Abstract
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.
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Acknowledgements
VG acknowledges the financial support of FAPESP Foundation, Brazil (Grant No. 2018/21330-2). The author AG thanks CNPq and FAPESP for the support. MO would like to thank for partial funding both from the Collaborative Research Center “Multiscale Simulation Methods for Soft Matter Systems” (TRR 146) through Project Number 233630050 and the single Project OB 96/48-1. Both projects are funded by the German Research Foundation (DFG).
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Grebenev, V.N., Grichkov, A.N., Oberlack, M. et al. Second-order invariants of the inviscid Lundgren–Monin–Novikov equations for 2d vorticity fields. Z. Angew. Math. Phys. 72, 129 (2021). https://doi.org/10.1007/s00033-021-01562-2
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DOI: https://doi.org/10.1007/s00033-021-01562-2
Keywords
- Lundgren–Monin–Novikov equations
- 2d vorticity field
- Conformal symmetries
- Multiscale representation of the invariants
- The second-order universal differential invariant