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

Grebenev V.N., Grichkov A.N., Oberlack M. and Wacławczyk M.

Zeitschrift für angewandte Mathematik und Physik

72, 2021, art. 129, 10.1007/s00033-021-01562-2

In Grebenev, Wac\l{}awczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33) the conformal invariance (CI) of the characteristic ${\bf X}_{1}(t)$ (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the $1$-point PDF $f_1({\bf x}_{1},\omega_{1},t)$, ${\bf x}_{1} \in D_1 \subset \Bbb R^2)$ 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 $D_1$. We define the conformal invariant differential form $ds^2 =  f_1\cdot\left({dX_{1}^{\it 1}}^2 + {dX_{1}^{\it 2}}^2\right)$ along the characteristic $\left.{\bf X}_{1}(t)\right|_{\omega_{1} = 0}$ together with the simple action functional ${\mathcal F}({\bf X}_{1},ds^2)$. We demonstrate that $G_{\mathcal Y}$, which is a subgroup of the group $G$ restricted on the variables ${\bf x}_{1}$ and $f_1$, gives rise to a symmetry transformations of  ${\mathcal F}({\bf X}_{1},ds^2)$. With this, we calculate the second-order universal differential invariant $J_2^{\mathcal Y}$ (or the multiscale representation of the invariants) of $G_{\mathcal Y}$ under the action on the zero-vorticity characterisctics. We show that ${\mathcal F}({\bf X}_{1},ds^2)$ is a scalar invariant and generates all differential invariants, which look like the quantities of different scales, from $J_2^{\mathcal Y}$  by the operators of invariant differentiation. It gives insight into the geometry of a flow domain nearby point ${\bf x}_{1}$ in the sense of Cartan.