Two-fluid plasma equations

Start with a kinetic distribution $f_s(\mathbf x,\mathbf v,t)$ for species $s$. Integrating over velocity gives number density; weighting by velocity gives particle flux and mean velocity. The zeroth kinetic moment yields continuity when ionization, recombination, and other particle sources are absent. The first moment yields species momentum balance. The pressure tensor is built from velocities relative to the species mean, so it represents random-motion stress rather than the bulk momentum flux counted a second time.

The material derivative follows that species' own mean velocity. Electron and ion derivatives need not be the same. Electric forces have opposite signs for their opposite charges; magnetic forces depend on their distinct velocities. The collision force densities exchange momentum between species and sum to zero for internal electron–ion collisions. Neglecting friction separately in one equation while retaining its reaction elsewhere would violate total momentum conservation unless an external momentum reservoir were specified.

For singly charged ions with $n_i\simeq n_e=n$, current is proportional to the difference of species velocities. Mass flow is instead weighted by species masses. Since ions are heavy, the centre-of-mass flow is usually close to ion flow even when electrons carry a substantial current. Quasineutrality constrains charge density, whereas current describes charge flux. Setting equal densities and then setting equal velocities without a scale argument discards electromagnetic dynamics that the two-fluid model was introduced to retain.

Maxwell's equations supply the fields and their coupling to both species. A quasineutral low-frequency reduction can replace the charge-separation equation while preserving current through Ampère's law in its appropriate ordering. Such a reduction cannot resolve an electron plasma oscillation that fundamentally needs Poisson's equation and electron inertia. Choosing which fields to eliminate is therefore a decision about the phenomena and scales of interest, not merely a way to save algebra.

Momentum evolution introduces the pressure tensor; pressure evolution introduces heat flux and higher moments. A finite list of exact moment equations therefore has more unknowns than equations. An isothermal closure fixes a temperature and sets scalar pressure proportional to density. An adiabatic scalar closure instead evolves $p_sn_s^{-\gamma_s}$ along the species flow when heat exchange and dissipation are negligible. Cold fluid equations discard pressure forces, which is justified by their size for the chosen process, not by the fluid label itself.

Magnetized weakly collisional populations can have different parallel and perpendicular pressures, while strong collisions can support a nearly isotropic local distribution. A scalar closure should therefore be checked against collision, gyro, streaming, and evolution timescales. Landau damping and detailed resonant particle trapping are generally not recovered by a simple local scalar closure. The diagnostic question is what physical information the closure removes and whether the intended calculation depends on that information.