Coherent vortices, being durable for some time, are often observed in nature. The vorticity vector is defined by the curl of the velocity vector of a fluid. In the absence of viscosity, the vorticity is frozen into the fluid (the Helmholtz's law). Preservation of the link and the knot type of vortex lines, Kelvin's circulation theorem and invariance in time of the helicity (=the Hopf invariant) are all consequences of the Helmholtz's law. An exposition is given to the significance of these topological invariance, based on the Euler-Poincare framework.

A steady incompressible Euler flow is characterized as an extremal of the total kinetic energy with respect to perturbations constrained to an isovortical sheet. An isovortical perturbation preserves vortex-line topology and is expressible most efficiently by the Lagrangian variables. I will show how topological ideas work in the variational formulation for deriving steady solutions of the Euler equations, and in analyses of their linear and nonlinear stability.