Momentum-scalar coupled turbulence with anomalous momentum and scalar diffusions. Part 2: With short-range external force and implementation in electrokinetic turbulence

Abstract

We extend the generalized anomalous diffusion framework established in Part I to the short-range forcing regime ($\beta >2/3$), where the multiscale-force dominated (MFD) subrange is intercalated after the inertial subrange, competing directly with the dissipation ranges. Focusing on electrokinetic (EK) turbulence as the prototypical example with $\beta =1$, we derive the relations for the velocity and scalar dissipation wavenumbers, $k_{MD}$ and $k_{SD}$, across all four subranges of the Quad-cascade process (inertial, constant-${\Pi }_u$, constant-${\Pi }_s$, and variable flux). By incorporating Golestanian's predicted anomalous diffusion regimes for electrolytes, we construct comprehensive phase diagrams showing how the relative magnitudes of $k_{MD}$ and $k_{SD}$ are governed by the scale-dependent anomalous Schmidt number $S{c}_Z$. We identify two new spectral subranges that emerge exclusively in this short-range forcing regime: (i) the convective-viscous subrange ($k_{MD}\ll k\ll k_{SD}$) of velocity spectrum for $S{c}_Z\gg 1$, where the scalar field drives a viscous flow yielding $E_u\sim k^{-\left(\frac{3}{2}+\frac{{\xi }_s}{2}\right)}$ with a stretched-exponential cutoff; and (ii) the diffusive-forcing subrange ($k_{SD}\ll k\ll k_{MD}$) of scalar spectrum for $S{c}_Z\ll 1$ and $\gamma \le \alpha$, where the scalar dissipation range determines the electric forcing, leading to $E_s\sim k^{{\xi }_u-2\alpha }$. These results provide a complete analytical map of EK turbulence under anomalous diffusion, revealing how external parameters such as electric field strength and ionic diffusivity determine the cascade topology.