New relations for energy flow in terms of vorticity

Abstract

Considering the vorticity formulation of the Euler equations, we partition the kinetic energy into its contribution from each pair of interacting vortices. We call this contribution the "interaction energy". We show that each contribution satisfies a reciprocity relation on triples of vortices: $\boldsymbol{A}$'s action on $\boldsymbol{B}$ changes the interaction energy between $\boldsymbol{B}$ and $\boldsymbol{C}$ in an equal and opposite way to the effect of $\boldsymbol{C}$'s action on $\boldsymbol{B}$ on the interaction energy between $\boldsymbol{A}$ and $\boldsymbol{B}$. This result is a curiously detailed accounting of energy flow, as contrasted to standard pointwise conservation laws in fluid dynamics. This result holds for all triples of points $\boldsymbol{A},\boldsymbol{B},\boldsymbol{C}$ in two dimensions; and in 3 dimensions for all points $\boldsymbol{A},\boldsymbol{C}$, and all closed vorticity streamlines $\boldsymbol{B}$. We show this result in 3 dimensions as a consequence of an interaction energy flow around $\boldsymbol{B}$ that is a function only of the triple $(\boldsymbol{A},\boldsymbol{b}\in \boldsymbol{B},\boldsymbol{C})$, a result which may be of independent interest.