Why do fusion reactors take on a doughnut-like shape?
Surprising implications from algebraic topology determine the shape of our plasma-confining machines.
Nuclear fusion holds the promise to generate energy in a clean, safe, and nearly inexhaustible way. The physical idea of fusion involves confining fuels at unearthly temperatures of approximately 150,000,000 degree Celsius which fusion reactions between atomic nuclei can happen. The fuels of interest, deuterium and tritium (isotopes of hydrogen), exist in the state of plasma. Clearly, containing these extremely hot plasmas with solid walls is unfeasible.
A plasma is an ionised gas comprising charged particles, both ions and electrons. Fortunately, the dynamics of charge particles are subject to constraints along magnetic field lines. This insight forms the basis of our current approach: constructing a magnetic bottle using powerful magnetic fields that effectively trap the plasma along these intangible field lines.
One of the most iconic magnetic confinement machine designs is the tokamak — a toroidally-shaped device, often likened to a doughnut. The name ‘tokamak’ is derived from the Russian acronym for ‘toroidal chamber with magnetic coils.’
Confining a plasma with magnetic field
Plasma physics is fundamentally rooted in the motion of charged particles, which are subject to the Lorentz force — a force that arises from the presence of (electro)magnetic fields. When a particle moves in alignment with the magnetic field, it remains unaffected by any force. However, when the particle moves across the field lines, the magnetic field exerts a push perpendicular to its motion, causing these charged particles to trace out a small gyrating path along the field lines. The dynamics of charged particles is therefore “trapped” along field lines.
This principle forms the basis for magnetic confinement of plasma. Our objective is to design a system in which these particles remain confined within the magnetic field lines for an extended duration. Let’s consider a linear configuration, such as a cylindrical setup. The challenge here is that the particles inevitably intersect with the system’s boundaries. Since there is no force acting along the field lines, there is no mechanism to prevent plasma particles from colliding with the vessel walls.
An attempt to address this issue, known as the magnetic mirror concept, capitalises on the mirror effect. It involves generating a region of significantly intensified magnetic field at the ends of the device, which can cause particles to reflect back into the central confining region. However, this reflective mechanism is effective only within a specific range of particle velocities and pitch angles. Particles outside these limits will escape, rendering magnetic mirrors intrinsically leaky.
Therefore, we need to create a configurations where the field lines close on themselves and do not end. Mathematically, we are looking for surfaces where these magnetic field vectors lie on in a tangential and continuous manner.
The hairy ball theorem and magnetic confinement
Let’s revisit our titular question: “Why do fusion reactors take on a doughnut-like shape?” It may seem more intuitive to adopt a spherical configuration for magnetic confinement, providing a straightforward means to close the magnetic field lines. So, why don’t we design our fusion machines as spheres? The answer to this question, surprisingly, lies in the realm of algebraic topology.
Suppose we were to attempt to establish magnetic fields across a spherical surface. While it is possible to create closed field lines, we encounter a fundamental challenge: the unavoidable existence of a field null — a point at which the magnitude of the field vector diminishes to zero. This property is governed by the hairy ball theorem, a result first proven by Henri Poincaré in 1885. This theorem asserts:
Given a ball with hairs all over it, it is impossible to comb the hairs continuously and have all the hairs lay flat without creating a cowlick.
Or slightly more formally:
For a three-dimensional continuous tangent vector field on the surface of an ordinary sphere, there must always be a point on the sphere where the field vanishes.
Here is a very nice short video illustrating this theorem by minutephysics:
The hairy ball theorem implies that there will be a point where the magnetic field reaches zero on a sphere. It’s at these null points that plasma can escape because there’s no magnetic confinement. Any shape that is topologically equivalent to a sphere will inevitably possess this trait, making it unsuitable for effective plasma confinement.
The most straightforward shape that overcomes this issue is that of a torus. A toroidal surface can be enveloped by a non-vanishing vector field, ensuring that magnetic fields are non-zero everywhere throughout its structure, thus offering an attractive and viable means to confine a plasma. This feature is inherently tied to the torus’s topological properties, specifically a topological invariant known as the Euler characteristic — a number that stays the same irrespective of how the shape undergoes continuous deformation, such as stretching or bending.
Formally, it can be demonstrated that the Euler characteristic of a torus is 0, in contrast to the value of 2 for a sphere. Building upon this insight, a more generalised version of the hairy ball theorem, referred to as the Poincaré–Hopf theorem, affirms the existence of a non-vanishing vector field on a torus, a shape with an Euler characteristic of 0. In contrast, this theorem precludes the possibility of such a non-vanishing vector field on a sphere due to the sphere’s non-zero Euler characteristic.
This is precisely why the majority of current fusion reactor designs opt for a toroidal shape. Fusion plasma confinement within tokamaks is an area of active and intensive research. Another design, known as stellarators, also adopts a toroidal shape and utilises the benefits of employing non-axisymmetric field coils. Nowadays, toroidal machines occupy the forefront of the quest for magnetic fusion energy. Significant strides have been made in advancing our understanding of the complex physics involved. The establishment of our current progress is made possible thanks to the fundamental discoveries and understanding of topology, which laid one of the theoretical foundations of building doughnut-shaped machines.
