Transport Coefficients

1.1 Introduction to Transport in Magnetized Plasmas

Transport coefficients describe how macroscopic quantities such as particle density, momentum, and energy are redistributed in a plasma through collisional processes. In unmagnetized plasmas, transport is isotropic and governed by collision frequencies. In magnetized plasmas, the Lorentz force confines particles to helical orbits, leading to a dramatic anisotropy: transport along the magnetic field (parallel) vastly exceeds transport across it (perpendicular).

The foundational framework was developed by S. I. Braginskii (1965), who derived a complete set of transport equations for a fully ionized, collisional plasma in a strong magnetic field. These equations provide expressions for resistivity, viscosity, thermal conductivity, and thermoelectric coefficients as tensors with distinct parallel, perpendicular, and cross (diamagnetic) components.

1.2 Spitzer Resistivity

The electrical resistivity of a fully ionized plasma was first calculated rigorously by Lyman Spitzer and collaborators. The parallel resistivity (along the magnetic field) is determined by the electron-ion collision time:

where the electron collision time is:

Substituting and collecting constants yields the Spitzer resistivity:

where T_e is in eV and ln Lambda is the Coulomb logarithm (typically 10-20 for fusion plasmas). Note the remarkable scaling: resistivity decreases with temperature as T^(-3/2), meaning hot plasmas become excellent conductors. At 10 keV, plasma is a better conductor than copper.

Parallel Conductivity

The parallel electrical conductivity is simply the inverse of resistivity:

This has the same form as Drude conductivity in a metal, but with the electron-ion Coulomb collision time replacing the electron-phonon scattering time. The factor 0.51 (for Z=1) appears when the full Fokker-Planck collision operator is used instead of a simple Krook model, accounting for the velocity dependence of the Coulomb cross section.

1.3 Perpendicular Diffusion

Perpendicular transport across the magnetic field is fundamentally different from parallel transport. Particles are tied to their field lines and can only migrate across the field through collisions that displace the guiding center by a gyroradius.

Classical (Collisional) Diffusion

In the classical random-walk picture, a particle steps one gyroradius per collision. The perpendicular diffusion coefficient is:

For ions, replace electron quantities with ion quantities. The key scaling is $$D_\perp \propto 1/B^2$$, showing that stronger magnetic fields improve confinement.

Bohm Diffusion

Experimentally, early fusion devices observed anomalously fast perpendicular transport, far exceeding classical predictions. David Bohm proposed a semi-empirical scaling:

The ratio of Bohm to classical diffusion is:

Bohm diffusion scales as 1/B (not 1/B^2), representing the worst case for magnetic confinement. It is now understood to arise from turbulent ExB transport driven by microinstabilities such as drift waves and ITG modes. Modern tokamaks achieve confinement between classical and Bohm levels.

1.4 Thermal Conductivity

Thermal conduction transports energy from hot to cold regions. The Braginskii parallel electron thermal conductivity is:

The numerical coefficient 3.16 comes from solving the Fokker-Planck equation with the full Landau collision operator for Z=1. For other charge states, the coefficient varies. The ion parallel thermal conductivity is:

Since $$\tau_e/\tau_i \sim (m_e/m_i)^{1/2}$$ and the mass ratio appears again in the denominator, the electron parallel heat conductivity exceeds the ion one by a factor of $$\sim (m_i/m_e)^{1/2} \approx 43$$ for hydrogen. This means electrons dominate parallel heat transport and set the equilibration rate along field lines.

The perpendicular thermal conductivities are much smaller:

The anisotropy ratio is $$\kappa_\parallel / \kappa_\perp \sim (\omega_c \tau)^2 \sim 10^{10}$$ in fusion plasmas, meaning heat flows almost exclusively along field lines.

1.5 Computational Examples

The following simulation plots Spitzer resistivity vs temperature and compares Bohm vs classical diffusion coefficients as a function of magnetic field strength.

1.6 Summary of Transport Scalings