Conformal Invariance of Characteristic Lines in a Class of Hydrodynamic Models

Abstract

This paper addresses the problem of the existence of conformal invariance in a class of hydrodynamic models. For this we analyse an underlying transport equation for the one-point probability density function, subject to zero-scalar constraint. We account for the presence of non-zero viscosity and large-scale friction. It is shown analytically, that zero-scalar characteristics of this equation are invariant under conformal transformations in the presence of large-scale friction. However, the non-zero molecular diffusivity breaks the conformal group (CG). This connects our study with previous observations where CG invariance of zero-vorticity isolines of the 2D Navier–Stokes equation was analysed numerically and confirmed only for large scales in the inverse energy cascade. In this paper, an example of CG is analysed and possible interpretations of the analytical results are discussed.

Keywords

• conformal symmetry
• Lundgren–Monin–Novikov equations
• method of characteristics