Magnetic Helicity, Spheromaks, Solar Corona Loops, and Astrophysical Jets

The following sections are included:

• Dedication
• Contents
• Preface

The following sections are included:

• Brief description of spheromaks
• History and time-line
  • Pre-1970: Antecedents of the spheromak
  • Advances in theory: Taylor relaxation and development of the theoretical model for the spheromak
  • The 1980’s: The spheromak investigated as a fusion confinement scheme
  • The 1990’s: Search for other applications and renaissance in confinement efforts
  • 2000 to time of writing: SSPX, HIT-SI, General Fusion, investigation of dynamics, inclusion of Hall term

This chapter develops certain basic concepts which underlie the analysis presented in later chapters.

Magnetic helicity is one of the central concepts underlying spheromak physics. Like many profound concepts, magnetic helicity can be understood from several different points of view. For example, magnetic helicity is closely related to both force-free currents and the topological theory of knots. Magnetic helicity is important for two reasons:

• (1) Magnetic helicity is a measure of the topological properties of the configuration and is more fundamental than any specific geometrical property.
• (2) Magnetic helicity is a robust invariant with respect to microscopic dissipative processes. This means that magnetic helicity is nearly perfectly conserved even when the system contains substantial small-scale turbulence and reconnection. Other invariants, energy in particular, are not robust and decay rapidly in the presence of small-scale turbulence and reconnection

The following sections are included:

• Introduction
• Helicity decay rate v. magnetic energy decay rate
• Derivation of the isolated Taylor state
• Relationship between helicity, energy, eigenvalue
  • Equality of poloidal and toroidal field energies in an isolated axisymmetric spheromak
• Cylindrical force-free states
• Ohmic Decay of an Isolated Taylor State
• Comparison of minimum energy states in a long cylinder
• Spheromaks in spherical geometry

An isolated spheromak decays with time. In order to sustain a spheromak there has to be some driving mechanism to replenish the energy and helicity consumed by dissipative processes. In a simply-connected system this driving mechanism is obtained by imposing a voltage across the ends of open field lines as shown in Fig. 5.1. The current driven by this voltage results in a power influx and it will be shown that there is also a helicity influx. Thus, analysis of a driven system involves consideration of the situation shown in Fig. 5.1 where open field lines intercept a gapped flux-conserving wall. It will be shown in Chapter 17 and following chapters that because magnetic flux is frozen into the plasma and because helicity depends quadratically on magnetic flux, injection of magnetic helicity typically necessitates injection of plasma. Furthermore, these later chapters will show that the system is not in equilibrium and instead has a well-defined dynamical behavior; this behavior is effectively the “turbulence” assumed in the relaxation model. Because these more complicated issues of plasma injection and dynamical behavior are not considered in the present chapter, it should be realized that the discussion in this chapter, while quite useful, tells only part of the story and so serves as an introductory outline for what happens in reality…

The following sections are included:

• Derivation of the MHD Energy Principle
• Relationship of the energy principle to Taylor states
• Beta limit

Spheromak configurations have been produced using quite diverse methods; this diversity validates the assertion that the spheromak is a minimum-energy state towards which any initial configuration will relax. As shown in Fig. 7.1, the common feature of all formation schemes is application of an electromotive force (EMF) along initially vacuum magnetic field lines so as to drive field-aligned plasma current. In the first edition of this book it was stated at this point…

The following sections are included:

• Room full of springs analogy
• Analogy to thermodynamics
• Classification of thermodynamic problems
• Analogy between lambda and temperature
• Strong and weak coupling
• Overview of next five chapters

The following sections are included:

• Flux, current, magnetic field, helicity and energy
• Experimental measurements
• Safety factor for a zero-β spheromak
  • Evaluation of q profile
  • Effect of flux conserver shape on q_wall
• Finite-β spheromak
  • Derivation of the Grad-Shafranov Equation
  • Finite-β spheromak solution
  • Boundary
  • Safety factor

We now examine the various ways a flux-conserving wall influences an isolated spheromak. Most of these wall properties are relevant to driven spheromaks as well…

In this chapter we consider a spheromak tightly coupled to a helicity source having specified λ. This situation is analogous to the thermodynamic system in thermal contact with a constant temperature bath shown in Fig. 8.1(b). The thermodynamic system comes to the same temperature as the bath, has negligible temperature gradients, and negligible net heat flux in equilibrium. The MHD system is similarly assumed to equilibrate to the same λ as the source. However, we also permit the existence of a small λ gradient and corresponding small helicity flux to make up for a small distributed helicity dissipation. Existing spheromak experiments appear to be too lossy to be characterized by this strong coupling model and are more consistent with the weak coupling situation discussed in the next chapter; in other words, the assumption of nearly uniform λ is not a good characterization for existing experiments. It is nevertheless quite instructive to analyze the strong coupling case because it can be solved in closed form with modest effort and the solutions reveal important conceptual relationships. Furthermore, it would be desirable to arrange an experiment to be in the strong coupling regime if possible…

Chapter 8 demonstrated that an analogy exists between λ and temperature, and in particular, showed that λ gradients drive helicity transport much like temperature gradients drive heat transport. The present chapter will show that λ gradients are closely related to kink stability, dynamo concepts, and helicity flow. It will also show that dynamo concepts have a strong dependence on geometrical issues such as symmetry and what happens at the wall.

The following sections are included:

• Overview
• Confinement on flux surfaces
• Confinement times
• Survey of transport mechanisms
  • Diffusive processes
  • Non-diffusive processes
  • Magnetic energy and magnetic helicity
  • Relationship between magnetic energy and thermal energy decay times
  • Dissipation in a single flux tube
• Experiments on transport in spheromaks
  • S-1 measurements
  • S-1 particle confinement measurement
  • Edge losses
  • CTCC-I gettering experiments
  • The CTX gettered, solid flux conserver experiment
  • Evidence for good confinement: hard X-rays on CTX
  • SSPX Conclusions: Conflict between good confinement and helicity injection
• Anomalous ion heating

While it is relatively easy to form a basic spheromak, it takes considerable effort to form a spheromak with high temperature, long life, and good confinement properties. This effort entails minimizing non-ideal phenomena that can easily overwhelm the desired spheromak attributes if left unattended. In this chapter we examine several practical issues which have been of critical importance for past spheromak experiments. Undoubtedly new issues will arise in future experiments as parameter regimes change.

Spheromaks are diagnosed in much the same way as tokamaks, stellarators, and reversed field pinches. This chapter surveys basic spheromak diagnostics and gives a detailed discussion of some of the more essential ones. Because spheromaks are relatively simple to build, a full complement of diagnostics can easily cost more than the spheromak itself.

Diagnostics measure gun voltage, gun current, internal magnetic fields, density, electron and ion temperatures, optical radiation, impurity concentrations, plasma motion, and x-ray emission. The ideal diagnostic would have low cost and provide unambiguous time- and space-resolved data requiring no deconvolution. Real diagnostics rarely approach this ideal and there is usually a substantial compromise between cost and performance.

The following sections are included:

• The spheromak as a fusion reactor
• Accelerated spheromaks
• Magnetized Target Fusion
• Tokamak Fuel injection
• Helicity injection current drive in tokamaks
• Colliding spheromaks to investigate magnetic reconnection
• Proposed additional spheromak applications
  • Pulsed high power X-ray radiation sources
  • Opening switches

While the Taylor relaxation principle predicts the final state to which a system will evolve, it avoids describing the dynamics by which this occurs and in particular, says nothing about density or pressure profile evolution. Not characterizing pressure and density profiles is logically inconsistent with assigning importance to confinement because confinement corresponds to existence of density and pressure gradients…

The following sections are included:

• Kink Instability of the Jet
• The kink instability as an agent of relaxation
• The Rayleigh-Taylor instability as an agent of reconnection
• Production of Hard X-rays
• Comparison of actual experimental observations with predictions of Taylor relaxation model

Taylor relaxation results from the postulate that a low β plasma will spontaneously undergo magnetic reconnection if this decreases the volume-integrated magnetic energy while conserving magnetic helicity. On the other hand, the essential feature of ideal MHD is that magnetic flux is perfectly frozen into the plasma (see Sec. 2.13) and such freezing-in would obviously prohibit reconnection. To circumvent this restriction, inclusion of finite resistivity in MHD was proposed as the means by which “unfreezing” of flux from the plasma could occur [267–271]. The central idea was that critical weak spots exist where resistivity allows diffusion of magnetic field across plasma and so provides an “unfreezing”. Although highly localized to the region around the weak spot, the reconnection would alter the topology of the entire system: for example, two distinct flux tubes could merge or one flux tube could divide into two flux tubes. Conceptual analogs are the cleaving of a diamond struck at just the right angle or the change of train trajectories on a railroad network when a switch operates between adjacent tracks so that initial tracks AB and CD become AC and BD where the first letter refers to the origin and the second letter to the destination…

The following sections are included:

• Overview
• Sun-Earth connection viewed as helicity flux/relaxation
• A simple linear-force free model of coronal loops
  • Setup of problem and derivation of solution
  • S-shapes
  • Flux tube bifurcation and breakup
  • Comparison of magnetic field, field lines, flux tubes
  • Relaxation and line tying
• Force-free state for arbitrary boundary conditions

The following sections are included:

• Boundary conditions
  • Solar surface boundary conditions
  • Electrostatic potential drop associated with the twisting of a magnetic flux loop
  • Relation to Alfvén waves
  • Motivation for the “gobble” finite beta model
• The gobble model
  • Stage 1: current ramp-up
  • Stage 2: ramped-up, constant current but no force balance
  • Stage 3: Stagnation and collimation
  • Hoop force and the gobble model
• Difference between finite and zero β models and consequences
  • Torus Instability
  • Co- and counter-helicity merging
• Solar loop simulation experiment
  • Loop expansion and collimation
  • Plasma upflow from footpoints
  • Magnetic measurements supporting the gobble model
  • Hoop force explanation for the dip seen in the bottom of Fig. 21.12

MHD is a fluid model based on the presumptions that particles have a Maxwellian velocity distribution function relative to some center of mass velocity so the distribution of particle velocities is summarized by just two parameters: the temperature and the center of mass velocity. Establishment of a Maxwellian velocity distribution requires collisions but in reality many plasmas are insufficiently collisional for this to happen. Instead particles will have complex trajectories determined by the Lorentz force equation or its equivalent, the associated Hamiltonian-Lagrangian system. Single particle motion is conventionally characterized by the guiding center approximation, an approximation that depends on a large separation between the scale of the Larmor radius of gyrating particles and fluid scale lengths. Furthermore, the guiding center approximation assumes that field lines are uniform and straight or that departures from being uniform and straight are small in which case weak field gradients or curvatures give slow drifts. The guiding center approximation additionally assumes a temporal ordering such that to lowest order a particle has an E ×B drift and then to next order the particle makes a polarization drift proportional to a time derivative of the electric field where the magnitude of this derivative is small. We show in this chapter two situations where the guiding center approximation fails. These failures occur when there is no longer a distinct separation between the microscopic particle orbit scales and the macroscopic fluid scales. The failures are related either directly to the helical nature of the magnetic field or indirectly via electric fields that are established as a result of the time-dependence of a helical magnetic field.

The following sections are included:

• Introduction to magnetic clouds
• Magnetic cloud solution to Grad-Shafranov equation in toroidal geometry

The following sections are included:

• Introduction
• Field and current topology in astrophysical jets
  • Inadequacy of Ideal MHD Ohm’s Law to Model Accretion
• Kepler v. Cyclotron Orbits
• Charged dust grains as a method for having a cyclotron frequency comparable to the Kepler frequency
• Weak ionization as a method for having an effective cyclotron frequency comparable to the Kepler frequency
• Inward Spiral Orbits of Zero-Canonical Angular Momentum Particles
• Meta-Particle Inward Spiral from the Point of View of Hall Ohm’s Law
• Accretion and removal of angular momentum via magnetic braking
  • Braking torque interpreted using conservation of canonical angular momentum
  • Braking torque from the MHD equation of motion
• Dynamo to generate poloidal magnetic field

The following sections are included:

• Appendix A: Vector Identities and Operators
• Appendix B: Bessel Orthogonality Relations
• Appendix C: Capacitor Banks
• Appendix D: Transmission lines, pulse forming networks, and transformers
  • Transmission lines
    • Wave propagation along a transmission line
    • Propagation velocity
    • Characteristic impedance
    • Matching, mismatching and reflected waves
  • Pulse forming networks
  • Transformers
    • Ideal Transformers
    • Transformer equations (resistive loads)
    • Non-ideal transformers
    • Pulse rise-time (high frequency response)
    • Low-frequency response
    • Air-core and iron-core transformers
    • Calculating inductance of iron core transformers
    • Measuring self and mutual inductances
• Appendix E: Selected Formulae

The following sections are included:

• Bibliography
• Index