## 1. Introduction

[2] Scalar mixing and spreading processes, which are typically represented by some diffusion or dispersion coefficient in transport equations, are fundamental to many geophysical systems and engineering applications. Mass transport driven by a concentration gradient is conventionally assumed to obey a diffusive process or is at least described by a diffusion-type equation. In a stratified flow field, the velocity profile due to shear stress enhances the mixing/spreading process resulting in so-called Fickian dispersion which is encapsulated in an effective dispersion coefficient.*Taylor* [1953] first showed that at some long enough time scale the mixing/spreading process through a tube follows Fickian behavior with a longitudinal dispersion coefficient. Later, *Fischer et al.* [1979] derived a corresponding longitudinal dispersion coefficient for a river also at a large time scale. *Güven et al.* [1984]analyzed horizontal transport through aquifers and showed that differences in stratified groundwater flow velocity caused by vertically-varying hydraulic conductivity would also lead to a Fickian dispersive process at some large enough scale. In such circumstances, the classical advection-dispersion (or diffusion) equation (ADE) is valid. However, at preasymptotic time scales, the classical ADE is invalid due to anomalous early arrival and persistent tails in breakthrough curves both in fractured and porous media [*Berkowitz*, 2002]. This phenomenon is referred to as non-Fickian behavior.

[3] Non-Fickian transport can be mathematically represented in many ways but one simple approach is to define a dynamic longitudinal dispersion coefficient (*D*) for the ADE. Several researchers have studied and quantified dynamic *D* using various approaches including spatial moment analysis [*Dentz and Carrera*, 2007], series expansion methods [*Gill and Sankarasubramanian*, 1970], center manifold description [*Mercer and Roberts*, 1990], and Lagrangian approach [*Haber and Mauri*, 1988]. These studies showed that *D* increases monotonically from its value at preasymptotic time scale to the value according to *Taylor*'s theory. Yet, the theoretical analysis by *Taylor* [1953] for a tube and *Fischer et al.* [1979] for a river at asymptotic time scales is not sufficiently complete.

[4] There are three key assumptions adopted by both *Taylor* [1953] and *Fischer et al.* [1979]: (1) the Peclet number is sufficiently large (i.e., advection dominated transport) so as to ignore longitudinal diffusion; (2) the longitudinal advective mass flux is balanced by transverse diffusive mass flux, and the gradient of the cross-sectional averaged concentration in the longitudinal direction is at steady-state; and (3) the gradient of the cross-sectional averaged concentration in the longitudinal direction is much greater than the gradient of concentration fluctuations. The validity of the above assumptions has since been ignored and subsequent studies have either retained these assumptions or circumvented them by following approaches that are not directly based on the complete transport equations. Here, we develop a more general theory that does not require the first two assumptions, and using this theory we derive a closed-form expression for the dynamic longitudinal dispersion coefficient. Afterwards, we analyze the time and length scales for distinguishing Fickian (asymptotic) and non-Fickian (preasymptotic) transport regimes.