Issues for FRCs

This page includes 3 issues for FRC's to be presented at the Snowmass conference by Loren Steinhauer: Transport, MHD and Boundary Physics.

Features of FRCs with respect to transport

FRCs have much in common with other magnetic fusion concepts, but several features are somewhat unique with regard to transport. (1) High-b . Consequently the turbulent fluctuations should be strongly electromagnetic. (2) Deep magnetic wells: field nulls at the O-point and the X-points; and deep magnetic wells (mirror ratio ~ 2) in the end regions of typically elongated FRCs. This causes a significant population of trapped particles. (3) Significantly sheared flows (probably). This may strongly suppress turbulence. (4) Possibility of low temperature gradients. It may be possible to thermally isolate SOL from the collection plates, which can be located remotely (natural divertor).

Possible "universal principle" of turbulence and transport

Despite the dynamical complexity of a two-fluid plasma, its overall behavior may be regulated by the presence of global invariants, namely the self-helicities for both electron and ion species. (These are generalizations of the magnetic helicity in which the vector potential A is replaced by the canonical momentum based on the fluid velocity.)

The self-helicities appear to be ruggedly invariant in a two-fluid on the same basis that the magnetic helicity in MHD ("one-fluid"). The existence of such invariants would have a major organizing effect on the transport.

Issues. Although the ruggedness of the self helicities has been defended on the basis of several arguments, it must be placed on a firmer footing by more advanced computations or experimental observation.

Resolution. (1) The ruggedness of the self helicities can be tested by a nonlinear two-fluid (or at least Hall-MHD) simulation with (a) resistive and viscous friction; and (b) small enough grid spacing to allow resolution of scales somewhat smaller than the ion collisionless skin depth. (2) The preservation of the self helicities in experiments needs to be verified by internal magnetic field and flow measurements.

Spontaneous flow shear

The role of flow shear in reducing transport is widely accepted. Flow shear is an essential feature of finite-b minimum energy states (MES) in a two-fluid. MES have minimum "organized" energy (fluid and magnetic) subject to constraints on the self helicities. Having no large toroidal field to suppress fast instabilities, FRCs may relax very rapidly to a MES, thereby spontaneously generating flow shear.

Issues. (1) Does strongly flow shear arise spontaneously in FRC experiments? (2) If so, why so in FRCs but (possibly) not in other configurations? (3) Is there a generalization of q including both magnetic fields and flow shear that affects transport?

Resolution. Two-fluid (or Hall-MHD) simulations will answer some of these questions. A more mature understanding of basic two-fluid stability is needed to consider a generalization of q.

"Metabolic" coupling of relaxation and transport.

If relaxation to a MES occurs, as it may over most of an FRC, then relaxation will couple closely with dissipative processes to produce a "metabolic" evolution of the configuration. Then the evolution results from a competition between dissipation (resistive, viscous), which evolves the plasma away from an MES, and fast instabilities that restore it. Although the dissipative processes themselves are local, the consequence is more-or-less nonlocal since the relaxation globalizes the effect of local transport. Such a metabolic process would give a physics basis for empirical "profile consistencies."

Issues. (1) When is the movement toward a lower energy state dominated by a fast, more-or-less continuous "metabolism" (e.g. ballooning), and when is it dominated by large scale intermittent events (low-order internal kinks)? (2) Does one equilibrium type tend to evolve toward another (e.g. will an FRC evolve slowly toward a spheromak) or are the equilibrium types "stable?" (3) Is a bifurcation between FRC and spheromak observed on TS-3 (Tokyo Un.) the consequence of metabolic evolution.

Resolution. Two-fluid (or Hall-MHD) simulations are required to resolve the question of which path the relaxations will take. Experimental measurements of density and magnetic field profiles are needed to verify the presence of MES profiles. The practical computation of 2D (axisymmetric) MES equilibria is needed to asses their evolutionary properties.

Free energy and most probable state of turbulence.

Although turbulence is complex, taking multiple forms, it may still be regulated by the governing principle of preserved invariants. This suggests the existence of a most probable state of turbulence governed by statistical mechanics principles. Such a state of turbulence would be characterized by particular spectral distributions both in k space and in terms of mode form. Since at each magnetic surface the local MES represents the state of minimal energy, it should be possible to define an unambiguous free energy (that in excess of the local minimum) which is available to drive a spectrum of modes; this energy would allow determination of the steady "saturated" level of the turbulence.

Issues. (1) What spectral and mode form distributions are predicted for a two-fluid? (2) Since a MES has uniform temperatures; what light is shed on the nature of ITG and ETG turbulence by the presence of invariant self helicities?

Resolution. The statistical theory of two-fluid turbulence needs to be developed, both in terms of the expected spectra and the consequent transport rates. Very few measurements of electrostatic and electromagnetic fluctuations have been made in FRCs: such measurements are necessary to corroborate any transport theory and to compare with global confinement predictions.

Edge effects on transport

Because of rapid (Alfvenic) losses of particles in the scrape-off layer (SOL) of an FRC, it is unlikely that the SOL structure will correspond to a MES. This will give rise to a source of free energy and therefore a "local instability drive" that will drive transport.

Issues. Do edge processes regulate the overall transport rate (bulk confinement time), i.e. does the tail (SOL) wag the dog? Do the unique outflow and electric field properties at the edge act as a source or sink for the invariants (self helicities) of the plasma as a whole; i.e. does the edge inject or extract helicity? What controls at the edge may affect the bulk confinement of the FRC?

Resolution. Improved, kinetic modelling of the SOL in FRCs needs to be done, especially to explain the (apparently) anomalously slow outflow of plasma.

Features of FRCs with respect to stability

FRCs have much in common with other magnetic fusion concepts, but some features are somewhat unique, especially with respect to MHD. (1) Truly high b ~ 1. The high b is inherent, set by equilibrium considerations rather than stability. FRCs offer vital insights into the physics of truly high-beta plasmas. (2) Simple magnetic topology. FRC have little or no toroidal field in FRCs; lack a rotational transform (q » 0); have no magnetic shear; have closed magnetic field lines; and magnetic surfaces are defined by axisymmetry rather than ergodic mapping of field lines. (3) Deep magnetic wells. FRCs have absolute field nulls at the O-point and the X-points; and significant magnetic wells (mirror ratio ~ 2) in the end regions of typically elongated FRCs. This promotes a significant population of trapped particles. (4) Remote conducting walls. Most experiments have relatively distant conducting walls (separation distance ~ size of plasma). (5) No current-driven instabilities. The absence of a parallel current makes FRC immune to this class of instabilities. (6) Purely diamagnetic current. All MHD instabilities in FRCs are driven by pressure gradient only.

Non MHD stability

Because of average "bad curvature" on internal field lines, static FRC equilibria are ideal-MHD unstable to a host of MHD modes. Familiar techniques such as current profile control, separatrix shaping, and nearby conducting walls are ineffective for producing stability. The stability observed in experiments can only be ascribed to non-standard effects such as high-speed flows, flow shear, two-fluid effects, and finite larmor radius (FLR) effects. Recent results suggest that FLR effects alone do not account for the stability of experiments.

Issues. (1) Which of the several non-ideal effects is the key player in stabilizing FRC experiments? (2) What are the stability limits for FLR in FRC, in particular, what is the maximum radius / ion gyroradius allowed for stability?

Resolution. Modes that threaten disruption (principally tilting) are global and thus have large macroscale. Thus MHD (single-fluid) models may be adequate to represent the behavior of these modes. Conspicuously missing (nearly) in previous modelling is flow shear, largely because of the lack of algorithms for computing flowing equilibria. The development of such methods therefore has high priority; after that, MHD stability modelling with flows can be brought to bear.

Two-fluid stability analysis

Two fluid dynamics is more complex than MHD because the ion and electron displacements need not be identical. Even so, it is regulated by invariants, the self helicities for each species (generalizations of the magnetic helicity where A is replaced by the canonical fluid momentum). In the ideal case these invariants have a local form.

Issues. (1) Since an MES has minimum energy subject to constraints, can it be shown that MESís are absolutely stable to all ideal two-fluid modes? (2) Does the existence of the global invariants foster a simpler formulation of ideal two-fluid stability?

Resolution. MHD stability has been developed to a sophisticated level. By contrast, two-fluid stability is relatively "raw" having at present only a variational principle. This is complicated by the fact that two-fluid analysis is inherently more complex. Two-fluid analysis needs maturation with, hopefully, intuitive paradigms and the development of concrete indices for stability.

Rapid restructuring processes

Theory predicts that the fastest instabilities in FRCs are fast ballooning modes concentrated in the high curvature regions. These are "benign" in that they may restructure the plasma through nonlinear processes toward a more stable state. The disruptive tilt mode (lowest order kink) has a growth rate several times smaller. Both ballooning and tilting modes are driven by pressure gradients.

Issues. (1) Will ballooning modes be fast enough to restructure the plasma into a more stable state before disruptive modes grow up? (2) If so, what is the key feature of the restructured plasma that produces stability? (3) The absence of a parallel current eliminates the associated possibility of disruptions; does this imply that FRCs can be disruption-free?

Resolution. Experimental observations on FRC stability are likely to remain ambiguous until adequate simulations can be performed. This will require a nonlinear two-fluid (or at least Hall-MHD) simulation with (a) both resistive and viscous friction; (b) small enough grid spacing to allow resolution of scales somewhat smaller than the ion collisionless skin depth, and (c) high enough Lundquist number that reconnections can take place on these short length scales.

Relaxation process

If the global self helicities for the electron and ion species are preserved, then the dynamical behavior, even if exceedingly complex will move toward a two-fluid minimum energy state (MES).

Issues. (1) Two-fluid MES are meaningful only if the self-helicities are roughly invariant. Their ruggedness has been defended on the basis of several arguments but needs the firmer footing of more advanced computations and more detailed experimental observations. (2) Toroidal plasmas with a strong "stabilizing" toroidal field may not relax to an MES because fast channels for relaxation (ideal modes) are blocked; can a quantitative prediction be made about when a plasma can and cannot relax to an MES?

Resolution. The ruggedness of the helicity invariants can be tested by the two-fluid simulations described earlier.

Spontaneous sheared flows

MESís of a two fluid are characterized by finite b and significant flow shear. Since the "arrow" of relaxation is toward an MES as an end-state, the proper sheared flows may be generated spontaneously.

Issues. (1) Do strongly sheared flows arise spontaneously in FRC experiments? If so, why do they appear in FRCs but (probably) not in other configurations? (2) Is there a generalization of q including both magnetic fields and flow shear that can be used as an index of stability in a flowing system? (3) Is there a "new class of plasma" with an approximate equipartition of between magnetic and flow energies and strong shear of the composite vorticity (fluid vorticity plus magnetic field)?

Resolution. A more mature two-fluid analysis will resolve some of these issues. Observations of flow shear is essential for confirmation on the experiment side.

Edge effects on stability

The rapid outflow of particles in the scrap-off layer (SOL) of an FRC make it unlikely that the SOL structure will correspond to a MES. This will give rise to a "local instability drive" that may destabilize the core plasma as well as drive faster transport.

Issues. Do edge processes regulate the overall behavior, i.e. does the tail (SOL) wag the dog? Do the unique outflow and electric field properties at the edge act as a source or sink for the invariants (self helicities) of the plasma as a whole; i.e. does the edge inject or extract helicity? What controls can be applied at the edge to influence its properties and affect the bulk confinement of the FRC

Resolution. A more mature two-fluid analysis will resolve some of these issues.

Observations of flows and electric fields in the SOL are needed to illuminate the phenomena there.

Features of FRC with reference to boundary physics

FRCs have much in common with other magnetic fusion concepts, but several features are rather unique. (1) Natural divertor. The generally favored method for isolating the hot core from the first wall is by a divertor. In FRCs there is a natural divertor with x-points on the geometric axis leading to spindle-like exhaust jets that extend outside the open coil system. (2) Remoteness of first walls. The primary source of impurity levels in the core is usually nearby first wall surfaces (limiters, antennas). FRCs have no close-fitting walls: the typical separation from the wall is comparable to the plasma radius. (3) Remote divertor collection. The natural divertor jets extend outside the confinement coil to walls that can be placed at an arbitrary distance. The ability to spread out the divertor jets to arbitrary size makes it easy to stay within reasonable heat flux levels at collection surfaces. There is no need for a "judicious" level of impurities in the edge layer to dissipate heat near divertor surfaces.

High level of thermal isolation

An open coil arrangement, as in an FRC, allows an arbitrary separation from heat collection surfaces, and greatly mitigates the pumping required to control recycling of deuterium or impurities. This may allow nearly complete thermal isolation of the plasma from cold boundaries. If so it may be possible that the edge layer energy losses will be purely convective (negligible recycling). This would allow the plasma boundary (separatrix) to run hot, as is the case in present FRC experiments. A hot boundary reduces the susceptibility to ITG/ETG turbulence and anomalous transport.

Issue. In near steady operation can recycling be reduced to such a level to allow "hot boundary" operation?

Resolution. A quiescent FRC experiment (>> msec) is needed to allow enough time for significant refluxing, and the conditions needed for reducing or eliminating it.

The properties of the jet plasma and the plasma/neutrals near the end walls in FRC experiments need to be measured.

Edge/core transport coupling
The core of an FRC may be a relatively stable minimum energy state (MES) based on two-fluid theory. However, because of rapid losses of particle in the scrape-off layer (SOL) it is unlikely that the SOL structure will conform to a MES. This will give rise to a source of free energy and therefore a "local instability drive" that will increase the transport rate.

Issues. (1) Do edge processes regulate the overall transport rate (bulk confinement time), i.e. does the tail (SOL) wag the dog? (2) Can the edge properties of an FRC be modified to reduce the transport of particles and energy such as has been done in tokamaks (H-mode)? (3) What active or passive controls can be applied at the edge to bulk confinement of the FRC? (4) Why is the outflow of plasma in the SOL of FRCs anomalously slow?

Resolution. The two-fluid relaxation theory (the basis for high-b MES) needs to reach a higher level of maturing before the intercoupling between a "relaxed" core and a "nonrelaxed" edge can be assessed. Improved, kinetic modelling of the SOL in FRCs needs to be done, especially to explain the (apparently) anomalously slow outflow of plasma.

Fueling and density control

Refueling in long-pulse experiments is accomplished by a combination of feeding gas to the plasma edge and deep injection of frozen D₂ pellets. In FRCs relaxation processes may simplify refueling. Bas