Magnetic Configuration

In magnetic confinement fusion, the shape into which the magnetic field lines are arranged (the magnetic configuration) determines whether the plasma can be held without escaping. This page starts from the intuition of how field lines trap particles, then works through why the field must be twisted into a doughnut shape, how that configuration is described mathematically, and how it is optimized.

Start with Intuition (High School)

Charged particles move by winding around magnetic field lines. When you scatter iron filings around a magnet, the lines become visible, and each of those individual field lines acts like a rail for an electrically charged particle. A particle cannot freely fly across the field lines; instead it spirals along them as it advances. This property of “difficulty crossing field lines” is the starting point for confining plasma with a magnetic field.

So could we just prepare straight field lines? Consider making a bundle of straight field lines inside a long solenoid (a cylindrical coil). Because particles wind around the field lines, they find it hard to escape sideways (across the field lines). However, along the field lines, that is, toward the two ends of the cylinder, they can slip away easily. This is the problem of end loss in linear devices. Imagine a bucket with no bottom, only side walls. Water does not leak out the sides, but it keeps draining out the bottom. The magnetic mirror scheme strengthens the field at both ends to put a “plug” in place, but even so it cannot completely stop the leakage from the ends.

So we change our approach. If it leaks because there are ends, then we can simply get rid of the ends. If we connect the two ends of the straight cylinder into a ring, the field lines become closed loops that go all the way around, and no matter how far a particle travels it never reaches an “end.” This is the idea of forming a torus (making a doughnut shape).

However, merely making a ring creates a new problem. When you form a doughnut shape, the field lines become crowded on the inside and sparse on the outside. When the field is stronger in some places and weaker in others, particles are slowly pushed up or down. This sideways slip is called drift. Positive particles flow upward and negative particles flow downward, so an electric imbalance builds up between top and bottom, and the force created by that imbalance then pushes the whole plasma outward until it hits the wall. In a simple doughnut, this destroys confinement in an instant.

The key to solving this difficulty is to twist the field lines. If you twist the field lines so that during one loop around they pass through both the upper and lower sides of the doughnut, a particle pushed upward is eventually carried to a lower section, where it is pushed in the opposite direction. The pushing on the way there and the way back cancel out, and no electric imbalance accumulates. This property of “twisting through top and bottom during one loop” is called the rotational transform. Both tokamaks and stellarators, though they achieve it in different ways, are all devices for creating these twisted field lines.

Understand the Physics (Undergraduate)

The magnetic field of a torus configuration is treated by dividing it broadly into two components. The component running in the toroidal direction of the doughnut (along the large ring) is called the toroidal field $B_t$, and the component that circulates the small cross section is called the poloidal field $B_p$. When these two are combined, the field lines spiral around the surface of the doughnut as they complete one loop.

The toroidal field is produced by external coils (toroidal field coils, TF coils). From Ampère’s law, with $R$ the distance from the central axis,

$$B_t(R) = \frac{\mu_0 N I}{2\pi R}$$

so it is inversely proportional to the major radius $R$. Here $\mu_0$ is the permeability of free space, $N$ is the number of coil turns, and $I$ is the coil current. Read out loud, this says the toroidal field is stronger on the inner side of the torus (small $R$) and weaker on the outer side (large $R$). This $1/R$ gradient is exactly the cause of the particle drift described in the previous section.

The drift velocity caused by the field gradient and the curvature of the field lines, with particle charge $q$ and mass $m$, is of order

$$v_D \sim \frac{m(v_\parallel^2 + \tfrac{1}{2}v_\perp^2)}{qB R}$$

where $v_\parallel$ is the velocity along the field line and $v_\perp$ is the perpendicular velocity. This expression shows that the drift is faster when the field is weak (small $B$) and the torus is small (small $R$). Because the direction of the drift reverses with the sign of the charge, ions and electrons separate up and down (for details of this drift motion, see Particle Motion).

The poloidal field is what cancels this drift. With $B_p$ present, the field lines travel while circling within the cross section, so a particle alternately passes through the upper half and the lower half. Positive charge that tried to accumulate at the top receives an oppositely directed drift in the lower section, so averaged over one loop the imbalance is resolved. How the poloidal field is produced is what gives rise to the difference between the schemes. A tokamak drives a large current through the plasma itself (the plasma current) and uses the field created by that current as $B_p$. A stellarator creates $B_p$ using only external twisted coils, without relying on a plasma current. The tokamak, which depends on the plasma current, easily produces strong confinement, but it must keep the current flowing and carries the danger of a disruption, in which the current is suddenly interrupted. The stellarator, which uses external coils, is suited to steady-state operation but has a complex coil shape.

The quantity expressing how much the field lines are twisted is the safety factor $q$. It is defined by how many times a field line loops in the toroidal direction while it loops once in the poloidal direction, and can approximately be written

$$q \approx \frac{r B_t}{R B_p}$$

where $r$ is the minor radius of the cross section and $R$ is the major radius. Read out loud, the stronger the toroidal field and the weaker the poloidal field, the larger $q$ is and the gentler the twist of the field lines. The rotational transform $\iota$ (iota) is its reciprocal $\iota = 1/q$; because this directly expresses “the number of twists in the poloidal direction,” it is often used in the stellarator field.

Field lines with the same $q$ fill a single nested tube-like surface within the doughnut. This surface is called a magnetic surface. They are nested in many layers like the skins of an onion, and on each layer the plasma pressure and temperature are nearly constant. Confinement can be restated as maintaining many of these magnetic surfaces without breaking them.

Deepen the Theory (Graduate)

The existence of magnetic surfaces can be understood from the equilibrium condition of ideal magnetohydrodynamics (ideal MHD). With pressure $p$, current density $\mathbf{j}$, and magnetic field $\mathbf{B}$, the force balance is

$$\nabla p = \mathbf{j} \times \mathbf{B}$$

Taking the dot product of this equation with $\mathbf{B}$ gives $\mathbf{B} \cdot \nabla p = 0$. Read out loud, this says the pressure does not change along a field line, that is, field lines run on surfaces of constant pressure (magnetic surfaces). Similarly, from $\mathbf{j} \cdot \nabla p = 0$, the current also flows on magnetic surfaces.

The equilibrium of a torus configuration, in the axisymmetric case, is collected into the Grad-Shafranov equation. Using the poloidal magnetic flux function $\psi$,

$$R^2 \nabla \cdot \left( \frac{\nabla \psi}{R^2} \right) = -\mu_0 R^2 \frac{dp}{d\psi} - F \frac{dF}{d\psi}$$

where $F = R B_t$ is a quantity related to the poloidal current. Solving this equation, the contours of constant $\psi$ become magnetic surfaces, and in a finite-pressure plasma the Shafranov shift appears, in which the centers of the magnetic surfaces are displaced outward.

When handling magnetic surfaces in this way, a coordinate system attached to the magnetic surfaces is more convenient than ordinary spatial coordinates. We introduce flux coordinates, which use the magnetic flux as the radial label and express positions on a magnetic surface by a poloidal angle and a toroidal angle. In particular, those in which the angles are chosen so that the field lines appear straight in the coordinates are called straight-field-line coordinates, and Hamada coordinates and Boozer coordinates are often used. In these coordinates the safety factor appears naturally as a constant for each magnetic surface, greatly simplifying the analysis of transport and stability.

A magnetic surface where $q$ becomes a rational number $m/n$ is called a rational surface. On a rational surface the field lines return to themselves and close after a finite number of loops, so resonant perturbations easily grow, and it becomes a site for MHD instabilities and the formation of magnetic islands. When the finite resistivity of the plasma reconnects field lines (magnetic reconnection), the tube around the rational surface tears apart to form an island-like structure, and inside the island the pressure flattens and confinement degrades locally. In particular, the neoclassical tearing mode (NTM) has a positive feedback in which a deficit of the spontaneously flowing bootstrap current grows the island, and it is controlled by supplementing the deficit current with electron cyclotron current drive (ECCD).

The radial rate of change of $q$ is called the magnetic shear, defined by $s = (r/q)(dq/dr)$. When the shear is strong, the direction of the twist shifts between radially adjacent magnetic surfaces, so many instability modes are stretched out and stabilized. In operation, $q_0$ at the center and $q_{95}$ near the plasma edge (the value at the 95% normalized-flux surface) are used as indicators; to avoid the internal kink mode we keep $q_0 > 1$, and to avoid the external kink mode we keep $q_{95} > 2$, in practice keeping around $q_{95} > 3$ with some margin.

At the plasma edge, the magnetic surfaces are intentionally opened to create a divertor configuration. By placing an X-point where the poloidal field is zero, outside it the field lines connect to a dedicated region of the wall (the divertor plates), where the heat and impurities are received and pumped out. Various schemes to reduce the heat load are studied, such as a single null with one X-point, a double null with two symmetric top-and-bottom X-points, and a snowflake configuration that spreads the flux with a second-order X-point.

Research Frontier (PhD)

The center of current magnetic configuration research is configuration optimization, which treats the configuration itself as a design variable to be optimized. Formerly the device shape was decided first and then the plasma was analyzed, but today the desired physics performance is specified first, and the coil shape and field distribution that satisfy it are worked out in reverse.

For stellarators, the discovery of quasi-symmetry accelerated this trend. Even if it is twisted and non-axisymmetric in real space, if the field strength is symmetric in some direction in Boozer coordinates, the momentum in that direction is conserved, and particles are confined as well as in a tokamak. There are types such as quasi-axisymmetry, quasi-helical symmetry, and quasi-poloidal symmetry, and large-scale numerical optimization has made it possible to design configurations that satisfy these to high accuracy. In helical devices Stellarator / Helical Scheme, demonstrating such optimized configurations is a major theme.

On the tokamak Tokamak Scheme side, advanced scenarios that actively shape the current distribution are being studied. A reversed shear configuration, in which $q$ has a minimum in the central region, forms an internal transport barrier (ITB) and dramatically improves confinement. A hybrid scenario that keeps the $q$ profile flat at the center aims for stable high-performance operation while avoiding sawtooth oscillations. The ultimate goal is fully non-inductive steady-state operation, which minimizes external current drive and supplies the necessary poloidal field with the bootstrap current alone.

The mathematical toolkit that supports optimization is also a frontier topic. Because the clean existence of magnetic surfaces is not itself guaranteed, the creation and suppression of stochastic regions, where islands and field lines wander randomly, is at issue. In recent years, methods that use the adjoint method and automatic differentiation to rapidly evaluate the sensitivity to the coil shape, and that use gradient methods to globally optimize the coils and plasma simultaneously, have developed. The problem of finding a configuration that simultaneously satisfies confinement, stability, and an engineering-feasible coil shape is still being actively researched.

Including schemes that suppress end loss with a different approach, such as mirror systems Mirror Scheme, the question of what shape of magnetic field can hold plasma most efficiently continues to lie at the very foundation of fusion research.

Check Your Understanding

Q1. Why can a linear magnetic field device not completely confine plasma?

A Because the field lines disappear over time B Because particles find it hard to escape across the field lines, but there is an end loss where they slip out along the field lines from both ends of the cylinder C Because the field is so strong that the particles break apart D Because in a straight field the particles do not wind around at all

Q2. Why does simply making a torus (doughnut shape) not achieve confinement?

A Because making a doughnut shape makes the field lines vanish completely B Because with no ends the particles accelerate too much and break through the wall C Because the toroidal field is stronger on the inside and weaker on the outside, causing a drift that separates ions and electrons up and down, and the resulting electric imbalance pushes the plasma outward D Because in a doughnut shape the field lines twist too much and the particles get tangled

Q3. Why can the rotational transform (the twisting of the field lines) solve the drift problem?

A Because the twisted field lines create an additional field that cancels the drift itself B Because the field lines twist so as to pass alternately through the upper and lower sides of the doughnut, so a drift received in one section is reversed in another section and cancels out over one loop C Because the twisting makes the particle velocity zero D Because twisting makes the field gradient disappear entirely

Q4. Regarding how the poloidal field is produced, which is the correct difference between a tokamak and a stellarator?

A A tokamak uses only external coils, while a stellarator uses the plasma current B Both create the poloidal field using only the plasma current C A tokamak uses the plasma current, so it has a disruption risk, while a stellarator uses only external coils, which suits steady-state operation but makes the coil shape complex D Both tokamaks and stellarators use only external coils, and the only difference is the number of coils

Q5. Why is quasi-symmetry considered important in configuration optimization?

A Because it makes it possible to create a perfectly axisymmetric shape in real space B Because if the field strength is symmetric in Boozer coordinates, momentum is conserved, and even a non-axisymmetric configuration achieves particle confinement comparable to an axisymmetric tokamak C Because it reduces the number of coils and makes the device cheaper D Because it increases the plasma current to strengthen confinement

Related Topics

• Particle Motion - The basics of the motion and drift of charged particles in a magnetic field
• Tokamak Scheme - The scheme that creates the poloidal field with the plasma current
• Stellarator / Helical Scheme - The scheme that twists the field lines using only external coils
• Mirror Scheme - A linear-system scheme that suppresses end loss with magnetic mirrors