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

and

α / 4 $\alpha /4$

, respectively. Focusing on the long-range external forcing or unforced turbulence, we derive analytical expressions for the kinetic energy spectrum

E u k ${E}_{u}\left(k\right)$

, the scalar spectrum

E s k ${E}_{s}\left(k\right)$

, the characteristic wavenumbers

k K = ε u 1 / 3 / c u 1 / γ − 2 3 ${k}_{K}={\left({\varepsilon }_{u}^{1/3}/{c}_{u}\right)}^{1/\left(\gamma -\frac{2}{3}\right)}$

(reciprocal of Kolmogorov scale) and

k S = ε u 1 / 3 / c s 1 / α − 2 3 ${k}_{S}={\left({\varepsilon }_{u}^{1/3}/{c}_{s}\right)}^{1/\left(\alpha -\frac{2}{3}\right)}$

(reciprocal of scalar dissipation scale) as functions of

γ $\gamma $

,

α $\alpha $

, turbulent dissipation rate

ε u ${\varepsilon }_{u}$

, diffusivities of momentum (

c u ${c}_{u}$

) and scalar (

c s ${c}_{s}$

), respectively. An anomalous Schmidt number

S c Z = k 0 γ − α c u / c s $S{c}_{Z}={k}_{0}^{\gamma -\alpha }{c}_{u}/{c}_{s}$

is defined to govern the cascade topology. It describes the ratio of diffusion times of scalar and momentum on the minimum wavenumber

k 0 ${k}_{0}$

. Superdiffusion (

γ < 2 $\gamma {< }2$

or

α < 2 $\alpha {< }2$

) is shown to counter-intuitively enlarge

k K ${k}_{K}$

and

k S ${k}_{S}$

, broadening the inertial range. The theory unifies the classical Kolmogorov–Obukhov–Corrsin–Batchelor scalings as special cases when

γ = α = 2 $\gamma =\alpha =2$

, and provides a foundation for understanding non-Fickian transport in complex turbulent systems.