Even-odd alternative dispersions and beyond. Part II. Noninertial and inertial particles, and, astrophysical chirality analogy

Abstract

Particle transports in carriers with even-odd alternating dispersions (introduced in Part I) are investigated. For the third-order dispersion as in Korteweg-de-Vries (KdV), such alternating dispersion has the effects of not only regularizing the velocity from forming shock singularity (thus the attenuation of particle clustering strength) but also symmetrizing the oscillations (thus the corresponding skewness of the particle densities), among others, as demonstrated numerically. The analogy of such dispersion effects and consequences (on particle transports in particular) with those of helicity in Burgers turbulence, addressed in the context of astrophysics and cosmology, is made for illumination and promoting models. Both dispersion and helicity regularize the respective systems, and both are shown to be transferred by the drag to the flows of the respective inertial particles carried by the latter and to similarly affect the particle clustering. A reward from studying particle transports is the understanding of the (asymptotic) $k^0$-scaling (equipartition among the wavenumbers, $k$s), before large-$k$ exponential decay, of the power spectrum of KdV solitons [resulting in the more general statement (valid beyond the KdV soliton and Burgers shock) that "a (one-dimensional) soliton is the derivative of a classical shock, just like the Dirac delta is the derivative of a step function"], motivated by the explanation of the the same scaling law of the particle densities as the apparent approximation of the Dirac deltas; while, the "shocliton" from the even-odd alternating dispersion in aKdV appears to be, indeed, $shock \oplus soliton$, accordingly the decomposition of the averaged odd-mode spectrum, from sinusoidal initial field, into a $k^{-2}$ part for the shock and a $k^0$-scaling part for the solitonic pulses, only the latter being contained in the averaged even-mode spectrum.