Lie Group Reconstruction of K41 and She-Leveque Scaling Laws in Incompressible Turbulence

The multi-scale self-similarity and intermittency in incompressible turbulence fundamentally stem from scaling symmetry and its breaking. Using Lie group and infinitesimal generator theory, this paper algebraically reconstructs the Kolmogorov K41 and She-Leveque (SL) scaling laws as an equivalent mapping of existing phenomenological models rather than a first-principles derivation. We show that K41 corresponds to the invariance (zero eigenvalue) of energy flux under the scaling generator, yielding a strictly linear character, whereas anomalous scaling reflects symmetry breaking. Furthermore, SL's hierarchical recursion is demonstrated to be equivalent to an eigenvalue difference equation of the prolonged generator acting on the hierarchy. This difference eigenvalue quantifies the hierarchy's ``nonlinear sensitivity'' to scale variations---akin to weight differences in representation theory---characterizing the degree of symmetry breaking. By solving this recursion with the geometric boundary conditions of 3D vortex tubes, the SL formula is reconstructed. Our framework reveals K41 as the flat spacetime of the turbulent mean field, and SL as the curved spacetime induced by vortex singularities, with Lie algebra providing the elegant mathematical language for this transition.