In Grebenev, Wacławczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33), the conformal invariance (CI) of the characteristic \({\varvec{X}}_{1}(t)\) (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the 1-point PDF \(f_1({\varvec{x}}_{1},\omega _{1},t)\), \({\varvec{x}}_{1} \in D_1 \subset {\mathbb {R}}^2)\) 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 \(\mathrm{d}s^2 = f_1\cdot \left( {\mathrm{d}X_{1}^{ 1}}^2 + {\mathrm{d}X_{1}^{ 2}}^2\right) \) along the characteristic \(\left. {\varvec{X}}_{1}(t)\right| _{\omega _{1} = 0}\) together with the simple action functional \({\mathcal F}({\varvec{X}}_{1},\mathrm{d}s^2)\). We demonstrate that \(G_{{\mathcal {Y}}}\), which is a subgroup of the group G restricted on the variables \({\varvec{x}}_{1}\) and \(f_1\), gives rise to a symmetry transformations of \({\mathcal {F}}({\varvec{X}}_{1},\mathrm{d}s^2)\). 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 characteristics. We show that \({{\mathcal {F}}}({\varvec{X}}_{1},\mathrm{d}s^2)\) 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 \({\varvec{x}}_{1}\) in the sense of Cartan.

在 Grebenev, Wacławczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33) 中，特征\({\varvec{X}}_{1}(t)\)（零-vorticity Lagrangian path) 的第一个方程（即对于 1-point PDF 的演化\(f_1({\varvec{x}}_{1},\omega _{1},t)\) , \( {\ varvec {X}} _ {1} \在D_1 \子集{\ mathbb {R}} ^ 2）\）的非粘性Lundgren的-莫宁-诺维科夫（LMN）方程2的d涡字段推导。特征方程所承认的无穷小算子产生一个无限维李伪群G，它共形作用于\(D_1\)。我们定义了共形不变微分形式\(\mathrm{d}s^2 = f_1\cdot \left( {\mathrm{d}X_{1}^{ 1}}^2 + {\mathrm{d}X_{1}^{ 2}} ^2\right) \)沿着特征\(\left. {\varvec{X}}_{1}(t)\right| _{\omega _{1} = 0}\)以及简单的动作功能\({\mathcal F}({\varvec{X}}_{1},\mathrm{d}s^2)\)。我们证明\(G_{{\mathcal {Y}}}\)是G组的一个子群，它限制在变量\({\varvec{x}}_{1}\)和\(f_1\ )，产生\({\mathcal {F}}({\varvec{X}}_{1},\mathrm{d}s^2)\)的对称变换。有了这个，我们计算二阶通用微分不变量\(J_2^{{\mathcal {Y}}}\)（或不变量的多尺度表示）\(G_{{\mathcal {Y}}}\)在零涡度特征的作用下。我们证明\({{\mathcal {F}}}({\varvec{X}}_{1},\mathrm{d}s^2)\)是一个标量不变量并生成所有微分不变量，看起来像不同尺度的量，来自\(J_2^{{\mathcal {Y}}}\)由不变微分的算子。它可以深入了解Cartan 意义上的点\({\varvec{x}}_{1}\)附近的流域的几何形状。