Momentum-scalar coupled turbulence with anomalous momentum and scalar diffusions. Part 1: Without external force and with long-range external force

Abstract: We present a theoretical model for momentum--scalar coupled turbulence in which both fields undergo anomalous diffusion, described by fractional biharmonic operators of orders $γ/4$ and $α/4$, respectively. Focusing on the long-range external forcing or unforced turbulence, we derive analytical expressions for the kinetic energy spectrum $E_u(k)$, the scalar spectrum $E_s(k)$, and the characteristic wavenumbers $$k_K = \left( \frac{ε_u^{1/3}}{c_u} \right)^{1/(γ- 2/3)}$$ (reciprocal of Kolmogorov scale) and $$k_S = \left( \frac{ε_u^{1/3}}{c_s} \right)^{1/(α- 2/3)}$$ (reciprocal of scalar dissipation scale) as functions of $γ$, $α$, turbulent dissipation rate $ε_u$, diffusivities of momentum ($c_u$) and scalar ($c_s$), respectively. An anomalous Schmidt number $Sc_Z = k_0^{γ- α} \frac{c_u}{c_s}$ is defined to governs the cascade topology. It describes the ratio of diffusion times of scalar and momentum on the minimum wavenumber $k_0$. Superdiffusion ($γ<2$ or $α<2$) is shown to counter-intuitively enlarge $k_K$ and $k_S$, broadening the inertial range. The theory unifies the classical Kolmogorov--Obukhov--Corrsin--Batchelor scalings as special cases when $γ=α=2$, and provides a foundation for understanding non-Fickian transport in complex turbulent systems.