Here we develop some basic ideas about random walk through a known Eulerian environment. We assume that particle motion is a sum of resolved or “macroscale” motion (the statistical mean motion – resolved on a finite grid); and a stochastic component representing the unresolved subgrid processes (the microscale). For exposition, we assume the mean motion is zero; it can be added later, without loss of generality.

The classic literature relates the ensemble of random walks to a macroscale diffusion process; the size of the walk and the macroscale diffusion coefficient are fundamentally linked. Practical extensions are reported in Hunter et al. [202], Proehl [365], Visser [464], Riddle [382], Skellam [420], Ross and Sharples [388]; and used in the applications by Fischer et al. [153], Dimou and Adams [117], Proehl [366], to name a few. The current frontier of Lagrangian work in the coastal ocean is reported in the LAPCOD volume [186]; the summary therein [302] is particularly useful.

The literature on timeseries analysis is immediately relevant and deeper than the exposition here. Box et al. [49] is a classic work in this area. There are several more introductory texts, for example, Chatfield [79]. In particular, these include the important topics related to identifying random walk models from observed timeseries.

We begin with a simple introductory development. Next we go into the more general continuous processes and address a hierarchy of random walk models with memory.

At the macroscale level, turbulence closure provides the essential link from the macroscale diffusivity to the random walk details. In Chapter 6 the closure problem is described.

Introduction: Discrete Drunken Walk

Here we provide a brief introduction to the simplest of random walks. Readers familiar with these developments should skip this section.

Consider the 1-D drunken walk process: in any time interval, move 1, 0, or−1 units with equal probability; do this repeatedly.