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A fluid element is an infinitesimal material parcel followed along a Lagrangian trajectory.

The MHD induction equation can be written in material form asIts $y$ component immediately givesFor the Cartesian magnetic flux function, the $x,z$ components of the MHD induction equation, or the $y$ component of the magnetic vector potential equation, giveThe additive function of time in $\psi$ has no effect on $\mathbf{B}$. Choose this gauge so that $C=0$. Then the flux label is materially conserved:This is magnetic flux freezing in the two-dimensional geometry.

To prove the absence of axial motion, construct an invariant solution with $u_y=0$. Along each Lagrangian trajectory, $\psi$ is fixed, and the assumed divergence givesThus $B_y$ remains a function of $\psi$ and time alone. Its gradient stays parallel to $\nabla \psi$, so the $y$ component of the Lorentz force remains zero. The axial momentum equation isbecause $(B_x{\partial }_x+B_z{\partial }_z)\psi =0$. With $u_y=0$ initially, it remains zero. Hence neither axial force nor axial motion is generated. This flux-surface preservation during magnetic-tube expansion argument assumes the smooth ideal evolution for which the initial-value problem is unique; the particular in-plane rising motion need not be calculated.

Parametrize a streamline by $\mathbf{r}(s)$ with $d\mathbf{r}/ds=\mathbf{u}(\mathbf{r}(s))$. The chain rule givesTherefore $H$ is constant along every streamline. A different regular parametrization merely multiplies this derivative by a scalar and gives the same conclusion. For time-dependent flow, the claim concerns streamlines at a fixed time, rather than Lagrangian trajectories.

The dyed particle satisfies the Lagrangian trajectory equationsFor the proposed approximation, $x=O({\epsilon }^2)$ andAlsoThe initial conditions hold, verifyingthrough order ${\epsilon }^2$.

Over one period, the periodic terms return to their initial values while the secular term changes $x$ by ${\epsilon }^2\pi$. Hence the dyed particle has Stokes driftto this order, despite the zero Eulerian mean velocity.