  If the vorticity field of an ideal fluid is tangent to a foliation,
additional conservation laws arise. For a class of zero-helicity vorticity
fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown
to be an invariant purely of the vorticity, becoming a higher-order
helicity-type invariant of the flow. GV non-zero gives both a global
topological obstruction to steady flow and, in a particular form, a local
obstruction. GV is interpreted as helical compression and stretching of vortex
lines. Examples are given where the value of GV is determined by a set of
distinguished closed vortex lines.
