The ordering behind ideal magnetohydrodynamics

A typical nonrelativistic MHD ordering uses macroscopic lengths much larger than the Debye length and ion gyroradius, frequencies much smaller than ion gyrofrequency, and characteristic speeds much smaller than light speed. It also requires an appropriate pressure closure. These conditions suppress bulk charge separation, finite-orbit corrections, and displacement current for the chosen motions. They do not establish isotropic pressure automatically: weakly collisional magnetized plasmas may need an anisotropic fluid description even at large spatial scales.

Single-fluid momentum uses total mass density and total pressure. Summing species momentum equations cancels internal collision forces and combines their magnetic terms into $\mathbf j\times\mathbf B$. Ion mass usually dominates inertia. Their small relative velocity can still supply the required current, so setting it identically to zero before using Ampère's law would be inconsistent. MHD keeps that electromagnetic effect while simplifying the separate inertial evolution of the species.

The electron momentum equation provides a generalized Ohm law containing electric and motional fields, a Hall term, electron pressure, inertia, and friction. Ideal MHD assumes the retained leading relation is $\mathbf E+\mathbf u\times\mathbf B=0$. This permits a finite laboratory electric field perpendicular to the magnetic field. It constrains that field in the moving-fluid frame. Good electrical conductivity suppresses a resistive contribution but does not alone remove Hall, pressure-tensor, or electron-inertial terms.

Combining this ideal relation with Faraday's law gives magnetic induction by fluid motion. A divergence-free initial field stays divergence-free under a sufficiently smooth solution. For an adiabatic scalar gas one may evolve $p/\rho^\gamma$ along the flow. Selecting that closure excludes heat conduction and dissipation at the retained order; ideal electrical behaviour by itself does not specify an equation of state, a temperature ratio, or the importance of thermal transport.

With electrical resistivity $\eta$ in ohm metres, magnetic diffusivity is $\eta_m=\eta/\mu_0$ in square metres per second. The magnetic Reynolds number $R_m=UL/\eta_m$ compares induction to resistive diffusion. A large value supports neglecting uniform-resistivity diffusion on length $L$ and speed $U$. It says less about a narrow current layer, because its smaller gradient scale increases the discarded diffusion term. The relevant length must belong to the feature being modelled.

For singly charged ions, let $d_i=\sqrt{m_i/(\mu_0ne^2)}$ and $v_A=B/\sqrt{\mu_0\rho}$. A Hall-to-motional-field estimate is $(d_i/L)(v_A/U)$. Thus $L\gg d_i$ is a useful sufficient geometric guide for Alfvénic flows, but a very slow flow can need a stricter comparison. Electron pressure and inertia require their own estimates. A convincing ideal reduction documents several small ratios; no single Reynolds number certifies every neglected process.