arXiv subjects

Allen H Boozer

Publications and source records attributed to Allen H Boozer.

At least 19 recordsLinked to original sources

arXiv2026

The toroidal flux and separatrix effects in tokamaks

An implication of Faraday's Law is thst the rate of change of the poloidal flux relative to the toroidal flux associated with a magnetic surface is given by the loop voltage. The emphasis in the tokamak literature on the poloidal almost to the exclusion of the toroidal magnetic flux has led to fundamental mistakes in understanding magnetic field evolution, especially as it affects plasma disuptivity. Designs of tokamak power plants include a divertor that is defined by a separatrix at the plasma edge. The use of the toroidal flux clarifies the descriptions of such plasmas. This paper explores the effects of a separatrix on plasma parameters that are used for assessing disruptivity: the edge safety factor and the internal inductance. The exploration of these effects includes an analytic model, which illustrates why the use of the poloidal instead of the toroidal flux introduces a significant definitional uncertainty in what is meant by the edge safety factor and the internal inductance.

physics.plasm-ph↗

arXiv2026

Pfirsch-Schl\"uter Current

The Pfirsch-Schl\"uter current is a current that flows along the magnetic field lines in a toroidal plasma equilibrium that is required to make the plasma current density divergence free in the presence of a plasma-pressure gradient. A distortion in the plasma shape is caused by the Pfirsch-Schl\"uter current, and it is desirable to minimize both the strength and the distance this current flows along the magnetic field lines. The Pfirsch-Schl\"uter current is localized within a half period of a stellarator when $d\ell/B$ integrated over the half period is the same for all lines in the magnetic surface. It is shown that within parts in a thousand this is the same condition as the distance $\ell_{s}$ required for a field line to cross the half period being the same for all lines in the surface. To make the $\ell_{s}$'s the same, the lines started on the small major radius side of the plasma must undergo wiggles to make their $\ell_{s}$ as long as those started on the outboard side. This is generally achieved using modular coils with a large helical component on the small major radius side but could be achieved with a central column carrying a helical current.

physics.plasm-ph↗

arXiv2026

Non-Ambipolarity of Microturbulent Transport

When exact magnetic surfaces are assumed to exist, the gyrokinetic theory of microturbulence gives the same radial transport for ions and electrons. But, exact magnetic surfaces do not exist in the presence of what is called electrostatic microturbulence. When the plasma pressure is non-zero, a turbulent electric potential is accompanied by a turbulent magnetic field, which splits the rational magnetic surfaces with which it resonates. If the magnetic field is assumed to have an ideal topology-conserving evolution, delta function current densities arise on resonant surfaces. The singularity of the current density allows islands to open quickly, but there is no singularity that allows a rapid closure. Islands remain and do not flutter into and out of existence. A relative rotation of the electron fluid in neighboring island chains produces a non-dissipative force that can lock the islands together and produce a non-ambipolar transport. At sufficient plasma pressure, the islands associated with different resonant rational surfaces can overlap. When this occurs some magnetic field lines will cross the entire radial region occupied by overlapping islands. The effect on the electron fluid is to create a viscosity-like force, which is dissipative and tends to remove gradients in the electron rotation. This also produces a non-ambipolar transport. Under many assumptions, the island locking force is larger than the viscosity-like force.

physics.plasm-ph↗

arXiv2026

Comment on Nuclear Fusion 66, 016012 (2026) and arXiv:2508.03561 by Richard Fitzpatrick, A Simple Model of Current Ramp-Up and Ramp-Down in Tokamaks

The paper Nuclear Fusion \textbf{66}, 016012 (2026) by Richard Fitzpatrick is based on fundamental errors in the physics of the evolution of the poloidal magnetic flux in tokamaks. With the certainty of Faraday's Law, the slippage of the poloidal relative to the toroidal flux is given by the loop voltage. Some of the errors may be endemic to the tokamak community, such as the focusing on a quantity $\psi$ that has a subtle relation to the poloidal flux while essentially ignoring the toroidal flux. Other errors are due to an entanglement of in principle separable concepts and unjustified assumptions. Although Nuclear Fusion allowed the use ``private communication" as a reference to bolster misquotes, the response must be in arXiv because Nuclear Fusion will not allow any mention of ``private communication" in a response. This comment also derives a new and simple expression for the loop voltage, which has the validity of the parallel component of Ohm's law.

physics.plasm-ph↗

arXiv2025

Magnetic Field Line Chaos, Cantori, and Turnstiles in Toroidal Plasmas

The mathematical concepts of chaos, cartori, and turnstiles underlie a number of areas of tokamak and stellarator physics. Nevertheless, these concepts have seldom explicitly appeared in publications on fusion plasmas. The absence of physical intuition about these concepts is responsible for misunderstandings and slows developments in a number of areas: magnetic reconnection, the most important electromagnetic correction to what are called electrostatic microinstabilities, non-resonant divertors in stellarators, disruptions and damage from runaway electrons in tokamaks. Physicists become interested in new mathematical concepts when they give insights into and solutions to practical problems. The importance of this review is not only in explaining chaos, cartori, and turnstiles as mathematical concepts but also in illustrating their significance through applications.

physics.plasm-ph↗

arXiv2025

Electron Inertia and Magnetic Reconnection

When electron inertia is the only non-ideal effect in the evolution of a magnetic field $\vec{B}$, the field lines of $\vec{B}$ reconnect, but the lines of a related field $\vec{\mathcal{B}}$ do not. $\vec{\mathcal{B}} \equiv \vec{B} + \vec{\nabla}\times \left( (c/\omega_{pe})^2\mu_0\vec{j} \right)$ with $\omega_{pe}$ the plasma frequency and $\vec{j}$ the current density. Although a full four-dimensional relativistic calculation of $\vec{\mathcal{B}}$ has been made, studies of $\vec{\mathcal{B}}$ have been focused on systems that depend on only two spatial coordinates. Three results are given: (1) A relatively simple demonstration in three dimensional space that the lines of $\vec{\mathcal{B}}$ do not reconnect when electron inertia is the only non-ideal effect. (2) The guiding center motion of charged particles is modified by a term that is proportional to $(c/\omega_{pe})^2$, which is smaller than the drifts proportional to the gyroradius unless the current density is extremely large. (3) In three dimensional space, the evolution velocity of $\vec{\mathcal{B}}$ is characteristically chaotic, which means neighboring streamlines separate exponentially on a timescale $\tau_u$. $\vec{\mathcal{B}}$ undergoes large scale reconnection on a timescale that is only an order of magnitude or two longer than $\tau_u$ unless all diffusive non-ideal effects, such as resistivity, are absolutely zero.

physics.plasm-ph↗

arXiv2025

Constraints on the magnetic field evolution in tokamak power plants

Forty-five years ago a coordinate system was shown to exist that gave simple but exact expressions whenever and wherever a toroidal plasma equilibrium $\vec{\nabla}p=\vec{j}\times\vec{B}$ exists. These coordinates, now called Boozer coordinates, which revolutionized the stellarator program, are also applicable to tokamaks. Here expressions for Faraday's Law, the safety factor, and the internal inductance are derived. Their constraints should be useful in the design of tokamak power plants and for the thoughtful allocation of resources to minimize the time and the cost to the achievement of practical fusion power. Simple explanations are obtained for (1) why disruptions in tokamaks are so common, (2) why current-profile control though difficult may be required, especially during plasma shutdowns, and (3) why only pulsed tokamaks seem possible. Lack of familiarity with Boozer coordinates can make simple but exact expressions appear naive. Complicated derivations with dubious assumptions have been interpreted as ``more rigorous.''

physics.plasm-ph↗

arXiv2025

Efficient analysis of magnetic field line behavior in toroidal plasmas

The confinement of plasmas in tokamaks and stellarators depends on magnetic field lines lying in nested toroidal surfaces. The transition near the plasma edge away from the lines lying in magnetic surfaces defines properties of divertors. The transition in time defines properties of disruptions. Divertor design and disruption analyses require a detailed understanding of these transitions. The use of a Fourier transform coupled with a Gaussian window function allows far more information to be extracted about these transitions using far shorter field line integrations than can be obtained using traditional methods based on Poincar\'e plots. The physics of divertors and disruptions is reviewed to clarify why the type of information that can be gained from more efficient methods of analysis is of central importance to the fusion program based on magnetic confinement.

physics.plasm-ph↗

arXiv2024

Magnetic reconnection and dynamos in the presence of plasma turbulence

Evolving magnetic fields are frequently embedded in plasmas that are turbulent. When the primary interest is in effects that are on a large scale compared to that of the turbulence, it is desirable to average over the turbulence to obtain equations for mean-field magnetohydrodynamics. An obvious constraint on the validity of the averaging is that large-scale quantities that evolve slowly using the exact evolution equations must remain slowly evolving in the mean-field theory. Magnetic helicity is the primary example of such a quantity and maintaining its slow evolution has been controversial in mean-field magnetohydrodynamics. A full theory of magnetic reconnection in turbulent plasmas is not the intent of this paper. What is the intent is to show how exact results from Maxwell's equations explain why fast reconnection is so ubiquitous and what constraints these results place on the theory of magnetic field evolution, including dynamos, whether the plasma is turbulent or not. These constraints are commonly broken in the reconnection literature, which has been heavily influenced by two-dimensional theory that is not applicable to three-dimensional problems.

physics.plasm-ph↗

arXiv2024

Use of current-potential patches to obtain fundamental improvements to the coils of magnetic fusion devices

A central issue in the design of tokamaks or stellarators is the coils that produce the external magnetic fields. The freedom that remains unstudied in the design of coils is enormous. This freedom could be quickly studied computationally at low cost with high reliability. In particular, the space between toroidal field or modular coils that block access to the plasma chamber could be increased by a large factor. The concept of current-potential patches, which was developed in Todd Elder's thesis, provides a method for separating the study of the feasibility of coils with attractive physics properties from the engineering design of specific coils. In addition to enhanced accessibility, coils can be designed for increased plasma-coil separation, insensitivity to coil position errors, and plasma control.

physics.plasm-ph↗

arXiv2024

Electric field effects during disruptions

Tokamak disruptions are associated with breaking magnetic surfaces, which makes magnetic field lines chaotic in large regions of the plasma. The enforcement of quasi-neutrality in a region of chaotic field lines requires an electric potential that has both short and long correlation distances across the magnetic field lines. The short correlation distances produce a Bohm-like diffusion coefficient $\sim T_e/eB$ and the long correlation distances $a_T$ produce a large scale flow $\sim T_e/eB a_T$. This cross-field diffusion and flow are important for sweeping impurities into the core of a disrupting tokamak. The analysis separates of the electric field in a plasma into the sum of a divergence-free, $\vec{E}_B$, and a curl-free, $\vec{E}_q$, part, a Helmholtz decomposition. The divergence-free part of $\vec{E}$ determines the evolution of the magnetic field. The curl-free part enforces quasi-neutrality, $\vec{E}_q=-\vec{\nabla}\Phi_q$. Magnetic helicity evolution gives the required boundary condition for a unique Helmholtz decomposition and an unfortunate constraint on steady-state tokamak maintenance.

physics.plasm-ph↗

arXiv2024

Needed computations and computational capabilities for stellarators

Stellarator plasmas are externally controlled to a degree unparalleled by any other fusion concept, magnetic or inertial. This control is largely through the magnetic fields produced by external coils. The development of fusion energy could be expedited by carrying out remarkably straight-forward computations to define strategies for exploiting this external control. In addition to these computations, which have a reliability limited only by competence, certain physics areas that affect the develop of stellarator power plants should have more intense study. The low cost and speed with which computations can be carried out relative to experiments has implications for the development of fusion. Computations should be used to develop a strategy that to the extent possible allows major issues to be circumvented. Required computations for this strategy are the subject of this paper.

physics.plasm-ph↗

arXiv2024

Current potential patches

A novel form of the current potential, a mathematical tool for the design of stellarators and stellarator coils, is developed. Specifically, these are current potentials with a finite-element-like basis, called \textit{current potential patches}. Current potential patches leverage the relationship between distributions of magnetic dipoles and current potentials to explore limits of the access properties of stellarator coil sets. An example calculation is shown using the Helically Symmetric Experiment (HSX) equilibrium, demonstrating the method's use in coil design and understanding the limits of the access properties of coil sets. Current potential patches have additional desirable properties such as of promoting sparse current sheet solutions and identifying crucial locations of shaping current placement. A result is found for the HSX equilibrium that shaping currents covering only 25\% of the winding surface is sufficient to produce the equilibrium to a good accuracy, provided a toroidal field is pre-supplied.

physics.plasm-ph↗

arXiv2024

Stellarators with enhanced tritium confinement and edge radiation control

A stellarator design is described with the purpose of achieving three goals: (1) Enhance the confinement time of tritium. (2) Have a sufficient density of high-Z impurities to radiate the thermal power escaping from the core while having an extremely low impurity density in the core. (3) Maintain a large fraction of the plasma in a burning plasma state with a 50/50 deuterium tritium ratio. Some features of this design could be used in tokamaks. Although having three confinement zones is natural for stellarators, it is not for tokamaks.

physics.plasm-ph↗

arXiv2023

Magnetic-field properties in non-axisymmetric divertors

Stellarator power plants require a plan for the removal of the particles and the heat that are exhausted across the plasma edge. Unless a flowing liquid metal can be used to carry the helium exhaust to places where it can be removed from the plasma chamber, the particle exhaust must be magnetically diverted into pumping chambers. Studies are required to determine how magnetic features relate to the required divertor properties, how these magnetic features can be produced, and how they can be controlled. General studies are clarified and simplified by the use of the magnetic field line Hamiltonian $\psi_p(\psi,\theta,\varphi)$ and a vector $\vec{x}(\psi,\theta,\varphi)$ that gives the point in space associated with each point in the $(\psi,\theta,\varphi)$ canonical coordinates, a flux and two angles. The non-resonant Fourier terms in $\psi_p$ can be removed by a canonical transformation, so only resonant Fourier terms can determine the field line properties in the plasma edge and divertor. This paper discusses the important divertor properties and explains how $\psi_p(\psi,\theta,\varphi)$ and $\vec{x}(\psi,\theta,\varphi)$ can be obtained numerically in a special form for any stellarator magnetic field, $\vec{B}(\vec{x})$. This form holds between an arbitrary magnetic surface and the chamber walls with the non-resonant terms eliminated. Studies based on variations in the terms in such derived field-line Hamiltonians can determine what magnetic features are mathematically possible and how they could be produced and controlled by the external magnetic field coils.

physics.plasm-ph↗

arXiv2022

Judgment of paradigms for magnetic reconnection in coronal loops

The traditional paradigm for magnetic field lines changing connections ignores magnetic field line chaos and requires an extremely large current density, $j_{max}\propto R_m$, flowing in thin sheets of thickness $1/R_m$, where $R_m$ is the magnetic Reynolds number. The time required for a general natural evolution to take a smooth magnetic field into such a state is rarely considered. Natural evolutions generally cause magnetic field lines to become chaotic. A fast change in field line connections then arises on the timescale defined by the evolution multiplied by a $\ln(R_m)$ factor, and the required maximum current density scales as $\ln(R_m)$. Even when simulations support the new paradigm based on chaos, they have been interpreted as supporting the old. How this could happen is an important example for plasma physics of Kuhn's statements about the acceptance of paradigm change and on Popper's views on the judgment of truth in science.

physics.plasm-ph↗

arXiv2022

Required toroidal confinement for fusion and omnigeneity

Deuterium-tritium (DT) burning requires a long energy confinement times compared to collision times, so the particle distribution functions must approximate local-Maxwellians. Non-equilibrium thermodynamics is applicable, which gives relations among transport, entropy production, the collision frequency, and the deviation from a Maxwellian. The distribution functions are given by the Fokker-Planck equation, which is an advection-diffusion equation. A large hyperbolic operator, the Vlasov operator with the particle trajectories as its characteristics, equals a small diffusive operator, the collision operator. The collisionless particle trajectories would be chaotic in stellarators without careful optimization. This would lead to rapid entropy production and transport -- far beyond what is consistent with a self-sustaining DT burn. Omnigeneity is the weakest general condition that is consistent with a sufficiently small entropy production associated with the thermal particle trajectories. Omnigeneity requires that the contours of constant magnetic field strength be unbounded in at least one of the two angular coordinates in magnetic surfaces and that there be a symmetry in the field-strength wells along the field lines. Even in omnigenous plasmas, fluctuations due to microturbulence can produce chaotic particle trajectories and the gyro-Bohm transport seen in many stellarator and tokamak experiments. The higher the plasma temperature above 10~keV, the smaller the transport must be compared to gyro-Bohm for a self-sustaining DT burn. The hot alphas of DT fusion heat the electrons. When the ion-electron equilibration time is long compared to the ion energy confinement time, a self-sustaining DT burn is not possible, which sets a limit on the electron temperature.

physics.plasm-ph↗

arXiv2022

Constraints on stellarator divertors from Hamiltonian mechanics

The design of any large stellarator requires a plan for the removal of the particles and heat that are exhausted across the plasma edge. This is called the divertor problem, for the particle exhaust must be diverted into pumping chambers. Although the physics of diverted plasmas has many subtleties, the magnetic field configuration between the plasma edge and the surrounding chamber walls is the foundation upon which divertor design is based. The properties of this magnetic configuration has both practical constraints and mathematical constraints from magnetic field lines obeying a 1~1/2 degree of freedom Hamiltonian. Constraints from plasma physics will also be discussed; they need to be integrated with the constraints from from Hamiltonian mechanics in conceptual designs of stellarator divertors.

physics.plasm-ph↗