Turbulence-informed kinetic theory of inertial-range fibre fragmentation

Slender fibres, including textile-derived microplastics, are abundant in aquatic environments and often extend beyond the Kolmogorov length scale. While breakup at dissipative scales has been characterised by velocity-gradient statistics, no closure existed for inertial-range spans where eddy turnover sets the clock. Here we develop a turbulence-informed kinetic theory of fibre fragmentation bridging turbulence forcing and slender-beam mechanics. First, we derive a load-to-curvature mapping showing that spanwise forcing generates peak bending moments scaling as ~ UL L² , with UL the velocity increment across fibre length L . Second, we construct a breakup hazard $h(L)$ from curvature-threshold exceedances over eddy-time blocks, which identifies a turbulence-defined critical span c . For L> c , breakup is eddy-time-limited, h(L)=O( ε 1/3L-2/3) with ε the mean turbulent energy dissipation rate, whereas for L< c , it is a rare-event process with h(L)∝ L5/3+α , α denoting the small correction from intermittency. Embedding this hazard in a self-similar binary kernel yields a closed population-balance equation for the fragment distribution $n(L,t)$ with sources and sinks. The framework produces explicit predictions: intermittency-corrected curvature scalings, critical spans set by material and flow parameters, start-up and halving times linked to surf-zone conditions and scaling profiles in the cascade. The steady-state bulk distribution on the subcritical branch, with vertical removal induced by horizontal convergence, follows n(L)∝ L-8/3-α L-2.7 , in striking agreement with the mean slope -2.68 observed for environmental microfibres in recent surveys. The reported variability of slopes is naturally explained in our framework by the coexistence of supercritical and subcritical branches together with L -dependent removal-driven sinks.