, some people prefer this backward method of returning data from functions.

The ‘help’ command only helps you if you already know whether an argument is a function, procedure, or data structure. Because you can index an array using parentheses rather than brackets, sometimes you can’t tell if an object is a function or an array by looking at the context!

Common blocks: if you ‘reference’ a common block (or load a save file for that matter) it pulls in the variables as named in the block/file and dumps them straight into the local namespace! This can overwrite other variables of the same name. It’s like python’s “from numpy import *” but for data structures. You can optionally supply a list of aliases for each of the variables you want to ‘import’ from the common block, but if you want the n-th one only, you have to assign junk names to the other n-1 variables that come before it! You can’t just index in, or even address it by name!

Automatically leaves you at whatever place the error occurred in your code. Good for debugging, terribly confusing though when you aren’t used to it, and expect to be back at global scope.

Imports modules on the fly when you call a function. Unlike python where you must name the module first (although you can be sloppy in python and import * from a package, which has a similar effect).

In conjunction with the problem of being able to declare a variable by passing it, you are able to pass an undefined variable, and the traceback indicates that the problem is occurring inside the called function, not the calling one! This one will gaslight you hardcore, because you assume that the incoming variable exists or otherwise your call would have failed, not the function evaluation!

The .compile command doesn’t work inside a .pro file, unlike an import statement in python. You have to use .compile at the interpreter or something @’ed into the interpreter. It’s not clear if there is a way to get around this restriction and specify which file you want to compile a procedure from, other than messing with the IDL path. So the scripting environment and the programming environment don’t behave the same.

If you slice an array down to length = one in some dimension, that dimension only goes away if it is on the end (ie, if you do x[*,i,*] you get back x[6,1,7], but x[*,*,i] =x[6,9] and x[i,*,*] =x[1,9,7].

If you make a mistake when broadcasting arrays, you won’t know it, because array combination just truncates to the smallest length along each dimension.

If your plots all show up red instead of whatever color they were supposed to be, make sure to state “device, decompose=0.” This allows color numbers to be interpreted in terms of a color table, rather than as hex representations of RGB colors. You’re probably using “loadct,39” to load the jet-like color table. Set “!p.color=0″ to set the text/axis color to black, then set ” !p.background=255″ to set the field of the plot to white, and finally, “!p.multi=0” to get rid of subplots.

Suppose you make an anonymous structure s that contains an array x[5,10], among others, and then you make an array of those structures: ss[i]=s for i=0 to 20. Then you look at ss.x and see a [5,10,20] array! However, you can’t index this array properly: the last dimension is an illusion. If you select any one of the elements from the first two dimensions (ie, ss.x[4,5]) you get back an array that is 20 long. But, you can’t do this: ss.x[*,*,0], because that last dimension is a phantom. You have to do this: ss[0].x[*,*] to access the array that way. So confusing!

Also, I just got an overflow from using 201*201 — the default for a string literal is too short!

If you do a contour overplot, you have to specify the levels for the second contour plot explicitly (can’t use nlevels!) or else it will default to the set of level values from the previous contour, which means you generally don’t see the second set of contours and have no idea why not (could have been the x/y range either!, or a color issue, or…)

Fusion power density

In this post, I’m going to consider the power density issue for fusion energy. The focus for now is on the technical obstacles, rather than the economic motivation for increasing power density. I’ll be referring back to the previous post summarizing the Jassby & Lidsky critiques.

Fuel choice

The deuterium-tritium (D-T) fusion reaction involves significant disadvantages due to tritium & high-energy neutrons. If we eliminated radioactivity and tritium, that would nullify all the political/societal issues, two of the three economic issues, and one of the purely technical issues, while simplifying another. Lidsky proposed changing research policy to look for reactor concepts that could use alternative fuels. So, what’s the catch?

The maximum fusion reaction power density (at a given reactor pressure) is ~50x more for D-T than for any other fuel choice. This is not only bad for the reactor economics, but may prevent the reactor from functioning at all. Lowering the power density at given conditions also makes it that much harder to achieve net energy gain. For instance, if a reactor using D-T had a gain of 25, switching to aneutronic fuel would make the gain <1, so the reactor would not produce power.

The next best fuel choice, deuterium – helium-3, requires mining the moon or the gas giants – this rules it out for the foreseeable future. It also only yields a 10x reduction in neutrons (or maybe more, but at a further cost to the power density). This reaction would eliminate concerns about tritium leakage and breeding. It might relieve some of the difficulty of materials selection, replacement rates, and radioactive waste, but the reactor would still be too radioactive to repair easily, and could still produce weapons material easily.

The deuterium-deuterium reaction has the 3^rd highest reaction rate. Like deuterium – helium-3, this choice also avoids tritium breeding and handling. Better yet, it doesn’t require exotic space mining. However, neutron production is only reduced by a factor of ~2 compared to D-T. (Possibly the neutron production could be lowered somewhat further, but again this would come at the expense of power density.) Pure deuterium reactors might be useful in hundreds/thousands of years if lithium (used for breeding tritium) becomes scarce – the oceans have enough deuterium for millions of years of consumption.

The 4^th reaction is hydrogen-boron. The peak power density is about 500x lower than D-T. In fact, it’s just barely above the power losses due to X-ray radiation – leaving very little room for any other losses to be allowed.[1] However, this reaction produces very little neutron radiation & activation, and does not involve tritium. Boron & hydrogen are both plentiful. This would be the ideal fusion fuel – if massive breakthroughs in plasma confinement could be made.

The up-shot is that in order to maximize the power density, D-T is the best choice. The next questions are, how much power density can we get, and how much do we need?

Power density: getting it

The thermal output power density of a light water reactor is around 50-100MW/m³, considering the volume of the pressure vessel — can fusion match this? The answer is ‘yes’ — at least in principle. This doesn’t imply that fusion can compete economically with fission, nor that pushing the power density this high optimizes the economics of fusion. Nonetheless, it’s an area where Lidsky’s critique doesn’t hold up any more.

Fusion power density scales as the plasma pressure squared: at the optimum temperature, the DT reaction yields 0.34 MW/(m³ bar²). For a magnetically-confined plasma, the plasma pressure is less than or equal to the magnetic field ‘pressure’ (energy density) — the ratio of the two pressures is called ‘beta.’ The magnetic pressure scales as the magnetic field squared, with a coefficient of about 4 bar/T².

The ‘beta’ ratio needs to be as high as possible — this favors concepts like the FRC, (100%), Z-pinch (100%), magnetic mirror (40-60%), spheromak (~40%), or reversed-field pinch (~25%), compared to tokamaks, which top out around 10%, and stellarators (1-5%). (The Z-pinch is unique because it doesn’t have external magnetic coils – a current flowing through plasma supplies the magnetic field. The maximum achievable field strength is not limited by the capability of superconductors.)

To put some numbers on the power density: Assuming the ‘beta’ ratio is 100%, then for a 5 T magnetic field, the maximum possible fusion power density is around 3400 MW/m³. (5 T is about the limit with existing ‘low temperature’ superconductors.) However, when averaging over the plasma volume, the achievable power density is perhaps 20% of this number, because the pressure rises gradually from the plasma edge to the center. Still, that’s around 680 MW/m³. Suppose the plasma is cylindrical with radius about 1 meter, and it has a shield of about 1.5 meters thickness surrounding it (to breed tritium, extract heat, and protect the magnets). The power averaged over the volume of the outer cylinder would be around 100 MW/m³.

Thus, it’s possible in principle to have comparable power density to a fission reactor, even using magnetically-confined fusion. Lidsky assumed a tokamak with ~5 T magnetic field (the limit given the existing superconductors at the time), which only has about ~10% ‘beta.’ Thus, the power density would be 100x lower, around 1 MW/m³.

The high-beta approach is one way to attack the power density problem. Higher magnetic field possible with new REBCO superconductors is another avenue. If 16 T is possible, as seems to be the case, then power density of a 10%-beta tokamak would be the same as a linear device with 5 T field and 100% beta – around the 100MW/m³ mark. Combining high-field superconductors with a 100%-beta reactor could potentially allow advanced fuels to reach power density near 100MW/m³as well.

Power density: dealing with it

In the majority of this section, I’m assuming we stuck with D-T fuel. I’ll address hydrogen-boron at the end of this section. For the D-T reaction, 20% of the fusion power is released as charged particles (helium nuclei), which heats the plasma. For our hypothetical 1-meter radius cylindrical plasma column, the ratio of the charged-particle power to wall surface area is about 20 MW/m². This is comparable to the heat load on re-entry, and only 1/3 of the heat flux at the surface of the sun! Beam dumps and divertors for tokamaks are required to withstand 10-20 MW/m². This appears to be close the limits of what is achievable with known materials. Also, the thermal conductivity of materials tend to degrade under neutron radiation, as the crystal structures become disorganized. Thus, even if we can produce comparable power density to a fission reactor, we may not be able to cope with the resulting heat flux. Is there a work-around?

For toroidal devices (tokamak, stellarator, RFP), the plasma is topologically trapped inside the coils, so the charged-particle portion of the power must[2] exit through the wall. Linear systems like the FRC and mirror get a free pass — the magnetic field lines can extend out of the cylindrical vessel and flare out, so that the heat is deposited over a larger surface area. Some of the power will still be radiated onto the vessel wall, but it might be as little as 10% of the total heat flux for D-T plasmas. (Note that in tokamak designs, the heat flux is concentrated at the divertor, so the problem is even worse than if the heat were distributed uniformly. It may be possible to spread the heat uniformly by intentionally introducing impurities to increase the X-ray radiation from the plasma, however.)

However, there’s another problem beyond heat flux. For our hypothetical D-T reactor, the neutron flux escaping the plasma is 80 MW/m². This translates to about 900 displacements per atom (dpa) per year at the first surface, for steel. Steel is likely to only survive about 100-200 dpa before needing replacement. Replacing the first wall several times per year is probably a show-stopper, as it: (1) eats into the capacity factor (2) increases operations & maintenance costs (3) results in a large volume of (low-level) radioactive waste.

Stellarator & tokamak designs typically call for ship-in-a-bottle robotic assembly of the blanket & plasma-facing components inside the cage formed by the magnet coils. The estimated time to repair/replace the first wall is in the range of months — clearly this cannot be done every month! Hence, existing tokamak designs are driven to low power density in order to prolong the life of the first wall. The ARC study proposed disassembling the magnets & lifting the inner components out in one piece — still far from simple & quick.)

Linear devices have an advantage from a maintenance perspective, compared to toroidal designs. However, even if replacement is quick & simple, it’s better to maximize the lifetime of the wall components, to reduce radioactive waste. Some materials may survive longer than others, but there are not many elements to chose from, in order to avoid producing high-level waste.

An obvious way forward is to replace the solid material walls with flowing liquid metal or liquid salt. The liquid should contain lithium for breeding tritium. A layer of 50 cm of liquid FLiBe salt would reduce the flux from 900 to around 10 dpa/yr, allowing a 20-year lifetime for the first solid surface. Another option is lead/lithium alloy. Liquid first walls solve both the neutron damage problem and the heat flux problem, if the flow is fast enough. There are several drawbacks, however:

Splashing of droplets into the plasma must be prevented — splashes could extinguish the plasma unexpectedly
Plasma sensors and actuators (RF antennas, particle beam or pellet injection, etc) would be hard to accommodate
If the liquid is metal, the magnetic field can increase drag and result in large energy consumption for pumping the liquid
There may be corrosion problems, especially for liquid metals, but also for salts
The coolant temperature must be kept low enough not to poison the plasma due to heightened vapor pressure — this restricts the thermodynamic efficiency of the turbines used to produce electricity.

Power handling: with hydrogen-boron

For the hydrogen-boron reaction, essentially all the charged-particle heat flux would emerge as X-rays hitting the wall. At 100MW/m², a liquid first surface is probably a necessity for this fuel as well, to handle the heat flux. The choice of liquid is more flexible, since tritium breeding is not required.

Fusion ‘Fuel Rods’

Fission reactors reduce the heat flux challenges they face by splitting the fuel up into many long, thin rods to reduce the volume-to-surface-area ratio. In principle, this could be done for fusion as well. The problem with splitting up the fusion plasma is that fusion gain is dependent on having good thermal insulation of the plasma – and reducing the volume-to-surface-area reduces the insulation value, so to speak. For fission reactors, this effect is actually beneficial, as it keeps the temperature at the center of the fuel rods below the melting point of the fuel, whereas fusion reactions need to be kept hot at the center. I don’t want to dive into the physics of plasma transport at this point in series, but for now I’ll say that it seems unlikely that the ‘fuel rod’ approach would work.

Conclusion

All of engineering is trade-offs. Optimizing individual components of a system in isolation doesn’t generally lead to the optimal system. The optimal fusion reactor might not involve pushing the power density all the way to the maximum. Nonetheless, it appears to be possible in theory to achieve power density comparable to a fission reactor, contrary to the assertions of critics. The trade-off is that liquid first walls would probably be required, to cope with the extreme neutron &/or heat fluxes produced. Liquid first walls have their own disadvantages. Power densities around 10 MW/m³ or less would be more feasible. It remains to be seen if this is sufficient to make fusion economical.

Footnotes

[1] (See Fig. 4 of “Fusion reactivity of the p-B¹¹ plasma revisited” by S.V. Putvinskiet al,Nuclear Fusion 59 076018 (2019))

[2] It might be possible to convert most of the plasma heat exhaust to some form of directed energy (the Carnot efficiency of a heat engine operating at thermonuclear temperature is > 99%), but it’s not been demonstrated for a thermal plasma.

Summary of fusion critiques

This is the third post in a series on nuclear fusion. The goal of the series is to address assumptions made in critiques of fusion energy, with an eye toward solutions. Lidsky’s and Jassby’s critiques overlap a bit. Here’s a combined list of the main arguments, broken into 3 categories. In this series, I’ll focus on the issues that impact the economics. The technical & societal/safety issues do feed into the economics, so I’ll comment on those places as they come up.

Summary of critiques

Economic issues:
1. Fusion reactors will have lower power density while being more complex, compared to fission reactors. Therefore, they will be larger, more expensive, and slower to construct, hence uneconomical.
2. Radiation damage will require frequent replacement of core parts of the reactor. This replacement, being complicated & robotic/remote, will be slow. Therefore, the down-time fraction of the plant will be large, and the operations & maintenance cost will be high, making it uneconomical.
3. The complexity of a fusion reactor makes accidents more likely. An accident that releases radioactivity would lead to decommissioning of the plant because repair would be impossible. Therefore fusion plants are economically risky.
Safety/societal issues:
1. Although fusion reactors will (with appropriate material choices) have less high-level radioactive waste, they will produce much more lower-level waste (as compared to fission reactors).
2. Fusion reactors can be used to breed plutonium (weapons material) from natural uranium. Tritium is also a weapons material.
3. Accidents & routine leakage will release radioactive tritium into the environment.
Primarily technical issues:
1. Tritium breeding has narrow margins & probably won’t be successful.
2. Fusion reactors require siphoning off some of the electricity output to sustain/control the reaction. This ‘drag’ will consume most/all of the output electricity (unless the plant is very large).
3. Materials that can withstand large heat fluxes and large neutron fluxes and don’t become highly radioactive are hard to find.

Assumptions used in critiques

There are a number of assumptions made in the critiques, some of which are unstated. The most consequential ones are:

The deuterium-tritium reaction will be used.
The reaction will be thermonuclear (ie, no cold fusion, muon catalysis, highly non-Maxwellian distributions, etc)
Reactors will be steady-state magnetic confinement devices.
Specifically, the magnetic confinement device will be a tokamak.
Magnetic field coils will be limited to about 5 Tesla field strength.
The reactor first wall will be solid material.

Not all the critiques depend on all the assumptions. I’ll indicate which assumptions are involved in each critique item. Notably, 7 of the 9 critiques involve the radioactivity & tritium usage of fusion. This motivates considering ‘aneutronic’ fusion reactions. However, aneutronic reactions produce orders of magnitude lower power density (all else being equal) versus deuterium-tritium. Thus, most fusion efforts focus on deuterium-tritium despite the radioactivity & tritium concerns. My next post will discuss the fuel cycle trade-off as part of the power density discussion.

Anti-perfectionism advice

I used to subscribe to the Mark Twain philosophy:

It is better to keep your mouth closed and let people think you are a fool than to open it and remove all doubt.
Mark Twain

It was an excuse I used to justify my perfectionism. Here are some antidotes that have helped me overcome this tendency & put my thoughts out here.

A book is never finished. It is only abandoned.
Honoré De Balzac

Ideas get developed in the process of explaining them to the right kind of person. You need that resistance, just as a carver needs the resistance of the wood.
Paul Graham, “Ideas“

Survey of energy costs for comparison with fusion

My previous post introduced Lidsky’s The Trouble With Fusion, which claims to show that fusion won’t be economically competitive with nuclear fission power. Of course, beating fission on cost isn’t even setting the bar very high. In this post, I’m taking a quick look at the competition & where the market is heading.

TL;DR: fusion needs to compete directly on wholesale electricity costs in the future. Current prices are around $40-50/(MW*hr), but fusion should be shooting for $20-25/(MW*hr) in order to take over rapidly and stay ahead of renewables. Fusion is capital intensive, like fission. To achieve this low target, fusion plants need to cost less than $2/W (which is about twice what combined-cycle gas turbine plants cost). A study of several innovative low-cost fusion approaches found costs in the range of $5-13/W, maybe reaching $2-6W when scaled up.

The on-ramp

Peter Thiel lays out an argument that diving directly into the wholesale electricity market is not wise:

Any big market is a bad choice, and a big market already served by competing companies is even worse. This is why it’s always a red flag when entrepreneurs talk about getting 1% of a $100 billion market. In practice, a large market will either lack a good starting point or it will be open to competition, so it’s hard to ever reach that 1%.
Zero to One, p54

It’s better to have a so-called “Tesla Roadster” or a “hair-on-fire use case:” some smaller market where your initially-more-expensive first-generation products can compete. This market needs to be sealed-off from the broader market. In other words, your product must have some special feature that prevents generic products from being substitutes. In terms of distinguishing features, neither fusion’s safety advantage over fission, nor its zero-carbon advantage over fossil fuels, are enough to induce utilities to pay a premium for fusion above the market rate. Even if a carbon tax were levied, fusion would still be on the same playing field as fission and renewables. Fusion, like fission, is primarily suited for baseload, not dispatchable power.

Thiel specifically addresses clean energy’s on-ramp:

Cleantech companies face the same problem: no matter how how much the world needs energy, only a firm that offers a superior solution for a specific energy problem can make money.
Finding small markets for renewable energy solutions will be tricky — you could aim to replace diesel as a power source for remote islands, or maybe build modular reactors for quick deployment at military installations in hostile territories.
Zero to one, p 170-171

Currently-envisioned fusion reactors are not suitable for remote locations — they are too large to be shipped pre-assembled, and the demanding assembly process calls for a high-tech supply chain that doesn’t exist in remote locations. Consequently, fusion start-ups typically target the wholesale electricity market directly (see for instance the Deployment section at Type One Energy’s home page). Some exceptions: Phoenix Nuclear is using low-gain fusion reactions to produce neutron sources for medical isotope production or neutron imaging. Similarly, TAE Technologies has a medical physics spin-off: boron neutron capture therapy. These low-flux applications could partly bridge the gap to fusion, but I suspect another intermediate will be necessary. Fusion-fission hybrid reactors have been explored, both for power as well as for breeding fission fuel or burning fission waste. The bottom line is that they are susceptible to melt-downs (being fission reactors), so they don’t have fusion’s safety advantage, and they have no advantages over fission breeder reactors. Industrial process heat and thermochemical fuel production are other uses for fusion power output, but fission could address these — yet it has not.

To summarize: fusion needs to compete on the wholesale energy market.

Energy costs

Here is a good source for energy costs. From this, I gather that the median LCOE of new nuclear energy is $80/(MW*hr), whereas natural gas combined cycle turbines (CCGT) is around $50/(MW*hr). Advanced nuclear might be as low as $40-60/(MW*hr). Renewable energy + storage is also in this territory, however. In sunny areas, LCOE as low as $30/(MW*hr) for photovoltaic-based electricity is already possible. Projections along the learning curve indicate that LCOE for PV as low as $10/(MW*hr) may be achieved by 2030. According to this article, storage might add $50/(MW*hr) on top. However, in the face of seasonal variations in renewables, it is cheaper to overproduce renewable energy rather than store it in batteries long-term. This opens the door for low-cost membrane-less hydrogen electrolyzers to step in. This online simulator is a fun way to explore zero-carbon energy possibilities. In particular, underground hydrogen storage looks like a promising way to deal with seasonal fluctuations. While the round-trip efficiency of electrolysis/compression/combustion of hydrogen is low, it doesn’t matter much since the energy will be inexpensive during times of over-production. The upshot is that while current electricity prices are in the $40-50/(MW*hr) range, renewables could put downward pressure on prices starting in the next decade.

This report shows that existing coal and nuclear can operate even if the cost of new plants is twice the going rate — but of course new plants won’t be built. This is the case right now. Basically, existing plants already have paid down their up-front costs, so they can operate below the cost of new generation. Because the capital costs of CCGT plants are only a small part of the LCOE (fuel is the dominant contribution), the gap between new & existing CCGT plants is smaller. In order to displace existing generation, the LCOE of new fusion generation would need to be around half the LCOE of new coal or nuclear — that is, around $40/(MW*hr). It would be better to significantly undercut even that price point, to stay ahead of renewables, to make a compelling argument for significant investment, and to allow some margin of error in case projections prove to be overly optimistic. Let’s set a goal of $20/(MW*hr) for fusion. What implications does that have?

Cost structure of nuclear energy

The fuel cost for fission is small — the capital cost dominates, followed by followed by operations & maintenance (O&M). Fusion will probably be similar, because fusion fuel is even less expensive than fission fuel. Using the formula in the above document and assuming a 25-year payback period, a 7%/yr financing rate, and 95% capacity factor, a capital cost of $1/W adds about $10/(MW*hr) to the LCOE. The cost estimates of nuclear fission plants are around $6/W, so the capital cost adds $60/(MW*hr) to a new plant, while the O&M expenses are around $20-30/(MW*hr). This is consistent with estimates of the LCOE of new nuclear.

A large combined-cycle gas turbine plant costs about $1/W or thereabouts — this is the lower limit for any turbine-based power plant. Suppose that we could produce a fusion power plant for $2/W. The resulting LCOE contribution is then around $20/(MW*hr), before we add any O&M costs. So, we’re hoping to get a fusion power plant that costs only about twice what it costs to build a CCGT plant, and has low maintenance costs, more like a gas plant than a fission plant. This is consistent with a DOE study that found that $2.2/W would be low enough to rapidly decarbonize the electricity market.

Cost studies of innovative fusion concepts

ARPA-E has spawned a program aiming to get the cost of a 1st-generation fusion plant down to around a few billion — see slide 16 of this presentation. A 2017 costing study was performed by Betchtel & others on behalf of the ARPA-E. The costing study report estimated $0.7 to 1.9 billion for 150 MW plants — about $5 to 13/W. Although the cost and size of the experiments are smaller than equivalent tokamak-based ones, the economics don’t work out because the power output is also low. The report speculates that $2-6/W might be possible if these designs are scaled up. Interestingly, the reactor itself only comprises 15-30% of the total plant cost in this study.

For comparison, the ARC design study found a cost of $5.3 billion for a reactor (not the full plant) that would produce 250 MW — about $20/W. They calculated $4.6 billion for the magnet support structure alone — this seems like an overestimate to me. They applied a factor of 100x in cost for that component compared to the cost of the raw steel. They estimated a cost of around $400 million for the raw materials, which is around $1.4/W.

I’m not aware of many recent cost estimates on fusion power plants. If you know of more, please send them my way.

Doing the impossible

All things which are proved to be impossible must obviously rest on some assumptions, and when one or more of these assumptions are not true then the impossibility proof fails—but the expert seldom remembers to carefully inspect the assumptions before making their “impossible” statements.
Richard Hamming, You and Your Research (1986), p182

We’re all familiar with the many dignitaries, including Kelvin and Edison, who declared heavier-than-air flying machines were impossible just a few years prior to the Wright brothers’ success at Kitty Hawk. In the field of nuclear fusion, there’s been a tendency recently to make references or analogies to Kitty Hawk. The question arises: is fusion at a Wright brothers moment, or a Da Vinci moment? (Da Vinci’s ideas for flight were ahead of their time — the Wright brothers’ were punctual.)

Rather than speculating on the future, here are a few examples of failed impossibility proofs related to nuclear fusion. I’ll tie it all together at the end with some further thoughts about the value of impossibility proofs as pointers to solutions.

Although it may not seem like it, the levitron magnetic top is quite relevant. The levitron consists of a small spinning ceramic magnet hovering above a larger one. There are no strings, no superconductors, and no electronics involved. According to Earnshaw’s theorem, static levitation using only permanent magnets is impossible. The inventor of the levitron, Roy Harrigan, was told by physicists that what he was trying to do was impossible — just another perpetual motion machine. Harrigan persisted, and eventually perfected the device. This is remarkable, because the levitron requires careful tuning — a great deal of patience must have been required. Once Harrigan demonstrated his invention, physicists were able to explain it: Earnshaw’s theorem is correct, but it only applies to static magnets — the levitron is actually in an orbit. (Source: Spin stabilized magnetic levitation, by Martin, Helfinger, & Ridgway, Am. J. Phys. 65, 4, 1997)

There’s a close analogy between the levitron and the field-reversed configuration (FRC) concept for nuclear fusion. Both the FRC and the levitron can be thought of as magnetic dipoles oriented the ‘wrong way’ in an external magnetic field. If it weren’t for rotation, the dipole would flip end-over-end to align with the external field. (In the FRC, the rotation in question is in the form of individual particles with orbits that are roughly the same diameter as plasma itself.) The FRC has several advantages over the tokamak. For instance, for a given external magnetic field strength, the FRC can support approximately 10 times the plasma pressure, leading to 100 times the fusion power density. TAE Technologies (my current employer) is pursuing this concept, as are others (such as Helion Energy and Compact Fusion Systems).

Here’s another example where a static plasma is unstable, but a moving one is stabilized: the shear-flow stabilized Z-pinch, pursued by Zap Energy. The Z-pinch is literally the textbook example of an unstable plasma configuration, because it is very simple to analyze, and it was one of the first attempted configurations for a nuclear fusion reactor. It has the advantage of not requiring external magnetic field coils — the magnetic field is generated by a current flowing through the plasma itself. The caveat is that the plasma must be in direct contact with electrodes at both ends which create the current — this is bad news for both the plasma and the electrodes! (Side note / self-promotion: the physics of the shear-flow Z-pinch is very similar to a phenomenon I discovered – flow stabilization of a cylindrical soap bubble. You can try this one at home.)

The violation of seemingly-innocuous assumptions is what links these examples. Earnshaw’s theorem assumes static levitation. The theorem is true — it just doesn’t apply to the levitron. Similarly, the ‘proof’ of instability of the FRC assumes that particle orbits are very small — a good assumption in other magnetic confinement configurations, but not for the lower magnetic fields involved in the FRC approach. In the case of the Z-pinch, the static plasma assumption seemed reasonable — the stabilizing effect of sheared flow was not appreciated for many years, so static plasma assumptions appeared to simply make the math easier.

…what is proved, by impossibility proofs, is lack of imagination.
J. S. Bell, On the Impossible Pilot Wave, Ref.TH.3315-CERN (1982)

Can we turn Hamming’s insight into a recipe for achieving the impossible? Bell’s quip gets to the heart of the problem: recognizing which assumptions to break, and in what manner, requires creativity, imagination, and perhaps intuition and luck as well. Not every assumption is critical, in that violating it will yield a solution. Also, there may be many ways in which an assumption could be violated or modified — not all of them productive. The productive paths are not always obvious, either. There’s also the question of which assumptions can be violated, practically-speaking.

Impossibility proofs are not without value — they rule out certain areas of search space, which can save time and effort exploring fruitless avenues. Assumptions simplify reasoning by removing variables — we can’t always work from first principles. Removing assumptions makes reasoning and calculations more difficult, so it’s important to be selective about which ones to remove. A useful impossibility proof is one whose assumptions are few, explicit, and necessary for the conclusion (that is, without the assumption a solution would become possible).

Returning to fusion energy: Lawrence Lidsky, former director of the MIT fusion research program, spelled out an ‘impossibility proof‘ for economically-competitive fusion energy in 1983. In a nutshell, Lidsky argues that fusion devices using the conventional deuterium-tritium fuel won’t be economically competitive with nuclear fission. For both fusion and fission, the dominant contribution to the cost of electricity is the cost of capital — the fuel costs are small, due to the enormous energy density of the fuel. On the other hand, fusion reactors will have lower power density (by 10 or 100 times) compared to fission, so they will be proportionately larger than an equivalent fission reactor core. Fusion reactors are also more complicated technology. Fusion reactions also cause more neutron damage to solid materials, requiring frequent maintenance. Therefore, fusion will be more expensive than fission — which is already not very economically competitive.

Lidsky’s critique of fusion has a large number of assumptions, some of them implicit, and not all of them necessary — in other words, it could be improved. I’m working on a series of posts addressing his assumptions, to see what can be learned from the arguments. My next post will be a look at contemporary energy economics, followed by consideration of how to achieve high power density in a fusion reactor.

Antifragile software?

Software systems are often fragile. One of the causes of software rot is changes to dependencies. In theory, updated versions of dependencies should bring improvements to the systems that use them. In practice, the result is often to introduce bugs.

Unison programming language seeks to address the robustness of software by freezing definitions.

Unison definitions are identified by content. Each Unison definition is some syntax tree, and by hashing this tree in a way that incorporates the hashes of all that definition’s dependencies, we obtain the Unison hash which uniquely identifies that definition. This is the basis for some serious improvements to the programmer experience: it eliminates builds and most dependency conflicts, allows for easy dynamic deployment of code, typed durable storage, and lots more.
When taken to its logical endpoint, this idea of content-addressed code has some striking implications. Consider this: if definitions are identified by their content, there’s no such thing as changing a definition, only introducing new definitions. That’s interesting. What may change is how definitions are mapped to human-friendly names.
A tour of Unison

What if it were possible to go beyond code that is merely robust to changes in the external environment, but improves monotonically as the environment changes? Can we have software that is antifragile? I think so. My idea is to hybridize the dispatch mechanism of Julia with a declarative language, taking some cues from Unison and automated/interactive theorem provers.

Julia’s dispatch system allows automatic use of more specific methods that are more computationally-efficient for particular problems — there’s potential here for software that grows better with time. The shortcoming of Julia’s approach: it’s up to subsequent developers to ensure semantic coherence of the set of methods that share a function name.

In order to ensure semantic correctness, functions must be identified by declarative specifications, like a Hoare triple. If this is done, then automatic substitution of methods can take place without causing errors. User-friendly names in the code are mapped to the function (declarative specification) using something like Unison’s hashing method; I call this hash table the ‘catalogue.’ Dispatch is generalized: the dispatcher receives the specifications and the argument types-tuple. It then select the method (AST) that implements the function (specification) for the specific types. There would be a ‘library’ of available methods, something like the ‘formal digital library’ of Nuprl or other theorem-provers. The library would include proofs (or some type of certificate) showing that methods obey the specifications associated to them. Annotations such as computational complexity could be added, to help the dispatcher select between various implementations according to performance.

You could program in either declarative or imperative style, when appropriate. In declarative style, you define a function using pre/post-conditions. If your specification matches an existing one (perhaps even partially), you could get a prompt with the name of the function(s). If not, you can assign a new function definition to the catalogue.

If the function is pre-existing, there are probably pre-existing methods also. In the case where there are no methods in the library matching a function definition & type signature, you could: revert to imperative programming and write the specific method, or invoke an automated (or interactive) implementation tool. The latter is made easier because the specifications of all existing functions are available.

The imperative approach proceeds as usual, but what happens at the end is different. Rather than the resulting AST being mapped directly to the user-friendly name, it is entered into the library along with the specifications it obeys and the type signature. (Also, either a proof or an ‘IOU’ stating that the new method satisfies the pre/post-conditions must be added to the library.) Then, in order to invoke the new method, a user-friendly name is associated to the function specifications in the catalogue. You then invoke the function by name, which refers to the specification, and then the dispatcher identifies the new implementation.

In theory, you never have to worry about regressions — it doesn’t matter if the dispatcher selects a different implementation (method) from the library in the future, because the specification ensures that the new method does the same thing as the old one. In fact, it’s likely that your code will improve in performance over time without you modifying it, because if someone comes up with a better implementation for your particular use case, then it will get used automatically. This is the real potential for ‘antifragile’ code.

In practice, this could go off the rails in two ways. First of all, the specifications might be incomplete. This would allow the dispatcher to substitute a non-equivalent method. If this occurs, it can be remedied by refining the specification such that it discriminates between the ‘old’ method that was giving the correct behavior, and any ‘new’ ones that do not. This provides a mechanism to progressively formalize your code — it’s not necessary to start out with a full formal specification. You can start out with something that works, written imperatively, and discover the specification as you go. This bypasses one of the major hurdles for adoption of formal methods for programming. Note that the refinement of the definition happens by the Unison mechanism, so that there are no unintended side-effects.

A second way regressions could occur: a library method doesn’t obey the specifications it claims to. This can happen if a certificate is accepted — demanding a proof eliminates this possibility. Allowing certificates is a compromise to ease adoption of the language — the proof can be supplied later (or, a disproof!). Proofs could exist to varying degrees of abstraction.

Other interesting side-effects of this programming system include the following. First, there is the possibility for automatic code ‘reuse.’ Imagine that you wrote a method to calculate the standard deviation of a set of numbers. You might have defined it directly in terms of summing over the squares of the numbers, normalizing by the cardinality of the set, and subsequently taking the square root. Later, you might define the ‘sum of squares’ function calling it ‘L2_norm’ for instance. If you return to the definition of the standard deviation, you could ‘refresh’ the representation of the definition, and the expression ‘L2_norm’ would appear in place of the longhand form. The method would have the same hash, because the AST would remain the same — the shorthand representation would be the only change.

Other benefits: program synthesis would become easier. The availability of the pre/post-conditions in the library of methods provides a synthesizer with a formalized semantics to grab onto. Human programmers could benefit as well from the ability to look at the list of properties satisfied by functions. Interactive tools like ‘semantic autocomplete’ for functions or methods might be possible.

Memory-prediction asymmetry

I recently read “Recognizing vs Generating.” The author poses the question, “both reading the textbook and writing proofs feel like they fit the definition ‘studying’, so why can’t the easier one work?” The answer: “Recognition does not necessarily imply understanding, and pushing for something you can Generate can help expose what you actually know.”

This brings to mind the “memory-prediction” framework promoted by Jeff Hawkins. The framework says that recognizing is accomplished by generating. There is a continual dialogue between predictions coming from top-level generative modeling and observations coming from the senses. Discrepancies are registered as surprises that leap from the unconscious to the conscious level:

If you have ever missed a step on a flight of stairs, you know how quickly you realize something is wrong. You lower your foot and the moment it “passes through” the anticipated stair tread you know you are in trouble. The foot doesn’t feel anything, but your brain made a prediction and the prediction was not met.
“On Intelligence”, p91

At first glance, the framework doesn’t accord well with an asymmetry between recognizing & generating. The asymmetry can be accommodated, by emphasizing the fact that most predictions are non-specific or ‘fuzzy.’ A fuzzy prediction doesn’t result in surprise if one of a set of expected outcomes occurs. Hawkins acknowledges this idea: “Prediction is not always exact. Rather, our minds make probabilistic predictions concerning what is about to happen. Sometimes we know exactly what is going to happen, other times our expectations are distributed among several possibilities.” (ibid, p92) Hawkins doesn’t make much of this point in the rest of the book, but it seems crucial to me. In particular, it explains the asymmetry between recognition and generation.

To return to the illustration of studying math: subjectively, I feel like I know what is going on when I read the proof, because I can see that the next line follows by application of a valid logical rule. (That is, the step is among the set of things consistent with my expectations.) Then, when I am called on to reproduce the step, I am surprised to find that I don’t know how — because my prediction is fuzzy, there are multiple reasonable things to do at each step, but I don’t know exactly which one I should do. If on the other hand, I know why each particular step in the proof was taken, then I can uniquely predict each one & reproduce the proof.

An aside about the mechanism of probabilistic predictions: it sounds difficult if you imagine that “probabilistic predictions” means calculating a probability distribution over all possible sensory experiences. However, all that is necessary is for the ‘prediction’ to be abstract — the more abstract it is, the larger the set of observations consistent with it, hence the wider the probability distribution that is implicitly associated with it. It’s not necessary to represent the probability distribution as a blurry activation pattern in the brain at the low-level sensory areas — it is more efficient to activate a single sharp, high-level abstract label which is functionally equivalent. The brain can then lazily evaluate the degree of surprise (ie, the probability) of whatever observation occurs, with respect to the expectation.

In this sense, a word is worth a thousand pictures: “brown dog” is consistent with a huge number of images. That phrase may not seem like a probability distribution — it seems pretty specific. However, from a certain perspective, it’s a blank distribution over all the possible attributes that an image of a dog could have, and in fact would be required to have in order to be made concrete. I may know a brown dog when I see one, but it doesn’t mean I can draw one, or that the one I’m imagining is much like the one you imagine.

This actually connects to statistical physics. There’s a well-defined procedure for constructing explicit probability distributions representing the situation where only a high-level abstraction (in this case, the expectation value or ‘moment’ of a function of the random variables) is known, and all other details of the distribution are uncertain. I suspect that the brain can accomplish something like this, in much the same way that dogs can do calculus. (TL;DR: of course they don’t, but they can approximate the solution to a problem that one could also pose and formally solve using calculus.)

As another aside about math & mental models, I’ve always thought of proofs or derivations like stories: the protagonist is stuck with this problem — our hero has the axioms and the rules (setup), she wants to get to the result, but doesn’t immediately know how (conflict). Then she remember this cool trick (like differentiating the integrand) and voila, problem solved (resolution). I suspect this framing helps with recall. It also puts the focus on why each step is done (the intuition for choosing the step), not just how (ie, the rule that justifies the step).

The potential for disruption of the scientific publishing industry

Alex, a former life-sciences PhD student (now in the software industry) has a great drawing of the ‘bulging pipeline’ where scientific results and careers are forced to stagnate due to the bottleneck imposed by the force of journals-as-gatekeepers. Here’s the key statement:

The real shame in academic publishing, if you ask me, isn’t Elsevier’s 35% profit margin on journal subscriptions. It’s the much larger amount of money, time and influence that is regressively taxed from the young scientists, to the old ones, in exchange for nothing but brand access. So long as journal access remains the yardstick that matters, then … I doubt that the overall structure of the ecosystem will change that much. It’s bad for science, and by extension, bad for all of us.
Can Twitter Save Science

Risk & innovation in science

Peter Shor made a remark on Sabine Hossenfelder’s blog about groupthink in physics:

It’s not just that scientists don’t want to move their butts, although that’s undoubtedly part of it. It’s also that they can’t. In today’s university funding system, you need grants (well, maybe you don’t truly need them once you have tenure, but they’re very nice to have).

So who decides which people get the grants? It’s their peers, who are all working on exactly the same things that everybody is working on. And if you submit a proposal that says “I’m going to go off and work on this crazy idea, and maybe there’s a one in a thousand chance that I’ll discover some of the secrets of the universe, and a 99.9% chance that I’ll come up with bubkes,” you get turned down.

But if a thousand really smart people did this, maybe we’d actually have a chance of making some progress. (Assuming they really did have promising crazy ideas, and weren’t abusing the system. Of course, what would actually happen is that the new system would be abused and we wouldn’t be any better off than we are now.)

So the only advice I have is that more physicists need to not worry about grants, and go hide in their attics and work on new and crazy theories, the way Andrew Wiles worked on Fermat’s Last Theorem.

He added:

Let me make an addendum to my previous comment, that I was too modest to put into it. This is roughly how I discovered the quantum factoring algorithm. I didn’t tell anybody I was working on it until I had figured it out. And although it didn’t take years of solitary toil in my attic (the way that Fermat’s Last Theorem did), I thought about it on and off for maybe a year, and worked on it moderately hard for a month or two when I saw that it actually might work.

So, people, go hide in your attics!

It’s true that many great innovations came about from working ‘in the attic:’ Einstein working as a clerk, Shannon developing information theory in secret at Bell Labs, J. S. Bell developing his theorem on his sabbatical, Shor’s work, and many more. While it may be the best move for an individual scientist to do given the current system, it is a suboptimal solution from a societal perspective — and we should not take the status quo as a boundary condition! Here’s my initial response:

Like you, I’d lay the blame on the poor quality of science management we have, not the scientists. The problem is the evident risk-averse strategy being pursued, squashing innovation, combined with unwillingness to take an appropriate amount of responsibility the decision-making process on the direction of research. The solution is to replace some of the present set of bureaucrats with people (such as venture capitalists) who have the experience and temperament to manage high-risk, high-reward endeavors (which is exactly what science is). Doing bootleg research may be the best strategy for individual scientists to pursue innovation given the current climate, but we need to treat the root of the problem, which is the current climate preventing innovation.

My inspiration came from the following article: The ‘feel-good’ horror of late-stage capitalism. Here’s the gist:

In the feel-good feel-bad story, irrefutable proof of an institutional failure is sold as a celebration of individual triumph. And it’s the desperate, cloying attempts to trumpet the latter as a means of obscuring the former that gives these pieces their distinct, acrid aftertaste.

We don’t need higher wages; just have an amazing CEO give you his car! Who cares if you can’t support a family on one job? The fix is simple: Get two more jobs!

Shor’s remarks constitute the same refrain, transposed to a different key. Our society is pushing risk onto individuals, when we should be socializing it, individual success stories to the contrary notwithstanding. We could benefit from a more risk-tolerant approach to science management, to supplement the more ‘business-as-usual’ approach.

I also take issue with Shor’s straw-man version of an alternative to the present system of grant review. Presently the fox is guarding the hen-house, but the solution isn’t to just fling the doors of the coop wide open. Instead, we need independent review from outside physics. Yes, this implies reviewers who aren’t likely to fully grasp the theory, which does create an information asymmetry and the potential for abuse — so it’s important to perform some due-diligence.

However, there will be dead-ends no matter what. Even the brightest of us may disagree about whether an approach will pan out — it’s research after all! The cost of funding a few wacky ideas along with one breakthrough may be worth it compared to the present approach of funding relatively staid approaches that almost certainly won’t result in a breakthrough.

I’m also concerned because the plausibility of claims of groupthink dovetails with the desires of ‘climate skeptics’ who would like to portray the scientific consensus on climate change as merely a result of a liberal echo chamber. To be frank, it’s not too difficult for me to imagine how that could happen, although I think in the case of climate change the evidence is really there.