# Plasma equilibria and stability

An important field of

plasma physics is the equilibria and stability of the

plasma.

*How's that for a circular definition?*
Magnetohydrodynamics (MHD) is a prominent fluid theory for an electromagnetic fluid, which is often used for plasmas. MHD theory is the simlest representation of a plasma, so MHD stability is a necesity for stable devices to be used for nuclear fusion, specifically magnetic fusion energy.

Beta is a measure of plasma pressure normalized to the magnetic field strength. See magnetohydrodynamics for a full definition.

MHD stability at high beta is crucial for a compact, cost-effective magnetic fusion reactor.
Fusion power density varies roughly as β^{2} at constant magnetic field, or as β N
4 at constant
bootstrap fraction in configurations with externally driven plasma current. (Here β N = β /(I/aB) is
the normalized beta.) In many cases MHD stability represents the primary limitation on beta and
thus on fusion power density. MHD stability is also closely tied to issues of creation and
sustainment of certain magnetic configurations, energy confinement, and steady-state operation.
Critical issues include understanding and extending the stability limits through the use of a
variety of plasma configurations, and developing active means for reliable operation near those
limits. Accurate predictive capabilities are needed, which will require the addition of new physics
to existing MHD models. Although a wide range of magnetic configurations exist, the underlying
MHD physics is common to all. Understanding of MHD stability gained in one configuration can
benefit others, by verifying analytic theories, providing benchmarks for predictive MHD stability
codes, and advancing the development of active control techniques.

The most fundamental and critical stability issue for magnetic fusion is simply that MHD
instabilities often limit performance at high beta. In most cases the important instabilities are long
wavelength, global modes, because of their ability to cause severe degradation of energy
confinement or termination of the plasma. Some important examples that are common to many
magnetic configurations are ideal kink modes, resistive wall modes, and neoclassical tearing
modes. A possible consequence of violating stability boundaries is a disruption, a sudden loss of
thermal energy often followed by termination of the discharge. The key issue thus includes
understanding the nature of the beta limit in the various configurations, including the associated
thermal and magnetic stresses, and finding ways to avoid the limits or mitigate the consequences.
A wide range of approaches to preventing such instabilities is under investigation, including
optimization of the configuration of the plasma and its confinement device, control of the internal
structure of the plasma, and active control of the MHD instabilities.

Ideal MHD instabilities driven by current or pressure gradients represent
the ultimate operational limit for most configurations. The long-wavelength kink mode and short-wavelength
ballooning mode limits are generally well understood and can in principle be avoided.
Intermediate-wavelength modes (n ~ 5–10 modes encountered in tokamak edge plasmas, for
example) are less well understood due to the computationally intensive nature of the stability
calculations. The extensive beta limit database for tokamaks is consistent with ideal MHD stability limits, yielding agreement to within about 10% in beta for cases where the internal profiles of the
plasma are accurately measured. This good agreement provides confidence in ideal stability
calculations for other configurations and in the design of prototype fusion reactors.
### Resistive Wall Modes

Resistive wall modes (RWM) develop in plasmas that require the
presence of a perfectly conducting wall for stability. RWM stability is a key issue for many
magnetic configurations. Moderate beta values are possible without a nearby wall in the tokamak,

stellarator, and other configurations, but a nearby conducting wall can significantly improve ideal
kink mode stability in most configurations, including the tokamak, ST, reverse field pinch (RFP), spheromak, and
possibly the FRC. In the advanced tokamak and ST, wall stabilization is critical for operation with
a large bootstrap fraction. The spheromak requires wall stabilization to avoid the low-m,n tilt and
shift modes, and possibly bending modes. However, in the presence of a non-ideal wall, the
slowly growing RWM is unstable. The resistive wall mode has been a long-standing issue for the
RFP, and has more recently been observed in tokamak experiments. Progress in understanding the
physics of the RWM and developing the means to stabilize it could be directly applicable to all
magnetic configurations. A closely related issue is to understand plasma rotation, its sources and
sinks, and its role in stabilizing the RWM.

### Resistive instabilities

Resistive instabilities are an issue for all magnetic configurations, since
the onset can occur at beta values well below the ideal limit. The stability of neoclassical tearing
modes (NTM) is a key issue for magnetic configurations with a strong bootstrap current. The
neoclassical tearing mode (NTM) is a metastable mode; in certain plasma configurations, a
sufficiently large deformation of the bootstrap current produced by a “seed island” can contribute
to the growth of the island. The NTM is already an important performance-limiting factor in many
tokamak experiments, leading to degraded confinement or disruption. Although the basic
mechanism is well established, the capability to predict the onset in present and future devices
requires better understanding of the damping mechanisms which determine the threshold island
size, and of the mode coupling by which other instabilities (such as sawteeth in tokamaks) can
generate seed islands.

### Configuration

The configuration of the plasma and its confinement device represent an
opportunity to improve MHD stability in a robust way. The benefits of discharge shaping and low
aspect ratio for ideal MHD stability have been clearly demonstrated in tokamaks and STs, and will
continue to be investigated in experiments such as DIII–D, C–Mod, NSTX, and MAST. New
stellarator experiments such as NCSX (proposed) will test the prediction that addition of
appropriately designed helical coils can stabilize ideal kink modes at high beta, and lower-beta tests
of ballooning stability are possible in HSX. The new ST experiments provide an opportunity to
test predictions that a low aspect ratio yields improved stability to tearing modes, including
neoclassical, through a large stabilizing “Glasser effect” term associated with a large Pfirsch-Schlüter
current. Neoclassical tearing modes can be avoided by minimizing the bootstrap current in
quasi-helical and quasi-omnigenous stellarator configurations. Neoclassical tearing modes are also
stabilized with the appropriate relative signs of the bootstrap current and the magnetic shear; this
prediction is supported by the absence of NTMs in central negative shear regions of tokamaks.
Stellarator configurations such as the proposed NCSX, a quasi-axisymmetric stellarator design,
can be created with negative magnetic shear and positive bootstrap current to achieve stability to the
NTM. Kink mode stabilization by a resistive wall has been demonstrated in RFPs and tokamaks,
and will be investigated in other configurations including STs (NSTX) and spheromaks (SSPX).
A new proposal to stabilize resistive wall modes by a flowing liquid lithium wall needs further
evaluation.

Control of the internal structure of the plasma allows more active
avoidance of MHD instabilities. Maintaining the proper current density profile, for example, can
help to maintain stability to tearing modes. Open-loop optimization of the pressure and current
density profiles with external heating and current drive sources is routinely used in many devices.
Improved diagnostic measurements along with localized heating and current drive sources, now
becoming available, will allow active feedback control of the internal profiles in the near future.
Such work is beginning or planned in most of the large tokamaks (JET, JT–60U, DIII–D,
C–Mod, and ASDEX–U) using rf heating and current drive. Real-time analysis of profile data
such as MSE current profile measurements and real-time identification of stability boundaries are
essential components of profile control. Strong plasma rotation can stabilize resistive wall modes,
as demonstrated in tokamak experiments, and rotational shear is also predicted to stabilize resistive
modes. Opportunities to test these predictions are provided by configurations such as the ST,
spheromak, and FRC, which have a large natural diamagnetic rotation, as well as tokamaks with
rotation driven by neutral beam injection. The Electric Tokamak experiment is intended to have a
very large driven rotation, approaching Alfvénic regimes where ideal stability may also be
influenced. Maintaining sufficient plasma rotation, and the possible role of the RWM in damping
the rotation, are important issues that can be investigated in these experiments.
### Feedback Control

Active feedback control of MHD instabilities should allow operation
beyond the “passive” stability limits. Localized rf current drive at the rational surface is predicted
to reduce or eliminate neoclassical tearing mode islands. Experiments have begun in ASDEX–U
and COMPASS-D with promising results, and are planned for next year in DIII–D. Routine use
of such a technique in generalized plasma conditions will require real-time identification of the
unstable mode and its radial location. If the plasma rotation needed to stabilize the resistive wall
mode cannot be maintained, feedback stabilization with external coils will be required. Feedback
experiments have begun in DIII–D and HBT-EP, and feedback control should be explored for the
RFP and other configurations. Physics understanding of these active control techniques will be
directly applicable between configurations.
Disruption Mitigation. The techniques discussed above for improving MHD stability are the
principal means of avoiding disruptions. However, in the event that these techniques do not
prevent an instability, the effects of a disruption can be mitigated by various techniques.
Experiments in
JT–60U have demonstrated reduction of electromagnetic stresses through operation at a neutral
point for vertical stability. Pre-emptive removal of the plasma energy by injection of a large gas
puff or an impurity pellet has been demonstrated in tokamak experiments, and ongoing
experiments in C–Mod, JT–60U, ASDEX–U, and DIII–D will improve the understanding and
predictive capability. Cryogenic liquid jets of helium are another proposed technique, which may
be required for larger devices. Mitigation techniques developed for tokamaks will be directly
applicable to other configurations.