Plasma Transport

The performance of fusion reactors depends critically on how well particles and energy are confined within the plasma. Plasma transport refers to the phenomena by which plasma particles and heat move across magnetic surfaces, and understanding and controlling this transport is key to realizing fusion energy.

This chapter systematically explains the physics of plasma transport, from classical transport theory to the latest gyrokinetic simulations. Understanding transport phenomena has a history of over a century and represents a field where achievements from statistical mechanics, plasma physics, and nonlinear dynamics converge.

Historical Development of Transport Theory

From Gas Kinetic Theory to Plasma Transport

The origins of plasma transport theory trace back to 19th-century gas kinetic theory. The molecular kinetic theory established by Maxwell and Boltzmann explained diffusion, thermal conduction, and viscosity in gases from microscopic molecular collisions.

In 1906, Lorentz proposed the “Lorentz gas” model in which electrons move through fixed ions, theoretically explaining electrical conduction in metals. This model became the prototype for plasma transport theory.

In the 1950s, Spitzer and Haerm rigorously calculated transport coefficients for fully ionized plasmas, completing “classical transport theory” based on Coulomb collisions. Their theory provides plasma electrical resistivity, thermal conductivity, and diffusion coefficients, and is still used as the foundational theory today.

The Advent of Tokamaks and Discovery of Anomalous Transport

In the late 1960s, when the Soviet T-3 tokamak demonstrated confinement performance surpassing previous devices, the measured transport greatly exceeded predictions from classical theory. This discovery of “anomalous transport” became the starting point for turbulent transport research.

In 1968, a team from the UK Culham Laboratory independently verified the T-3 performance, and the results led to the acceleration of tokamak research worldwide. However, it also became clear that the physics of transport that determines confinement performance was not yet understood.

Establishment of Neoclassical Transport Theory

In the 1970s, Galeev, Sagdeev, Rosenbluth, Hinton, and others theoretically studied collisional transport in toroidal magnetic configurations. They incorporated geometric effects of particle orbits (banana orbits) and established “neoclassical transport theory.”

The most important achievement of neoclassical theory was the theoretical prediction of the bootstrap current. In 1971, Bickerton, Connor, and Taylor predicted a toroidal current that flows spontaneously due to pressure gradients, which was experimentally confirmed at JET in 1988.

Development of Turbulent Transport Theory

From the 1980s to the 1990s, theories of drift wave instabilities and the turbulent transport they cause developed. Through the research of Horton, Tang, Waltz, and others, the physics of ion temperature gradient (ITG) modes, trapped electron modes (TEM), and electron temperature gradient (ETG) modes became clear.

In 1982, H-mode (high confinement mode) was discovered at the ASDEX tokamak. This brought the new physical concept of turbulence suppression by $E \times B$ sheared flow, and research on transport barriers began in earnest.

The Era of Gyrokinetic Simulations

From the late 1990s, improvements in computing power made large-scale simulations based on gyrokinetics possible. Codes such as GYRO, GS2, GENE, and GTC were developed, opening the path to “first-principles prediction” of turbulent transport.

Since the 2000s, quantitative comparison between simulation results and experiments (verification and validation, V&V) has progressed, providing a theoretical foundation for ITER performance predictions.

Classical Transport Theory

Classical transport is transport phenomena arising solely from binary Coulomb collisions between particles. It theoretically describes how charged particles in a uniform magnetic field diffuse and conduct heat.

Larmor Motion and Collisions

A charged particle in magnetic field $\mathbf{B}$ gyrates around magnetic field lines with Larmor radius $\rho_L$:

$$\rho_L = \frac{mv_\perp}{|q|B}$$

where $m$ is the particle mass, $v_\perp$ is the velocity component perpendicular to the magnetic field, $q$ is the charge, and $B$ is the magnetic field strength. Using the thermal velocity $v_{th} = \sqrt{2T/m}$:

$$\rho_i = \frac{\sqrt{2m_i T_i}}{e B}, \quad \rho_e = \frac{\sqrt{2m_e T_e}}{e B}$$

For typical fusion plasma parameters ($T = 10$ keV, $B = 5$ T):

$$\rho_i \approx 3 \, \text{mm}, \quad \rho_e \approx 0.07 \, \text{mm}$$

The ratio of ion to electron Larmor radii is:

$$\frac{\rho_i}{\rho_e} = \sqrt{\frac{m_i}{m_e}} \approx 43 \quad (\text{for hydrogen})$$

Because of this, ions tend to move across larger distances than electrons.

Collision Frequency and Mean Free Path

The Coulomb collision frequency is derived from the collision cross section and plasma parameters. The electron-ion collision frequency is:

$$\nu_{ei} = \frac{4\sqrt{2\pi}}{3} \frac{n_e Z^2 e^4 \ln\Lambda}{(4\pi\varepsilon_0)^2 m_e^{1/2} T_e^{3/2}}$$

where $\ln\Lambda$ is the Coulomb logarithm. This is the logarithm of the ratio of the minimum collision parameter (Landau length $r_L = e^2/(4\pi\varepsilon_0 T)$) to the maximum collision parameter (Debye length $\lambda_D$):

$$\ln\Lambda = \ln\left(\frac{\lambda_D}{r_L}\right) = \ln\left(\frac{12\pi n \lambda_D^3}{Z}\right)$$

In fusion plasmas, $\ln\Lambda \approx 15-20$. Numerically:

$$\nu_{ei} \approx 2.9 \times 10^{-12} \frac{n_e Z \ln\Lambda}{T_e^{3/2}} \quad [\text{s}^{-1}]$$

where $n_e$ is in $\text{m}^{-3}$ and $T_e$ is in eV.

The electron mean free path is:

$$\lambda_{mfp,e} = \frac{v_{th,e}}{\nu_{ei}} \approx 10^4 \frac{T_e^2}{n_{20} Z \ln\Lambda} \quad [\text{m}]$$

At $T_e = 10$ keV and $n = 10^{20}$ m$^{-3}$, $\lambda_{mfp,e} \sim 10$ km, which is much longer than the plasma diameter (several meters).

The ion-ion collision frequency is:

$$\nu_{ii} = \frac{4\sqrt{\pi}}{3} \frac{n_i Z^4 e^4 \ln\Lambda}{(4\pi\varepsilon_0)^2 m_i^{1/2} T_i^{3/2}} \approx \sqrt{\frac{m_e}{m_i}} \nu_{ei}$$

The electron-electron collision frequency is similarly defined:

$$\nu_{ee} \approx \nu_{ei} / Z$$

Classical Diffusion Coefficient

When particles diffuse due to collisions in a uniform magnetic field, the diffusion coefficient is derived from random walk arguments. Particles move across field lines by approximately $\rho_L$ each collision, with time between collisions $\tau_c = 1/\nu$.

$$D_c \sim \frac{(\Delta x)^2}{\Delta t} \sim \frac{\rho_L^2}{\tau_c} = \nu \rho_L^2$$

More precisely, the classical diffusion coefficient for electrons is:

$$D_{c,e} = \frac{\nu_{ei} \rho_e^2}{2} = \frac{\nu_{ei} m_e T_e}{e^2 B^2}$$

For ions:

$$D_{c,i} = \frac{\nu_{ii} \rho_i^2}{2} = \frac{\nu_{ii} m_i T_i}{e^2 B^2}$$

From the mass dependence, $D_{c,i} / D_{c,e} \sim \sqrt{m_i/m_e}$, so ion diffusion dominates.

Numerically, for typical tokamak parameters ($T = 10$ keV, $B = 5$ T, $n = 10^{20}$ m$^{-3}$):

$$D_{c,i} \sim 10^{-4} \, \text{m}^2/\text{s}$$

This is about $10^4$ times smaller than the observed diffusion coefficient ($\sim 1$ m$^2$/s). This enormous discrepancy is at the heart of the “anomalous transport” problem.

Classical Thermal Conduction

The thermal conductivity $\chi$ follows similar scaling. In Braginskii’s transport theory, the thermal conductivity perpendicular to the magnetic field is:

$$\chi_\perp = n \nu \rho^2$$

In the direction along the field lines (parallel direction), since there is no constraint by the magnetic field:

$$\chi_\parallel = n \lambda_{mfp} v_{th} = \frac{n v_{th}^2}{\nu}$$

where $\lambda_{mfp} = v_{th}/\nu$ is the mean free path.

The ratio of perpendicular to parallel is:

$$\frac{\chi_\parallel}{\chi_\perp} \sim \left(\frac{\lambda_{mfp}}{\rho}\right)^2 = \left(\frac{v_{th}}{\nu \rho}\right)^2 = \left(\frac{\omega_c}{\nu}\right)^2$$

where $\omega_c = eB/m$ is the cyclotron frequency. In fusion plasmas, $\omega_c/\nu \sim 10^8$, and heat is conducted extremely quickly along field lines.

Braginskii Transport Equations

Braginskii derived a complete set of fluid equations describing macroscopic transport in magnetized plasmas. The momentum transport equation:

$$m_i n_i \frac{d\mathbf{v}_i}{dt} = -\nabla p_i - \nabla \cdot \boldsymbol{\pi}_i + e n_i (\mathbf{E} + \mathbf{v}_i \times \mathbf{B}) + \mathbf{R}_{ie}$$

where $\boldsymbol{\pi}$ is the viscous stress tensor and $\mathbf{R}_{ie}$ is the electron-ion friction force.

The heat transport equation:

$$\frac{3}{2} n_s \frac{dT_s}{dt} + p_s \nabla \cdot \mathbf{v}_s = -\nabla \cdot \mathbf{q}_s + Q_s$$

The heat flux vector $\mathbf{q}$ is separated into parallel and perpendicular components:

$$\mathbf{q} = -\chi_\parallel \nabla_\parallel T - \chi_\perp \nabla_\perp T + \chi_\wedge \hat{\mathbf{b}} \times \nabla T$$

The third term is the diamagnetic heat flux, representing heat transport in the direction perpendicular to both the temperature gradient and the magnetic field.

Spitzer Resistivity

The electrical resistivity of plasma was theoretically derived by Spitzer:

$$\eta_\parallel = \frac{m_e \nu_{ei}}{n_e e^2} = \frac{4\sqrt{2\pi}}{3} \frac{Z e^2 \ln\Lambda}{(4\pi\varepsilon_0)^2 T_e^{3/2}}$$

Numerically:

$$\eta_\parallel \approx 2.8 \times 10^{-8} \frac{Z \ln\Lambda}{T_e^{3/2}} \quad [\Omega \cdot \text{m}]$$

At $T_e = 10$ keV, $\eta \sim 10^{-9}$ $\Omega \cdot \text{m}$, which shows lower resistivity than copper (approximately $10^{-8}$ $\Omega \cdot \text{m}$).

The resistivity perpendicular to the magnetic field is:

$$\eta_\perp = \frac{\eta_\parallel}{1 + (\omega_{ce}/\nu_{ei})^2} \approx \frac{\eta_\parallel}{(\omega_{ce}/\nu_{ei})^2}$$

In magnetized plasmas, since $\omega_{ce}/\nu_{ei} \gg 1$, the perpendicular resistivity is orders of magnitude smaller than the parallel resistivity.

This extremely low resistivity is the reason why Ohmic heating efficiency decreases at high temperatures, and is the basis for requiring auxiliary heating. The Ohmic heating power density:

$$P_{OH} = \eta j^2 \propto T^{-3/2}$$

Neoclassical Transport Theory

Neoclassical transport is collisional transport theory in toroidal magnetic configurations. In tokamaks and stellarators, since the magnetic field strength varies spatially, particle orbits become complex, requiring corrections from classical theory.

Non-uniformity of Toroidal Magnetic Field

The magnetic field strength in a tokamak varies inversely with the major radius $R$:

$$B(\theta) \approx \frac{B_0 R_0}{R_0 + r\cos\theta} \approx B_0 (1 - \epsilon \cos\theta)$$

where $\epsilon = r/R_0$ is the inverse aspect ratio and $\theta$ is the poloidal angle. The field is weak on the outboard side ($\theta = 0$) and strong on the inboard side ($\theta = \pi$).

The ratio of this field variation (mirror ratio) is:

$$\frac{B_{max}}{B_{min}} = \frac{1 + \epsilon}{1 - \epsilon} \approx 1 + 2\epsilon$$

Classification of Particles: Passing and Trapped

Due to the non-uniformity of the magnetic field, particles are classified into two types:

Passing particles have sufficient velocity $v_\parallel$ along the field lines to circulate around the torus.

Trapped particles are reflected at the high-field inboard side and are confined to the outboard region. This is due to the magnetic mirror effect.

The condition for particle trapping is derived from conservation of the magnetic moment $\mu = mv_\perp^2/(2B)$. From energy conservation:

$$\frac{1}{2}m v_\parallel^2 + \mu B = \text{const} = E$$

Particles are reflected at points where $v_\parallel = 0$. The trapping condition:

$$\mu B_{max} = E \quad \Rightarrow \quad \frac{v_\perp^2}{v^2} > \frac{B_{max} - B_{min}}{B_{max}} \approx 2\epsilon$$

Assuming an isotropic distribution in velocity space, the fraction of trapped particles $f_t$ is:

$$f_t = \sqrt{1 - \frac{B_{min}}{B_{max}}} \approx \sqrt{2\epsilon}$$

For a typical tokamak with $\epsilon = 0.3$, approximately 77% are passing particles and approximately 23% are trapped particles.

Banana Orbits

Trapped particles trace banana-shaped orbits in the poloidal cross-section while being reflected in the high-field region. This characteristic shape arises from the combination of magnetic gradient drift and curvature drift.

The tips (reflection points) of banana orbits are located on the inboard (high-field) side. The width of the orbit (banana width) is:

$$\Delta_b \approx \frac{q\rho_p}{\sqrt{\epsilon}}$$

where $\rho_p = mv/(eB_p)$ is the poloidal Larmor radius and $q$ is the safety factor. Since the poloidal field $B_p$ is typically 1/10-1/5 of the toroidal field:

$$\rho_p = \frac{B_T}{B_p} \rho \approx \frac{R}{r} q \rho$$

Therefore, the banana width is:

$$\Delta_b \approx \frac{q^2}{\sqrt{\epsilon}} \frac{R}{r} \rho$$

The banana width for ions can reach several centimeters, 10-100 times larger than the Larmor radius.

The time to complete one banana orbit (bounce time) is:

$$\tau_b = \frac{2\pi}{\omega_b} = \frac{qR}{\sqrt{\epsilon} v_{th}}$$

Collisionality Regimes

Neoclassical transport is classified into three regimes based on the ratio of collision frequency to the frequency at which particles complete a banana orbit (bounce frequency).

The bounce frequency is:

$$\omega_b = \frac{\sqrt{\epsilon} v_{th}}{qR}$$

The normalized collisionality is defined as:

$$\nu^* = \frac{\nu}{\omega_b} = \frac{\nu q R}{\sqrt{\epsilon} v_{th}}$$

This dimensionless parameter determines whether particles collide before completing a banana orbit.

Banana Regime (Collisionless Regime): $\nu^* < 1$

This is the regime where trapped particles can complete banana orbits without collisions. It is realized in high-temperature, low-density plasmas.

The diffusion coefficient is enhanced by the effect of trapped particles. Since the fraction of trapped particles is $f_t \sim \sqrt{\epsilon}$ and the step size is the banana width $\Delta_b \sim q\rho_p/\sqrt{\epsilon}$:

$$D_{banana} \sim f_t^2 \Delta_b^2 \nu \sim \epsilon^{-3/2} q^2 D_c$$

The neoclassical diffusion coefficient in the banana regime is:

$$D_{nc,\perp}^{banana} = -\frac{c_1}{\epsilon^{3/2}} \frac{q^2 \rho_p^2 \nu}{\epsilon} \frac{1}{n} \frac{dn}{dr}$$

where $c_1$ is a numerical coefficient ($\sim 1$).

The thermal conductivity shows similar scaling:

$$\chi_i^{banana} \approx 1.35 \frac{q^2}{\epsilon^{3/2}} \rho_i^2 \nu_{ii}$$

The enhancement factor over classical values, $q^2/\epsilon^{3/2}$, is approximately 20 for typical tokamak parameters ($q \sim 2$, $\epsilon \sim 0.3$).

Plateau Regime (Intermediate Regime): $1 < \nu^* < \epsilon^{-3/2}$

In the regime of moderate collisionality, the diffusion coefficient shows a “plateau” independent of $\nu$ due to resonance effects.

In this regime, particles that are detrapped (trapped particles becoming passing particles) by collisions are in dynamic equilibrium with those being re-trapped. Transport is dominated by “resonant” particles, namely those passing near $v_\parallel \approx 0$:

$$D_{plateau} \sim \frac{q v_{th} \rho^2}{R}$$

Transport in this regime is dominated by the resonant interaction between motion parallel to field lines and collisions.

Collisional Regime (Pfirsch-Schluter Regime): $\nu^* > \epsilon^{-3/2}$

This is the regime where collisionality is high and particles cannot move freely along field lines. It is realized in low-temperature, high-density plasmas.

In this regime, banana orbits do not form, and particles are localized by collisions. However, pressure-driven flow due to the toroidal geometry (Pfirsch-Schluter flow) causes additional transport:

$$D_{PS} = (1 + 2q^2) D_c$$

The enhancement factor over classical values, $(1 + 2q^2)$, reflects the effect of flow due to toroidal geometry.

Unified Expression for Neoclassical Transport Coefficients

Hinton and Hazeltine derived an expression that uniformly describes each regime:

$$D_{nc} = D_c \left[ 1 + 2q^2 + \frac{\epsilon^{3/2}}{\sqrt{\nu^*}} + \frac{f(\nu^*)}{\nu^*} \right]$$

where $f(\nu^*)$ is a function that smoothly interpolates between collisionality regimes.

For more precise calculations, the Hirshman-Sigmar model or the NCLASS code are used. These include multi-species effects, impurities, and finite aspect ratio effects.

Bootstrap Current

One of the most important achievements of neoclassical theory is the discovery of the bootstrap current. This is a toroidal current that flows spontaneously due to pressure gradients, occurring without external drive.

Physical Mechanism

In the banana regime, trapped particles on average flow in one direction along field lines. This parallel flow is driven by pressure gradients.

The physical explanation is as follows. When there is a pressure gradient, the particle density differs between the inboard and outboard sides of banana orbits. Particles reflected at the tips of banana orbits (inboard side) spend more time on the outboard path of their orbit.

This density asymmetry gives trapped particles a net toroidal precession, which drives current. Furthermore, friction between trapped and passing particles induces a parallel flow in passing particles as well.

Bootstrap Current Density

The bootstrap current density in the banana regime is:

$$j_{bs} = -c_{bs} \frac{\sqrt{\epsilon}}{B_p} \frac{dp}{dr}$$

where $c_{bs}$ is a numerical coefficient (typically 2-4) and $dp/dr$ is the pressure gradient.

In a more detailed multi-species expression:

$$j_{bs} = -\frac{\sqrt{\epsilon}}{B_p} \left[ L_{31} \frac{dp_e}{dr} + L_{32} \frac{dp_i}{dr} + L_{34} n_e e \frac{dT_e}{dr} \right]$$

The coefficients $L_{31}, L_{32}, L_{34}$ depend on collisionality and impurity concentration and are calculated using Hirshman-Sigmar theory.

Bootstrap Current Fraction

The fraction of bootstrap current to total plasma current is:

$$f_{bs} = \frac{I_{bs}}{I_p} \sim \frac{\sqrt{\epsilon} \beta_p}{q}$$

where $\beta_p = 2\mu_0 \langle p \rangle / B_p^2$ is the poloidal beta.

In high $\beta_p$ operation, $f_{bs} > 0.7$ can be achieved, which is extremely important for steady-state tokamak reactors because it can significantly reduce external current drive power.

At JT-60U, $f_{bs} \approx 0.8$ was achieved, and for ITER’s advanced operation scenarios, $f_{bs} \approx 0.5$ is assumed. For future steady-state reactors, $f_{bs} > 0.9$ is targeted.

Particle Pinch Effects

Neoclassical theory also predicts particle fluxes in the direction opposite to density gradients (inward pinch).

Ware Pinch

When a toroidal electric field $E_\phi$ is present, trapped particles drift toward the plasma center. This Ware pinch is:

$$\Gamma_{Ware} = -n_e \frac{2\epsilon^{3/2}}{\sqrt{\nu^*}} \frac{E_\parallel}{B_p}$$

where $E_\parallel$ is the toroidal electric field.

Physically, the banana orbits of trapped particles are accelerated by the toroidal electric field, and changes in orbit shape cause inward drift.

Thermoelectric Pinch

When a temperature gradient exists, particles tend to drift toward higher temperature regions. This is similar to the thermoelectric effect:

$$V_{pinch} = -\frac{D_{nc}}{n} \frac{C_T}{L_T}$$

where $C_T$ is the thermoelectric coefficient and $L_T$ is the temperature scale length.

These pinch effects contribute to the formation of density profiles and central density peaking.

Anomalous Transport and Turbulent Transport

Transport observed in experiments often greatly exceeds predictions from neoclassical theory, and this additional transport is called anomalous transport. Its main cause is turbulence in the plasma.

Discovery of Anomalous Transport and Challenges

From tokamak experiments in the late 1960s, it became clear that measured confinement times were orders of magnitude shorter than predictions from neoclassical theory. This difference is called “anomalous transport.”

The anomalous diffusion coefficient is typically:

$$D_{anom} / D_{nc} \sim 10 - 100$$

For ion heat transport in particular, there are cases close to neoclassical values, but electron heat transport is always dominated by anomalous transport.

General Theory of Turbulent Transport

Transport due to turbulence is expressed as correlations between fluctuations of electric field and density/temperature. Particle flux:

$$\Gamma_{turb} = \langle \tilde{n} \tilde{v}_E \rangle = \left\langle \tilde{n} \frac{\tilde{\mathbf{E}} \times \mathbf{B}}{B^2} \right\rangle$$

Heat flux:

$$Q_{turb} = \frac{3}{2} \langle \tilde{p} \tilde{v}_E \rangle$$

where $\tilde{v}_E = \tilde{\mathbf{E}} \times \mathbf{B} / B^2$ is the $E \times B$ drift velocity, and $\tilde{n}$, $\tilde{p}$ are density and pressure fluctuations. The tilde denotes fluctuating components.

For transport to be non-zero, there must be a phase difference between fluctuations. If density fluctuations and potential fluctuations are in phase, the $E \times B$ flow merely moves density peaks sideways, producing no net transport.

The turbulent diffusion coefficient from quasilinear theory is:

$$D_{turb} \sim \frac{|\tilde{\phi}|^2 \tau_c k_\perp^2}{B^2}$$

where $\tilde{\phi}$ is the electrostatic potential fluctuation, $\tau_c$ is the correlation time, and $k_\perp$ is the perpendicular wave number.

Bohm Diffusion and Gyro-Bohm Diffusion

As a typical scaling for turbulent transport, there is the Bohm diffusion coefficient:

$$D_B = \frac{1}{16} \frac{T}{eB} \approx 6.25 \times 10^{-2} \frac{T}{B} \quad [\text{m}^2/\text{s}]$$

where $T$ is in eV. This coefficient was obtained from studies of ionized gas in magnetic fields during World War II (“Bohm diffusion”) and corresponds to cases where the turbulence correlation length is of order the system size.

In Bohm scaling, the energy confinement time is:

$$\tau_E^{Bohm} \sim \frac{a^2}{D_B} \propto \frac{a^2 B}{T}$$

Since it is proportional to the square of device size $a$, the advantage of larger devices becomes limited.

As a more realistic scaling, when the correlation length is of order the ion Larmor radius:

$$D_{gB} = \frac{\rho_i}{L_T} D_B = \frac{\rho_i}{L_T} \frac{T}{16eB}$$

This is gyro-Bohm diffusion, where $L_T = T/(dT/dr)$ is the temperature scale length.

An important consequence of gyro-Bohm scaling is that the diffusion coefficient is proportional to $\rho^* = \rho_i/a$:

$$D_{gB} \propto \rho^* T$$

The energy confinement time is:

$$\tau_E^{gyroBohm} \sim \frac{a^2}{D_{gB}} \propto \frac{a^3 B}{T \rho_i} \propto \frac{a^3 B^2}{\sqrt{T}}$$

For large devices, $\rho^*$ becomes small, so gyro-Bohm scaling gives favorable predictions for fusion reactors. With Bohm scaling, ITER-sized devices would be necessary, but with gyro-Bohm scaling, there is the possibility of achieving $Q = 10$ with smaller devices.

Critical Temperature Gradient and Stiff Transport

Turbulent transport has a threshold, and transport increases sharply when the temperature gradient exceeds a certain critical value. The normalized temperature gradient:

$$R/L_T = -\frac{R}{T} \frac{dT}{dr}$$

When this exceeds the critical value $(R/L_T)_c$, turbulence develops.

The heat flux is expressed as:

$$Q = n \chi \frac{dT}{dr}$$

where the thermal conductivity $\chi$ is:

$$\chi = \chi_0 \left[ \frac{R/L_T - (R/L_T)_c}{(R/L_T)_c} \right]^\alpha H(R/L_T - (R/L_T)_c)$$

$H$ is the Heaviside function, and $\alpha$ is typically 1-2.

This “stiff” transport property tends to keep the temperature profile near the critical gradient. Even if heating power is increased, the central temperature does not rise much, and the gradient is maintained. This is called “profile stiffness.”

Consequences of stiff transport:

• Central temperature does not rise much even with central heating
• Edge temperature (pedestal) determines central temperature
• Gradient enhancement through transport barriers is key to confinement improvement

Drift Waves and Microinstabilities

Drift waves are low-frequency instabilities that appear universally in plasmas with density or temperature gradients. These are the primary drivers of turbulent transport.

Basic Physics of Drift Waves

When a pressure gradient $\nabla p$ and magnetic field $\mathbf{B}$ coexist, diamagnetic drift occurs:

$$\mathbf{v}_* = \frac{\mathbf{B} \times \nabla p}{qnB^2}$$

The drift direction is opposite for electrons and ions (due to opposite charge signs).

The electron diamagnetic drift velocity is:

$$v_{*e} = -\frac{T_e}{eB L_n}$$

where $L_n = n/(dn/dr)$ is the density scale length. The negative sign indicates that electron drift is upward in the poloidal direction perpendicular to both the pressure gradient and magnetic field.

Linear Theory of Drift Waves

When there is a density fluctuation $\tilde{n}$, charge separation occurs and an electric field $\tilde{E}$ is generated. The $E \times B$ drift due to this field propagates the density fluctuation, becoming a drift wave.

Assuming adiabatic electron response (electrons equilibrate instantaneously along field lines):

$$\frac{\tilde{n}_e}{n_0} = \frac{e\tilde{\phi}}{T_e}$$

Combining this with the ion continuity equation yields the drift wave dispersion relation:

$$\omega = \omega_* = \frac{k_\perp T}{eB L_n}$$

where $k_\perp$ is the wave number perpendicular to the magnetic field.

Drift waves propagate in the electron diamagnetic drift direction, and the frequency is much lower than the ion cyclotron frequency ($\omega \ll \omega_{ci}$).

Pure drift waves are stable, but various effects cause destabilization.

Ion Temperature Gradient Mode (ITG)

The ion temperature gradient mode (ITG) is a type of drift wave that becomes unstable when the ion temperature gradient exceeds a critical value. It is the most important instability dominating ion heat transport in the core region.

Physical Mechanism of ITG Mode

The driving mechanism of ITG instability can be understood as follows. In toroidal magnetic configurations, ions undergo drift motion due to magnetic field curvature and gradient. This drift destabilizes fluctuations on the outboard (low-field) side, which is called the “bad curvature” region.

When a temperature gradient exists, ions coming from high-temperature regions have higher energy than ions in low-temperature regions. Since curvature drift is proportional to energy, temperature fluctuations cause charge separation, driving the instability.

ITG Mode Dispersion Relation

The ITG dispersion relation in fluid approximation is:

$$\omega^2 + \omega \omega_{*i} \left( 1 + \eta_i \right) - \omega_{*i}^2 \eta_i \frac{v_{ti}^2 k_\parallel^2}{\omega^2} = 0$$

where:

• $\omega_{*i} = k_\perp T_i / (eB L_n)$: ion diamagnetic frequency
• $\eta_i = L_n / L_{Ti}$: ratio of temperature gradient to density gradient
• $L_{Ti} = T_i / (dT_i/dr)$: ion temperature scale length

A more complete description requires gyrokinetic dispersion relations that include trapped particle effects, finite Larmor radius effects, and magnetic shear effects.

ITG Instability Threshold

The ITG mode becomes unstable when $\eta_i$ exceeds a critical value:

$$\eta_i > \eta_{i,c}$$

Typically $\eta_{i,c} \approx 1-2$. Expressed in normalized temperature gradient:

$$\frac{R}{L_{Ti}} > \left( \frac{R}{L_{Ti}} \right)_c \approx 4-6$$

The critical gradient depends on the following parameters:

Magnetic shear $\hat{s} = (r/q)(dq/dr)$: positive shear has a stabilizing effect

$$\left( \frac{R}{L_T} \right)_c \approx \frac{4}{3} \left( 1 + \tau \right) \left( 1 + \frac{2\hat{s}}{q} \right)$$

where $\tau = Z_i T_e / T_i$.

Shafranov shift: In high $\beta$ plasmas, the magnetic axis shifts outward, which has a stabilizing effect.

Plasma shape: Positive triangularity provides stabilization.

ITG Mode Growth Rate

The maximum growth rate above the threshold is:

$$\gamma_{max} \approx \frac{v_{ti}}{R} \sqrt{\frac{R}{L_{Ti}} - \left( \frac{R}{L_{Ti}} \right)_c}$$

Or normalized:

$$\gamma_{max} \approx \omega_{*i} \sqrt{\eta_i - \eta_{i,c}}$$

The growth rate is maximum at wave numbers of ion Larmor radius scale, $k_\perp \rho_i \sim 0.3$. This shows that ITG turbulence is an ion-scale phenomenon.

ITG Turbulence Saturation Mechanisms

When the linear instability grows, it saturates due to nonlinear effects. Main saturation mechanisms:

Zonal flow generation: ITG turbulence generates zonal flows through nonlinear interactions. The shear of zonal flows destroys turbulent eddies and suppresses turbulence levels.

Mode coupling: Energy transfers between modes of different wave numbers, eventually reaching dissipation scales.

Shearing: Magnetic shear stretches turbulent structures, effectively causing damping.

Transport by ITG Turbulence

Heat transport by ITG turbulence after nonlinear saturation is:

$$\chi_i^{ITG} \sim \frac{\gamma}{k_\perp^2} \sim \frac{\rho_i^2 v_{ti}}{L_{Ti}} \sqrt{\eta_i - \eta_{i,c}}$$

This follows gyro-Bohm scaling.

In quasilinear theory:

$$Q_i = n_i \chi_i \frac{dT_i}{dr}, \quad \chi_i = \chi_0 \left( \frac{R}{L_{Ti}} - \left(\frac{R}{L_{Ti}}\right)_c \right)$$

where $\chi_0$ is a constant depending on device size and plasma parameters, typically:

$$\chi_0 \sim \frac{\rho_i^2 c_s}{R}$$

$c_s = \sqrt{T_e/m_i}$ is the ion sound speed.

Trapped Electron Mode (TEM)

The trapped electron mode (TEM) is a mode that becomes unstable when the precession of trapped electrons resonates with drift waves.

Physical Mechanism of TEM

In toroidal magnetic configurations, trapped electrons precess (drift) in the toroidal direction while tracing banana orbits. This precession frequency is:

$$\omega_{de} = \frac{m_e v_{th,e}^2}{e B R} \left( \cos\theta + \hat{s} \theta \sin\theta \right) k_\theta$$

where $\hat{s}$ is the magnetic shear and $\theta$ is the poloidal angle.

When the precession of trapped electrons resonates with the phase velocity of drift waves ($\omega \approx \omega_{de}$), energy exchange occurs between electrons and the mode, driving the instability.

This resonance mechanism is similar to the inverse process of Landau damping. When the electron distribution function has a gradient, energy transfer from resonant electrons to the mode becomes possible.

TEM Dispersion Relation

The dispersion relation including trapped electrons is:

$$1 + \tau + \frac{\omega_{*e}}{\omega} \left[ 1 + \eta_e \left( \frac{\omega - \omega_{de}}{\omega_{de}} - \frac{3}{2} \right) \right] f_t \mathcal{F}(\omega/\omega_{de}) = 0$$

where:

• $\tau = T_e/T_i$: temperature ratio
• $\eta_e = L_n/L_{Te}$: electron temperature gradient parameter
• $f_t \approx \sqrt{2\epsilon}$: trapped particle fraction
• $\mathcal{F}$: resonance factor (complex function)

TEM Instability Threshold and Characteristics

TEM is driven by density gradient or electron temperature gradient.

Threshold for density gradient-driven TEM:

$$\frac{R}{L_n} > \left( \frac{R}{L_n} \right)_c$$

For electron temperature gradient-driven TEM:

$$\eta_e > \eta_{e,c}$$

Important characteristics of TEM:

Stabilization by collisions: When collision frequency is high, trapped electrons are detrapped and resonance is broken.

$$\gamma_{TEM} \propto \frac{1}{1 + \nu_{eff}/\omega_{de}}$$

where $\nu_{eff}$ is the effective collision frequency.

Mode frequency: TEM propagates in the electron diamagnetic direction, opposite to ITG.

$$\omega_{TEM} \sim \omega_{*e}$$

Transport by TEM

Transport by TEM turbulence primarily affects the electron channel:

$$\chi_e^{TEM} \sim \frac{\gamma}{k_\perp^2} \sim D_{gB} \sqrt{\frac{R}{L_n} - \left(\frac{R}{L_n}\right)_c}$$

ITG and TEM compete or coexist, and the dominant mode depends on the ratio of temperature to density gradients:

• $\eta_i > \eta_e$ and large temperature gradient: ITG dominates
• Large density gradient: TEM dominates

TEM is particularly important for particle transport and plays a major role in determining density profiles.

Electron Temperature Gradient Mode (ETG)

The electron temperature gradient mode (ETG) is an instability that can be called the electron version of ITG, developing at electron Larmor radius scales.

Characteristics of ETG

The ETG mode has the following characteristics:

Wavenumber scale: $k_\perp \rho_e \sim 0.1-1$, i.e., $k_\perp \rho_i \sim 10-100$

Frequency: scales with electron thermal velocity

$$\omega \sim \frac{v_{te}}{L_{Te}}$$

Threshold:

$$\frac{R}{L_{Te}} > \left( \frac{R}{L_{Te}} \right)_c \approx 4-6$$

Similar threshold conditions to ITG, but with electron parameters substituted.

ETG Dispersion Relation

Assuming adiabatic ion response (ions cannot follow the high frequency of ETG):

$$\omega^2 + \omega \omega_{*e} (1 + \eta_e) - \omega_{*e}^2 \eta_e \frac{v_{te}^2 k_\parallel^2}{\omega^2} = 0$$

This has the same form as ITG but with electron parameters substituted.

Scaling of ETG Transport

In simple scaling, transport due to ETG follows electron gyro-Bohm scaling:

$$\chi_e^{ETG} \sim \frac{\rho_e^2 v_{te}}{L_{Te}} \sim \left( \frac{\rho_e}{\rho_i} \right)^2 \chi_i^{ITG} \sim \frac{m_e}{m_i} \chi_i^{ITG}$$

Due to the mass ratio factor $m_e/m_i \approx 1/1836$, ETG transport is naively expected to be small.

ETG Streamers and Transport Enhancement

However, ETG turbulence can form elongated structures called “streamers” along field lines, potentially enhancing transport.

Streamers are structures that are very long in the field line direction ($k_\parallel \rightarrow 0$) and at electron Larmor radius scale in the perpendicular direction. Due to this structure:

$$\chi_e^{ETG} \sim D_{gB,e} \left( \frac{L_n}{\rho_e} \right)^\alpha, \quad \alpha \sim 0.5-1$$

Due to the scale factor $(L_n/\rho_e)^\alpha$, ETG transport can be larger than naive predictions.

The relationship between ETG and electron transport is still an active research topic. In particular, the extent to which electron transport is due to ETG and how important coupling effects from ITG are is being debated.

Multi-scale Turbulence Interactions

In real plasmas, ITG ($k_\perp \rho_i \sim 0.3$) and ETG ($k_\perp \rho_e \sim 0.3$) coexist, and energy may cascade between different scales.

These “multi-scale turbulence” interactions include:

Cross-scale coupling: Large-scale (ITG) flows modulate small-scale (ETG) turbulence

Energy cascade: Energy transfer from ITG scales to ETG scales, or vice versa

Due to computational resource constraints, multi-scale simulations are challenging, but are being pursued in cutting-edge research. Multi-scale simulations with codes such as Trinity have demonstrated the importance of inter-scale interactions in electron transport.

Gyrokinetic Simulations

For quantitative predictions of plasma turbulence and transport, large-scale numerical simulations based on gyrokinetics are essential.

Fundamentals of Gyrokinetics

Gyrokinetics is a theory that averages out cyclotron motion and tracks the motion of guiding centers. This averaging eliminates high-frequency cyclotron oscillations and greatly improves computational efficiency.

Time scale separation:

$$\omega_{ci} \gg \omega \sim \omega_* \sim \omega_{turb}$$

where $\omega_{ci}$ is the ion cyclotron frequency, $\omega_*$ is the diamagnetic frequency, and $\omega_{turb}$ is the turbulence frequency.

Spatial scale separation:

$$\rho_i \ll L_n, L_T, a$$

It is assumed that the Larmor radius is much smaller than the plasma scale lengths.

Gyrokinetic Equation

The time evolution of the gyro-averaged distribution function $F(\mathbf{R}, v_\parallel, \mu, t)$ is:

$$\frac{\partial F}{\partial t} + \dot{\mathbf{R}} \cdot \nabla F + \dot{v}_\parallel \frac{\partial F}{\partial v_\parallel} = C(F)$$

where:

• $\mathbf{R}$: guiding center position
• $v_\parallel$: parallel velocity
• $\mu = mv_\perp^2/(2B)$: magnetic moment (adiabatic invariant)
• $C(F)$: collision operator

The guiding center equations of motion are:

$$\dot{\mathbf{R}} = v_\parallel \hat{\mathbf{b}} + \mathbf{v}_E + \mathbf{v}_{\nabla B} + \mathbf{v}_\kappa$$

where each drift is:

$$\mathbf{v}_E = \frac{\mathbf{E} \times \mathbf{B}}{B^2}, \quad \mathbf{v}_{\nabla B} = \frac{\mu}{m \Omega} \hat{\mathbf{b}} \times \nabla B, \quad \mathbf{v}_\kappa = \frac{v_\parallel^2}{\Omega} \hat{\mathbf{b}} \times \boldsymbol{\kappa}$$

$\boldsymbol{\kappa} = \hat{\mathbf{b}} \cdot \nabla \hat{\mathbf{b}}$ is the field line curvature.

The time evolution of parallel velocity:

$$m \dot{v}_\parallel = -\mu \nabla_\parallel B + q E_\parallel$$

Gyro-averaging Operator

Finite Larmor radius effects are represented by the gyro-averaging operator $\langle \cdot \rangle_\alpha$:

$$\langle \phi \rangle_\alpha = \frac{1}{2\pi} \oint \phi(\mathbf{R} + \boldsymbol{\rho}) d\alpha$$

where $\boldsymbol{\rho}$ is the Larmor radius vector and $\alpha$ is the gyro-phase.

In Fourier space:

$$\langle \tilde{\phi}_k \rangle = J_0(k_\perp \rho) \tilde{\phi}_k$$

$J_0$ is the Bessel function of the first kind. For $k_\perp \rho \ll 1$, $J_0 \approx 1$; for $k_\perp \rho \sim 1$, finite Larmor radius effects become important.

$\delta f$ Method

The distribution function is separated into equilibrium and fluctuating parts:

$$F = F_0 + \delta f$$

By solving the evolution equation for the fluctuation $\delta f$, computational efficiency is greatly improved:

$$\frac{\partial \delta f}{\partial t} + \dot{\mathbf{R}} \cdot \nabla \delta f + \dot{v}_\parallel \frac{\partial \delta f}{\partial v_\parallel} = -\delta \dot{\mathbf{R}} \cdot \nabla F_0 + C(\delta f)$$

Advantages of the $\delta f$ method:

• Significant reduction in statistical noise
• Accurate representation of equilibrium distribution
• Computation focused on fluctuation level ($\delta f/F_0 \sim 1\%$)

Major Gyrokinetic Codes

Major codes developed worldwide include:

GYRO/CGYRO (General Atomics, USA)

• Continuum method (Eulerian method)
• Local and global simulations possible
• Electromagnetic fluctuation support
• One of the most widely validated codes

GS2 (Culham/Oxford, UK)

• Flux tube method
• Linear and nonlinear calculations
• Contributed to spherical tokamak research on NSTX

GENE (IPP Garching, Germany)

• Continuum method, highly efficient algorithms
• Both tokamak and stellarator compatible
• Global simulation capability

GTC/GTS (Princeton/UCI, USA)

• Particle method (Lagrangian method)
• Global simulations
• From ion scales to Alfven waves

ORB5 (EPFL/IPP, Switzerland/Germany)

• Particle method, global and electromagnetic simulations
• High energy particle physics
• Wendelstein 7-X stellarator research

GT5D/GKNET (JAEA, Japan)

• Particle method, full $f$ method
• Accurate treatment of collision effects
• Analysis of the JT-60 series

Flux Tube Method and Global Method

The flux tube (local) method solves turbulence in a small region on a magnetic surface:

• Uses periodic boundary conditions
• Computational domain is $\Delta r \ll a$
• Gradients are assumed locally constant
• High computational efficiency

Advantages:

• Suitable for detailed study of turbulence physics
• Easy parameter scans
• Optimal for verification and validation

Limitations:

• Does not include $\rho^* = \rho_i/a$ effects
• Cannot directly simulate global structures (transport barriers, etc.)
• Does not include profile evolution

The global method solves turbulence over the entire plasma (or large portion):

• Requires radial boundary conditions
• Includes $\rho^*$ effects
• Can simulate self-formation of transport barriers

Computational cost is high, but global effects and transport barrier formation can be simulated.

Comparison of Simulations and Experiments

Gyrokinetic simulations have succeeded in quantitatively reproducing transport in many tokamak experiments:

$$\chi_i^{sim} / \chi_i^{exp} \sim 0.5 - 2$$

This “verification and validation (V&V)” effort has built confidence in predictive capability.

Validated examples:

• DIII-D L-mode transport (GYRO)
• JET H-mode core transport (GENE)
• ASDEX Upgrade ion thermal transport (GS2)

Remaining challenges:

• Quantitative prediction of electron transport
• Momentum transport (intrinsic rotation)
• Impurity transport
• Turbulence in pedestal region

Transport Barriers

Transport barriers are regions where turbulence is locally suppressed and steep pressure gradients are formed. This significantly improves confinement performance.

Discovery of H-mode

In November 1982, at the ASDEX tokamak in Germany, Wagner and colleagues were conducting experiments increasing heating power. They discovered a phenomenon where plasma confinement suddenly improved by about a factor of two when a threshold power was exceeded.

This new confinement state was named H-mode (high confinement mode), and the previous state became known as L-mode (low confinement mode).

The discovery of H-mode is one of the most important discoveries in fusion research history. ITER’s basic operating mode is H-mode, and all current fusion reactor designs assume H-mode operation.

Edge Transport Barrier (ETB)

In H-mode, a transport barrier (Edge Transport Barrier, ETB) forms at the plasma edge.

L-H Transition

When heating power exceeds a certain threshold (L-H transition power $P_{LH}$), the plasma transitions from L-mode to H-mode. The scaling law based on the international database:

$$P_{LH} \approx 2.15 \times 10^{-2} \bar{n}_e^{0.78} B_T^{0.77} S^{0.98} \quad [\text{MW}]$$

where $\bar{n}_e$ is the line-averaged density ($10^{20}$ m$^{-3}$), $B_T$ is the toroidal field (T), and $S$ is the plasma surface area (m$^2$).

From this scaling, the L-H transition power for ITER is predicted to be approximately 50-80 MW.

The L-H transition typically occurs on extremely short time scales of several milliseconds. Just before the transition, oscillations in $D_\alpha$ emission (Limit Cycle Oscillations) are sometimes observed.

ETB Formation Mechanism

The main mechanism for ETB formation is turbulence suppression by $E \times B$ sheared flow.

Radial electric field shearing rate:

$$\omega_{E \times B} = \frac{1}{B} \frac{dE_r}{dr}$$

When this shearing rate exceeds the turbulence growth rate:

$$\omega_{E \times B} > \gamma_{turb}$$

Turbulent cells are stretched and fragmented, suppressing transport.

Physically, the sheared flow stretches turbulent eddies and shortens their correlation length. As the correlation length decreases, transport by eddies is reduced.

The radial electric field is determined by force balance (radial momentum balance):

$$E_r = \frac{1}{n_i Z_i e} \frac{dp_i}{dr} - v_\theta B_\phi + v_\phi B_\theta$$

The first term is the diamagnetic term, and the second and third terms are related to plasma rotation.

Pedestal Physics

When the ETB forms, a pedestal with a steep pressure gradient appears at the plasma edge. The pedestal height (temperature and density at the top) has a decisive influence on core confinement.

Pedestal Structure

The pedestal region typically has:

• Width: several cm ($\Delta_{ped}/a \sim 3-8\%$)
• Temperature gradient: $dT/dr \sim 50-200$ keV/m
• Density gradient: $dn/dr \sim 10^{21}$ m$^{-4}$

Values at the pedestal top:

$$T_{ped} \sim 1-5 \text{ keV}, \quad n_{ped} \sim 5-10 \times 10^{19} \text{ m}^{-3}$$

Importance of Pedestal Height

Due to stiff transport, core temperature strongly depends on pedestal temperature:

$$T_0 \approx T_{ped} + \Delta T_{core}$$

where $\Delta T_{core}$ is the temperature rise due to the critical gradient. The higher the pedestal temperature, the higher the central temperature.

Since fusion power is a strong function of temperature:

$$P_{fus} \propto n^2 \langle \sigma v \rangle \propto T^2 \quad (\text{in the 10-20 keV range})$$

Pedestal height directly determines fusion reactor performance.

EPED Model

The EPED model developed by Snyder and colleagues is a theoretical model for predicting pedestal height. This model combines two constraint conditions:

1. Peeling-ballooning stability limit: the pedestal pressure gradient is limited by MHD stability
2. Kinetic ballooning mode (KBM) critical gradient: determines pedestal width

From these conditions, pedestal height $p_{ped}$ and pedestal width $\Delta$ are determined self-consistently:

$$\beta_{p,ped} \approx c_1 \epsilon^{1/2} \left( \frac{\Delta}{\rho_{\theta,i}} \right)^{0.8}$$

The EPED model has been validated on many experiments (DIII-D, JET, ASDEX-U, etc.) and is used for ITER pedestal predictions.

ELM (Edge Localized Mode)

In H-mode, periodic bursts (ELMs) occur when the pedestal pressure gradient reaches the MHD stability limit.

Physics of ELMs

ELMs are triggered by peeling-ballooning instability:

Peeling mode: an external kink mode driven by current gradient (bootstrap current)

Ballooning mode: a mode localized in bad curvature regions, driven by pressure gradient

When these modes couple and become unstable in the pedestal region, an ELM occurs.

Types of ELMs

Type-I ELM (Giant ELM):

• Most common
• Expels 5-15% of pedestal energy
• Period: 10-100 Hz
• Expulsion time: $\sim$ 1 ms

Type-II ELM (Grassy ELM):

• Observed in high triangularity plasmas
• Smaller and more frequent than Type-I

Type-III ELM:

• Occurs near L-H transition threshold
• Small and frequent

ELM-free H-mode:

• Maintained without ELMs
• Has impurity accumulation issues

Heat Load from ELMs

Type-I ELMs impose large instantaneous heat loads on the divertor. For ITER:

$$\Delta W_{ELM} \sim 10-20 \text{ MJ}$$

Heat flux to the surface:

$$q_{ELM} \sim 10-100 \text{ GW/m}^2$$

This exceeds the ablation threshold of tungsten and significantly shortens divertor lifetime. Therefore, ELM suppression or mitigation is mandatory for ITER.

ELM Control Techniques

Resonant Magnetic Perturbation (RMP):

• Apply small non-axisymmetric magnetic fields with external coils
• Increase transport through magnetic islands at pedestal edge
• ELM suppression demonstrated at DIII-D, KSTAR, ASDEX-U

Pellet ELM pacing:

• Periodically inject small solid fuel pellets
• Trigger controlled small ELMs
• Prevent large ELMs

Small ELM/QH mode:

• Achieved with high rotation or RMP
• Maintains ELM-free or small ELM states

Internal Transport Barrier (ITB)

Transport barriers can also form in the plasma interior. Internal Transport Barriers (ITB) are formed mainly under the following conditions.

Reversed Magnetic Shear Configuration

In configurations where the safety factor profile $q(r)$ has a minimum at the center ($q_{min}$), ITBs are easily formed near $q_{min}$:

$$\hat{s} = \frac{r}{q} \frac{dq}{dr} < 0 \quad (\text{negative magnetic shear})$$

Reasons why negative shear promotes ITB:

• Increased critical gradient for ITG/TEM
• Reduced trapped particle orbit width
• Promoted development of zonal flows

Negative shear configurations are achieved through heating during early current ramp-up or off-axis current drive.

Weak Magnetic Shear Configuration

ITBs are also formed in weak shear configurations where $\hat{s} \approx 0$ over a wide region. In this case, turbulence suppression effects at rational surfaces (surfaces where $q$ is rational) become important.

ITB by Rotation Shear

Radial shear in plasma rotation also contributes to ITB formation:

$$\omega_{v\phi} = \frac{d(R v_\phi)}{dr}$$

Toroidal rotation from NBI heating has turbulence suppression effects.

Pressure Gradient in ITB

In ITBs, gradients significantly exceeding the critical gradient are achieved:

$$\frac{R}{L_T} \gg \left( \frac{R}{L_T} \right)_c$$

In strong ITBs at JT-60U, $R/L_T \sim 20-30$ was achieved, reaching several times the L-mode critical gradient ($\sim 6$).

Typical ITB performance:

• Confinement improvement factor: $H_{98} \sim 1.5-2$
• Central temperature: over 40 keV (JT-60U)
• Equivalent fusion $Q$: 1.25 (JT-60U, D-D equivalent)

Combined Transport Barriers

Forming both ETB and ITB simultaneously can lead to 3-4 times the L-mode confinement performance.

Stored energy is:

$$W = \int \frac{3}{2}(n_e T_e + n_i T_i) dV \propto n T V$$

When both central temperature (ITB) and edge temperature (ETB pedestal) are high, $W$ increases dramatically.

However, combined barrier operation also has challenges:

• MHD stability (due to high $\beta$)
• Impurity accumulation (due to reduced transport)
• Steady-state maintenance (current profile control)

Confinement Time Scaling Laws

Confinement time is expressed as scaling laws (proportionality relations) as functions of device and plasma parameters. These form the foundation of fusion reactor design.

Definition of Energy Confinement Time

Energy confinement time $\tau_E$ is defined as the ratio of stored energy to loss power:

$$\tau_E = \frac{W}{P_{loss}} = \frac{\int \frac{3}{2}(n_e T_e + n_i T_i) dV}{P_{in} - \frac{dW}{dt}}$$

In steady state, $dW/dt = 0$, so:

$$\tau_E = \frac{W}{P_{in}}$$

where $P_{in}$ is the power input to the plasma (Ohmic heating + auxiliary heating - radiation loss).

Derivation of Scaling Laws

Empirical scaling laws are derived by regression analysis of data from numerous tokamak experiments. The International Tokamak Physics Activity (ITPA) database contains thousands of discharge data.

The power law form:

$$\tau_E = C_0 \prod_i x_i^{\alpha_i}$$

where $x_i$ are device parameters ($I_p, R, a, \kappa, \bar{n}, B_T, A, P$) and $\alpha_i$ are power exponents.

Taking logarithms:

$$\ln \tau_E = \ln C_0 + \sum_i \alpha_i \ln x_i$$

This becomes a linear regression problem, and the coefficients $\alpha_i$ are determined by least squares.

L-mode Scaling (ITER89P)

As the standard scaling law for L-mode (low confinement mode), ITER89P is widely used:

$$\tau_E^{ITER89P} = 0.048 \, I_p^{0.85} R^{1.2} a^{0.3} \kappa^{0.5} \bar{n}_{20}^{0.1} B_T^{0.2} A^{0.5} P^{-0.5}$$

Units are: $I_p$ (MA), $R, a$ (m), $\bar{n}_{20}$ ($10^{20}$ m$^{-3}$), $B_T$ (T), $P$ (MW), $\tau_E$ (s)

Characteristic dependencies:

• $P^{-0.5}$: confinement degrades with increasing heating power (power degradation)
• $I_p^{0.85}$: strong dependence on plasma current
• $R^{1.2}$: dependence on device size
• $A^{0.5}$: isotope effect (heavier isotopes have better confinement)

H-mode Scaling

For H-mode operation, there are several versions of scaling laws based on international databases.

IPB98(y,2)

The most widely used H-mode scaling:

$$\tau_E^{IPB98(y,2)} = 0.0562 \, I_p^{0.93} R^{1.97} a^{0.58} \kappa^{0.78} \bar{n}_{19}^{0.41} B_T^{0.15} A^{0.19} P^{-0.69}$$

Density is in $10^{19}$ m$^{-3}$ units.

ITER’s design values are based on this scaling.

ELMy H-mode Scaling

Scaling specialized for H-mode with Type-I ELMs:

$$\tau_E^{ELMy} = 0.0365 \, I_p^{0.97} R^{1.93} a^{0.57} \kappa^{0.67} \bar{n}_{19}^{0.41} B_T^{0.08} A^{0.20} P^{-0.63}$$

The H-mode confinement improvement factor is:

$$H_{98} = \frac{\tau_E}{\tau_E^{IPB98(y,2)}} \approx 1 \quad (\text{by definition})$$

Excellent H-mode discharges achieve $H_{98} \sim 1.0-1.5$.

Physics of Power Degradation

The $P^{-0.5}$ to $P^{-0.69}$ dependence in scaling laws is called “power degradation” and reflects the nature of turbulent transport.

Assuming stiff transport, the temperature gradient is fixed at the critical value:

$$\frac{dT}{dr} = \frac{T}{L_T} = \frac{T}{\xi a}$$

where $\xi = L_T/a$ is the normalized temperature scale length, determined by the critical gradient.

Heat flux:

$$Q \propto \chi (T_c - T_{edge}) / a, \quad \chi \propto T^{3/2}$$

Solving for steady state with $P = Q$:

$$T \propto P^{2/5}$$

Therefore:

$$\tau_E = W / P \propto n T / P \propto P^{-3/5}$$

This is consistent with observed scaling.

Confinement Improvement Factor H

For evaluation of operational performance, the ratio to the scaling law (confinement improvement factor) is used:

$$H = \frac{\tau_E^{exp}}{\tau_E^{scaling}}$$

$H > 1$ indicates better confinement than predicted, $H < 1$ indicates worse confinement than predicted.

ITER’s baseline operation scenario assumes $H_{98(y,2)} = 1.0$.

For advanced operation scenarios:

• Hybrid scenario: $H_{98} \sim 1.2-1.4$
• Steady-state operation scenario: targeting $H_{98} \sim 1.5$

ITER Predictions

Substituting ITER parameters ($I_p = 15$ MA, $R = 6.2$ m, $a = 2.0$ m, $\kappa = 1.7$, $\bar{n}_{19} = 10$, $B_T = 5.3$ T, $P = 50$ MW) into IPB98(y,2):

$$\tau_E^{ITER} \approx 3.7 \, \text{s}$$

With this confinement time, the triple product required for achieving fusion gain $Q = 10$ is realized.

Dimensionless Parameter Scaling

For physical understanding, scaling in dimensionless parameters is also studied. Key dimensionless parameters:

Normalized Larmor radius:

$$\rho^* = \frac{\rho_i}{a} \propto \frac{\sqrt{T}}{Ba}$$

Normalized collisionality:

$$\nu^* = \frac{\nu_{ii} qR}{\epsilon^{3/2} v_{ti}} \propto \frac{nR}{T^2}$$

Normalized pressure (beta):

$$\beta = \frac{2\mu_0 nT}{B^2}$$

The dimensionless scaling form:

$$\omega_{ci} \tau_E \propto \rho^{*-\alpha} \nu^{*-\beta} \beta^{-\gamma}$$

Or:

$$\tau_E B \propto \rho^{*-\alpha} \nu^{*-\beta} \beta^{-\gamma}$$

From experimental data:

• $\alpha \approx 2.7$: close to gyro-Bohm scaling (Bohm would be $\alpha = 3$)
• $\beta \approx 0.01-0.3$: weak dependence on collisionality
• $\gamma \approx 0.9$: inverse dependence on pressure

Isotope Effect

Scaling laws show a dependence of $A^{0.2-0.4}$, and better confinement is expected for deuterium-tritium plasmas than for hydrogen.

Isotope mass ratio:

$$\frac{\tau_E(D-T)}{\tau_E(H)} \approx \left(\frac{2.5}{1}\right)^{0.2-0.4} \approx 1.2-1.5$$

D-T experiments at JET confirmed this isotope effect. The physical understanding is that the mass dependence of the ion Larmor radius:

$$\rho_i \propto \sqrt{m_i}$$

affects turbulence scales.

Zonal Flows and Turbulence Self-regulation

Nonlinear interactions in drift wave turbulence spontaneously generate poloidal zonal flows.

Structure of Zonal Flows

Zonal flows are axisymmetric ($n = 0$) flows that are uniform on magnetic surfaces:

$$\phi_{ZF} = \tilde{\phi}(r) e^{i n \phi}, \quad n = 0, m = 0$$

They vary radially and flow in the poloidal ($\theta$) direction. Typical radial wavelengths are several $\rho_i$.

Zonal flows have the following properties:

• Axisymmetric, so they do not directly cause particle or heat transport
• Receive energy from turbulence
• Destroy turbulent eddies through shearing

Zonal Flow Generation Mechanism

Zonal flows are generated by Reynolds stress from drift wave turbulence:

$$\frac{\partial v_{ZF}}{\partial t} = -\frac{\partial}{\partial r} \langle \tilde{v}_r \tilde{v}_\theta \rangle - \gamma_d v_{ZF}$$

The first term is driving by Reynolds stress, and $\gamma_d$ is the damping rate (mainly collisional damping).

From the perspective of nonlinear mode coupling, three-wave interaction:

$$\mathbf{k}_1 + \mathbf{k}_2 = \mathbf{k}_{ZF} = (k_r, 0, 0)$$

transfers energy from drift waves to zonal flows.

Turbulence Suppression by Zonal Flows

The turbulence suppression effect by zonal flow shear:

$$\omega_{ZF} = \frac{1}{B} \frac{d^2 \phi_{ZF}}{dr^2} = \frac{1}{B} \frac{dv_{ZF}}{dr}$$

When this shearing rate exceeds the turbulence growth rate ($\omega_{ZF} > \gamma$), turbulence is suppressed.

Physically, zonal flow shear stretches turbulent eddies and shortens their correlation length. This reduces the transport efficiency of turbulence.

A positive correlation between zonal flow strength and confinement performance exists and has been confirmed in gyrokinetic simulations.

Geodesic Acoustic Mode (GAM)

As an oscillatory component of zonal flows, there is the Geodesic Acoustic Mode (GAM). Frequency:

$$\omega_{GAM} \approx \sqrt{2} \frac{c_s}{R} \sqrt{1 + \frac{1}{2q^2}}$$

where $c_s = \sqrt{T_e/m_i}$ is the ion sound speed.

GAM is a finite-frequency zonal flow that oscillates due to compressibility effects (geodesic curvature).

GAM characteristics:

• $n = 0$, $m = 0$ (same symmetry as zonal flow)
• Finite frequency ($\omega \sim c_s/R \sim 10-100$ kHz)
• Coherent oscillations
• Observed in the edge region

Turbulence-Zonal Flow Interaction

The interaction between turbulence and zonal flows is important as a self-regulation mechanism for plasma turbulence.

Predator-prey model: Diamond and colleagues proposed a simple model for turbulence energy $E$ and zonal flow energy $U$:

$$\frac{dE}{dt} = \gamma E - \alpha EU - \beta E^2$$ $$\frac{dU}{dt} = \alpha EU - \gamma_d U$$

$\gamma$ is the turbulence growth rate, $\alpha$ is the zonal flow generation efficiency, $\beta$ is the nonlinear saturation of turbulence, and $\gamma_d$ is the zonal flow damping.

This model predicts oscillatory coexistence (limit cycle) of turbulence and zonal flows, explaining the limit cycle oscillations observed before the L-H transition.

History of Confinement Improvement

The history of fusion research is also a history of continuous improvement in confinement performance.

Early Tokamaks (1950s-1970s)

In the late 1950s, the T-3 tokamak was built in the Soviet Union. The electron temperature $T_e \sim 1$ keV and confinement time $\tau_E \sim 10-20$ ms verified by the UK Culham team in 1968 greatly surpassed other devices of the time.

In the 1970s, tokamak construction progressed around the world:

• PLT (USA): 1975, achieved $T_i \sim 6$ keV with auxiliary heating
• T-10 (USSR): developed electron cyclotron heating
• TFR (France): advanced plasma diagnostic techniques

Era of Medium-sized Tokamaks (1980s-1990s)

The discovery of H-mode at ASDEX in 1982 was revolutionary. Confinement improved by about a factor of two, becoming the foundation for fusion reactor design.

In 1983, Goldston published the L-mode scaling law, showing that confinement time strongly depends on plasma current.

The three major tokamaks JET (Europe), TFTR (USA), and JT-60 (Japan) were built, achieving plasma parameters close to reactor conditions:

• JET: achieved Q = 0.6 in D-T experiments in 1991
• TFTR: 10.7 MW fusion power in D-T in 1994
• JT-60: equivalent Q = 1.25 (extrapolated from D-D experiments)

Discovery of Advanced Confinement Modes (1990s-2000s)

In the 1990s, various improved confinement modes were discovered:

VH-mode (Very High confinement): discovered at DIII-D, achieved $H_{89} \sim 2.5$

Reversed shear ITB: discovered at JT-60U and DIII-D, strong transport barrier in the core

Steady-state high $\beta$ operation: hybrid scenario developed at DIII-D

Modern Achievements (2000s onward)

Since the 2000s, understanding of confinement physics has deepened:

Quantitative comparison between gyrokinetic simulations and experiments has progressed, improving the predictive capability of turbulent transport.

Development of ELM control techniques (such as RMP) has progressed, and application to ITER is planned.

Long-duration H-mode discharges (over 100 seconds) have been achieved in superconducting tokamaks like KSTAR and EAST.

Current State of the Art

Typical parameters achieved in current tokamaks:

Device	$T_0$ (keV)	$n_0$ ($10^{20}$ m$^{-3}$)	$\tau_E$ (s)	$n T \tau$ (m$^{-3}$ keV s)
JET	12	0.8	0.9	$0.9 \times 10^{21}$
JT-60U	14	0.6	0.8	$0.7 \times 10^{21}$
DIII-D	20	0.5	0.3	$0.3 \times 10^{21}$

There is approximately 3-5 times shortfall from the Lawson criterion ($n T \tau > 3 \times 10^{21}$), but ITER’s scale is expected to achieve this.

Importance for Fusion

Understanding and controlling transport phenomena is the central challenge of fusion reactor development.

Confinement Performance and Reactor Size

The smaller the transport, the higher the temperature and density that can be achieved with the same heating power. The Lawson criterion:

$$n T \tau_E > 3 \times 10^{21} \, \text{m}^{-3} \cdot \text{keV} \cdot \text{s}$$

The device size required to achieve this is inversely proportional to confinement performance.

Fusion power:

$$P_{fus} \propto n^2 \langle \sigma v \rangle V \propto \frac{W^2}{\tau_E^2} \langle \sigma v \rangle V$$

The $\tau_E$ required to achieve $Q = P_{fus}/P_{in}$ is:

$$\tau_E \propto \frac{1}{Q^{1/2}}$$

If the $H$ factor doubles, the required device volume can be reduced to about 1/4.

Reactor Economics

The cost of fusion reactors strongly depends on device size. Capital cost:

$$C \propto R^{\alpha}, \quad \alpha \approx 2-3$$

Improved confinement enables:

• Reduction in superconducting magnet quantity
• Reduction in vacuum vessel and shield volume
• Reduction in building size

leading to reduced power generation costs.

For example, if the H factor improves from 1.0 to 1.5, the device size to achieve the same fusion output can be reduced by about 30%, potentially reducing costs by about 50%.

Bootstrap Current and Steady-state Operation

For steady-state operation of fusion reactors, minimizing external current drive is important. If a bootstrap current fraction $f_{bs} > 0.7$ is achieved:

$$P_{CD} = P_{CD,0} (1 - f_{bs})$$

External current drive power $P_{CD}$ can be significantly reduced, improving the recirculating power fraction (recirculated power/total output).

Steady-state tokamak reactor designs target $f_{bs} \sim 0.8-0.9$. This enables economical operation even with low current drive efficiency $\eta_{CD}$.

Burn Control

In burning plasmas, alpha particle heating becomes dominant:

$$P_\alpha = \frac{1}{4} n_D n_T \langle \sigma v \rangle E_\alpha$$

where $E_\alpha = 3.5$ MeV is the alpha particle energy.

Transport determines the temperature profile and directly affects the fusion reaction rate. The property of temperature gradients being fixed at critical values (stiff transport) may work favorably for burn control.

Thermal feedback: When temperature rises, the reaction rate increases, and heating further increases - a positive feedback exists. However, due to stiff transport, additional heating is mainly converted to heat flux toward the edge, preventing runaway of the central temperature.

Impurity Transport

In fusion reactors, impurities sputtered from the wall and helium ash exhaust are important. Impurity flux:

$$\Gamma_Z = -D_Z \frac{dn_Z}{dr} + V_Z n_Z$$

The sign of the convection term $V_Z$ determines whether impurities accumulate (inward) or are expelled (outward).

In neoclassical transport, inward pinch is generally predicted:

$$V_{neo} = -\frac{D_{neo}}{L_n} \left( Z - 1 + \frac{Z L_n}{L_T} \right)$$

When the temperature gradient is large, high-$Z$ impurities tend to accumulate in the center.

Turbulent transport is generally diffusive and transports impurities outward. The balance between turbulent diffusion and neoclassical pinch is important.

If high-$Z$ impurities like tungsten accumulate in the core, radiation losses increase and burning cannot be sustained:

$$P_{rad} = n_e n_Z L_Z(T_e)$$

The radiative cooling rate $L_Z$ for tungsten is very large at $T_e \sim 1-5$ keV, and even trace amounts ($n_Z/n_e \sim 10^{-5}$) become a serious problem.

Helium Ash Exhaust

Helium produced by D-T reactions must be exhausted as “ash” after depositing 20% of its energy into the plasma.

Helium accumulation rate depends on transport:

$$\frac{dn_{He}}{dt} = S_{He} - \frac{n_{He}}{\tau_{He}^*}$$

where $S_{He}$ is the production rate from fusion reactions and $\tau_{He}^*$ is the effective confinement time.

In steady-state operation:

$$\frac{n_{He}}{n_e} = \frac{S_{He} \tau_{He}^*}{n_e} \propto \frac{P_{fus}}{n_e} \tau_{He}^*$$

$\tau_{He}^*/\tau_E \sim 5-10$ is required. Diffusive transport by turbulence contributes to helium exhaust.

Summary

Plasma transport has evolved from classical collision theory, through neoclassical theory incorporating toroidal effects, to turbulent transport theory. This development is the result of over a century of interaction between theory and experiment.

Key findings are as follows:

Classical transport predicts diffusion at Larmor radius scales, but greatly underestimates experimental values. Basic transport coefficients such as Spitzer resistivity are accurately predicted from classical theory.

Neoclassical transport in toroidal magnetic configurations is enhanced by a factor of $q^2/\epsilon^{3/2}$ over classical values and includes banana orbit effects. The most important achievement is the prediction of the bootstrap current, which is key to steady-state reactor operation.

Anomalous transport (turbulent transport) determines actual confinement performance, with drift wave instabilities such as ITG, TEM, and ETG as the main drivers. The concepts of critical temperature gradient and stiff transport are important for temperature profile determination and burn control.

Gyrokinetic simulations enable quantitative predictions of turbulent transport and contribute to ITER performance predictions. Codes such as GYRO, GENE, and GTC show quantitative agreement with many experiments.

Transport barriers (H-mode pedestal, ITB) significantly improve confinement performance and enhance fusion reactor feasibility. Turbulence suppression by $E \times B$ sheared flow is the main physics mechanism.

Empirical scaling laws form the design foundation for ITER and support the achievement of $Q = 10$. Dependence close to gyro-Bohm scaling provides favorable predictions for large devices.

Zonal flows are important as a turbulence self-regulation mechanism and contribute to the L-H transition and determination of transport levels in simulations.

Future challenges include quantitative prediction of electron transport, complete understanding of pedestal physics, elucidation of multi-scale turbulence, and verification of transport in burning plasmas. Experiments at ITER are expected to significantly advance this understanding.

Related Topics

• Plasma Physics Overview - Fundamentals of plasma parameters
• Magnetohydrodynamics (MHD) - Plasma equilibrium and stability
• Charged Particle Motion - Single particle orbits
• Plasma Instabilities - Instabilities driving turbulence
• Tokamak - The main confinement approach utilizing transport barriers
• ITER - The international project based on H-mode operation