Second-order invariants of the inviscid Lundgren–Monin–Novikov equations for 2d vorticity fields

In Grebenev, Wacławczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33), the conformal invariance (CI) of the characteristic XX1(t) (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the 1-point PDF f1(xx1,ω1,t), xx1∈D1⊂R2) of the inviscid Lundgren–Monin–Novikov (LMN) equations for 2d vorticity fields was derived. The infinitesimal operator admitted by the characteristics equation generates an infinite-dimensional Lie pseudo-group G which conformally acts on D1. We define the conformal invariant differential form ds2=f1⋅(dX112+dX212) along the characteristic XX1(t)|ω1=0 together with the simple action functional F(XX1,ds2). We demonstrate that GY, which is a subgroup of the group G restricted on the variables xx1 and f1, gives rise to a symmetry transformations of F(XX1,ds2). With this, we calculate the second-order universal differential invariant JY2 (or the multiscale representation of the invariants) of GY under the action on the zero-vorticity characteristics. We show that F(XX1,ds2) is a scalar invariant and generates all differential invariants, which look like the quantities of different scales, from JY2 by the operators of invariant differentiation. It gives insight into the geometry of a flow domain nearby point xx1 in the sense of Cartan.