The development of humankind's ultimate energy source, *nuclear fusion*, has proceeded slowly but surely over the course of the last 60 years. Moreover, the
perceived need for such an energy source has never been more acute than it is at present. Of all of the plasma confinement schemes that have been attempted over the years, *magnetic confinement*, by which a
thermonuclear plasma equilibrium is contained by a strong magnetic field, seems to be the most practical. Moreover, by far and away the most successful magnetic confinement device is the *tokamak*.

A tokamak is a device whose purpose is to confine a thermonuclear plasma on a set of *axisymmetric*, nested, toroidal magnetic flux-surfaces generated by a combination of electrical currents flowing in external field-coils, and currents induced within the plasma itself by transformer action.
Confinement is possible because, although heat and particles stream along magnetic field-lines very rapidly, they can only diffuse across magnetic flux-surfaces comparatively slowly.

Unlike most naturally occurring plasmas (e.g., the solar wind), tokamak plasmas are extremely quiescent. (Of course, this is by design.) Tokamak plasma discharges usually last tens of millions of Alfvén times. [The Alfvén time is the typical timescale on which Alfvén waves traverse the plasma, and also on which ideal magnetohydrodynamical (MHD) instabilities grow, and is of order a 1/10 th of a microsecond in conventional tokamak plasmas.]

Tokamak plasmas are sometimes terminated by violent events known as *disruptions*. One major class of disruption is caused by the plasma discharge crossing an ideal-MHD stability boundary. However, such disruptions are easy to avoid, because the locations of the stability boundaries in operational space can be calculated very accurately.

The overwhelming majority of disruptions that are not caused by crossing ideal stability boundaries are associated with *tearing modes*. Tearing modes are slowly growing, macroscopic instabilities of tokamak plasmas that tear and reconnected magnetic field-lines at various resonant surfaces in the plasma to produce radially-localized magnetic island chains. Tearing modes are driven by radial current and pressure gradients within the plasma, and can be unstable even when the plasma is ideally stable. Tearing modes degrade plasma confinement because heat and particles can flow very rapidly from one (radial) side of a magnetic island chain to another by streaming along magnetic field-lines, rather than having to slowly diffuse across magnetic flux-surfaces.
Tearing modes in tokamak plasmas generally saturate at fairly low amplitudes (such that the associated magnetic island chains have radial extents that are a few percent of the plasma minor radius), and can persist over a large fraction of the lifetime of the plasma discharge.

Tearing modes in tokamak plasmas usually rotate rapidly (at many kilo-radians per second) as a consequence of plasma flows induced by the radial density and temperature gradients in the plasma. However, tearing modes that grow to comparatively large amplitudes tend to *slow down*, due to eddy currents induced in the vacuum vessel surrounding the plasma, and eventually *lock* (i.e., become stationary in the laboratory frame) to static imperfections in the externally generated magnetic field known as *error-fields*. Such tearing modes often trigger disruptions. In fact, there is a very clear correlation between the occurrence of so-called *locked modes* and disruptions.

Tearing modes are generally driven unstable by radial current and pressure gradients within tokamak plasmas. However, there exists a particularly virulent class of
tearing modes, known as *neoclassical tearing modes*, that is driven by the loss of the neoclassical bootstrap current inside the separatrix of a magnetic island chain, consequent on the flattening of the
plasma pressure profile within the separatrix.

Tearing modes in tokamak plasmas are very poorly described by conventional single-fluid resistive-MHD, because of the relatively low collisionality of such plasmas, combined with the significantly different drift velocities of the various plasma species. Tearing modes are also not always well described by linear analysis, which becomes invalid as soon as the radial widths of the magnetic island chains at the various resonant surfaces exceed the (very narrow) linear layer widths.

The aim of this book is to outline a realistic, comprehensive, self-consistent, analytic theory of tearing mode dynamics in tokamak plasmas. The theory in question models the plasma as a multi-component fluid (a kinetic approach would be infeasible), and makes extensive use of asymptotic matching methods.

Chapter 1 estimates the typical plasma parameters needed to achieve thermonuclear fusion in a conventional tokamak, and goes on to give a general overview of tearing modes in tokamaks.

Chapter 2 outlines the fundamental fluid theory that underpins the analysis of tearing mode
dynamics in tokamak plasmas. This task is complicated by the low collisionality of tokamak plasmas, which requires a so-called *neoclassical* closure of the parallel (to the magnetic field) dynamics, as well as by the presence of small-scale plasma turbulence, which necessitates a phenomenological closure of the perpendicular dynamics.

Chapter 3 introduces an approximation that forms the basis of much of the analysis in this book by which a tokamak plasma is treated as a periodic cylinder. The chapter also introduces the fundamental asymptotic matching method that underpins all tearing mode theory.

In Chapter 4, a reduced drift-MHD model is extracted from the fundamental fluid equations derived in Chapter 2 by, first, neglecting all specifically neoclassical terms in the equations, and by, second, removing from the remaining equations the irrelevant physics of compressible-Alfvén waves.

Chapter 5 uses the reduced drift-MHD model of Chapter 4 to determine all linear response regimes of a resonant layer interacting with a rotating magnetic perturbation.

In Chapter 6, the linear response theory of Chapter 5 is employed to estimate the linear growth-rates, rotation frequencies, and resonant layer thicknesses of tearing modes in tokamak plasmas.

Chapter 7 uses the linear response theory of Chapter 5 to determine the critical error-field amplitude above which such a field is able to introduce a locked magnetic island chain into a tokamak plasma.

Chapter 8 employs the reduced drift-MHD model of Chapter 5 to determine the nonlinear response of a magnetic island chain to a rotating magnetic perturbation.

In Chapter 9, the nonlinear response theory of Chapter 8 is used to analyze the growth, saturation, and rotation of nonlinear tearing modes in tokamak plasmas.

Chapter 10 employs the nonlinear response theory of Chapter 8 to investigate the braking of a magnetic island chain's rotation when it interacts electromagnetically with a resistive vacuum vessel.

In Chapter 11, a neoclassical reduced drift-MHD model is extracted from the fundamental fluid equations derived in Chapter 2 by, first, retaining all specifically neoclassical terms in the equations, and by, second, removing from the equations the irrelevant physics of compressible-Alfvén waves.

Chapter 12 uses the neoclassical reduced drift-MHD model of Chapter 11 to investigate the physics of neoclassical tearing modes.

In Chapter 13, the neoclassical reduced drift-MHD model of Chapter 11 is employed to analyze the locking of a rotating magnetic island chain to a static error-field.

Finally, Chapter 14 generalizes the analysis of the book to take the true toroidal geometry of tokamak plasmas into account.

The author would like to express his gratitude to the teachers, colleagues, and students with whom he has interacted over the years in his quest to gain a more complete understanding of tearing mode dynamics in tokamak plasmas. These include R.J. Hastie, J.W. Connor, J.B. Taylor, T.C. Hender, A.W. Morris, G.M. Fishpool, C.G. Gimblett, H.R. Wilson, C.M. Roach, A. Thyagaraja and D.A. Gates at the UKAEA Culham Laboratory; J.A. Wesson, P.G. Carolan and M.F.F. Nave at the Joint European Torus (JET); T.H. Jensen, R.J. La Haye, J.T. Scoville, M.S. Chu and C. Paz-Solden at General Atomics; F.L. Waelbroeck, R. Hazeltine, P.G. Watson, F. Militello, E.P. Yu, E. Rossi, A.J. Cole, R.L. White, R. Carrera, W.L. Rowan, R.D. Bengston, E.R. Solano, P.H. Edmonds, H. Gasquet, G. Cima and A.J. Wooten at the University of Texas at Austin; J.D. Callen, C.C. Hegna, B.E. Chapman, D. Craig and S.C. Prager at the University of Wisconsin-Madison; M.E. Mauel, G.A. Navratil, A.M. Garofalo, S.A. Sabbagh and D.A. Maurer at Columbia University; J.P. Freidberg, I.H. Hutchinson, R.S. Granetz, S.M. Wolfe and A. Hubbard at MIT; A.H. Glasser and J.M. Finn at Los Alamos National Laboratory; K.M. McGuire, J. Bialek, M. Okabayashi, H.R. Strauss, A.O. Nelson, A.H. Reiman, J.-K. Park, S. Kim, A. Bhattacharjee, D.P. Brennan, N.C. Logan, Q.M. Hu and R. Nazikian at Princeton Plasma Physics Laboratory; A.I. Smolyakov at the University of Saskatchewan; S.C. Guo, D.F. Escande, P. Zanca and P. Martin at the Consorzio RFX, Padua; F. Porcelli and D. Grasso at the Politecnico di Torino; J. Lee at the Korea Institute of Fusion Energy, Daejeon; K.H. Finken at Jülich; and H. Zohm and S. Günter at the Max-Planck-Institut für Plasmaphysik, Garching.