The Two-Fluid Plasma Equations

Separate momentum and continuity equations for electrons and ions, coupled through electromagnetic fields and collisions.

Why Two Fluids

Single-fluid MHD treats the plasma as one conducting medium and cannot describe effects that differ between species: charge separation, Hall physics, electron inertia, and separate temperatures. The two-fluid model keeps electrons and ions as distinct fluids, each with its own density n_s, velocity u_s, and pressure p_s, s labeling species.

The Equations

For each species the continuity equation is dn_s/dt + div(n_s u_s) = 0. The momentum equation is m_s n_s (du_s/dt) = q_s n_s (E + u_s x B) - grad p_s + R_s, where R_s is the collisional friction between species and du_s/dt is the convective derivative. An energy or pressure equation closes each fluid, and Maxwell's equations supply E and B self-consistently with the charge and current densities summed over species.

Generalized Ohm's Law

Subtracting the electron and ion momentum equations, weighted by charge, yields a generalized Ohm's law. It contains the resistive term eta J, the Hall term J x B / (n e), the electron-pressure term grad p_e / (n e), and the electron-inertia term. Dropping all but the resistive term recovers resistive MHD; keeping the Hall term gives Hall MHD.

Scales and Application

Two-fluid effects matter at the ion skin depth and ion gyroradius scales, and they are essential for fast magnetic reconnection, drift waves, and edge physics. Electron and ion temperatures decouple whenever collisional equilibration is slower than heating or transport, which is common in hot fusion plasmas. In the Hyperion breeder and in the D-3He burner alike, separate ion and electron energy channels are standard modeling practice; these are simulation results, not measurements.