= Fast-time electron-MHD limit of Hall magnetohydrodynamics
{title2=$l\ll d_i,\quad\tau\ll l/v_A$}
For field amplitudes of order the guide field, the <ion> Lorentz-force response over time $\tau$ is $\Delta u/v_A=O(v_A\tau/l)\ll1$. The electron-current velocity is of order $v_Ad_i/l$, larger than a bulk velocity of order $v_A$. Thus the leading <Hall magnetohydrodynamics> induction equation becomes the closed <electron magnetohydrodynamics> equation, while <ions> remain fixed to leading order on this fast scale. The first nonzero <ion> response need not have zero acceleration. This limit retains the fast branch of <incompressible Hall-MHD wave dispersion>, not the slow branch; very oblique modes must separately satisfy the fast-time ordering.
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