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

The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids