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)
SN - 0962-8444
VL - 476
JO - Proceedings of the Royal Society A: Mathematical and Physical Sciences
JF - Proceedings of the Royal Society A: Mathematical and Physical Sciences
IS - 20190851
ER -