Nonreactive solute transport in soil columns: classical and fractional-calculus modeling

Vertical nonreactive solute transport data collected in three laboratory soil columns (made out of sediment samples from the Pampean aquifer located southeast of the Buenos Aires province) are contrasted with the explicit solutions of two model 1D linear PDEs: the classical advection–dispersion equation (ADE), and a fractional advection–dispersion equation (FADE) which has proven to be a useful modeling tool for highly inhomogeneous media exhibiting nontrivial scaling laws. Whereas two of the samples turn out to be quite homogeneous (thus requiring a fractional-derivative order γ → 2), the third one is best described by a FADE with fractional-derivative order γ = 1.68. This example illustrates the FADE’s ability to reveal self-similar geometric structures inside the sample.Fil: Benavente, Micaela Andrea. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Deza, Roberto Raul. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Físicas de Mar del Plata. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Instituto de Investigaciones Físicas de Mar del Plata; ArgentinaFil: Grondona, Sebastian. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Mascioli, S.. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; ArgentinaFil: Martinez, Daniel Emilio. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; Argentin

Finite Larmor radius effects on non-diffusive tracer transport in a zonal flow

Finite Larmor radius (FLR) effects on non-diffusive transport in a prototypical zonal flow with drift waves are studied in the context of a simplified chaotic transport model. The model consists of a superposition of drift waves of the linearized Hasegawa-Mima equation and a zonal shear flow perpendicular to the density gradient. High frequency FLR effects are incorporated by gyroaveraging the ExB velocity. Transport in the direction of the density gradient is negligible and we therefore focus on transport parallel to the zonal flows. A prescribed asymmetry produces strongly asymmetric non- Gaussian PDFs of particle displacements, with L\'evy flights in one direction but not the other. For zero Larmor radius, a transition is observed in the scaling of the second moment of particle displacements. However, FLR effects seem to eliminate this transition. The PDFs of trapping and flight events show clear evidence of algebraic scaling with decay exponents depending on the value of the Larmor radii. The shape and spatio-temporal self-similar anomalous scaling of the PDFs of particle displacements are reproduced accurately with a neutral, asymmetric effective fractional diffusion model.Comment: 14 pages, 13 figures, submitted to Physics of Plasma

An inverse Sturm-Liouville problem with a fractional derivative

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order $\alpha\in(1,2)$ of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of Computational Physic

Fractional dynamics of coupled oscillators with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise interaction. The corresponding term in dynamical equations is proportional to $1/|n-m|^{\alpha+1}$. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order $\alpha$, when $0<\alpha<2$. We consider few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on $\alpha$. The presence of fractional derivative leads also to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.Comment: 34 pages, 18 figure