Plasma Instabilities

In fusion plasmas, various instabilities disrupt plasma confinement and limit performance. Understanding and controlling these instabilities is critically important for realizing fusion reactors. This chapter provides a detailed discussion of major instabilities in tokamak plasmas and their control methods.

Classification of Plasma Instabilities

Plasma instabilities are broadly classified into two categories based on their spatial scale and physical mechanism.

MHD Instabilities

MHD (magnetohydrodynamic) instabilities are macroscopic instabilities described within the framework of treating plasma as a single conducting fluid. The typical spatial scale is on the order of the plasma minor radius $a$, and the growth time is characterized by the MHD time scale (Alfven time).

$$\tau_A = \frac{a}{v_A} = a\sqrt{\frac{\mu_0 \rho}{B^2}}$$

Here, $v_A$ is the Alfven velocity, $\rho$ is the plasma mass density, and $B$ is the magnetic field strength. In typical tokamaks, $\tau_A \sim 1 \, \mu\text{s}$.

MHD instabilities are further classified by their driving mechanism as follows.

Current-driven modes are instabilities caused by the spatial distribution of plasma current. Kink modes, tearing modes, and resistive wall modes fall into this category. When the plasma current has a component parallel to the magnetic field lines, the distribution of magnetic field line twist (safety factor $q$) determines the instability.

Pressure-driven modes are instabilities caused by plasma pressure gradients. Ballooning modes and interchange modes belong to this category. When the pressure gradient becomes steep, a force acts to push the plasma across the magnetic field lines, causing instability.

Microinstabilities

Microinstabilities are microscopic instabilities where particle kinetic effects play an essential role. The typical spatial scale is on the order of the ion Larmor radius $\rho_i$, and the growth time is on the order of the inverse diamagnetic drift frequency.

$$\rho_i = \frac{m_i v_{ti}}{eB}, \quad v_{ti} = \sqrt{\frac{2T_i}{m_i}}$$ $$\omega_*^{-1} \sim \frac{a}{\rho_i} \cdot \tau_A$$

Representative microinstabilities include the ion temperature gradient (ITG) mode, trapped electron mode (TEM), and electron temperature gradient (ETG) mode. These are the primary causes of anomalous transport and determine energy confinement performance.

The ITG mode is an instability driven by the ion temperature gradient. When the critical gradient is exceeded, it grows rapidly and increases the ion thermal diffusion coefficient. The critical gradient length is characterized by:

$$\left(\frac{R}{L_{T_i}}\right)_{\text{crit}} \approx \frac{4}{3}\left(1 + \frac{T_i}{T_e}\right)$$

Here, $L_{T_i}^{-1} = -\nabla T_i / T_i$ is the ion temperature gradient length.

The TEM is an instability involving the precessional drift motion of electrons trapped in the toroidal magnetic field. It is driven by electron density gradients or electron temperature gradients and becomes particularly important when the density gradient is steep.

The ETG mode is an instability driven by the electron temperature gradient, corresponding to the electron version of the ITG mode. Although its spatial scale is very small, on the order of the electron Larmor radius $\rho_e$, it contributes to electron heat transport.

Basic Theory of MHD Instabilities

Ideal MHD Equations

The ideal MHD equation system is the fundamental set of equations describing the macroscopic behavior of plasma. It consists of mass conservation, momentum conservation, the induction equation, and the adiabatic condition.

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$ $$\rho \frac{d\mathbf{v}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla p$$ $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B})$$ $$\frac{d}{dt}\left(\frac{p}{\rho^\gamma}\right) = 0$$

Here, $\mathbf{v}$ is the plasma flow velocity, $\mathbf{J}$ is the current density, $\mathbf{B}$ is the magnetic field, $p$ is the pressure, and $\gamma$ is the ratio of specific heats.

Energy Principle

The energy principle provides a powerful method for determining MHD stability. For a plasma displacement vector $\boldsymbol{\xi}(\mathbf{r})$, the change in potential energy $\delta W$ of the system is evaluated.

$$\delta W[\boldsymbol{\xi}] = \frac{1}{2} \int_V d^3r \left[ \frac{|\mathbf{Q}|^2}{\mu_0} + \gamma p |\nabla \cdot \boldsymbol{\xi}|^2 + \boldsymbol{\xi}_\perp \cdot \nabla p \, \nabla \cdot \boldsymbol{\xi}_\perp^* + \mathbf{J}_\parallel \cdot (\boldsymbol{\xi}_\perp^* \times \mathbf{Q}) \right]$$

Here, $\mathbf{Q} = \nabla \times (\boldsymbol{\xi} \times \mathbf{B})$ is the perturbed magnetic field, $\boldsymbol{\xi}_\perp$ is the displacement component perpendicular to the magnetic field lines, and $\mathbf{J}_\parallel$ is the parallel current.

The stability condition is expressed as follows:

$$\delta W[\boldsymbol{\xi}] > 0 \quad \text{for all admissible } \boldsymbol{\xi}$$

If this condition is satisfied, the equilibrium state is stable. If a displacement exists for which $\delta W < 0$, the mode corresponding to that displacement is unstable.

Each term in the energy principle has physical meaning.

The first term $|\mathbf{Q}|^2/\mu_0$ represents the restoring force against bending and compression of the magnetic field, always contributing to stabilization. This is called “magnetic field stiffness.”

The second term $\gamma p |\nabla \cdot \boldsymbol{\xi}|^2$ is the restoring force against plasma compression and contributes to stabilization.

The third term $\boldsymbol{\xi}_\perp \cdot \nabla p \, \nabla \cdot \boldsymbol{\xi}_\perp^*$ is the pressure gradient drive term, which, combined with curvature effects, causes destabilization.

The fourth term $\mathbf{J}_\parallel \cdot (\boldsymbol{\xi}_\perp^* \times \mathbf{Q})$ is the current drive term and is the driving force for kink instability.

Mode Analysis

In plasma with toroidal symmetry, perturbations can be Fourier expanded.

$$\boldsymbol{\xi}(\mathbf{r}, t) = \sum_{m,n} \boldsymbol{\xi}_{m,n}(r) \exp[i(m\theta - n\phi - \omega t)]$$

Here, $m$ is the poloidal mode number, $n$ is the toroidal mode number, $\theta$ is the poloidal angle, and $\phi$ is the toroidal angle.

The condition for resonance between the mode and magnetic field lines is given by:

$$q(r_s) = \frac{m}{n}$$

Here, $r_s$ is the position of the resonant surface (rational surface). At the resonant surface, magnetic field lines trace closed periodic orbits, and perturbations accumulate, making instabilities more likely to occur.

Kink Instability

Kink instability is a current-driven instability where the plasma column deforms helically. When current flows parallel to the magnetic field lines, the plasma becomes unstable due to the attractive force between parallel currents.

Kink Instability in Cylindrical Plasma

Consider the simplest model in cylindrical plasma. A helical displacement with $m = 1$ and wavelength $\lambda = 2\pi/k$ is applied to a plasma column with uniform current distribution and radius $a$.

When the plasma column shifts from the magnetic axis, the poloidal magnetic field distribution becomes asymmetric. Due to the attractive force between currents, a further force acts in the direction of the shift, amplifying the displacement.

The growth rate $\gamma$ of this instability is evaluated using the Alfven velocity as:

$$\gamma \sim \frac{v_A}{R} \sqrt{1 - nq}$$

For $n = 1$, the system becomes unstable when $q < 1$.

Kruskal-Shafranov Limit

The Kruskal-Shafranov limit is known as the condition for suppressing kink instability. The safety factor $q$ represents the twist of magnetic field lines and is defined as:

$$q = \frac{r B_\phi}{R B_\theta}$$

In cylindrical plasma, $q > 1$ is a necessary condition for kink stability. In tokamaks, the following condition using the safety factor at the plasma surface $q_a$ serves as a practical stability condition:

$$q_a = \frac{2\pi a^2 B_\phi}{\mu_0 R I_p} > 2 \sim 3$$

Here, $I_p$ is the plasma current. In modern tokamaks, operation typically maintains $q_a \geq 3$.

Internal Kink Mode

The internal kink mode is an $m/n = 1/1$ mode that resonates with the $q = 1$ surface inside the plasma. The plasma core inside the $q = 1$ surface displaces helically.

The stability of the internal kink mode depends on the plasma pressure distribution and current distribution. In ideal MHD theory, the existence of the $q = 1$ surface itself is a necessary condition for instability.

The growth rate is evaluated as:

$$\gamma \sim \omega_A \left(\frac{r_1}{R}\right)^{1/2} \left|\frac{dq}{dr}\right|_{r_1}^{-1/2}$$

Here, $\omega_A = v_A/R$ is the Alfven frequency, and $r_1$ is the radius of the $q = 1$ surface.

The internal kink mode is a fundamental mode that causes sawtooth oscillations and fishbone oscillations.

External Kink Mode

The external kink mode is a mode whose resonant surface is located outside the plasma boundary. The entire plasma deforms helically, producing large displacement at the plasma surface.

A characteristic of the external kink mode is that interaction with the vacuum region and conducting wall is important. When an ideal conducting wall exists, magnetic perturbations are shielded at the wall, suppressing the instability.

The stability limit without a wall (no-wall limit) is given by:

$$\beta_N^{\text{no-wall}} \approx 4 l_i$$

Here, $\beta_N = \beta / (I_p/aB)$ is the normalized beta, and $l_i$ is the internal inductance.

On the other hand, the stability limit with an ideal wall sufficiently close (ideal-wall limit) is higher:

$$\beta_N^{\text{ideal-wall}} \approx 4 l_i + C$$

Here, $C$ is a coefficient that depends on the wall position.

Ballooning Mode

The ballooning mode is a pressure-driven instability that locally bulges on the outboard side (low-field side) of toroidal plasma.

Physical Mechanism

In toroidal plasma, the curvature of magnetic field lines varies with location. On the outboard side of the torus, the center of curvature is outside the plasma, which is called “bad curvature.” On the inboard side, the center of curvature is inside the plasma, called “good curvature.”

In the bad curvature region, centrifugal effects cause lighter plasma (high temperature, low density) to move outward and heavier plasma (low temperature, high density) to move inward. When a pressure gradient exists, this effect drives instability.

Considering plasma motion along magnetic field lines, plasma alternately passes through good curvature and bad curvature regions. Overall stability is determined by the average along the magnetic field lines.

Ballooning Equation

The ballooning equation for high $n$ modes is formulated as the following eigenvalue problem:

$$\frac{d}{d\theta} \left[ \frac{|\nabla\psi|^2}{B^2} \frac{dX}{d\theta} \right] + \left[ \frac{\omega^2}{\omega_A^2} \frac{|\nabla\psi|^2}{B^2} - \frac{2\mu_0}{B^4} (\mathbf{B} \times \nabla\psi) \cdot \boldsymbol{\kappa} \frac{dp}{d\psi} \right] X = 0$$

Here, $\psi$ is the magnetic surface label, $X$ is the mode amplitude, and $\boldsymbol{\kappa}$ is the magnetic field line curvature vector.

This equation is an ordinary differential equation along the poloidal angle $\theta$, solved with boundary conditions from $\theta = -\infty$ to $\theta = +\infty$.

Stability Boundary

Ballooning stability is often represented in an $s$-$\alpha$ diagram. Here, $s$ is the magnetic shear and $\alpha$ is the normalized pressure gradient.

$$s = \frac{r}{q} \frac{dq}{dr}$$ $$\alpha = -\frac{2\mu_0 R q^2}{B^2} \frac{dp}{dr}$$

The first stability region is the low $\alpha$ region, where magnetic shear provides stabilization. When $\alpha$ exceeds the critical value, it becomes unstable, entering the ballooning unstable region.

$$\alpha_{\text{crit}} \approx 0.6 \, s \quad \text{(large shear limit)}$$

However, with further increase in $\alpha$, one can enter the second stability region. In the second stability region, stabilization occurs through the magnetic well effect.

Beta Limit

The ballooning mode is one of the major modes that determine the tokamak beta limit. The beta limit considering toroidal effects is given by the Troyon limit:

$$\beta_{\text{max}} = g \frac{I_p}{aB}$$

Here, $g \approx 0.028 \sim 0.035$ is the Troyon coefficient. Using the normalized beta $\beta_N$, this is expressed as:

$$\beta_N = \frac{\beta}{I_p/aB} \lesssim 3.5$$

Beyond this limit, ballooning modes or kink modes become unstable, and plasma confinement degrades.

Tearing Mode and Magnetic Islands

The tearing mode is a resistive instability where magnetic field line reconnection occurs due to the finite electrical resistivity of plasma.

Classical Tearing Mode

In ideal MHD, magnetic field lines are frozen into the plasma, and topology is conserved. However, in plasma with finite resistivity, a diffusion term is added to the induction equation.

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \frac{\eta}{\mu_0} \nabla^2 \mathbf{B}$$

Here, $\eta$ is the electrical resistivity.

Near the resonant surface $q = m/n$, magnetic field lines are cut and reconnect through magnetic diffusion, forming a new topology. This is the essence of the tearing mode.

In tearing mode analysis, the plasma region is divided into “outer region” and “inner region.” In the outer region, ideal MHD applies, while in the inner region (resistive layer), resistive effects become important.

The instability parameter $\Delta'$ obtained from the outer region solution is defined as:

$$\Delta' = \lim_{\epsilon \to 0} \left[ \frac{1}{\psi} \frac{d\psi}{dr} \right]_{r_s - \epsilon}^{r_s + \epsilon}$$

Here, $\psi$ is the perturbed magnetic flux. The tearing mode is unstable when $\Delta' > 0$.

The growth rate is given by:

$$\gamma = C \left(\frac{\Delta'}{\tau_R}\right)^{3/5} \tau_A^{-2/5}$$

Here, $\tau_R = \mu_0 a^2 / \eta$ is the magnetic diffusion time, and $C$ is a numerical coefficient. In typical tokamaks, $\tau_R \sim 10^3$ s.

Magnetic Island Formation and Structure

Through the development of the tearing mode, magnetic islands are formed at the resonant surface. Magnetic islands are regions surrounded by closed magnetic field lines with a different topology from the main plasma.

The magnetic island width $w$ is given by:

$$w = 4 \sqrt{\frac{r_s q}{m} \frac{B_r}{\left|dB_\theta/dr\right|_{r_s}}}$$

Here, $B_r$ is the radial component of the perturbed magnetic field.

Inside the magnetic island, magnetic field lines are connected, so temperature and density become homogenized. This flattening causes the pressure gradient to locally vanish, degrading confinement performance.

The growth dynamics of magnetic islands are described by the modified Rutherford equation:

$$\frac{\tau_R}{r_s^2} \frac{dw}{dt} = \Delta'(w)$$

Here, $\Delta'(w)$ is the effective $\Delta'$ that depends on the island width.

Neoclassical Tearing Mode (NTM)

The neoclassical tearing mode (NTM) is a tearing mode driven by neoclassical transport effects. It is one of the major instabilities in high-beta plasma.

Physical Mechanism

In toroidal plasma, bootstrap current flows due to trapped particle effects. The bootstrap current density is proportional to the pressure gradient.

$$J_{bs} = -\frac{c_1 \sqrt{\epsilon}}{B_\theta} \frac{dp}{dr}$$

Here, $\epsilon = r/R$ is the inverse aspect ratio, and $c_1$ is a numerical coefficient.

When a magnetic island forms, temperature and pressure become homogenized inside the island, and the pressure gradient vanishes. This causes a deficit in the bootstrap current inside the island.

The deficit in bootstrap current disturbs the equilibrium current distribution, which becomes the driving force for the tearing mode. Unlike the classical tearing mode, the NTM can become unstable even under conditions where $\Delta' < 0$.

Modified Rutherford Equation

The growth of NTM is described by the modified Rutherford equation:

$$\frac{\tau_R}{r_s^2} \frac{dw}{dt} = \Delta' + \frac{\Delta'_{bs}}{w} - \frac{\Delta'_{GGJ}}{w^2}$$

Here, $\Delta'_{bs}$ is the bootstrap current drive term, and $\Delta'_{GGJ}$ is the Glasser-Greene-Johnson term (a stabilizing term due to finite island width effects).

The bootstrap drive term is given by:

$$\Delta'_{bs} = -\frac{\mu_0 r_s}{B_\theta^2} \frac{\langle J_{bs} \cdot B \rangle}{\langle B \rangle}$$

The $\Delta'_{GGJ}$ term represents a stabilizing effect that becomes important when the island width is small.

$$\Delta'_{GGJ} = \Delta'_{bs} \cdot w_{crit}$$

Here, $w_{crit}$ is the critical island width, approximately equal to the ion poloidal Larmor radius $\rho_{\theta i}$.

Critical Island Width and Seed Magnetic Island

A characteristic behavior of NTM is the threshold effect. When the island width is smaller than the critical value $w_{crit}$, the stabilizing effect exceeds the driving effect, and the island does not grow.

However, if a “seed magnetic island” exceeding the critical island width is formed by some cause, the NTM begins to grow. Major sources of seed magnetic islands include:

1. Sawtooth crashes
2. ELMs
3. External magnetic field perturbations
4. Error fields

When the NTM grows, it reaches a saturated island width $w_{sat}$.

$$w_{sat} = \frac{\Delta'_{bs}}{|\Delta'|}$$

The larger the saturated island width, the more severe the degradation of confinement performance.

Rational Surfaces of NTM

Important rational surfaces for NTM in tokamaks include the $q = 3/2$ surface and the $q = 2$ surface.

The $m/n = 3/2$ NTM occurs in the peripheral region of the plasma and causes confinement degradation.

The $m/n = 2/1$ NTM occurs further outward, and when it grows, it can lead to major disruptions.

Resistive Wall Mode (RWM)

The resistive wall mode (RWM) is an instability that couples the external kink mode with the resistive effects of a conducting wall.

Ideal Wall and Resistive Wall

The external kink mode is stabilized by an ideal conducting wall. This is because the perturbation magnetic field from the mode is completely shielded at the ideal wall.

However, real conducting walls have finite electrical resistivity. The perturbation magnetic field penetrates the wall to a skin depth and transmits through the wall over time. The wall time constant is given by:

$$\tau_w = \mu_0 \sigma_w d_w r_w$$

Here, $\sigma_w$ is the wall conductivity, $d_w$ is the wall thickness, and $r_w$ is the wall radius. For an aluminum wall, $\tau_w \sim 10$ ms.

The growth rate of the resistive wall mode is evaluated as:

$$\gamma \sim \frac{1}{\tau_w} \frac{\beta_N - \beta_N^{\text{no-wall}}}{\beta_N^{\text{ideal-wall}} - \beta_N}$$

The RWM becomes unstable in the region between the no-wall limit $\beta_N^{\text{no-wall}}$ and the ideal-wall limit $\beta_N^{\text{ideal-wall}}$.

Rotational Stabilization

Plasma rotation has the effect of stabilizing the RWM. When the plasma rotates relative to the wall, momentum is exchanged through the interaction between the mode and the wall, causing resistive damping.

The critical rotation frequency required for stabilization is evaluated as:

$$\Omega_{crit} \sim \frac{\gamma_{RWM}}{\omega_A} \omega_A$$

Here, $\gamma_{RWM}$ is the RWM growth rate in stationary plasma, and $\omega_A$ is the Alfven frequency.

Experimentally, it has been shown that the RWM can be stabilized if the toroidal rotation velocity is a few percent of the Alfven velocity.

Active Control

Active control of the RWM is important for high-beta operation. Major control methods include the following.

In feedback control with external coils, correction magnetic fields are applied from external coils according to the mode amplitude detected by sensors. The RWM can be stabilized by appropriately setting the feedback gain.

In control with internal coils, coils installed inside the vacuum vessel are used to achieve higher frequency response.

ELM (Edge Localized Mode)

ELM (Edge Localized Mode) is an MHD instability localized in the edge transport barrier (pedestal) of H-mode plasma.

H-mode and Pedestal

In H-mode (high confinement mode) plasma, a transport barrier forms at the plasma periphery, and a pedestal structure with steep pressure gradients develops.

In the pedestal region, electron density, electron temperature, and ion temperature change rapidly at the plasma periphery. The pedestal width $\Delta_{ped}$ is typically several times the ion poloidal Larmor radius.

$$\Delta_{ped} \sim (5 \sim 10) \rho_{\theta i}$$

The pedestal pressure $p_{ped}$ is an important parameter that determines core confinement performance.

$$W_{\text{plasma}} \sim p_{ped} \cdot V$$

Here, $W_{\text{plasma}}$ is the plasma stored energy, and $V$ is the plasma volume.

Peeling-Ballooning Mode

The steep pressure gradient and high current density in the pedestal region excite peeling-ballooning instability.

The peeling mode is a current-driven external kink mode driven by bootstrap current in the pedestal region.

The ballooning mode is an instability driven by the pressure gradient.

Pedestal stability is represented in the $j$-$\alpha$ diagram (current density-pressure gradient space). As the pedestal grows, it reaches the stability limit, and an ELM occurs.

Classification of ELMs

There are several types of ELMs, each with different physical mechanisms and characteristics.

Type I ELM is a typical large-scale ELM. It is associated with ideal MHD peeling-ballooning modes and occurs when the heating power exceeds a threshold. The frequency is approximately 10-100 Hz, and the energy released per ELM is 5-15% of the pedestal stored energy.

$$\Delta W_{ELM} \sim (0.05 \sim 0.15) \cdot 3 p_{ped} V_{ped}$$

The ELM frequency increases with heating power.

$$f_{ELM} \propto P_{heat} - P_{LH}$$

Here, $P_{LH}$ is the threshold power for L-H mode transition.

Type II ELM, also called “grassy ELM,” is a small amplitude, high frequency ELM. It is observed under conditions of strong shaping effects (high triangularity) or high pedestal collisionality.

Type III ELM is a small-scale ELM associated with resistive ballooning modes. It occurs when the heating power is close to the threshold or when the edge resistivity is high. The frequency is higher than Type I, and the energy released per event is about 10% of Type I.

Heat Load from ELMs

When an ELM occurs, pedestal energy is released to the divertor through the scrape-off layer (SOL). The instantaneous heat load on the divertor from ELMs is a serious problem.

The instantaneous heat flux is evaluated as:

$$q_{ELM} = \frac{\Delta W_{ELM}}{A_{wet} \cdot \tau_{ELM}}$$

Here, $A_{wet}$ is the wetted area (the area receiving heat load), and $\tau_{ELM}$ is the ELM duration.

In ITER-class devices, the heat flux from Type I ELMs may exceed the tolerance limits of divertor materials. Therefore, ELM control is essential.

Sawtooth Oscillations

Sawtooth oscillations are periodic MHD phenomena universally observed in tokamak plasma with a $q = 1$ surface.

Observational Characteristics

Sawtooth oscillations are named because the time variation of the central electron temperature shows a sawtooth-like waveform.

1. Ramp phase: Central temperature gradually increases
2. Crash phase: Central temperature rapidly decreases
3. Recovery phase: Heat re-accumulates in the center

This process repeats periodically. The typical period is 10-100 ms.

During the crash, heat is rapidly transported from inside to outside the $q = 1$ surface. The inversion radius $r_{inv}$ corresponds to the position of the $q = 1$ surface.

$$r_{inv} \approx r_1 \quad (q(r_1) = 1)$$

Physical Mechanism: Internal Kink Mode

The driving source of sawtooth oscillations is the $m/n = 1/1$ internal kink mode.

When the plasma core is heated, the central temperature rises and the resistivity decreases. This causes the current distribution to peak, resulting in $q(0) < 1$. When a $q = 1$ surface exists, the internal kink mode becomes unstable.

Ideal MHD stability is evaluated by the Bussac condition:

$$\delta W_{MHD} = \delta \hat{W} - \frac{\xi_0^2}{2} \left(1 - \frac{1}{q_0}\right)$$

Here, $q_0 = q(0)$ is the safety factor on the magnetic axis, and $\xi_0$ is the displacement amplitude on the magnetic axis. When $q_0 < 1$, $\delta W_{MHD}$ can become negative.

Crash Models

Several models have been proposed for the detailed mechanism of sawtooth crashes.

In the Kadomtsev reconnection model, when the $m/n = 1/1$ mode grows, helical magnetic flux reconnects, and the core magnetic surfaces are expelled outward. After complete reconnection, $q(0) = 1$ is restored.

In the quasi-interchange model, magnetic surfaces of the core and periphery are exchanged through ideal MHD magnetic field line interchange (quasi-interchange).

The partial reconnection model explains that the crash does not lead to complete reconnection, ending with $q(0) < 1$. This is consistent with experimental observations.

Determination of the Period

The sawtooth period $\tau_{ST}$ is determined by the following factors:

1. Rate of decrease of $q(0)$ due to central heating
2. Time until destabilization of the internal kink mode
3. Delay due to resistive effects and diamagnetic effects

Approximately, the following scaling holds:

$$\tau_{ST} \propto \tau_R^{1/3} \tau_A^{2/3}$$

However, experimentally, the period can be significantly extended due to diamagnetic stabilization effects.

Monster Sawteeth

In high-temperature plasma or under strong heating conditions, diamagnetic effects and fast particle effects stabilize the internal kink mode, and the sawtooth period is remarkably extended. Such long-period sawteeth are called “monster sawteeth.”

Monster sawtooth crashes are accompanied by larger energy release than normal and have a higher risk of generating seed magnetic islands for NTM.

Fast Particle Instabilities

Fusion plasma contains fast ions generated by heating and fusion reactions. These fast particles excite characteristic instabilities.

Fishbone Oscillations

Fishbone oscillations are $m/n = 1/1$ internal kink modes driven by energetic particles. The name derives from the time variation of soft X-ray signals showing a fishbone-like shape.

The driving mechanism is the resonance between the precessional drift motion of fast ions and the internal kink mode. The resonance condition is given by:

$$\omega = \omega_d = \frac{E}{eR B_\theta} \frac{m}{q} \left(\frac{1}{q} - 1\right)$$

Here, $\omega$ is the mode frequency, $\omega_d$ is the precessional drift frequency, and $E$ is the energy of the fast ion.

The dispersion relation for fishbone oscillations is given by:

$$\delta \hat{W}_{MHD} + \delta \hat{W}_{hot}(\omega) = 0$$

Here, $\delta \hat{W}_{hot}$ is the contribution from fast particles, a complex quantity that depends on frequency.

$$\delta \hat{W}_{hot} = \int d^3v \, F_h \frac{\omega_{*h}}{\omega - \omega_d + i\nu}$$

Here, $F_h$ is the distribution function of fast particles, $\omega_{*h}$ is the diamagnetic frequency of fast particles, and $\nu$ is the damping rate.

The condition for instability has a threshold in the fast particle beta value $\beta_h$.

$$\beta_h > \beta_h^{crit}$$

When fishbones occur, fast particles are redistributed outside the $q = 1$ surface, reducing heating efficiency.

Toroidal Alfven Eigenmode (TAE)

TAE (Toroidal Alfven Eigenmode) is an eigenmode of Alfven waves in toroidal magnetic fields, excited by fast particles.

The dispersion relation of Alfven waves in uniform plasma is given by:

$$\omega = k_\parallel v_A = \frac{n - m/q}{R} v_A$$

In toroidal plasma, adjacent poloidal modes ($m$ and $m+1$) couple. This coupling creates a gap in the continuous spectrum, and discrete eigenmodes can exist within the gap.

The TAE frequency is given by:

$$\omega_{TAE} \approx \frac{v_A}{2qR}$$

The resonance condition with fast particles is:

$$\omega_{TAE} = n \omega_\phi + p \omega_\theta - l \omega_b$$

Here, $\omega_\phi$ is the toroidal transit frequency, $\omega_\theta$ is the poloidal transit frequency, $\omega_b$ is the bounce frequency, and $l$ is an integer.

The main resonance with passing particles ($l = 0$) is given by:

$$v_\parallel = \frac{v_A}{3}$$

When TAE becomes unstable, fast particles are transported radially, causing reduced heating efficiency and localized heat loads on the first wall.

Reversed Shear Alfven Eigenmode (RSAE)

RSAE (Reversed Shear Alfven Eigenmode) is an Alfven eigenmode observed in reversed magnetic shear configurations. It is also called “Alfven cascade.”

In reversed shear configurations, there is a minimum point $q_{min}$ in the $q$ profile. When $q_{min}$ passes through a rational number, the RSAE frequency varies with time, and characteristic chirping (frequency sweeping) is observed.

$$\omega_{RSAE} \approx \frac{v_A}{R} \left| n - \frac{m}{q_{min}} \right|$$

EPM (Energetic Particle Mode)

EPM (Energetic Particle Mode) is a mode driven by fast particles that exists within the Alfven continuous spectrum. Unlike TAE, it grows while subject to continuum damping.

EPM becomes unstable when the fast particle beta value is sufficiently high to overcome continuum damping. The frequency approaches the characteristic frequency of fast particles.

Disruption

Disruption is a violent MHD instability phenomenon in which the plasma discharge suddenly terminates. Understanding and avoiding disruptions is one of the most important issues for fusion reactors.

Classification of Disruptions

Major disruption is a large-scale collapse in which the plasma current is completely lost. It consists of a thermal quench followed by a current quench.

Minor disruption is accompanied by partial degradation of confinement but the plasma is maintained.

Vertical displacement event (VDE) is a phenomenon where the plasma displaces vertically and contacts the wall.

Progression of Disruption

Disruption progresses through the following stages.

In the precursor phase, MHD instability (mainly the $m/n = 2/1$ mode) grows and magnetic islands expand. When locking (mode rotation stops) occurs, the risk of disruption increases.

In the thermal quench, the plasma stored energy is lost on a time scale of 1-10 ms. The electron temperature drops rapidly, and impurities are released from the wall.

$$\tau_{TQ} \sim 1 \sim 10 \, \text{ms}$$

In the current quench, the plasma current decays on a time scale of 10-100 ms.

$$\tau_{CQ} \sim 10 \sim 100 \, \text{ms}$$

During the current quench, the following problems occur:

1. Electromagnetic forces: Induced currents flow in the vacuum vessel, generating large electromagnetic forces
2. Halo currents: Current flows to the wall through the halo region around the plasma, generating asymmetric forces
3. Runaway electrons: A high-energy runaway electron beam is formed, causing localized damage

Density Limit

The density limit is a phenomenon where radiation collapse causes disruption when the plasma density exceeds a threshold. The Greenwald density limit is given by:

$$n_{GW} = \frac{I_p}{\pi a^2} \times 10^{20} \, \text{m}^{-3}$$

Here, $I_p$ is the plasma current in MA, and $a$ is the minor radius in m.

The physical mechanism of the density limit is the breakdown of power balance due to radiation cooling at the edge. As density increases, radiation losses increase and edge temperature decreases. Resistivity increases, the current distribution changes, and MHD instabilities are excited.

Beta Limit

When the beta limit is exceeded, ballooning modes or kink modes become unstable, leading to disruption. The Troyon limit is as described earlier.

$$\beta_N \lesssim 3.5$$

In actual operation, beta values sufficiently below this limit are maintained.

Runaway Electrons

When electron temperature drops rapidly during the thermal quench, resistance to the electric field increases. Some electrons are accelerated to relativistic energies because acceleration by the electric field exceeds deceleration by collisions. These are runaway electrons.

The critical electric field for runaway electrons is given by:

$$E_c = \frac{n_e e^3 \ln \Lambda}{4\pi \varepsilon_0^2 m_e c^2}$$

The energy of the runaway electron beam can reach tens of MeV, and collision with the wall causes severe damage.

Instability Control Methods

Control of plasma instabilities is essential for stable operation of fusion reactors. Major control methods are described below.

Electron Cyclotron Current Drive (ECCD)

ECCD is a method of driving current locally using electron cyclotron waves. It is widely used to suppress NTM magnetic islands.

The resonance condition for electron cyclotron waves is given by:

$$\omega = n\Omega_{ce} - k_\parallel v_\parallel$$

Here, $\Omega_{ce} = eB/m_e$ is the electron cyclotron frequency, and $n$ is the harmonic number. Usually, $n = 2$ (second harmonic) is used.

The current drive efficiency is evaluated as:

$$\zeta = \frac{n_e R I_{CD}}{P_{EC}} \sim 0.01 \sim 0.03 \, \text{A/W/m}^2$$

For NTM control, the ECCD beam is directed at the O-point (center of the island) of the magnetic island. By driving current inside the island, the bootstrap current deficit is compensated, suppressing island growth.

For successful control, beam aiming accuracy is important. Real-time systems have been developed that detect the position of the magnetic island and adjust mirrors to track the beam.

Resonant Magnetic Perturbation (RMP)

RMP is a method of applying helical magnetic fields from external coils to change the magnetic structure of the pedestal region. It is mainly used for ELM control.

RMP coils generate magnetic field components with toroidal mode numbers $n = 3$ or $n = 4$. This magnetic field resonates at rational surfaces in the pedestal region, forming ergodic magnetic field regions or small magnetic islands.

Modes of ELM control by RMP include ELM suppression (complete disappearance of ELMs) and ELM mitigation (increased ELM frequency, reduced size).

The ELM suppression window depends on $q_{95}$ (safety factor at the 95% magnetic surface) and is achieved only in specific ranges.

$$q_{95} = 3.5 \pm 0.1, \, 4.1 \pm 0.1, \, ...$$

Challenges of RMP include density pump-out (decrease in edge density) and effects on energy confinement.

Pellet Injection

Solid hydrogen pellet injection is used for both particle supply and ELM control.

ELM pacing is a method of inducing ELMs by high-frequency pellet injection, generating small-scale ELMs continuously. The energy released per ELM can be reduced.

$$\Delta W_{ELM} \propto \frac{1}{f_{ELM}}$$

Killer pellets are a method of rapidly injecting large amounts of impurities (argon, neon, etc.) for disruption mitigation. Heat is distributed over a wide area by radiation cooling, suppressing the generation of runaway electrons.

Active MHD Control

Active control of MHD instabilities uses feedback systems combining sensors and actuators.

Sensors include magnetic probes (magnetic field perturbation detection), soft X-ray arrays (mode structure detection), and electron cyclotron emission (temperature fluctuation detection).

Actuators include external coils (magnetic field application), electron cyclotron waves (local current drive), and neutral beam injection (momentum and current drive).

The feedback loop for NTM control has the following configuration:

1. Detect $m/n$ mode with magnetic probes
2. Determine magnetic island position and phase through real-time analysis
3. Aim ECCD beam at the island O-point
4. Adjust driven current to suppress the island

The feedback for RWM control has the following configuration:

1. Detect RWM amplitude with magnetic probes
2. Separate growing mode by phase-sensitive detection
3. Apply correction magnetic field from external coils
4. Adjust gain for stabilization

Current Profile Control

Optimization of the current distribution is a fundamental control method that prevents many instabilities.

By keeping the central safety factor $q(0)$ greater than 1, sawtooth oscillations can be avoided. Off-axis current drive (ECCD, NBCD) is effective for this.

By optimizing the shear of the $q$ profile, the stability of NTM and ballooning modes can be improved.

$$s = \frac{r}{q} \frac{dq}{dr} > 0 \quad \text{(positive shear condition)}$$

Reversed magnetic shear configurations are favorable for the formation of internal transport barriers (ITB) but carry the risk of double tearing modes.

Future Outlook

Instability control in fusion reactors is expected to develop further toward ITER and future demonstration reactors (DEMO).

In ITER, RMP coils for ELM control, ECCD systems for NTM control, and disruption mitigation systems are planned. A comprehensive control system is needed to stably maintain 15 MA of plasma current.

In DEMO, since long-duration steady-state operation is required, emphasis is placed on securing “stable operating regions” that do not generate instabilities rather than active control of instabilities.

Machine learning-based disruption prediction is also a rapidly developing field in recent years. Neural networks trained on past disruption events are making it possible to detect precursors early and take avoidance measures.

Related Topics

• MHD - Fundamentals of magnetohydrodynamics and equilibrium
• Charged Particle Motion - Fundamentals of drift motion
• Transport - Anomalous transport and microinstabilities
• Heating Principles - Details of ECCD and NBI
• Tokamak - Practical instability control
• ITER - Implementation of disruption mitigation systems