Magnetic Configuration

In magnetic confinement fusion, the structure of magnetic field lines (magnetic configuration) determines plasma confinement performance and stability. This page provides a detailed explanation of the fundamental theory of magnetic configurations in toroidal devices through advanced configuration design, including mathematical descriptions.

Fundamental Theory of Magnetic Configuration

Toroidal Coordinate System

To describe the magnetic field in toroidal devices, we introduce the toroidal coordinate system $(r, \theta, \zeta)$ in addition to cylindrical coordinates $(R, \phi, Z)$:

• $R$: Distance from the torus symmetry axis (major radius direction)
• $\phi$: Toroidal angle (circumferential direction)
• $Z$: Vertical coordinate
• $r$: Distance from the magnetic axis (minor radius direction)
• $\theta$: Poloidal angle (angle within the minor cross-section)
• $\zeta$: Toroidal angle (synonymous with $\phi$ but used in flux coordinates)

The relationship between cylindrical and toroidal coordinates is given by:

$$R = R_0 + r\cos\theta, \quad Z = r\sin\theta$$

where $R_0$ is the major radius of the magnetic axis.

Toroidal Magnetic Field

The toroidal magnetic field $B_\phi$ is the magnetic field component along the circumferential direction of the torus (donut shape). It is generated by toroidal field coils (TF coils) and derived from Ampere’s law.

When current $I$ flows through $N$ coils, from the line integral:

$$\oint \mathbf{B} \cdot d\mathbf{l} = 2\pi R B_\phi = \mu_0 N I$$

Therefore, the magnetic field strength is inversely proportional to the distance $R$ from the central axis of the torus:

$$B_\phi = \frac{\mu_0 N I}{2\pi R} = \frac{B_0 R_0}{R}$$

where $B_0 = \mu_0 N I / (2\pi R_0)$ is the magnetic field strength at the magnetic axis.

The magnetic field is stronger on the inboard side ($R < R_0$) and weaker on the outboard side ($R > R_0$). This magnetic field gradient causes particle drift.

$$\frac{1}{B}\frac{\partial B}{\partial R} = -\frac{1}{R}$$

The $\nabla B$ drift velocity is given by:

$$\mathbf{v}_{\nabla B} = \frac{m v_\perp^2}{2 q B^2} \frac{\mathbf{B} \times \nabla B}{B}$$

Since ions and electrons drift in opposite directions, a toroidal field alone causes charge separation and generates an electric field $E_Z$. The $\mathbf{E} \times \mathbf{B}$ drift from this electric field causes the entire plasma to flow outward.

Poloidal Magnetic Field

The poloidal magnetic field $B_\theta$ is the magnetic field component that circulates around the minor cross-section of the torus. This magnetic field component cancels the charge separation caused by drift, enabling plasma confinement.

In a tokamak, the plasma current $I_p$ generates the poloidal magnetic field:

$$B_\theta(r) = \frac{\mu_0 I_p(r)}{2\pi r}$$

where $I_p(r)$ is the total current flowing within radius $r$. Using the current density distribution $j(r)$:

$$I_p(r) = \int_0^r j(r') \cdot 2\pi r' dr'$$

Confinement devices are classified by the method of generating the poloidal magnetic field:

Device	Poloidal Field Generation	Characteristics
Tokamak	Plasma current	Axisymmetric, requires current sustainment
Stellarator	External helical coils	Non-axisymmetric, no current required
RFP	Plasma current ($B_\theta \sim B_\phi$)	Weak toroidal field
Heliotron	External helical coils	Continuous helical coils

Helical Magnetic Field Lines

The combination of toroidal and poloidal magnetic fields causes field lines to spiral around the torus surface. The pitch angle $\alpha$ indicating the direction of field line progression is:

$$\tan\alpha = \frac{B_\theta}{B_\phi}$$

The field line equation is given by:

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

From this equation, the poloidal angle $d\theta$ traversed while advancing $d\phi$ in the toroidal direction is obtained:

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

Due to the helical structure, particles alternately pass through the inboard and outboard sides of the torus, and the $\nabla B$ drift is canceled on time average.

Flux Coordinates and Mathematical Description of Magnetic Surfaces

Poloidal Flux Function

In axisymmetric systems, the magnetic field can be described using a flux function $\psi$. The poloidal flux $\psi$ is defined as the magnetic flux of the toroidal field passing through a disk centered on the magnetic axis:

$$\psi = \frac{1}{2\pi} \int_S \mathbf{B} \cdot d\mathbf{S}$$

Assuming axisymmetry, the magnetic field is expressed as:

$$\mathbf{B} = \nabla\psi \times \nabla\phi + F(\psi)\nabla\phi$$

where the first term represents the poloidal component and the second term represents the toroidal component. $F(\psi) = RB_\phi$ is the poloidal current function.

In component form:

$$B_R = -\frac{1}{R}\frac{\partial\psi}{\partial Z}, \quad B_Z = \frac{1}{R}\frac{\partial\psi}{\partial R}, \quad B_\phi = \frac{F(\psi)}{R}$$

Toroidal Flux Function

The toroidal flux $\chi$ is defined as the magnetic flux passing through the poloidal cross-section:

$$\chi = \frac{1}{2\pi} \oint_{S_p} \mathbf{B} \cdot d\mathbf{S}$$

In flux coordinate systems, $\psi$ or $\chi$ is used as the radial coordinate.

Flux Coordinate System

Flux coordinates $(\psi, \theta^*, \zeta)$ are a coordinate system with magnetic surfaces as coordinate surfaces:

• $\psi$: Radial coordinate (flux function)
• $\theta^*$: Poloidal angle (defined in flux coordinates)
• $\zeta$: Toroidal angle

Field lines are represented as straight lines in flux coordinates. The contravariant representation of the magnetic field is:

$$\mathbf{B} = \nabla\psi \times \nabla\theta^* + \frac{1}{q}\nabla\zeta \times \nabla\psi$$

Alternatively, introducing a coordinate $\alpha$ along field lines:

$$\mathbf{B} = \nabla\psi \times \nabla\alpha, \quad \alpha = \theta^* - \frac{\zeta}{q}$$

where $\alpha$ is called the field line label, which distinguishes each field line on the same magnetic surface.

Boozer Coordinates

Boozer coordinates $(\psi, \theta_B, \zeta_B)$ are special flux coordinates chosen so that contours of magnetic field strength $B$ coincide with coordinate lines.

Definition condition for Boozer coordinates:

$$B^2 = B^2(\psi, \theta_B - N\zeta_B)$$

where $N$ is the toroidal mode number (or period number of quasi-symmetry).

The covariant representation of the magnetic field in Boozer coordinates is:

$$\mathbf{B} = \beta(\psi, \theta_B, \zeta_B)\nabla\psi + I(\psi)\nabla\theta_B + G(\psi)\nabla\zeta_B$$

where $I(\psi)$ and $G(\psi)$ are functions related to toroidal and poloidal currents, respectively.

Advantages of Boozer coordinates:

1. Simplified calculation of particle longitudinal invariants
2. Suitable for neoclassical transport analysis
3. Easy evaluation of quasi-symmetry

Properties of Magnetic Surfaces

A magnetic surface is a surface formed by field lines circulating around the torus. In flux coordinates, it is represented as a surface where $\psi = \text{const}$.

In ideal MHD equilibrium, magnetic surfaces coincide with isobaric surfaces (flux functions):

$$\mathbf{B} \cdot \nabla p = 0 \quad \Rightarrow \quad p = p(\psi)$$

Also, current flows on magnetic surfaces:

$$\mathbf{j} \cdot \nabla\psi = 0$$

The shape of magnetic surfaces is characterized by the Shafranov shift $\Delta$, which displaces the magnetic axis outward from the geometric center:

$$\Delta(r) = \Delta_0 + \int_0^r \frac{\beta_p(r') + l_i(r')/2}{r'} r' dr'$$

where $\beta_p$ is the poloidal beta and $l_i$ is the internal inductance.

Grad-Shafranov Equation

Derivation

The Grad-Shafranov (GS) equation determines the magnetic field configuration satisfying the magnetohydrodynamic equilibrium condition $\nabla p = \mathbf{j} \times \mathbf{B}$ in axisymmetric toroidal equilibrium.

Substituting Ampere’s law $\mu_0 \mathbf{j} = \nabla \times \mathbf{B}$:

$$\mu_0 \nabla p = (\nabla \times \mathbf{B}) \times \mathbf{B}$$

Substituting $\mathbf{B} = \nabla\psi \times \nabla\phi + F\nabla\phi$ in an axisymmetric system and rearranging yields the GS equation:

$$R^2 \nabla \cdot \left(\frac{\nabla\psi}{R^2}\right) = -\mu_0 R^2 \frac{dp}{d\psi} - F\frac{dF}{d\psi}$$

Expanding this:

$$R \frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial\psi}{\partial R}\right) + \frac{\partial^2\psi}{\partial Z^2} = -\mu_0 R^2 \frac{dp}{d\psi} - F\frac{dF}{d\psi}$$

Or, defining the operator $\Delta^*$:

$$\Delta^* \psi \equiv R \frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial\psi}{\partial R}\right) + \frac{\partial^2\psi}{\partial Z^2} = -\mu_0 R^2 p'(\psi) - FF'(\psi)$$

Boundary Conditions and Solution Methods

The GS equation is an elliptic partial differential equation solved with boundary conditions:

1. Fixed boundary problem: Specify the plasma boundary shape
2. Free boundary problem: Determine the boundary from external coil currents

Numerical methods include finite element methods, finite difference methods, and Green’s function methods.

Representative equilibrium codes:

Code	Developer	Features
EFIT	General Atomics	Reconstruction from experimental data
VMEC	PPPL	3D free boundary equilibrium
HELENA	JET	High-precision fixed boundary
CHEASE	EPFL	High-precision equilibrium calculation

Analytical Solution: Solov’ev Solution

Analytical solutions of the GS equation can be obtained by assuming special pressure and current distributions. For Solov’ev equilibrium:

$$p'(\psi) = \text{const} = A, \quad FF'(\psi) = \text{const} = C$$

The GS equation becomes:

$$\Delta^* \psi = -\mu_0 R^2 A - C$$

The solution is obtained in the form:

$$\psi = \frac{A}{8}(R^2 - R_0^2)^2 + \frac{C}{2}(R^2 - R_0^2) + D_1 + D_2 R^2 + D_3(R^4 - 4R^2 Z^2)$$

Beta Value and Shafranov Shift

The beta value characterizing plasma pressure is the ratio of plasma pressure to magnetic pressure:

$$\beta = \frac{\langle p \rangle}{B^2/(2\mu_0)}$$

Toroidal beta $\beta_t$ and poloidal beta $\beta_p$:

$$\beta_t = \frac{2\mu_0 \langle p \rangle}{B_0^2}, \quad \beta_p = \frac{2\mu_0 \langle p \rangle}{\langle B_\theta \rangle^2}$$

Higher beta values result in larger Shafranov shift, displacing the magnetic axis outward:

$$\Delta_0 \approx \frac{a^2}{R_0}\left(\beta_p + \frac{l_i}{2}\right)$$

Rotational Transform and Safety Factor

Definition of Rotational Transform

The rotational transform $\iota$ represents the poloidal angle traversed while field lines make one toroidal circuit around the torus:

$$\iota = \frac{1}{2\pi}\oint \frac{d\theta}{d\phi} d\phi = \frac{1}{2\pi}\oint \frac{r B_\phi}{R B_\theta} d\phi$$

In flux coordinates, simply:

$$\iota(\psi) = \frac{d\chi}{d\psi}$$

The stellarator community uses rotational transform $\iota$, while the tokamak community conventionally uses the safety factor $q = 1/\iota$.

Physical Meaning of the Safety Factor

The safety factor $q$ represents the number of toroidal circuits made while field lines complete one poloidal circuit:

$$q = \frac{1}{\iota} = \frac{1}{2\pi}\oint \frac{R B_\theta}{r B_\phi} d\theta$$

In the cylindrical approximation (large aspect ratio limit):

$$q(r) = \frac{r B_\phi}{R_0 B_\theta(r)}$$

More precisely, as an average over the magnetic surface:

$$q(\psi) = \frac{1}{2\pi} \oint \frac{\mathbf{B} \cdot \nabla\phi}{\mathbf{B} \cdot \nabla\theta^*} d\theta^*$$

Current Distribution and q Profile

The safety factor distribution is determined by the current density distribution $j(r)$. In the cylindrical approximation:

$$q(r) = \frac{2\pi r^2 B_\phi}{\mu_0 R_0 I_p(r)}$$

A typical current distribution model:

$$j(r) = j_0 \left(1 - \frac{r^2}{a^2}\right)^\nu$$

Then:

$$q(r) = q_a \frac{(1 + \nu)(r/a)^2}{1 - (1 - r^2/a^2)^{1+\nu}}$$

where $q_a$ is the safety factor at the boundary.

The central safety factor $q_0$ is:

$$q_0 = \lim_{r \to 0} q(r) = \frac{q_a}{1 + \nu}$$

For ohmic heated plasmas, $\nu \approx 1$ (parabolic distribution) is typical, giving $q_0 \approx q_a/2$.

Safety Factor and MHD Stability

The safety factor is directly related to plasma MHD stability:

$q$ Value	State	Physical Explanation
$q_0 < 1$	Internal kink mode unstable	$m=1, n=1$ mode
$q_0 = 1$	Sawtooth oscillations occur	Periodic relaxation phenomenon
$q_{95} < 2$	External kink mode unstable	Global deformation
$q_{95} > 3$	Stable operation region	Normal operating conditions

where $q_{95}$ is the safety factor at the surface containing 95% of the poloidal flux.

Kruskal-Shafranov limit:

$$q_a > 1 \quad \text{(external kink stability condition)}$$

In actual tokamak operation, maintaining $q_{95} > 3$ is standard.

Rational Surfaces and Mode Coupling

Magnetic surfaces where the safety factor is a rational number $q = m/n$ ($m$, $n$ are coprime integers) are called rational surfaces.

Characteristics of rational surfaces:

1. Field lines close after $n$ toroidal and $m$ poloidal circuits
2. Resonant response to perturbations
3. Become growth points for instabilities

Resonance condition for mode $(m, n)$:

$$m - nq(r_s) = 0 \quad \Rightarrow \quad r_s: \text{resonant surface location}$$

Typical rational surfaces and associated instabilities:

Rational Surface	$q$ Value	Associated Instability
$q = 1$	1	Internal kink, sawtooth
$q = 3/2$	1.5	Neoclassical tearing mode
$q = 2$	2	Tearing mode, external kink
$q = 3$	3	Edge instabilities

Magnetic Shear and Its Effects

Definition of Magnetic Shear

Magnetic shear $s$ is a dimensionless parameter representing the radial variation of the safety factor:

$$s = \frac{r}{q} \frac{dq}{dr} = \frac{d \ln q}{d \ln r}$$

In flux coordinates:

$$\hat{s} = \frac{\rho}{q} \frac{dq}{d\rho}$$

where $\rho = \sqrt{\psi_N}$ (normalized minor radius).

In a standard tokamak, $s > 0$ (positive shear), with $s = 0$ at the center and $s \sim 1-2$ at the edge.

Stabilizing Effect of Magnetic Shear

Magnetic shear suppresses many instabilities. Physical mechanism:

1. Field line pitch differs for each magnetic surface
2. Perturbation phases become misaligned (phase mixing)
3. Spatial growth of perturbations is suppressed

Ballooning mode stability condition (first stability region):

$$s > s_{\text{crit}}(\alpha) \quad \text{where} \quad \alpha = -q^2 R \frac{d\beta}{dr}$$

where $\alpha$ is the normalized pressure gradient.

Stronger shear improves stability limits, but excessively strong shear can degrade particle confinement.

Local Shear and Global Shear

Magnetic shear can vary locally. It is modified especially by plasma rotation and bootstrap current:

$$s_{\text{eff}} = s_0 + \Delta s_{\text{rotation}} + \Delta s_{\text{bootstrap}}$$

Shear modification by bootstrap current:

$$\Delta s_{\text{bootstrap}} \propto -\frac{q^2}{r} \frac{dp}{dr} \sqrt{\epsilon}$$

where $\epsilon = r/R$ is the inverse aspect ratio.

Magnetic Islands and Resistive Instabilities

Formation of Magnetic Islands

Magnetic islands are structures formed when magnetic field line reconnection occurs near rational surfaces due to finite plasma resistivity.

The linear growth rate of tearing modes is:

$$\gamma = 0.55 \left(\frac{\eta}{\mu_0}\right)^{3/5} \left(\frac{k_\parallel v_A}{\Delta'}\right)^{-2/5} (\Delta')^{4/5}$$

where $\eta$ is resistivity, $v_A$ is Alfven velocity, and $\Delta'$ is the stability parameter.

Definition of $\Delta'$:

$$\Delta' = \frac{1}{\tilde{\psi}(r_s)} \left[\frac{d\tilde{\psi}}{dr}\bigg|_{r_s^+} - \frac{d\tilde{\psi}}{dr}\bigg|_{r_s^-}\right]$$

For $\Delta' > 0$, the tearing mode is unstable.

Structure of Magnetic Islands

The magnetic island width $w$ is determined by the perturbed flux $\tilde{\psi}$:

$$w = 4\sqrt{\frac{r_s \tilde{\psi}}{m B_\theta'}}$$

where $B_\theta' = dB_\theta/dr$.

Helical flux function within the magnetic island:

$$\psi^* = \frac{B_\theta'}{2r_s}(r - r_s)^2 + \tilde{\psi}\cos(m\theta - n\zeta)$$

The separatrix (boundary between island and exterior) is defined by $\psi^* = \tilde{\psi}$.

Island O-point (center) and X-point (saddle point):

$$\text{O-point}: \quad r = r_s, \quad m\theta - n\zeta = \pi \pmod{2\pi}$$ $$\text{X-point}: \quad r = r_s, \quad m\theta - n\zeta = 0 \pmod{2\pi}$$

Saturation of Magnetic Islands and Confinement Degradation

As magnetic islands grow, plasma pressure flattens within the island. This causes:

1. Local loss of confinement
2. Increased radial transport of heat and particles
3. Plasma performance degradation

Degradation of energy confinement time:

$$\frac{\Delta \tau_E}{\tau_E} \approx -C \left(\frac{w}{a}\right)^2$$

Typically, significant degradation occurs for $w/a > 0.1$.

Neoclassical Tearing Modes

NTM Generation Mechanism

Neoclassical Tearing Modes (NTM) are resistive instabilities driven by bootstrap current deficit.

Bootstrap current density:

$$j_{BS} = -\frac{2.44}{\sqrt{1 - \epsilon^2}} \frac{p'}{B_\theta} \sqrt{\epsilon}$$

When pressure flattens within a magnetic island, bootstrap current also decreases locally. This current deficit provides positive feedback that further grows the magnetic island.

Rutherford Equation

The time evolution of magnetic island width is described by the modified Rutherford equation:

$$\tau_R \frac{dw}{dt} = r_s \left[\Delta' + \Delta'_{BS} - \Delta'_{GGJ} - \Delta'_{pol}\right]$$

Meaning of each term:

• $\Delta'$: Classical tearing stability parameter
• $\Delta'_{BS}$: Bootstrap current term (destabilizing)
• $\Delta'_{GGJ}$: Glasser-Greene-Johnson term (stabilizing)
• $\Delta'_{pol}$: Poloidal flux term (stabilizing)

Bootstrap term:

$$\Delta'_{BS} = \frac{\beta_p \sqrt{\epsilon}}{w}$$

GGJ term (small island limit):

$$\Delta'_{GGJ} = -\frac{w_d}{w^2}$$

where $w_d$ is the critical island width.

NTM Threshold and Saturation

NTM requires a seed island. Growth begins when the seed island width $w_{\text{seed}}$ exceeds the critical width $w_c$:

$$w_c \approx w_d = 1.3 \rho_{\theta i} \sqrt{\beta_p}$$

where $\rho_{\theta i}$ is the ion poloidal Larmor radius.

The saturated island width from $dw/dt = 0$:

$$w_{\text{sat}} \approx \frac{\beta_p \sqrt{\epsilon}}{|\Delta'| + w_d/w_{\text{sat}}}$$

Typically $w_{\text{sat}}/a \sim 0.05-0.15$.

NTM Control

Electron cyclotron current drive (ECCD) is effective for NTM control:

$$\Delta'_{ECCD} = -\frac{\eta_{CD} j_{ECCD} w}{B_\theta' w^2/4r_s}$$

ECCD deposits localized current at the O-point of the magnetic island to compensate for the lost bootstrap current.

NTM control requirements for ITER:

Parameter	Required Value
Power deposition width	$< 5$ cm
Aiming accuracy	$< 2$ cm
Response time	$< 1$ s

Boundary Configuration: Limiter and Divertor

Limiter Configuration

The limiter configuration defines the plasma boundary using solid structures.

Characteristics:

1. Simple magnetic field structure
2. Last closed flux surface (LCFS) contacts the limiter
3. Short scrape-off layer
4. Difficult impurity control

Heat load at the limiter:

$$q_{\parallel} = \frac{P_{\text{SOL}}}{2\pi R \lambda_q \cdot 2\pi}$$

where $\lambda_q$ is the e-folding length of the scrape-off layer.

Challenges of limiter configuration:

• High-Z impurities easily enter the plasma
• Heat load concentration
• Difficulty in helium ash exhaust

Fundamentals of Divertor Configuration

The divertor configuration uses a magnetic field structure with X-points to separate the exhaust region from the main plasma.

Magnetic field structure characteristics:

• Existence of separatrix
• Magnetic field null at X-point
• Formation of private region

Magnetic field near the separatrix:

$$B_R \approx B'_{RZ}(R - R_X), \quad B_Z \approx B'_{ZR}(Z - Z_X)$$

Zero magnetic field condition at X-point:

$$B_R(R_X, Z_X) = B_Z(R_X, Z_X) = 0$$

Single Null Configuration

The Single Null (SN) configuration with one X-point is the most common divertor configuration.

Lower Single Null (LSN):

• X-point located at the bottom
• Favorable for debris falling by gravity
• Easier H-mode transition when ion $\nabla B$ drift is toward the X-point

Upper Single Null (USN):

• X-point located at the top
• Used for special operation scenarios

Field expansion near the X-point:

$$\psi \approx \psi_X + \frac{1}{2}(\psi_{RR}(R-R_X)^2 + \psi_{ZZ}(Z-Z_X)^2)$$

Double Null Configuration

The Double Null (DN) configuration has two X-points symmetrically positioned above and below.

Advantages:

• Heat load distributed across four divertor legs
• Vertical symmetry of magnetic surfaces
• Compatible with high triangularity plasmas

Challenges:

• Difficult magnetic balance control between X-points
• Heat load concentration on one side with slight asymmetry
• Asymmetric behavior during H-L transitions

Balance condition:

$$\psi_{\text{upper}} = \psi_{\text{lower}} \quad \text{(magnetically symmetric)}$$

Double null coefficient:

$$\sigma = \frac{\delta\psi_{\text{upper}} - \delta\psi_{\text{lower}}}{\delta\psi_{\text{upper}} + \delta\psi_{\text{lower}}}$$

$\sigma = 0$ for perfect symmetry, $|\sigma| = 1$ corresponds to single null.

Snowflake Configuration

The Snowflake configuration has zero magnetic field gradient near the X-point, forming a second-order null.

Condition:

$$B_R = B_Z = \frac{\partial B_R}{\partial R} = \frac{\partial B_Z}{\partial Z} = 0$$

Magnetic field expansion near the X-point:

$$B \propto r^2 \quad \text{(for a normal X-point, } B \propto r \text{)}$$

Advantages:

1. Increased wetted area due to flux expansion
2. Increased connection length (enhanced radiative cooling)
3. Reduced heat load

Experimental devices: Demonstrated at TCV (Switzerland) and NSTX (USA)

Snowflake configuration variations:

Configuration	X-point Structure	Features
SF-plus	Two X-points close together	Heat load distribution
SF-minus	Secondary X-point on private side	Favorable for exhaust
Exact SF	Perfect second-order null	Ideal but difficult to control

Super-X Divertor

The Super-X divertor extends the outer divertor leg in the major radius direction.

Design principle:

$$\frac{R_t}{R_u} > 1.5 \quad \text{(target radius/upstream radius)}$$

Advantages:

1. Reduced total flux density at target
2. Increased wetted area
3. Enhanced radiative losses
4. Easier detachment achievement

Demonstration at MAST-U:

$$\frac{B_{\text{target}}}{B_{\text{upstream}}} = \frac{R_u}{R_t} \cdot \frac{B_{\phi,u}}{B_{\phi,t}}$$

Heat load reduction factor in Super-X:

$$f_{\text{reduction}} \approx \left(\frac{R_t}{R_u}\right)^2$$

Comparison of Advanced Divertor Configurations

Configuration	Flux Expansion	Connection Length	Controllability	Practicality
Standard SN	1	1	High	Demonstrated
DN	1	1	Medium	Demonstrated
Snowflake	2-3	2-3	Low	Research stage
Super-X	2-4	2-4	Medium	Being demonstrated
X-divertor	1.5-2	1.5	Medium	Research stage

Advanced Tokamak Configurations

Reversed Shear Configuration

The Reversed Shear (negative central shear, NCS) configuration has a minimum safety factor in the plasma core.

$$\frac{dq}{dr}\bigg|_{r < r_{\min}} < 0, \quad \frac{dq}{dr}\bigg|_{r > r_{\min}} > 0$$

q profile shape:

$$q(r) = q_0 + (q_{\min} - q_0)\left(\frac{r}{r_{\min}}\right)^{\alpha_1} + (q_a - q_{\min})\left(\frac{r - r_{\min}}{a - r_{\min}}\right)^{\alpha_2}$$

Advantages of reversed shear configuration:

1. Formation of internal transport barriers (ITB)
2. High confinement performance
3. High bootstrap current fraction
4. Compatibility with steady-state operation

Physical mechanism:

• $E \times B$ shearing rate increases in the reversed shear region
• Transport barrier formation through turbulence suppression
• Ion thermal diffusivity reduced to neoclassical level

$$\chi_i \to \chi_i^{\text{neo}} \quad \text{(inside ITB)}$$

Internal Transport Barrier (ITB)

An ITB is a region where radial heat and particle transport are locally suppressed.

ITB formation condition (empirical rule):

$$\omega_{E\times B} > \gamma_{\text{max}}$$

where $\omega_{E\times B}$ is the $E \times B$ shearing rate and $\gamma_{\text{max}}$ is the maximum linear growth rate.

$E \times B$ shearing rate:

$$\omega_{E\times B} = \frac{RB_\theta}{B}\frac{\partial}{\partial r}\left(\frac{E_r}{RB_\theta}\right)$$

The ITB footpoint (barrier location) is often locked to a rational surface near the $q_{\min}$ surface.

Hybrid Scenario

The hybrid scenario is an intermediate operating mode between H-mode and advanced tokamak.

Characteristics:

• Flat q profile in the core ($q_0 \sim 1$)
• Avoidance of sawtooth oscillations
• High beta value with good confinement
• Moderate bootstrap current fraction

q profile:

$$q_0 \approx 1.0-1.1, \quad q_{95} \approx 4-5$$

Normalized beta:

$$\beta_N = \frac{\beta_t a B_0}{I_p} \approx 2.5-3.5$$

Hybrid vs. Advanced Tokamak:

Parameter	Hybrid	Advanced Tokamak
$q_0$	$\sim 1$	$> 1.5$
ITB	Weak or none	Strong
$f_{BS}$	30-50%	50-80%
Pulse duration	Long pulse	Steady state

Fully Non-Inductive Current Drive

The ultimate goal of the advanced tokamak is fully non-inductive operation through a combination of bootstrap current and external current drive.

Current balance:

$$I_p = I_{BS} + I_{CD} + I_{OH}$$

Steady-state operation condition:

$$I_{OH} = 0, \quad I_p = I_{BS} + I_{CD}$$

Bootstrap current fraction:

$$f_{BS} = \frac{I_{BS}}{I_p} = C_1 \epsilon^{1/2} \beta_p q$$

To achieve high bootstrap current fraction:

• High beta operation
• High q profile
• High inverse aspect ratio

Current drive efficiency:

$$\eta_{CD} = \frac{n_e R I_{CD}}{P_{CD}} \quad [\text{A/W/m}^2]$$

Efficiency of each current drive method:

Method	$\eta_{CD}$ [10^20 A/W/m^2]	Position Control
NBI	0.2-0.4	Low
ECCD	0.1-0.2	High
LHCD	0.3-0.5	Medium
ICCD	0.05-0.1	Low

Comparison of Tokamak and Stellarator Configurations

Characteristics of Tokamak Configuration

The tokamak has an axisymmetric configuration and generates the poloidal magnetic field through plasma current.

Axisymmetry of magnetic field:

$$\frac{\partial \mathbf{B}}{\partial \phi} = 0$$

Characteristics:

• Good particle orbits due to axisymmetry (neoclassical transport)
• Requires plasma current sustainment (challenge for steady-state operation)
• Disruption risk
• Relatively simple coil geometry
• Demonstrated high confinement performance

Neoclassical transport coefficient (banana regime):

$$D_{neo} \approx q^2 \rho_i^2 \nu_{ii} \epsilon^{-3/2}$$

Characteristics of Stellarator Configuration

The stellarator has a non-axisymmetric configuration and forms the complete magnetic configuration using only external coils.

Three-dimensionality of magnetic field:

$$B = B(\psi, \theta, \zeta) \neq B(\psi, \theta)$$

Characteristics:

• No plasma current required (inherently capable of steady-state operation)
• No disruptions
• Complex three-dimensional coil geometry
• Requires quasi-symmetry optimization

Rotational transform in stellarators:

$$\iota = \iota_{\text{ext}} + \iota_{\text{plasma}}$$

The plasma current contribution $\iota_{\text{plasma}}$ is small, and $\iota_{\text{ext}}$ dominates.

Quasi-Symmetry Optimization

Modern stellarators optimize particle confinement using the concept of quasi-symmetry.

Quasi-symmetry condition:

$$B = B(\psi, M\theta_B - N\zeta_B)$$

where $M$, $N$ are integers characterizing the symmetry.

Types of quasi-symmetry:

Quasi-Symmetry	$(M, N)$	Features	Representative Device
Quasi-Axisymmetric (QA)	$(1, 0)$	Tokamak-like orbits	NCSX (cancelled)
Quasi-Helically Symmetric (QH)	$(1, N)$	Helical direction symmetry	HSX
Quasi-Isodynamic (QI)	-	Closed $	B

Improvement of neoclassical transport through quasi-symmetry:

$$D_{neo}^{QS} \ll D_{neo}^{\text{conventional stellarator}}$$

Configuration Selection and Reactor Design

Design Requirements

In fusion reactor design, the choice of magnetic configuration affects the following factors:

1. Confinement performance (energy confinement time)
2. Stability (beta limits, disruption avoidance)
3. Steady-state operability (current drive efficiency, bootstrap current)
4. Engineering feasibility (coil manufacturing, maintainability, neutron shielding)
5. Economics (device size, construction cost, power generation efficiency)

Comparison of Key Parameters

Parameter	Tokamak	Stellarator	Spherical Tokamak
$\beta$ limit	3-5%	3-5%	10-40%
$\tau_E$ scaling	Demonstrated	Under development	Being verified
Disruption	Yes	No	Yes
Steady-state	Requires current drive	Inherently possible	Requires current drive
Coil complexity	Medium	High	Low

Future Reactor Design

Currently, tokamaks are the most advanced in research, and both ITER and DEMO adopt the tokamak configuration.

ITER design parameters:

Parameter	Value
$R_0$	6.2 m
$a$	2.0 m
$B_0$	5.3 T
$I_p$	15 MA
$q_{95}$	3.0
Configuration	Single Null

Meanwhile, stellarators are attractive as power reactors from the perspective of steady-state operation and disruption avoidance. The results from Wendelstein 7-X will influence future directions.

Spherical tokamaks are being developed by private fusion ventures (such as Tokamak Energy) due to their potential for high-beta, compact designs.

Summary

Magnetic configuration is the most important factor determining confinement and stability of fusion plasmas. The main concepts explained on this page are summarized below:

1. Flux coordinates and Boozer coordinates are the foundation for mathematical description of magnetic configurations
2. The Grad-Shafranov equation determines axisymmetric equilibrium
3. The safety factor q profile governs MHD stability
4. Magnetic shear suppresses many instabilities
5. Magnetic islands and NTM cause confinement degradation and require active control
6. Divertor configurations (especially advanced divertors) are important for solving heat exhaust problems
7. Advanced tokamak configurations (reversed shear, hybrid) provide pathways to steady-state operation

Based on these understandings, burning plasma experiments at ITER and subsequent power reactor designs are being advanced.

Related Topics

• Confinement Methods: Overview - Overview of confinement methods
• Tokamak - The most advanced magnetic confinement method
• Stellarator/Helical - Magnetic confinement method with excellent steady-state capability
• MHD Equilibrium and Stability - Mechanical equilibrium of plasmas
• Particle Motion - Motion of charged particles in magnetic fields
• Heating and Current Drive - Methods for plasma heating and current drive