TY - JOUR

T1 - The Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids

AU - Machon, Thomas J

PY - 2020/7/8

Y1 - 2020/7/8

N2 - 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 ≠ 0 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.

AB - 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 ≠ 0 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.

U2 - 10.1098/rspa.2019.0851

DO - 10.1098/rspa.2019.0851

M3 - Article (Academic Journal)

VL - 476

JO - Proceedings of the Royal Society A: Mathematical and Physical Sciences

JF - Proceedings of the Royal Society A: Mathematical and Physical Sciences

SN - 0962-8444

IS - 20190851

ER -