The Braginskii Equations
The classic collisional two-fluid transport equations with explicit magnetized transport coefficients.
A collisional closure
Braginskii derived a set of fluid equations for a collisional, magnetized plasma by expanding the kinetic equation in the ratio of gyroradius to mean free path. For each species he gives continuity, momentum, and energy equations plus explicit expressions for the viscosity, thermal conductivity, resistivity, and thermoelectric coefficients.
Anisotropic transport
Because a magnetic field strongly restricts cross-field motion, every transport coefficient splits into parallel, perpendicular, and cross (diamagnetic) components. Parallel heat conduction is enormous while perpendicular conduction is suppressed by roughly the ratio (gyrofrequency times collision time) squared. This anisotropy is the defining feature of the Braginskii closure.
q_e = -kappa_parallel grad_parallel T - kappa_perp grad_perp T + kappa_cross (b x grad T)Extra force terms
The Braginskii momentum and heat equations contain thermal-force terms, the friction between electrons and ions, and the diamagnetic heat flux. These give rise to the Nernst effect and the Ettingshausen effect, important in dense, collisional edge and divertor plasmas.
How it is solved numerically
- Edge and divertor codes (of the SOLPS/UEDGE family) solve Braginskii fluid equations on field-aligned meshes
- The huge parallel-to-perpendicular conductivity ratio demands implicit or operator-split time stepping
- Flux limiters cap parallel heat flux where the collisional expansion breaks down
The core assumption is collisionality: the ordering fails in the hot, low-collision core, where gyrokinetics or neoclassical theory takes over. But in the cooler edge and divertor of a device like the Hyperion breeder, Braginskii transport is the workhorse for predicting heat loads on plasma-facing components.