An SO(3) Gauge Theory of Turbulence with Spontaneous Symmetry Breaking

Abstract

Fully developed isotropic turbulence exhibits a dual nature: a continuous, scale-invariant energy cascade coexists with discrete, intense vortex filaments. We show that this duality arises from a spontaneously broken SO(3) gauge symmetry. By identifying the specific angular momentum $\mathbf{L}=\mathbf{r}\times \mathbf{u}$ as a non-Abelian gauge connection and the radial velocity $u_r$ as a Higgs field, the turbulent vacuum is described by the SO(3) Georgi-Glashow model. When the radial strain condenses, the symmetry breaks SO(3) $\to$ U(1), generating a topological mass gap $M_W=gv$. This gap partitions the energy into a massless U(1) sector (the solenoidal background) that sustains the Kolmogorov cascade, and a massive SO(3)/U(1) sector that is confined to vortex filaments. Using high-resolution DNS data (JHTDB, $R{e}_{\lambda }\approx 433$), we empirically verify three key predictions: (i) the energy spectra obey a strict 1:2 equipartition over the inertial range, with a sharp divergence at $M_W\approx 40$; (ii) the radial Higgs field extracted around isolated vortex cores follows the exact BPS monopole profile $H(r)=\coth (r/\eta )-\eta /r$ with $\eta =0.0093$ domain units and the VEV $v=0.338$, identifying the ubiquitous "worms" as macroscopic 't Hooft-Polyakov monopoles; (iii) the Wilson loop computed from the velocity field exhibits a clean area law $⟨W_C⟩\sim e^{-\sigma A}$ with string tension $\sigma =0.303\pm 0.009$, directly confirming the confining nature of the turbulent vacuum.