Radial transport calculations for tandem mirrors

A simple one-dimensional (radial) model describing end loss and radial transport in a tandem mirror is developed. The model enables a self-consistent calculation of various ambipolar potential profiles and the center-cell density profile in the presence of either conducting or insulating end walls. Results of the computation reproduce many features seen in the Tandem Mirror Experiment (TMX). Signatures of radial transport are delineated, and the effect on confinement is studied over a range of parameters. Optimal and critical values in parameter space are also investigated.

A time-dependent version of the steady-state radial transport model in symmetric tandem mirrors in Ref. [1] has been coded up and first tests performed. Our code, named SYMTRAN, is an adaptation of the earlier SPHERE code for spheromaks, now modified for tandem mirror physics. Motivated by Post's new concept of kinetic stabilization of symmetric mirrors, it is an extension of the earlier TAMRAC rate-equation code omitting radial transport [2], which successfully accounted for experimental results in TMX. The SYMTRAN code differs from the earlier tandem mirror radial transport code TMT in that our code is focused on axisymmetric tandem mirrors and classical diffusion, whereas TMT emphasized non-ambipolar transport in TMX and MFTF-B due to yin-yang plugs and non-symmetric transitions between the plugs and axisymmetric center cell. Both codes exhibit interesting but different non-linear behavior.

The radial transport of tungsten ions in a fusion plasma of the HELIAS stellarator with five magnetic field periods is studied by means of a new numerical code. The code solves guiding center equations for test particles (tungsten ions) with the use of a Runge–Kutta integrating scheme. Coulomb scattering of the tungsten ions on the background plasma particles (electrons, deuterons and tritons) is simulated by means of a discretized collision operator based on the binomial distribution and presented in terms of pitch-angle scattering and energy slowing down and scattering. The coronal model is used to determine the mean charge state of the tungsten ion ensemble ⟨Z(Te, ne)⟩ as a function of background electron temperature and density. Two plasma configurations with and without ergodic confinement regions and both with finite plasma pressure of β = 3% are considered. The nonergodic configuration possesses closed nested magnetic surfaces throughout the entire confinement volume. The ergodic magnetic field configuration is represented through additional magnetic field perturbations. Comparative analysis of the radial transport is performed for a time interval greater by a factor of 15 than the energy confinement time τE = 1.62 s required for the HELIAS reactor. In spite of the fact that the tendency of impurities to penetrate towards the plasma core is observed in both cases, the stochastic scenario exhibits reduced inward impurity flux.

<p>The ion radiation belts just above the surface of the giant planets Jupiter and Saturn have recently been observed for the first time with Juno and Cassini. The relevant physical processes differ from Earth’s inner proton belt. Jupiter’s innermost ion belt consists of protons, oxygen, and sulfur ions. A comparison of Juno particle and plasma data with numerical modeling supports that these ions are occasionally transported from the magnetosphere across the main ring of Jupiter. It has been suggested earlier that this ring is populated through the stripping of energetic neutral atoms that are produced in the magnetosphere. This process is found to be too slow to populate the belt. After radial transport, the new ions lose energy in the tenuous ring halo inward of the main ring. This gives rise to an unusual spectral shape that rises from 100keV to 1MeV. Neutralization of the ions in the ring grains acts slower and eventually removes <100keV ions until the next transport across the ring.</p><p>Saturn’s innermost belt differs from Jupiter’s and Earth’s inner belts in the sense that Saturn’s rings are too dense and extended to allow radial transport of magnetospheric ions into the innermost belt. Saturn’s ion belts are therefore thought to be exclusively populated by cosmic ray tertiary particles from the CRAND process. While the source is different, the losses are similar as at Jupiter, namely interaction with the tenuous D-ring and the planetary exosphere. This interaction shows in the proton pitch angle distribution and has been used to constrain the scale height of Saturn’s exosphere that is difficult to do otherwise.</p>

Experimental radial ion transport rates and diffusion coefficients are presented for the Constance-B magnetic mirror [Phys. Rev. Lett. 58, 1853 (1987)]. The transport experiments are performed by measuring steady state equilibrium radial profiles of plasma density, ionization source, end loss current, electric field, electron temperature, and ion temperature. A charge coupled device (CCD) camera system [Rev. Sci. Instrum. 60, 2835 (1989)] is used to measure the two-dimensional radial density, source, and electron temperature profiles. End loss diagnostics including movable Faraday cups, electrostatic end loss analyzers, and an ion time-of-flight analyzer [Rev. Sci. Instrum. 59, 601 (1988)] are used to measure radial profiles of potential and ion temperature. The ion confinement time perpendicular to the magnetic field is found to be an order of magnitude shorter than predicted by classical and neoclassical transport theories. The radial profiles of the perpendicular diffusion coefficient (D⊥) are presented for hydrogen, helium, and argon plasmas. The coefficients are a factor of 10 larger than the maximum classical and neoclassical coefficients in all three plasmas. Plasma fluctuations resulting from whistler mode microinstability [Phys. Rev. Lett. 59, 1821 (1987)] as well as nonaxisymmetric potentials are suggested as possible explanations for the experimentally measured radial transport rate.

This paper is largely devoted to tandem mirror and field-reversed mirror experiments at the Lawrence Livermore Laboratory (LLL), and briefly summarizes results of experiments in which field-reversal has been achieved. In the tandem experiment, high-energy, high-density plasmas (nearly identical to 2XIIB plasmas) are located at each end of a solenoid where plasma ions are electrostatically confined by the high positive poentials arising in the end plug plasma. End plug ions are magnetically confined, and electrons are electrostatically confined by the overall positive potential of the system. The field-reversed mirror reactor consists of several small field-reversed mirror plasmas linked together for economic reasons. In the LLL Beta II experiment, generation of a field-reversed plasma ring will be investigated using a high-energy plasma gun with a transverse radial magnetic field. This plasma will be further heated and sustained by injection of intense, high-energy neutral beams.

The formation of plug potential is usually observed in a tandem mirror experiment without neutral beam injection in the plug region [Phys. Plasmas 2, 2321 (1995)]. The formation requires only the electron cyclotron resonance heating experimentally. In this Letter the plug potential is analytically shown to be created in a tandem mirror with passing ions, mirror trapped ions in the thermal barrier, passing electrons and φ-trapped electrons in a plug potential but without high energy sloshing ions.

Results from the Tandem Mirror Experiment-Upgrade (TMX-U) demonstrate that radial transport in the plasma volume can be affected by the boundary conditions at the end wall. Measurements in which end-wall plates, mapping to the plasma core, are switched from floating to grounded during plugging operation demonstrate that this floating increases the build-up rate of the central cell plasma (by increasing the radial confinement time), steepens the core density profile, and affects the plasma throughout the entire cross section.

A mechanical device for enhancing the halo density in the TMX-U tandem mirror

This paper presents a Hammir tandem mirror confinement performance analysis based on Realta Fusion’s first-of-a-kind model for axisymmetric magnetic mirror fusion performance. This model uses an integrated end plug simulation model including, heating, equilibrium and transport combined with a new formulation of the plasma operation contours (POPCONs) technique for the tandem mirror central cell. Using this model in concert with machine learning optimization techniques, it is shown that an end plug utilizing high temperature superconducting magnets and modern neutral beams enables a classical tandem mirror pilot plant producing a fusion gain Q > 5. The approach here represents an important advance in tandem mirror design. The high-fidelity end plug model enables calculations of heating and transport in the highly non-Maxwellian end plug to be made more accurately. The detailed end plug modelling performed in this work has highlighted the importance of classical radial transport and neutral beam absorption efficiency on end plug viability. The central cell POPCON technique allows consideration of a wide range of parameters in the relatively simple near-Maxwellian central cell, facilitating the selection of more optimal central cell plasmas. These advances make it possible to find more conservative classical tandem mirror fusion pilot plant operating points with lower temperatures, neutral beam energies and end plug performance requirements than designs in the literature. Despite being more conservative, it is shown that these operating points have sufficient confinement performance to serve as the basis of a viable fusion pilot plant provided that they can be stabilized against magnetohydrodynamic and trapped particle modes.

The path to practical fusion power through plasma confinement in magnetic fields, if it is solely based on the present front-runner, the tokamak, is clearly long, expensive, and arduous. The root causes for this situation lie in the effects of endemic plasma turbulence and in the complexity the tokamak's ''closed'' field geometry. The studies carried out in the investigations described in the attached reports are aimed at finding an approach that does not suffer from these problems. This goal is to be achieved by employing an axisymmetric ''open'' magnetic field geometry, i.e. one generated by a linear array of circular magnet coils, and employing the magnetic mirror effect in accomplishing the plugging of end leakage. More specifically, the studies were aimed at utilizing the tandem-mirror concept in an axisymmetric configuration to achieve performance superior to the tokamak, and in a far simpler system, one for which the cost and development time could be much lower than that for the tokamak, as exemplified by ITER and its follow-ons. An important stimulus for investigating axisymmetric versions of the tandem mirror is the fact that, beginning from early days in fusion research there have been examples of axisymmetric mirror experiments where the plasma exhibited crossfield transport far below the turbulence-enhanced rates characteristic of tokamaks, in specific cases approaching the ''classical'' rate. From the standpoint of theory, axisymmetric mirror-based systems have special characteristics that help explain the low levels of turbulence that have been observed. Among these are the facts that there are no parallel currents in the equilibrium state, and that the drift surfaces of all of the trapped particles are closed surfaces, as shown early on by Teller and Northrop. In addition, in such systems it is possible to arrange that the radial boundary of the confined plasma terminates without contact with the chamber wall. This possibility reduces the probability of so-called ''temperature-gradient'' instabilities, known to be endemic to closed systems. Finally, the open-ended nature of the field readily allows the control of the radial potential distribution, a circumstance that has been shown, for example in the Gamma 10 tandem-mirror experiment at Tsukuba Japan, to suppress drift-type instability modes. Standing against all of these attractive properties of axisymmetric mirror-based systems is the fact, shown early on, that such systems are prone to MHD ''interchange'' instabilities, one in which the plasma column drifts transversely, at a rate far above classical transport. Observed early on, the ''cure'' that was universally adopted, as first demonstrated in the famous ''Ioffe experiment'', was to abandon axisymmetry and employ so-called ''magnetic-well'' fields, ones in which the field increases radially and axially from its interior, strongly suppressing the MHD interchange mode, up to plasma ''beta'' values approaching unity, observed in the 2X2B experiment. When the tandem mirror concept was introduced in 1976 every experiment that was constructed employed various combinations of non-axisymmetric coil configurations (''Baseball,'' and ''Yin-Yang'' coils) to create the magnetic fields. But it came at a heavy price: non-axisymmetric fields gave rise to new non-classical loss channels, and the complexity of the fields introduced difficult engineering problems. It was well recognized at the time that it would be highly advantageous to preserve axisymmetry of the tandem mirror coils, but there was no apparent way to stabilize the ubiquitous MHD interchange mode. A decade later a way to accomplish this end was analyzed theoretically, and, a few years later successfully demonstrated experimentally, in the Gas Dynamic Trap (GDT) experiment at Novosibirsk. The concept: the presence of a sufficient amount of plasma on the expanding field lines outside the end mirrors of a mirror machine can act as an ''anchor,'' MHD stabilizing the interior, confined, plasma. Moreover, Ryutov's theory showed that the pressure of this anchor plasma could be orders of magnitude smaller than that of the confined plasma, and still be able to stabilize it. In the GDT, which operates in a collision-dominated region (as opposed to the near-collisionless mode of a tandem mirror), the effluent plasma, though much lower in density than that of the confined plasma, is sufficient to stabilize the central plasma, up to plasma beta values of 40 percent. Furthermore, once MHD stabilized, the confined plasma in the GDT exhibited no signs of plasma turbulence or enhanced cross-field transport, even in the presence of a substantial population of high energy ions produced by neutral-beam injection.

Ion-cyclotron fluctuations in the tandem mirror experiment have wave characteristics consistent with the Alfv\'en ion-cyclotron instability. The oscillations are coherent, have small azimuthal mode numbers, and have frequencies as low as 12% below the diamagnetically depressed, minimum ion-cyclotron frequency in the end cell. Polarization is in the direction of ion gyration, and azimuthal propagation is in either the electron or the ion diamagnetic direction, with the former prevailing.

In the Tandem Mirror Experiment (TMX), the central-cell losses provide the warm unconfined plasma necessary to stabilize the drift-cyclotron loss-cone instability in the end cells. This places a theoretical limit on central-cell confinement, which is expressed as a limit on the end-cell to central-cell density ratio. As this density ratio increases in a TMX experiment, large increases of end-cell ion-cyclotron-frequency plasma fluctuations are observed. These fluctuations cause the central-cell confinement to decrease, in agreement with a theoretical model.

We describe a field reversed configuration (FRC) experiment featuring in-vessel θ-pinch coils and open-field-line plasmas confined in a tandem mirror. Two FRCs, formed near the west and the east mirror throats of a central cell, are ejected toward the mid-plane for colliding and merging. Each FRC consists of four groups of pulsed power supplies and four groups of coils, having diameters 35, 35, 40, and 45 cm. The rise time of the main reversal field is 7.15 μs, and the maximum voltage is 40 kV with total currents of 416 kA, corresponding to a magnetic field of 1690 G. The total capacitive stored energy is 115.2 kJ. A fast pulse gas injection system was designed and tested to inject neutral gas into the FRC formation region with controlled directions. The successful installation of the θ-pinch coils inside the vacuum vessel offers greater freedom for diagnostics and control instruments as well as preserving magnetic tandem mirror configuration. The magnetic field reversal is confirmed by internal magnetic field measurements. The plasma temperature, density, and lifetime are, respectively, ∼100 eV, ∼3.0 × 1018 m-3, and ∼300 μs for the current operating conditions.

Loss regions in the plasma particles' velocity space are decreased by the electrostatic potentials created on both sides of the plasma, which results in the improvement of the axial confinement in the tandem mirror. We found experimentally that the ion loss region was in contact mainly with both a plug potential bounce region and the outer mirror throat bounce region in which ions were bounced near the outer mirror throat of the plug/barrier cell. These bounce ions play an important role in tandem mirror confinement. The outer mirror throat bounce region is caused by the relatively high potential in the neighborhood of the outer mirror throat. The core plasma's radial potential profile was controlled by changing the potential of coaxially separated end plates, and it was found that control of the radial potential profile was useful for retardation of the radial loss of the bounce ions. We observed a hump structure on the energy distribution function of the end-loss ions, and found that the enhancement of the end-loss ions was caused by ion flow from the trapped region to the loss region due to Alfvn Ion Cyclotron (AIC) fluctuations. Although the ion diffusion due to the fluctuations enhances the axial ion loss, the axial loss can be reduced by creation of a higher confining potential.