Nonasymptotic Behavior of a Common Eulerian Approximation for Transport in Random Velocity Fields

Lagrangian methods of analyzing solute transport in random velocity fields deal with the motion of indivisible “solute particles” through such fields, whereas Eulerian methods treat the solute concentration, C, as the dependent variable in a stochastic mass balance equation. Both approaches lead to nonlinear stochastic differential equations which are difficult to solve analytically without approximation. A particular Eulerian approximation which has become popular in recent years derives from the premise that, in mildly fluctuating velocity fields, terms involving the products of fluctuating quantities can be disregarded in comparison to terms which involve only one such quantity. This leads to a non‐Fickian low‐order approximation for the ensemble mean concentration 〈C〉, where the dispersive flux is not proportional to ∇〈C〉, but is given instead by a convolution integral. The spatial moments of 〈C〉 based on this approximation are shown to be in conflict with those obtained from a first‐order Lagrangian analysis. We explore this apparent contradiction and conclude that the reason lies in the nonasymptotic nature of the Eulerian approximation whereby terms neglected are of the same order as terms retained.