  We derive the spin Euler equation for ideal flows by applying the spherical
Clebsch mapping. This equation is based on the spin vector rather than the
velocity. It enables a feasible Lagrangian study of fluid dynamics, as the
isosurface of a spin-vector component is a vortex surface and material surface
in ideal flows. The spin Euler equation is also equivalent to a special case of
the Landau-Lifshitz equation with a specific effective magnetic field,
revealing a possible connection between ideal flow and magnetic crystal. We
conduct direct numerical simulations of three ideal flows of the vortex knot,
vortex link and modified Taylor-Green flow by solving the spin Euler equation.
The evolution of the Lagrangian vortex surface illustrates that the regions
with large vorticity are rapidly stretched into spiral sheets. We establish a
non-blowup criterion for the spin Euler equation, suggesting that the Laplacian
of the spin vector must diverge if the solution forms a singularity at some
finite time. The DNS result exhibits a pronounced double-exponential growth of
the maximum norm of Laplacian of the spin vector, showing no evidence of the
finite-time singularity formation if the double-exponential growth holds at
later times. Moreover, the present criterion with Lagrangian nature appears to
be more sensitive than the Beale-Kato-Majda criterion in detecting the flows
that are incapable of producing finite-time singularities.
