= Ideal magnetohydrodynamics
{title2=ideal MHD}
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= Ideal MHD
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<Ideal magnetohydrodynamics> describes a conducting <fluid> whose electromagnetic evolution neglects <electrical resistivity>. Its defining electric-field relation is
$$
\mathbf E+\mathbf u\times\mathbf B=0,
$$
so the <electric field> vanishes in the local nonrelativistic <fluid> rest frame. Together with <Faraday's law> this gives the <ideal magnetohydrodynamic induction equation>
$$
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B),
\qquad \nabla\cdot\mathbf B=0.
$$
The <Lorentz force density> couples the field back to <fluid> <momentum>. <Lecture notes: Astrophysical fluid dynamics by Gordon I. Ogilvie>[Sections 3.1.2–3.1.3, equations 42–48]{ref}
For sufficiently smooth fields, the induction equation implies <magnetic flux freezing>: flux through a <material surface> moving with the <fluid> is conserved. Stretching the <fluid> can therefore amplify the <magnetic field> while preserving its connectivity. <Magnetic tension> supports <Alfvén waves>, providing a direct dynamical consequence of the coupling.
A large <magnetic Reynolds number> $\mathrm{Rm}=UL/\eta$ motivates neglecting diffusion on the chosen scales. Thin <current sheets> can still make nonideal terms important locally. The ideal induction condition alone does not set <fluid> <viscosity>; mechanical dissipation and <magnetic diffusion> are different approximations.
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