The ideal Craik-Leibovich equations

We show that the Craik-Leibovich (CL) theory of Langmuir circulation in an ideal incompressible fluid driven by rapidly fluctuating surface waves due to the wind may be formulated in terms of Eulerian mean fluid variables as a Hamiltonian system. This formulation is facilitated by first determining Hamilton's principle for the CL equations. The CL Hamilton's principle is similar to that for a fluid plasma, driven by a rapidly varying external electromagnetic field via “ J · A” minimal coupling, after averaging the plasma action over the fast phase of the (single frequency) driving field. This similarity leads to a precise analogy between the CL vortex force and the Lorentz force on an electrically charged fluid due to an exernally imposed electromagnetic field. We determine the effect of this force on the inflection point criterion and the Richardson number criterion for stability of planar CL flows. The Noether symmetries of Hamilton's principle for the CL equations (under fluid particle relabeling) lead to conservation laws for Eulerian mean potential vorticity and helicity, and generate the steady Eulerian mean flows as canonical transformations. The generalized Lagrangian mean theory is discussed from the same viewpoint.