Search

Plasma turbulence can enhance the diffusion of both magnetic fields and plasmas. But, any theory must be consistent with their evolution equations having the mathematical form of advection-diffusion equations. Advection-diffusion equations have a remarkable feature. When the diffusion is extremely weak but non-zero, its magnitude makes only a small difference even when varied over sixteen orders of magnitude. This is true when the advective velocity is chaotic -- the exponential separation of neighboring streamlines -- as natural flows generally are. Even highly turbulent flows have in addition to eddies large-scale coherent flows, such as the Gulf Stream in the Atlantic, which dominates their transport effects on longest scales. Basic physics and easily understood mathematics place important constraints, which are explained.

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.

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.

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.

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.

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.

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.

The mathematics and physics of each of the three aspects of magnetic field evolution -- topology, energy, and helicity -- is remarkably simple and clear. When the resistivity $\eta$ is small compared to an imposed evolution, $a/v$, timescale, which means $R_m\equiv\mu_0va/\eta>>1$, magnetic field line chaos dominates the evolution of field-line topology in three-dimensional systems. Chaos has no direct role in the dissipation of energy. A large current density, $j_\eta\equiv vB/\eta$, is required for energy dissipation to be on a comparable time scale to the topological evolution. Nevertheless, chaos plus Alfv\'en wave damping explain why both timescales tend to be approximately an order of magnitude longer than the evolution timescale $a/v$. Magnetic helicity is injected onto tubes of field lines when boundary flows have vorticity. Chaos can spread but not destroy magnetic helicity. Resistivity has a negligible effect on helicity accumulation when $R_m>>1$. Helicity accumulates within a tube of field lines until the tube erupts and moves far from its original location., Comment: arXiv admin note: text overlap with arXiv:2009.08779 by other authors

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.

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.

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.

Fast magnetic reconnection is defined by the topology of the magnetic field lines changing on a timescale that is approximately an order of magnitude longer than the topology-conserving ideal-evolution timescale. Fast reconnection is an intrinsic property of Faraday's law when the evolving magnetic field depends non-trivially on all three spatial coordinates and is commonly observed -- even when the effects that allow topology breaking are arbitrarily small. The associated current density need only be enhanced by a factor of approximately ten and flows in thin but broad ribbons along the magnetic field. These results follow from the variation in the separation of neighboring pairs of magnetic field lines, which in an ideal evolution typically increases exponentially with time, and the existence of a spatial scale below which magnetic field lines freely change their identities due to non-ideal effects such as resistivity. Traditional reconnection theory ignores exponentially large variations and relies on the current density reaching a magnitude that is exponentially larger than is actually required. Here, an analysis of the behavior of magnetic field lines in the neighborhood of an arbitrarily chosen line is used to obtain more precise and rigorous results on intrinsic reconnection. The maximum parallel kinetic energy of collisionless charged particles is shown to have an exponential increase in time during a generic magnetic evolution.

A new type of stellarator divertor is found. It has features of both a nonresonant divertor (A. Punjabi and A. H. Boozer, Phys. Plasmas 27, 012503 (2020)) as well as a resonant divertor. It has the outermost confining surface with sharp edges and large islands outside the outermost surface. For this reason, we have called it hybrid divertor. This divertor can be configured by adjusting the currents in external coils which produce nonresonant perturbations. We have simulated this divertor using the method developed in (A. H. Boozer and A. Punjabi, Phys. Plasmas 25, 092520 (2018)). The simulation shows that the footprints have fixed locations on the wall and are stellarator symmetric. The magnetic field lines leave and enter the outermost surface through three magnetic turnstiles. The probability exponents of the three turnstiles are 2.1, 2.25, and 4.3. The hybrid divertor confines larger plasma volume, has higher average shear, larger footprints, lower average density of strike points, lower maximum density of strike points, and longer loss-times than the nonresonant stellarator divertor. The hybrid divertor is robust against small changes in the rotational transform and large changes in the shape parameter that controls the sharp edges on the outermost confining surface.

An efficient numerical method of studying nonresonant stellarator divertors was introduced in Boozer and Punjabi [Phys. Plasmas 25, 092505 (2018)]. This method is used in this paper to study a different magnetic field model of a nonresonant divertor. The most novel and interesting finding of this study is that diffusive magnetic field lines can be distinguished from lines that exit through the primary and the secondary turnstile, and that below some diffusive velocity, all lines exit through only the primary turnstile. The footprints of each family are stellarator symmetric and have a fixed location on the wall for all velocities. The probability exponent of the primary turnstile is d1 = 9/4 and that of the secondary turnstile is d2 = 3/2. This study also addresses the issues of an inadequate separation of the chamber walls from the outermost confining magnetic surface and a marginal step size of the numerical integrations that could compromise the interpretation of the earlier results [Boozer and Punjabi, Phys. Plasmas 25, 092505 (2018)]. The previous value of d1 = 2 is within the error bar of d1 = 9/4 estimated here.

Much emotion is expended on the dangers of carbon dioxide, but solutions require reason and recognition of facts: (1) The cost of developing options is approximately a thousand times less than their deployment. (2) Timescales involve two questions: (a) How quickly can an option be demonstrated? (b) How quickly can the required equivalent of thousands of units be built. Two questions are implied: (1) What options would most fundamentally change the carbon-dioxide problem? (2) For each option, how could it be demonstrated most quickly? An option of fundamental importance is direct air capture of carbon dioxide. The option that appears most attractive for carbon-free energy production is the stellarator fusion concept, which is poised for a rapid demonstration., Comment: To be published "Stellarator News"

Non-resonant stellarator divertors have magnetic flux tubes, called magnetic turnstiles, that cross cantori, which are fractal remnants of destroyed invariant tori with holes, that lie outside the outermost confining surface. The exiting and entering flux tubes can be adjacent as is generally expected but can also have the unexpected feature of entering or exiting at separate locations of the cantori. Not only can there be two types of turnstiles, but pseudo turnstiles can also exist. A pseudo turnstile is formed when a cantorus has a sufficiently large, although limited, radial excursion to strike a surrounding chamber wall. The existence of non-adjacent and adjacent turnstiles and pseudo turnstiles resolves issues that arose in earlier simulations of nonresonant stellarator divertors [A. Punjabi and A. H. Boozer, Phys. Plasmas 27, 012503 (2020)]., Comment: 15 pages, 7 figures

The operation of ITER will require reliable simulations in order to avoid major damage to the device from disruptions. Disruptions are the sudden breakup of magnetic surfaces across the plasma volume -- a fast magnetic reconnection. This reconnection can be caused by the growth of perturbations outside of the plasma core causing an ideal perturbation to the core. This causes an increasing ratio of the maximum to the minimum separation, $\Delta_{max}/\Delta_{min}$, between neighboring magnetic surfaces. Magnetic reconnection becomes a dominant process when magnetic field lines can quickly interchange connections over a spatial scale $a_r$. This occurs when $\Delta_{max}/\Delta_{min}\gtrsim a_r/\Delta_d$, where $\Delta_d$ is the scale over which non-ideal effects make magnetic field lines indistinguishable. Traditional reconnection theory is fundamentally different. It is a study of the steady-state cancellation of oppositely directed magnetic field components across a thin layer. During more than sixty years, mathematical implications of Faraday's Law have been derived that clarify and constrain the physics of fast magnetic reconnection. These are reviewed because they are not commonly known but are needed to understand and to place in context how an ideal magnetic evolution can cause reconnection to quickly become a dominant process no matter how small $\Delta_d/a_r$ may be.

This paper is focused on three points: (1) Overcoming obstacles to tokamak power plants may require a configuration modification as large as that of a stellarator. (2) The demonstrated reliability of the computational design of stellarators should change fusion strategy. (3) Deployment of carbon-free energy sources is mandated by the thirty-year doubling of carbon dioxide emissions. Carbon-free energy options must be developed and fully deployed within a few doubling times. Unit size and cost of electricity are only relevant in comparison to alternative worldwide energy solutions. Intermittency, site specificity, waste management, and nuclear proliferation make fusion attractive as the basis for a carbon-free energy system compared to the alternatives. Nonetheless, fusion is not an option for deployment until a power plant has successfully operated. A critical element in a minimal time and risk program is the use of computational design as opposed to just extrapolation. Only the stellarator has an empirical demonstration of the reliable computational design through large changes in configuration properties and scale. The computational design of stellarators should proceed while the inventions necessary for a more tokamak-like power plant are sought. The cost of computational design is extremely small, but adequate time is required for the development of ideas that maximize attractiveness and minimize risk. Rapid power-plant construction without many intermediate steps may seem risky, but the price is small compared to the cost of trillions of dollars for each year's delay in addressing carbon-dioxide emissions.

The fraction of the tritium that is burned during one pass through a DT fusion system, $f_{tb}$, is a central issue for success of fusion energy. Reducing the tritium fraction, $f_t$, in a $DT$ burning plasma below a half increases the burn fraction, $f_{tb}\propto1/f_t$ but also the required confinement to achieve a burn $nT\tau_E\propto1/f_t(1-f_t)$. A doubling of the fractional burn entails only a 4/3 enhancement of the required $nT\tau_E$. The energy confinement time $\tau_E$ in tokamaks and stellarators is empirically gyro-Bohm with an approximate factor of two between the best and worse results used to construct scaling laws. Gyro-Bohm is also the approximate level of transport needed in a power plant. What has received little study are $\tau_t/\tau_E$, the ratio of the tritium to the energy confinement time, and $\tau_\alpha/\tau_E$, the ratio of the alpha particle to the energy confinement time. The tritium burn fraction is proportional to $\tau_t/\tau_E$, so the larger the better. The contamination of the plasma by helium ash is proportional to $\tau_\alpha/\tau_E$, so the smaller the better., Comment: This is a short note on a critical issue in DT fusion energy, the tritium burn-fraction

Steering tokamak plasmas is commonly viewed as a way to avoid disruptions and runaway electrons. Plasma steering sounds as safe as driving to work but will be shown to more closely resemble driving at high speed through a dense fog on an icy road. The long time required to terminate an ITER discharge compared to time over which dangers can be foreseen is analogous to driving in a dense fog. The difficulty of regaining plasma control if it is lost resembles driving on an icy road. Disruptions and runaways are associated with three issues -- a solution to one tends to complicate the solution to the other two: loss of plasma position control, excessive heat deposition, and wall melting due to runaway electrons. All three risks must be addressed for ITER to achieve its mission and essentially eliminated before tokamak power plants can be deployed., Comment: Accepted by the journal Nuclear Fusion

In laboratory and natural plasmas of practical interest, the spatial scale $\Delta_d$ at which magnetic field lines lose distinguishability differs enormously from the scale $a$ of magnetic reconnection across the field lines. In the solar corona, plasma resistivity gives $a/\Delta_d\sim10^{12}$, which is the magnetic Reynold number $R_m$. The traditional resolution of the paradox of disparate scales is for the current density $j$ associated with the reconnecting field $B_{rec}$ to be concentrated by a factor of $R_m$ by the ideal evolution, so $ j\sim B_{rec}/\mu_0\Delta_d$. A second resolution is for the ideal evolution to increase the ratio of the maximum to minimum separation between pairs of arbitrarily chosen magnetic field lines, $\Delta_{max}/\Delta_{min}$, when calculated at various points in time. Reconnection becomes inevitable where $\Delta_{max}/\Delta_{min}\sim R_m$. A simple model of the solar corona will be used for a numerical illustration that the natural rate of increase in time is linear for the current density but exponential for $\Delta_{max}/\Delta_{min}$. Reconnection occurs on a time scale and with a current density enhanced by only $\ln(a/\Delta_d)$ from the ideal evolution time and from the current density $B_{rec}/\mu_0a$. In both resolutions, once a sufficiently wide region, $\Delta_r$, has undergone reconnection, the magnetic field loses static force balance and evolves on an Alfv\'enic time scale. The Alfv\'enic evolution is intrinsically ideal but expands the region in which $\Delta_{max}/\Delta_{min}$ is large.

Mitigation of the multiple risks associated with disruptions and runaway electrons in tokamaks involves competing demands. Success requires that each risk be understood sufficiently that appropriate compromises can be made. Here the focus is on the interaction of short timescale magnetic-perturbations with the structure in ITER that is closest to the plasma, blanket modules covered by separated beryllium tiles. The effect of this tiled surface on the perturbations and on the forces on structures are subtle. Indeterminacy can be introduced by tile-to-tile shorting. A determinate subtlety is introduced because electrically separated tiles can act as a conducting surface for magnetic perturbations that have a normal component to the surface. A practical method for including this determinate subtlety into plasma simulations is developed. The shorter the timescales and the greater the localization, particularly in the toroidal direction, the more important the magnetic effects of the tiles become.

The prominence of nulls in reconnection theory is due to the expected singular current density and the indeterminacy of field-lines at a magnetic null. Electron inertia changes the implications of both features. Magnetic field lines are distinguishable only when their distance of closest approach exceeds a distance $\Delta_d$. Electron inertia ensures $\Delta_d\gtrsim c/\omega_{pe}$. The lines that lie within a magnetic flux tube of radius $\Delta_d$ at the place where the field strength $B$ is strongest are fundamentally indistinguishable. If the tube, somewhere along its length, encloses a point where $B=0$,vanishes, then distinguishable lines come no closer to the null than $\approx (a^2c/\omega_{pe})^{1/3}$, where $a$ is a characteristic spatial scale of the magnetic field. The behavior of the magnetic field lines in the presence of nulls is studied for a dipole embedded in a spatially constant magnetic field. In addition to the implications of distinguishability, a constraint on the current density at a null is obtained, and the time required for thin current sheets to arise is derived.

During tokamak disruptions the profile of the net parallel current is observed to flatten on a time scale that is so fast that it must be due to a fast magnetic reconnection. After a fast magnetic reconnection has broken magnetic surfaces, a single magnetic field line covers an entire volume and not just a magnetic surface. The current profile, given by $K\equiv\mu_0j_{||}/B$, relaxes to a constant within that volume by Alfv\'en waves propagating along the chaotic magnetic field lines. The time scale for this relaxation determines the commonly observed disruption phenomena of a current spike and a sudden drop in the plasma internal inductance. An efficient method for studying this relaxation is derived, which allows a better understanding of the information encoded in the current spike and the associated sudden drop in the plasma internal inductance. Implications for coronal heating are also discussed., Comment: 4 pages, no figures

Although magnetic reconnection takes place in three-dimensional space, reconnection theory has focused on two-dimensional models for more than sixty years. Well-posed three-dimensional mathematics associated with the theory of fluid mixing provides a predictive and compelling explanation for why fast magnetic reconnection is prevalent---exponentially large variations in the separations between magnetic field lines. The proofs have been simplified to remove any rational reason to maintain a focus on two dimensional models, which fail to represent the mathematical properties of three-dimensional space., Comment: arXiv admin note: text overlap with arXiv:1407.6069

The increasing level of atmospheric carbon dioxide has driven public discourse throughout the world. An immediate implementation of carbon-free energy sources is demanded with little discussion of costs, technical constraints on the sources, or implications of high residual levels of carbon dioxide. Residual carbon-dioxide can be removed from the air, but the cost to remove the carbon-dioxide produced by human activity during a year is thought to be trillions of dollars---otherwise it remains in the atmosphere for centuries. Economic considerations may limit wind and solar sources to less than 40\% of the electricity production. Fission or fusion may be the only choice for most of the rest. Development costs are orders of magnitude smaller than implementation costs, which are tens of trillions of dollars for fission. A needless delay in the development of fusion has enormous financial implications. As will be shown stellarators are better positioned than any other concept for a fast path to fusion. A computationally derived conceptual design for a stellarator reactor may allow final design and construction to be initiated without the delay of intermediate generations of experiments. The most urgent issue is the development of conceptual designs., Comment: 18 pages, 3 figures

Null and X-points are not themselves directly important to magnetic reconnection because distinguishable field lines do not approach them closely. Even in a collision-free plasma, magnetic field lines that approach each other on a scale $c/\omega_{pe}$ become indistinguishable during an evolution. What is important is the different regions of space that can be explored by magnetic field lines that pass in the vicinity of null and X-points. Traditional reconnection theories made the assumption that the reconnected magnetic flux must be dissipated or diffused by an electric field. This assumption is false in three dimensional systems because an ideal evolution can cause magnetic field lines that cover a large volume to approach each other within the indistinguishability scale $c/\omega_{pe}$. When the electron collision time $\tau_{ei}$ is short compared to the evolution time of the magnetic field $\tau_{ev}$, the importance of $c/\omega_{pe}$ is replaced by the resistive time scale $\tau_\eta=(\eta/\mu_0)L^2$ with $L$ the system scale. The magnetic Reynolds number is $R_m\equiv\tau_\eta/\tau_{ev}$ is enormous in many reconnection problems of interest. Magnetic flux diffusion implies the current density required for reconnection to compete with evolution scales as $R_m$ while flux mixing implies the required current density to compete scales as $\ln R_m$.

This paper describes a new and efficient method of defining an annular region of a curl-free magnetic field with specific physics and coil properties that can be used in stellarator design. Three statements define the importance: (1) Codes can follow an optimized curl-free initial state to a final full-pressure equilibrium. The large size of the optimization space of stellarators, approximately fifty externally-produced distributions of magnetic field, makes success in finding a global optimum largely determined by the starting point. (2) The design of a stellarator is actually improved when the central region of the plasma has rapid transport with the confinement provided by a surrounding annulus of magnetic surfaces with low transport. (3) The stellarator is unique among all fusion concepts, inertial as well as magnetic, in not using the plasma itself to provide an essential part of its confinement concept. This permits reliable computational design, which opens a path to faster, cheaper, and more certain achievement of fusion energy.

Plasmoid theory uses two-coordinate models to explain fast magnetic reconnection, which occurs on an Alfv\'enic, not a resistive time scale, in plasmas that are evolving in the near-ideal limit. The primary application has been to three-dimensional naturally-occurring plasmas. A magnetic field representation that applies to all magnetic fields allows features of magnetic evolution in toroidal and natural plasmas to be compared. This comparison illustrates why plasmoid models of magnetic field evolution are not robust solutions to Maxwell's equations. Despite five-hundred papers on plasmoid models, they provide a problematic description of fast magnetic reconnection.

Mathematics demonstrates that an ideally evolving magnetic field has an exponentially increasing sensitivity to non-ideal effects for all but truly exceptional evolutions. On a time scale that depends only logarithmically on the magnitude of non-ideality, an evolving magnetic field will generally reach a state of fast magnetic reconnection. The effects of fast magnetic reconnection proceed at a rate determined by Alfv\'enic, not resistive, physics. The best known of these effects are associated with the transfer of magnetic field energy to the plasma and the conservation of magnetic helicity, which limits the energy transfer. As will be shown, weak non-ideality implies helicity conservation in regions bounded by magnetic surfaces or rigid perfect conductors. But, weak non-ideality makes the rapid transfer of energy subtle. This transfer can be understood by the drive for Alfv\'en waves and two other effects, which are present even in an ideal evolution, an effective parallel electric field $\mathcal{E}_{||}$, which can accelerate particles despite the particle acceleration due to the true parallel electric field $E_{||}$ being negligible, and a coefficient $\nu_K$, which gives a rate for exponentiation of the kinetic energy of particle motion along the magnetic field. Classical theories of reconnection are two-dimensional and exclude the exponentially increasing sensitivity that is robustly present for magnetic fields in three-dimensional space.

Regardless of how small non-ideal effects may be, phenomena associated with changes in magnetic field line connections are frequently observed to occur on an Alfv\'enic time scale. Since it is mathematically impossible for magnetic field line connections to change when non-ideal effects are identically zero, an ideal evolution must naturally lead to states of unbounded sensitivity to non-ideal effects. That such an evolution is natural is demonstrated using Lagrangian coordinates based on the flow velocity of the magnetic field lines. The Lagrangian representation of an evolving magnetic field is highly constrained when neither the magnetic field strength nor the forces exerted by the magnetic field increase exponentially with time. The development of a state of fast reconnection consistent with these constraints (1) requires a three-dimensional evolution, (2) has an exponentially increasing sensitivity to non-ideal effects, and (3) has a parallel current density, which lies in exponentially thinning but exponentially widening ribbons, with a magnitude that is limited to a slow growth. The implication is that exponential growth in sensitivity is the cause of fast magnetic reconnection when non-ideal effects are sufficiently small. The growth of the non-ideal effect of the resistivity multiplied by the parallel current density is far too slow to be competitive., Comment: 13 pages, no figures

The adjoint Fokker-Planck equation method is applied to study the runaway probability function and the expected slowing-down time for highly relativistic runaway electrons, including the loss of energy due to synchrotron radiation. In direct correspondence to Monte Carlo simulation methods, the runaway probability function has a smooth transition across the runaway separatrix, which can be attributed to effect of the pitch angle scattering term in the kinetic equation. However, for the same numerical accuracy, the adjoint method is more efficient than the Monte Carlo method. The expected slowing-down time gives a novel method to estimate the runaway current decay time in experiments. A new result from this work is that the decay rate of high energyelectrons is very slow when E is close to the critical electric field. This effect contributes further to a hysteresis previously found in the runaway electron population.

Magnetic fields without a direction of continuous symmetry have the generic feature that neighboring field lines exponentiate away from each other and become stochastic, hence the ideal constraint of preserving magnetic field line connectivity becomes exponentially sensitive to small deviations from ideal Ohm's law. The idea of breaking field line connectivity by stochasticity as a mechanism for fast reconnection is tested with numerical simulations based on reduced magnetohydrodynamics equations with a strong guide field line-tied to two perfectly conducting end plates. Starting from an ideally stable force-free equilibrium, the system is allowed to undergo resistive relaxation. Two distinct phases are found in the process of resistive relaxation. During the quasi-static phase, rapid change of field line connectivity and strong induced flow are found in regions of high field line exponentiation. However, although the field line connectivity of individual field lines can change rapidly, the overall pattern of field line mapping appears to deform gradually. From this perspective, field line exponentiation appears to cause enhanced diffusion rather than reconnection. In some cases, resistive quasi-static evolution can cause the ideally stable initial equilibrium to cross a stability threshold, leading to formation of intense current filaments and rapid change of field line mapping into a qualitatively different pattern. It is in this onset phase that the change of field line connectivity is more appropriately designated as magnetic reconnection. Our results show that rapid change of field line connectivity appears to be a necessary, but not a sufficient condition for fast reconnection., Comment: Accepted for publication in Astrophysical Journal

The kinematic regime of the magnetic dynamo neglects the backreaction of the magnetic field on the flow. For small magnetic diffusivity, in the early stage of evolution, there is an ideal phase where dissipative effects can also be neglected. We estimate the magnitude of the energy dissipation term (Ohmic heating), taking into account differential constraints on chaotic flows. We find that the period of ideal evolution is roughly doubled over an estimate without constraints. The helicity generation terms are exponentially smaller than the energy dissipation, so that large quantities of energy are dissipated before any helicity can be created. Helicity flow is exponentially larger than net helicity generation. The constraints also lead to the existence of a singular initial condition for the magnetic field for which sizable amounts of helicity can potentially be created., Comment: 12 pages, ReVTeX 4. Submitted to Physics of Plasmas

Constraints are found on the spatial variation of finite-time Lyapunov exponents of two and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection-diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current., Comment: 34 pages, 15 figures. Uses RevTeX, epsfig, ams packages, and psfrag. Published version

The advection-diffusion equation is studied via a global Lagrangian coordinate transformation. The metric tensor of the Lagrangian coordinates couples the dynamical system theory rigorously into the solution of this class of partial differential equations. If the flow has chaotic streamlines, the diffusion will dominate the solution at a critical time, which scales logarithmically with the diffusivity. The subsequent rapid diffusive relaxation is completed on the order of a few Lyapunov times, and it becomes more anisotropic the smaller the diffusivity. The local Lyapunov time of the flow is the inverse of the finite time Lyapunov exponent. A finite time Lyapunov exponent can be expressed in terms of two convergence functions which are responsible for the spatio-temporal complexity of both the advective and diffusive transports. This complexity gives a new class of diffusion barrier in the chaotic region and a fractal-like behavior in both space and time. In an integrable flow with shear, there also exist fast and slow diffusion. But unlike that in a chaotic flow, a large gradient of the scalar field across the KAM surfaces can be maintained since the fast diffusion in an integrable flow is strictly confined within the KAM surfaces., Comment: 21 pages, 8 figures

We consider the design criteria of a chemical mixing device based on a chaotic flow, with an emphasis on the steady-state devices. The merit of a reactor, defined as the $Q$-factor, is related to the physical dimension of the device and the molecular diffusivity of the reactants through the local Lyapunov exponents of the flow. The local Lyapunov exponent can be calculated for any given flow field and it can also be measured in experimental situations. Easy-to-compute formulae are provided to estimate the $Q$-factor given either the exact spatial dependence of the local Lyapunov exponent or its probability distribution function. The requirements for optimization are made precise in the context of local Lyapunov exponents.

From a kinematical point of view, the geometrical information of hamiltonian chaos is given by the (un)stable directions, while the dynamical information is given by the Lyapunov exponents. The finite time Lyapunov exponents are of particular importance in physics. The spatial variations of the finite time Lyapunov exponent and its associated (un)stable direction are related. Both of them are found to be determined by a new hamiltonian of same number of degrees of freedom as the original one. This new hamiltonian defines a flow field with characteristically chaotic trajectories. The direction and the magnitude of the phase flow field give the (un)stable direction and the finite time Lyapunov exponent of the original hamiltonian. Our analysis was based on a $1{1\over 2}$ degree of freedom hamiltonian system.