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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References
Cartan, E.: La Théoric des Groupes Finis et Continus et la Géometrie Differentielle traittée par le Méthode du Repére Mobile. Gauthier-Villars, Paris (1937)
Olver, P.J., Pohjanpelto, J.: Differential invariant algebra of Lie pseudo-groups. Adv. Math. 222(5), 1746–1792 (2009)
Hubert, E., Olver, P.J.: Differential invariants of conformal and projective surfaces. SIGMA 3, 97–115 (2007)
Grebenev, V.N., Wac��awczyk, M., Oberlack, M.: Conformal invariance of the Lungren–Monin–Novikov equations for vorticity fields in 2D turbulence. J. Phys. A: Math. Theor. 50, 435502 (2017)
Grebenev, V.N., Wacławczyk, M., Oberlack, M.: Conformal invariance of the zero-vorticity Lagrangian path in 2D turbulence. J. Phys. A: Math. Theor. 52, 335501 (2019). https://doi.org/10.1088/1751-8121/ab2f61
Wacławczyk, M., Grebenev, V.N., Oberlack, M.: Conformal invariance of characteristic lines in a class of hydrodynamic models. Symmetry 12, 1482 (2020). https://doi.org/10.3390/sym12091482
Friedrich, R., Daitche, A., Kamps, O., Lülff, J., Michel Voßkuhle, M., Wilczek, M.: The Lundgren–Monin–Novikov hierarchy: kinetic equations for turbulence. C. R. Physique 13, 929–953 (2012)
Rivera, M.K., Aluie, H., Ecke, R.E.: The direct enstrophy cascade of two-dimensional soap film flows. Phys. Fluids 26, 055105 (2014)
Ouellette, N.T.: Turbulence in two dimensions. Phys. Today 65, 68–69 (2012)
Bernard, D., Boffetta, G., Celani, A., Falkovich, G.: Conformal invariance in two-dimensional turbulence. Nat. Phys. 2(2), 124–128 (2006)
Falkovich, G., Musacchio, S: Conformal invariance in inverse turbulent cascades (2010). arXiv:1012.3868
Bernard, D., Boffetta, G., Celani, A., Falkovich, G.: Inverse turbulent cascades and conformally invariant curves. Phys. Rev. Lett. 98, 024501–504 (2007)
Lie, S.: Klassifikation und integration von gewöhnlichen differentntialgleichungen zwischen \(x\), \(y\), die eine gruppe von transformationen gestatten I. II. Math. Ann. 32, 213–281 (1888)
Beffa, M.G.: Relative and absolute differential invariants for conformal curves. J. Lie Theory 13, 213–245 (2003)
Ovsyannikov, L.V.: Group Analysis of Differential Equations. Nauka, Moscow (1978)
Tresse, A.: Sur les invariants différentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)
Simon, U.: The Pick invariant in equaffine differential geometry. Abh. Math. Sem. Univ. Hamburg 1983, 225–228 (1983)
Thalabard, S., Bec, J.: Turbulence of generalised flows in two dimensions. J. Fluid Mech. 883 A49 (2020)
Sabitov, K.: Quasi-conformal mappings of a surface generated by its isometric transformation and bendings of the surface onto itself. Fundamentalnaya i prikladnaya matematika 1, 281–288 (1995)
Kontsevich, M., Suhov, Y.: On Malliavin measure and SLE and CFT. Proc. Steklov Inst. Math. 258(1), 100–146 (2007)
Eisenhart, L.H.: Continuous Group of Transformations. Prienceton University Press, Prienceton (1933)
Wacławczyk, M., Oberlack, M.: Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation. J. Math. Phys. 54, 072901 (2013). https://doi.org/10.1063/1.4812803
Olver, P., Sapiro, G., Tannenbaum, A.: Differential invariant signatures and flows in computer vision: a symmetry group approach. In: ter Haar, Romeny B.M. (ed.) Geometry-Driven Diffusion in Computer Vision. Computational Imaging and Vision, vol. 1. Springer, Dordrecht (1994)
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