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 Eu(k), the scalar spectrum Es(k), and the characteristic wavenumbers kK = ( εu1/3/cu )1/(γ− 2/3) (reciprocal of Kolmogorov scale) and kS = ( εu1/3/cs )1/(α− 2/3) (reciprocal of scalar dissipation scale) as functions of γ, α, turbulent dissipation rate εu, diffusivities of momentum (cu) and scalar (cs), respectively. An anomalous Schmidt number ScZ = k0γ− α cu/cs is defined to governs the cascade topology. It describes the ratio of diffusion times of scalar and momentum on the minimum wavenumber k0. Superdiffusion (γ<2 or α<2) is shown to counter-intuitively enlarge kK and kS, 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.
Versions
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BibTeX
@misc{zhao2026momentumscalar,
title = {{Momentum-scalar coupled turbulence with anomalous momentum and scalar diffusions. Part 1: Without external force and with long-range external force}},
author = {Wei Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.23946},
eprint = {2608.23946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23946},
}
RIS
TY - UNPB
TI - Momentum-scalar coupled turbulence with anomalous momentum and scalar diffusions. Part 1: Without external force and with long-range external force
AU - Wei Zhao
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.23946
UR - https://arxiv.org/abs/2608.23946
AB - 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 = ( ε_u^1/3/c_u )^1/(γ− 2/3) (reciprocal of Kolmogorov scale) and k_S = ( ε_u^1/3/c_s )^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 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.
ER -