R. Fitzpatrick, Tearing Mode Dynamics in Tokamak Plasmas (IOP Publishing 2023)

A research monograph on the physics of tearing modes in tokamak plasmas. [HTML] [eBOOK]


A tokamak is a device whose purpose is to confine a thermonuclear plasma on a set of nested toroidal magnetic flux-surfaces generated by a combination of electrical currents flowing in external field-coils, and currents induced within the plasma by transformer action. (The plasma essentially acts as a single-turn secondary coil in a transformer curcuit.) 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 typically 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.) Next-generation tokamaks, such as ITER, will feature discharges that last tens of billions of Alfven times. It is clear that, under normal circumstances, and unlike most naturally-occurring plasmas, the Alfven time is not a particulaly important timescale in 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 instabilities 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 side of a magnetic island chain to another by streaming along magnetic field-lines, rather than having to slowly diffuse across magnetic flux-surfaces. Fortunately, 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).

Tearing modes in tokamak plasma 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 very poorly 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.

Unless something goes seriously wrong, the widths of magnetic island chains in tokamak plasmas evolve very slowly in time. In the short-lived linear regime, the typical evolution timescale is a factor of S to the power 3/5 larger than the Alfven time, where S (typically in excess of a hundred million) is the dimensionless Lundquist number. In the nonlinear regime, the typical evolution timescale is a factor of S larger than the Alfven time. (It is actually physically impossible for magnetic reconnection to proceed on a longer timescale than this.)

Tearing modes in tokamak plasmas usually rotate rapidly (many krad per second) as a consequence of plasma flows induced by the radial density and temperature gradients in the plasma, as well as flows induced by the toroidal momentum imparted to the plasma by means of unbalanced neutral beam injection (the most common plasma heating scheme). However, tearing modes that grow to comparatively large ampliutudes 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 stationary 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.

Error-fields are small (typically a few Gauss) non-axisymmetric perturbations in the equilibrium magnetic field generated by field-coil misalignements. Error-fields are sometimes deliberately applied to tokamak plasmas (by running currents in external field coils) in order to control an annoying instability known as an edge-localized mode (ELM). Deliberately applied error-fields are usually referred to as resonant magnetic perturbations (RMPs).

Modeling the dynamics of tearing modes in tokamak plasmas, in the presence of a resistive wall, error-fields, and RMPs, is very challenging, because it inevitably involves modeling the discharge for tens, or even hundereds, of millions of Alfven times. In ITER, it will involve modeling the discharge for tens, or even hundereds, of billions of Alfven times. In either case, performing the simulation with a coventional, finite-difference/finite-element, toroidal, two-fluid, neoclassical, resistive-MHD code is out of the question. So, what do we do instead??


The Extended Perturbed Equilibrium Code (EPEC) models tearing mode dynamics in toroidal tokamak plasmas, in the presence of a ressitive wall and externally-generated resonant magnetic perturbations (RMPs), using the highly efficient asymptotic matching approach. The code takes two-fluid effects, neoclassical physics, and neutral particles into account. The amplitudes and phases of the reconnected fluxes at the various resonant (with the RMP) surfaces in the plasma are evolved in time in conjuction with the poloidal and toroidal plasma rotation profiles. EPEC is capable of modeling a whole tokamak plasma discharge on an ordinary desktop computer in a matter of hours. Performing an equivalent calculation using a conventional, finite-difference/finite-element, toroidal, two-fluid, neoclassical, resistive-MHD code would take months (if not years!) on a supercomputer.

EPEC OMFIT Module Documentation

Documentation for the EPEC OMFIT module.

EPEC Documents

EPEC Research Results

Richard Fitzpatrick