Tearing Mode Dynamics in Tokamak Plasmas

Author:            Richard Fitzpatrick

Publisher:         IOP Publishing

Publication Date:  July, 2023

ISBN:              978-0-7503-5367-0

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 Alfven times. [The Alfven time is the typical timescale on which Alfven 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.

1. Introduction.

2. Plasma Fluid Theory.

3. Cylindrical Tearing-Mode Theory.

4. Reduced Resonant Response Model.

5. Linear Resonant Response Model.

6. Linear Tearing-Mode Stability.

7. Error-Field Penetration in Tokamak Plasmas.

8. The Nonlinear Resonant Response Model.

9. Nonlinear Tearing Mode Stability.

10. Rotational Braking in Tokamak Plasmas.

11. The Nonlinear Neoclassical Resonant Response Model.

12. Neoclassical Tearing Modes.

13. Mode Locking in Tokamak Plasmas.

14. Toroidal Tearing Modes.

A. Neoclassical Theory.