There’s an interesting new study, published on February 23, 2022, that discusses a way to make nuclear fusion devices called stellarators more efficient by applying equations used all the way away in systems biology.
The Wikipedia article about stellarators is surprisingly well-written; I’ve often found that I’ve had to bring my undergraduate engineering lessons to bear to understand the physics articles. Not here. Let me quote at length from the sections describing why physicists need stellarators, which also serves to explain how these machines work.
Heating a gas increases the energy of the particles within it, so by heating a gas into hundreds of millions of degrees, the majority of the particles within it reach the energy required to fuse. … Because the energy released by the fusion reaction is much greater than what it takes to start it, even a small number of reactions can heat surrounding fuel until it fuses as well. In 1944, Enrico Fermi calculated the deuterium-tritium reaction would be self-sustaining at about 50,000,000º C.
Materials heated beyond a few tens of thousand degrees ionize into their electrons and nuclei, producing a gas-like state of matter known as plasma. According to the ideal gas law, like any hot gas, plasma has an internal pressure and thus wants to expand. For a fusion reactor, the challenge is to keep the plasma contained. In a magnetic field, the electrons and nuclei orbit around the magnetic field lines, confining them to the area defined by the field.
A simple confinement system can be made by placing a tube inside the open core of a solenoid.
A solenoid is a wire in the shape of a spring. When an electric current is passed through the wire, it generates a magnetic field running through the centre.
The tube can be evacuated and then filled with the requisite gas and heated until it becomes a plasma. The plasma naturally wants to expand outwards to the walls of the tube, as well as move along it, towards the ends. The solenoid creates magnetic field lines running down the center of the tube, and the plasma particles orbit these lines, preventing their motion towards the sides. Unfortunately, this arrangement would not confine the plasma along the length of the tube, and the plasma would be free to flow out the ends.
The obvious solution to this problem is to bend the tube around into a torus (a ring or donut) shape.
A nuclear fusion reactor of this shape is called a tokamak.
Motion towards the sides remains constrained as before, and while the particles remain free to move along the lines, in this case, they will simply circulate around the long axis of the tube. But, as Fermi pointed out, when the solenoid is bent into a ring, the electrical windings would be closer together on the inside than the outside. This would lead to an uneven field across the tube, and the fuel will slowly drift out of the center. Since the electrons and ions would drift in opposite directions, this would lead to a charge separation and electrostatic forces that would eventually overwhelm the magnetic force. Some additional force needs to counteract this drift, providing long-term confinement.
[Lyman] Spitzer’s key concept in the stellarator design is that the drift that Fermi noted could be canceled out through the physical arrangement of the vacuum tube. In a torus, particles on the inside edge of the tube, where the field was stronger, would drift up. … However, if the particle were made to alternate between the inside and outside of the tube, the drifts would alternate between up and down and would cancel out. The cancellation is not perfect, leaving some net drift, but basic calculations suggested drift would be lowered enough to confine plasma long enough to heat it sufficiently.
These calculations are not simple because this how a stellarator can look:
When a stellarator is operating and nuclear fusion reactions are underway, impurities accumulate in the plasma. These include ions that have formed but which can’t fuse with other particles, and atoms that have entered the plasma from the reactor lining. These pollutants are typically found at the outer layer.
An additional device called a diverter is used to remove them. The heavy ions that form in the reactor plasma are also called ‘fusion ash’, and the diverter is the ashtray.
It works like a pencil sharpener. The graphite is the plasma and the blade is the diverter. It scrapes off the wood around the graphite until the latter is fully exposed and clean. But accomplishing this inside a stellarator is easier said than done.
In the image above, let’s isolate just the plasma (yellow stuff), slice a small section of it and look at it from the side. Depending on the shape of the stellarator, it will probably look like a vertical ellipse, an elongated egg – a blob, basically. By adjusting the magnetic field near the bottom of the stellarator, operators can change the shape of the plasma there to pinch off its bottom, making the overall shape more like an inverted droplet.
At the bottom-most point, called the X-point, the magnetic field lines shaping the plasma intersect with each other. At least, some magnetic field lines intersect with each other while others move towards each other without fully criss-crossing, but which are in contact with the surface of the reactor. (In the image below, the boundary between these two layers of the plasma is called the separatrix.)
Diverter plates are installed near this crossover point to ‘drain’ the plasma moving along the non-intersecting fields.
In the new study, physicists addressed the problem of diverter overheating. The heat removed at the diverter is considered to be ‘waste’ and not a part of the fusion reactor’s output. The primary purpose here is to take away the impure plasma, so the cooler it is, the longer the diverter will be able to operate without replacement.
The researchers used the Large Helical Device in Gifu, Japan, to conduct their tests. It is the world’s second largest stellarator (the first is the Wendelstein 7-X). Their solution was to stop heating the plasma just before it hit the diverter plates, in order to allow the ions and electrons to recombine into atoms. The energy of the combined atom is lower than that of the free ions and electrons, so less heat reaches the diverter plates.
How to achieve this cooling? There were different options, but the physicists resorted to arranging additional magnetic coils around the stellarator such that, just before the plasma hit the diverter, its periphery would detach into a smaller blob that, being separated from the overall plasma, could cool. These smaller blobs are called magnetic islands.
When they ran tests with the Large Helical Device, they found that the diverter removed heat from the plasma chamber in short bursts, instead of continuously. They interpreted this to mean the magnetic islands didn’t exist in a steady state but attached and detached from the plasma at a regular frequency. The physicists also found that they could model the rate of attachment using the so-called predator-prey equations.
These are the famous Lotka-Volterra equations. They describe how the populations of two species – one predator and one prey – vary over time. Say we have a small ecosystem in which crows feed on worms. As they do, the crow population increases, but due to overfeeding, the population of worms dwindles. This forces the crow population to shrink as well. But once there are fewer crows around, the number of worms increases again, which then allows more crows to feed on worms and become more populous. And so the cycle goes.
Similarly, the researchers found that the Lotka-Volterra equations (with some adjustments) could model the attachment frequency if they assumed the magnetic islands to be the predators and an electric current in the plasma to be the prey. This current is the product of electrons moving around in the plasma, which the authors call a “bootstrap current”.
When the strength of the bootstrap current increases, the magnetic island expands. At the same time, the confining magnetic field resists the expansion, forcing the current to dwindle. This allows the island to shrink as well, receding from the field. But then this allows the bootstrap current to increase once more to expand the island. And so the cycle goes.
The researchers reported in their paper that while they observed a frequency of 40 Hz (i.e. 40 times per second) in the Large Helical Device, the equations on paper predicted a frequency of around 20 Hz. However, they have interpreted to mean there is “qualitative agreement” between their idea and their observation. They also wrote that they expect the numbers to align once they fine-tune their math to account for various other specifics of the stellarator’s operation.
They eventually aim to find a way to control the attachment rate so that the diverters can operate for as long as possible – and at the same time take away as much ‘useless’ energy from the plasma as possible.
I also think that, ultimately, it’s a lovely union of physics, mathematics, biology and engineering. This is thanks in part to the Lotka-Volterra equations, which are a specific form of the Kolmogorov model. This is a framework of equations and principles that describes the evolution of a stochastic process in time. A stochastic process is simply one that depends on variables whose values change randomly.
In 1931, the Soviet mathematician Andrei Kolmogorov described two kinds of stochastic processes. In 1949, the Croatian-American mathematician William Feller described them thus:
… the “purely discontinuous” type of … process: in a small time interval there is an overwhelming probability that the state will remain unchanged; however, if it changes, the change may be radical.
… a “purely continuous” process … there it is certain that some change will occur in any time interval, however small; only, here it is certain that the changes during small time intervals will be also small.
Kolmogorov derived a pair of ‘forward’ and ‘backward’ equations for each type of stochastic process, depending on the direction of evolution we need to understand. Together, these four equations have been adapted to a diverse array of fields and applications – including quantum mechanics, financial options and biochemical dynamics.
Featured image: Inside the Large Helical Device stellarator. Credit: Justin Ruckman, Infinite Machine/Wikimedia Commons, CC BY 2.0.