TEARING MODES IN TOKAMAK PLASMAS
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
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 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
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
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
of Alfven times. In ITER, it will involve modeling the discharge for tens, or even hundereds, of
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.
- This presentation outlines
the basic philosophy of the EPEC code.
- The EPEC code has been implemented as a module in the OMFIT integrated modeling
- The EPEC source code can be found here. (The development branch is
currently the relevant branch.)
- The OMFIT source code can be found here. (The EPEC_refactored branch is
currently the relevant branch.)
EPEC OMFIT Module Documentation
Documentation for the EPEC
- Complete description of the physics model implemented in the EPEC code [PDF].
- Details of the algorithms employed in the FLUX submodule [PDF].
- Details of the cylindrical tearing mode analysis used in the FLUX submodule to determine the parameters Delta', Sigma', Delta_w, and Sigma_w. [PDF].
- Details of the algorithms employed in the NEOCLASSICAL submodule [PDF].
- Details of the algorithms employed in the PHASE submodule [PDF].
- Details of the algorithms employed in the RESCALE submodule [PDF].
EPEC Research Results
- R. Fitzpatrick, SangKyeun Kim, and Jaehyun Lee,
Modeling of q95 windows for the suppression of edge localized modes by resonant magnetic
perturbations in the KSTAR tokamak,
Phys. Plasmas 28, 082511 (2021)
- R. Fitzpatrick,
Further modeling of q95 windows for the suppression of edge localized modes by resonant magnetic
perturbations in the DIII-D tokamak,
Phys. Plasmas 28, 022503 (2021)
- R. Fitzpatrick,
Modeling q95 windows for the suppression of edge localized modes by resonant magnetic
perturbations in the DIII-D tokamak.
Phys. Plasmas 27, 102511 (2020)
- R. Fitzpatrick, and A.O. Nelson,
An improved theory of the response of DIII-D H-mode discharges to static resonant magnetic perturbations and its implications for the suppression of edge localized modes,
Phys. Plasmas 27, 072501 (2020)