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Chapter 1
Introduction
1.1Brief description of spheromaks
Plasmas are gases composed of free electrons and ions. Typically, the electron and ion charge densities are nearly the same so that the plasma is an approximately neutral electrically conducting gas that is subject to electrical and magnetic forces in addition to the usual hydrodynamic forces. The process by which an ordinary gas is transformed into plasma is called ionization. For most plasmas, ionization takes place when free electrons strike neutral atoms with sufficient force to eject bound electrons, thereby creating more free electrons and ions. In order for this process to occur, there must be some free electrons with kinetic energy exceeding the binding energy of the most weakly bound outer electron in a neutral atom. This means that plasmas typically have an electron temperature of at least a few electron volts (1 eV = 11,604 K). Plasmas occur naturally in space environments (e.g., the solar corona, Earth’s magnetosphere, the aurora) but must be created in the laboratory using artificial means.
If one wishes to trap a laboratory plasma, then some kind of confinement scheme is required, because otherwise the plasma will quickly convect to the surrounding walls and recombine. Substantial effort has been directed during the past half century towards developing devices that use magnetic fields to confine plasmas. These magnetic confinement schemes can be understood at many levels of sophistication, but ultimately are based on the magnetic force F = qv × B acting on individual charged particles.
Spheromaks are a toroidal confinement configuration where the magnetic field is produced almost entirely by currents flowing in the plasma. The spheromak configuration is defined as an axisymmetric magnetohydrodynamic equilibrium with (i) a simply connected bounding surface, (ii) both toroidal and poloidal magnetic fields, and (iii) at least some closed poloidal flux surfaces. What distinguishes spheromaks from other toroidal configurations is that the toroidal magnetic field in spheromaks vanishes at the bounding surface (i.e., at the wall). Therefore no external coils link the spheromak and so the spheromak manages to have an internal toroidal field while still being simply connected. In contrast, tokamaks, reversed field pinches (RFP’s), and stellarators all have finite toroidal magnetic field at the wall; this corresponds to having external coils linking the plasma. Field reversed configurations (FRC’s) have zero toroidal magnetic field everywhere and so, like spheromaks, do not have coils linking the plasma. Thus, spheromaks manage to have a toroidal field without having toroidal field coils; FRC’s do not have toroidal field coils but also do not have a toroidal field.
Figure 1.1 compares spheromak topology to the other toroidal confinement methods and Table 1.1 lists the similarities and differences. The device complexity increases going down the table; this is also obvious from Fig. 1.1. All devices except the stellarator use a toroidal current to produce the poloidal field required for confinement; the poloidal field in the stellarator is created by external helical coils so that current-free operation is obtained at the expense of loss of axisymmetry. The FRC is the simplest device but, having no toroidal field is MHD-unstable and also has a field null on the magnetic axis.
Table 1.1:Comparison of topologies of various toroidal confinement devices.
According to the magnetohydrodynamic (MHD) point of view, plasma is modeled as an electrically conducting fluid and confinement involves balancing the outward force of hydrodynamic pressure against the inward force resulting from the interaction between the magnetic field and the electric current in the plasma. This balancing is most effective when the magnetic field lines in the plasma form nested surfaces called flux surfaces. The existence of flux surfaces means that any field line traces out a surface in three-dimensional space and does not fill up a volume.
Fig. 1.1:Comparison between various toroidal confinement devices. FRC’s and spheromaks have simply connected vacuum chambers, others have doubly connected vacuum chambers.
A point of view complementary to MHD and also more physically correct is provided by Hamiltonian-Lagrangian theory which shows that if there is symmetry about an axis, then confinement results from the conservation of canonical angular momentum for each particle. In this case, particle trajectories are restricted to surfaces on which the canonical angular momentum is a constant and confinement is akin to a spinning top standing upright because of conservation of angular momentum. Both the microscopic Hamiltonian-Lagrangian point of view and the macroscopic magnetohydrodynamic point of view arrive at the same conclusion because as particle mass goes to zero, invariance of canonical angular momentum becomes equivalent to the existence of flux surfaces. Thus, symmetry is important for confinement whether one uses the MHD point of view or the single particle point of view.
Flux surfaces are formed from the magnetic field produced by the combined effect of internal plasma currents and external coil currents. The various schemes for producing flux surfaces can be categorized according to the extent to which the flux surfaces are prescribed by plasma or external currents. Flux surfaces in stellarators are produced entirely by currents in external coils which link the toroidal plasma: these precision-engineered helical coils create accurate flux surfaces minimally affected by the plasma because the plasma is nearly current-free. Flux surfaces in tokamaks are prescribed by the dominantly toroidal internal current profile; the reason the plasma current is dominantly toroidal is because large external coils linking the plasma produce a strong toroidal magnetic field which provides stabilization against kinks. Flux surfaces in RFP’s result from the interaction between the small toroidal field produced by coils linking the plasma and poloidal flux directly injected by induction. The coil-produced toroidal field can be considered as a seed field which is considerably modified by plasma instabilities.
Spheromaks are closely related to RFP’s but have no coils linking the plasma so that flux surfaces are entirely the consequence of instabilities. Since the spheromak configuration results from spontaneous instabilities, spheromaks have the notable advantage of not having to be as precisely engineered as tokamaks, stellarators, or RFP’s. The tendency to form spontaneously also suggests that spheromak-like configurations should occur in nature, and indeed, certain space and solar plasmas are closely related to spheromaks.
The question often arises whether a spheromak is a device or a plasma configuration. This question is reasonable, because the nomenclature ‘tokamak’ refers to the device, not the plasma, and yet one often hears spheromaks referred to as the plasma configuration. The reason for this semantic ambiguity is that there is no unique way for making spheromak configurations because spheromak plasmas form spontaneously given the appropriate initial conditions. What is important is the plasma configuration and not the device.
A traditional way for dealing with a complicated three-dimensional problem is to reduce the problem to a simplified one- or two-dimensional version which contains the essential phenomenology but because of the reduced dimensionality is much more amenable to analysis. This traditional method cannot be applied to spheromaks, because spheromaks are intrinsically three dimensional and, in particular, involve helical geometry.
Spheromaks result from plasma self-organization and represent a minimum energy state towards which the plasma evolves. The study of spheromaks is relevant to a wide range of topics including thermonuclear fusion, solar physics, magnetospheric physics, astrophysics, magnetic reconnection, topology, self-organization, inaccessible states, magnetic turbulence, Ohm’s law, magnetohydrodynamics, vacuum techniques, pulse power engineering, and various diagno...
