Dual Theory of MHD Turbulence

We present an exact analytic solution for decaying incompressible magnetohydrodynamic (MHD) turbulence. Our solution reveals a dual formulation in terms of two interacting Euler ensembles--one for hydrodynamic and another for magnetic circulation. This replaces empirical scaling laws with an infinite set of power terms with calculable decay exponents, some of which appear as complex-conjugate pairs related to the Riemann zeta function. A key result of our analysis is the explicit dependence of the solution on the Prandtl number ($\mathrm{Pr} = \nu/\eta$), leading to a phase transition at $\mathrm{Pr} = 1$. In the $\mathrm{Pr}<1$ regime, turbulence is dominated by hydrodynamic fluctuations, while for $\mathrm{Pr}>1$, two distinct solutions emerge: a metastable one in which magnetic fluctuations grow with $\mathrm{Pr}$ and a stable one where they remain balanced with hydrodynamic fluctuations. We compare our theoretical predictions with recent direct numerical simulations (DNS) and discuss their implications for astrophysical plasmas, fusion devices, and laboratory MHD experiments. Our results provide a rigorous mathematical framework for understanding MHD turbulence and its dependence on fundamental parameters, offering a new perspective on turbulence in highly conducting fluids.