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Write the physical magnetic field as ${\mathbf{B}}_{\mathrm{phys}}$ and use the constant mass density ${\rho }_0$ to introduce Alfvén velocity units, $\mathbf{B}={\mathbf{B}}_{\mathrm{phys}}/\sqrt{{\mu }_0{\rho }_0}$. 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 toThe 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 $p$.

The viscous penetration depth is the distance over which a time-periodic shear velocity driven by an oscillating boundary decays by $1/e$ in a semi-infinite Newtonian fluid. The momentum diffusion equation gives $u=U_0{e}^{-z/\delta }\cos (\omega t-z/\delta )$. Both attenuation and phase lag are controlled by $z/\delta$. The corresponding fluid-displacement amplitude has the same spatial decay, because dividing each velocity amplitude by the common angular frequency does not change their ratio.

Vortex stretching changes the strength of vorticity when a fluid velocity varies along the direction of the local vorticity. For an inviscid, constant-density incompressible flow, the vorticity equation reduces to $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}$. Thus a flow that stretches a vortex in its axial direction can amplify its vorticity without an applied rotational body force.

An ideal thin vortex tube gives a geometric explanation. Incompressibility preserves the volume of a material segment, so stretching its length reduces its cross-sectional area. Conserved circulation then requires a larger mean axial vorticity. Compression reverses that effect. The Burgers vortex adds a competing mechanism: radial momentum diffusion balances the axial stretching and selects a finite core size.