= Solution
Write the physical <magnetic field> as $\mathbf B_{\rm phys}$ and use the constant <mass density> $\rho_0$ to introduce <Alfvén velocity> units, $\mathbf B=\mathbf B_{\rm phys}/\sqrt{\mu_0\rho_0}$. If $P$ is the fluid's mechanical <pressure>, the <magnetohydrodynamic total pressure> per unit <mass density> is $p=P/\rho_0+B^2/2$. The <ideal magnetohydrodynamic equations> then reduce to
$$
\boxed{
\begin{aligned}
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u
&=-\nabla p+(\mathbf B\cdot\nabla)\mathbf B,\\
\partial_t\mathbf B+(\mathbf u\cdot\nabla)\mathbf B
&=(\mathbf B\cdot\nabla)\mathbf u,\\
\nabla\cdot\mathbf u&=0,\qquad \nabla\cdot\mathbf B=0.
\end{aligned}}
$$
The first equation is <momentum conservation>: the <magnetic pressure> has been included in $p$, leaving <magnetic tension> explicitly. The second is the <ideal magnetohydrodynamic induction equation>, expressing <magnetic flux freezing>. The last two equations impose <incompressibility> and the <solenoidal magnetic-field constraint>.
The model assumes a smooth conducting-fluid continuum with constant <mass density> and constant <magnetic permeability>, nonrelativistic <velocity>, and scales on which a single-fluid description is appropriate. Negligible <electrical resistivity> removes <magnetic diffusion>; negligible <viscosity> removes viscous <momentum diffusion>. Equivalently, the <magnetic Reynolds number> and the <Reynolds number> are large on the scales being retained. The low-frequency approximation neglects the <displacement current>; Hall and other kinetic corrections to <Ohm's law> are absent. <Incompressibility> is an imposed approximation, commonly justified when the <Mach number> is small. There is no independently evolved thermodynamic <pressure>: $p$ enforces the <incompressibility condition>. We omit external body forces; a conservative one could instead be absorbed into $p$.
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