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Sawtooth Instability

  • Ian T. Chapman
Chapter
Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP, volume 83)

Abstract

Sawtooth oscillation results in a periodic relaxation of the tokamak core plasma. These periodic oscillations consist of a quiescent period, during which the density and temperature increase, followed by a rapid collapse in the core pressure, which is often preceded by the growth of a helical magnetic perturbation. The period between these rapid sawtooth collapses is expected to increase in the presence of alpha particles in burning fusion plasmas. However, long sawtooth periods have been observed to increase the likelihood of triggering neoclassical tearing modes (see chapter 8) at lower plasma pressures; these instabilities in turn can then significantly degrade the plasma confinement. Consequently, recent efforts have focussed on developing methods to deliberately trigger short sawtooth periods to avoid seeding NTMs while retaining the benefits of core impurity expulsion. The main sawtooth control tools involve driving localised currents to change the safety factor profile or tailoring the fast ion distribution.

Keywords

Neutral Beam Injection Kink Mode Magnetic Shear Electron Cyclotron Resonance Heating Lower Hybrid Current Drive 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Glossary

I. Greek Symbols

\( \beta \)

Plasma beta, ratio of pressure to magnetic pressure, \( \beta = 2\mu_{0} p/B^{2} \)

\( \epsilon \)

Inverse aspect ratio, \( \epsilon = r/R_{0} \)

\( \epsilon_{1} \)

Inverse aspect ratio of \( q = 1 \) surface, \( \epsilon_{1} = r_{1} /R_{0} \)

\( \eta \)

Resistivity

\( \gamma \)

Linear growth rate

\( \gamma_{ad} \)

Adiabatic index, ratio of specific heats

\( \Gamma \)

Phase space

\( \lambda \)

Pitch angle, \( \lambda = v_{{\parallel }} /v \)

\( \mu \)

Magnetic moment, \( \mu = Mv_{ \bot }^{2} /2B \)

\( \mu_0 \)

Permeability

\( \Phi \)

Scalar potential

\( \Phi_{ad} \)

Third adiabatic invariant of motion

\( \phi \)

Toroidal angle

\( \dot{\phi } \)

Toroidal precession frequency

\( \rho_{i,e} \)

Ion/Electron Larmor radius, \( \rho_{i,e} = Mv_{ \bot } /eB \)

\( \hat{\rho } \)

Average Larmor radius, \( \hat{\rho } = \sqrt {\rho_{i}^{2} + \rho_{e}^{2} } \)

\( \psi_{t} \)

Toroidal flux

\( \psi_{p} \)

Poloidal flux

\( \psi \)

Radial coordinate in toroidal geometry

\( \psi_{*} \)

Helical flux

\( \psi_{1} \)

Flux at the \( q = 1 \) surface

\( \theta \)

Poloidal angle

\( \omega \)

Mode frequency

\( \tilde{\omega } \)

Doppler shifted mode frequency, \( \tilde{\omega } = \omega - \Omega_{\phi } (r_{1} ) \)

\( \omega_{b} \)

Bounce frequency, \( \omega_{b} = 2\pi /\tau_{b} \)

\( \omega_{c} \)

Cyclotron frequency, \( \omega_{c} = eB/M \)

\( \omega_{A} \)

Toroidal Alfv\( e^{\prime} \)n frequency, \( \omega_{A} = v_{A} /R_{0} \)

\( \omega_{*i} \)

Diamagnetic frequency of thermal plasma ions, \( \omega_{*i} = (T_{i} dp_{i} /dr)/(eBp_{i} r_{1} ) \)

\( \omega_{*h} \)

Diamagnetic frequency of hot ions, \( \omega_{*h} = (\partial f_{h} /\partial P_{\phi }^{0} )/(\partial f_{h} /\partial {\mathcal{E}}^{0} ) \)

\( \omega_{dh} \)

Drift frequency of hot ions, \( \omega_{dh} \approx cE_{h} /4eBR_{o} r_{1} \)

\( \langle \omega_{dh} \rangle \)

Bounce averaged magnetic drift frequency of hot ions

\( \Omega_{\phi } \)

Toroidal plasma rotation

\( \Omega_{E} \)

Toroidal plasma rotation caused by electric potential, \( \Omega_{E} = q\Phi^{'} /rB_{0} \)

\( \Delta \Omega \)

Flow shear, \( \Delta \Omega = \Omega_{E} (r) - \Omega_{E} (r_{1} ) \)

\( {\kern 1pt} \xi {\kern 1pt} \)

Fluid displacement

\( \xi_{0} \)

Fluid displacement at the magnetic axis

\( \xi_{1} \)

Fluid displacement at the \( q = 1 \) surface

\( \xi_{a} \)

Fluid displacement at the plasma edge

\( \tau_{A} \)

Alfv\( e^{\prime} \)n time, \( \tau_{A} = \sqrt {3R} /v_{A} \)

\( \tau_{s} \)

Sawtooth period

\( \tau_{E} \)

Energy confinement time

\( \tau_{\eta } \)

Resistive diffusion time

\( \chi \)

Pitch angle, \( \chi = v_{ \bot }^{2} B_{0} /v^{2} B \)

\( \zeta \)

Toroidal coordinate, \( \zeta = q\theta - \phi \)

\( \dot{\zeta } \)

Toroidal precession frequency

II. Roman Symbols

\( a \)

Minor radius of the plasma edge

\( {\mathbf{A}} \)

Vector potential

\( B_{\phi } \)

Toroidal magnetic field strength

\( B_{\theta } \)

Poloidal magnetic field strength

\( {\mathbf{B}} \)

Magnetic field

\( e \)

Charge of particle

\( {\mathbf{E}} \)

Electric field

\( {\mathcal{E}}_{i} \)

Energy of \( i \)th particle

\( f_{h} \)

Hot minority ion distribution function

\( f_{i} \)

Thermal ion distribution function

\( f_{0} \)

Initial distribution function

\( \delta f_{h,i} \)

Perturbed hot or thermal ion distribution function

\( \delta f_{hk} \)

Perturbed fast ion distribution function due to kinetic effects

\( \delta f_{hf} \)

Perturbed fast ion distribution function due to fluid effects

\( {\mathbf{j}} \)

Current density

\( {\mathbf{k}} \)

Wave vector

\( m \)

Poloidal mode number

\( M \)

Particle mass

\( n_{e} \)

Electron number density

\( n_{i} \)

Ion number density

\( n \)

Toroidal mode number

\( p \)

Plasma pressure

\( P_{\zeta ,\phi } \)

Canonical angular momentum

\( q \)

Safety factor, \( q = 1/2\pi \int B_{\phi } /RB_{\theta } {\rm{d}}s \)

\( R \)

Major radius

\( R_{0} \)

Major radius at magnetic axis

\( r \)

Minor radius

\( r_{1} \)

Minor radius at \( q = 1 \) surface

\( s \)

Magnetic shear, \( S = r/qdq/dr \)

\( s_{1} \)

Magnetic shear at \( q = 1 \) surface

\( T_{i} \)

Temperature of ions

\( T_{e} \)

Temperature of electrons

\( v_{A} \)

Alfv\( e^{\prime} \)n speed, \( v_{A} = B_{0} /\sqrt {\mu_{0} \rho_{0} } \)

\( {\mathbf{v}}_{h} \)

Fast particle velocity

\( v_{\phi } \)

Toroidal speed of plasma

\( {\mathbf{v}} \)

Particle velocity

\( {\mathbf{v}}_{||} \)

Particle velocity parallel to the magnetic field

\( {\mathbf{v}}_{ \bot } \)

Particle velocity perpendicular to the magnetic field

III. Potential Energy Terms

\( \delta W \)

Perturbed potential energy

\( \delta W_{MHD} \)

Perturbed potential energy due to MHD terms only

\( \delta W_{KO} \)

Perturbed potential energy due to collisionless thermal ions

\( \delta W_{core} \)

\( \delta W_{core} = \delta W_{MHD} + \delta W_{KO} \)

\( \delta W_{h} \)

Perturbed potential energy due to collisionless energetic ions

\( \delta W_{hf} \)

Perturbed potential energy due to hot ion adiabatic terms only

\( \delta W_{hk} \)

Perturbed potential energy due to hot ion non-adiabatic terms only

\( \delta W^{t} \)

Perturbed potential energy due to trapped hot ions only

\( \delta W^{p} \)

Perturbed potential energy due to passing hot ions only

\( \hat{\delta }W \)

Potential energy normalised as per [61], \( \hat{\delta }W = \mu_{0} \delta W/6\pi^{2} \xi_{0}^{2} \varepsilon_{1}^{4} R_{0} B^{2} \)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CCFECulham Science CentreOxfordshireUK

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