Introduction

Sediments are dense particles. In still water they sink to the bottom. The introduction of bottom turbulence enables particle motion. At low turbulence, particles will bump or roll along the bottom. Higher turbulence will lift particles off the bottom. Once lifted, they are advected in the fluid flow, and dispersed by 3-D turbulent mixing. Near bottom, both turbulent diffusivity and horizontal advection increase with height; higher particles will move faster with larger random walks.

Bottom turbulence will have a sorting effect, with lighter particles moving higher and faster; and more often. The sorting will result in greater horizontal displacement for particles that fly high and long. When the shear is removed, the turbulence decays and particles sink.

We consider a single noncohesive particle and describe its motion. It is assumed that a 3-D circulation field is available, providing velocity, bottom stress, and eddy viscosity. Consistent with common practice, the computational bottom is slightly above the true bottom, with the fluid slipping horizontally above an idealized boundary layer, wherein the law of the wall governs. As particle motion is initiated by turbulence in this layer, it will be necessary to follow particle motions within and across it.

We will present the 3-D case first. Then, we will turn to the additional simplifications/assumptions necessary to deal with the vertically averaged case. The analysis introduced in Section 5.4 is developed more fully here for the bottom boundary.

Sediment dynamics is a mature field, and the coupling with hydraulic models is well established. The use of 3-D particle tracking is relatively new; comprehensive implementations include the PTM (MacDonald et al. [291]), PSED (Davies [102])and Sand Track (Soulsby et al. [428, 429]).

There will be several notational conventions. They will be introduced as needed; a summary is included in Section 9.15

Three States: P, M, B

In Figure 9.1 we illustrate the general arrangement of things. An individual particle will be in one of three states:

P: The pelagic or oceanic state. The Coastal Ocean Model is the gridded area at the top; it provides fluid velocity and bottom stress.

Introduction

Oil is a particular case of a chemically active substance. Of interest here is its oceanic transport, chemical transformations, and fate. In the present context (high abundance, small space/time scales) oil is not “natural” in the sea, but is introduced unintentionally by accidents – spills, exploration/production blowouts – or clandestinely by dumping of waste products.

Oil is not a pure substance, but a mixture of hundreds of specific compounds. For the commercially common oil products, there are large databases of chemical composition and properties. Some are maintained by government agencies, some by transboundary corporations or their associations; some are embedded in operational models. Crude oils are distinctive and discernible, by origin; refinery products are quite specific about chemical properties, and subject to commercial norms of identity and quality.

Generally, oil will be a buoyant liquid mixture, immiscible in water, with density, viscosity, and other physical properties sensitive to its weathering history. In the sea it will occur either as a surface slick or as subsurface droplets. It will be transported by oceanic motion while weathering. Evaporation tends to account for the lion's share of most surface spills, unless a slick makes landfall early or emulsifies. Exposure in the environment will alter the chemical mix and its bulk properties. Destinations for present purposes include the sea floor, mingled with the benthos; the shoreline; in the biota; in the atmosphere as evaporation products. Some subsurface oil will dissolve or biodegrade in the sea.

We use parcels to represent discrete quantities of oil in the environment. A parcel may be floating at the surface as a slick, or submerged as an aggregation of identical small droplets. Each parcel moves as a Lagrangian particle; its content is subject to the operative processes summarized in Figure 10.1. A surface parcel could be thought of as a “spillet” – a physically well-defined subdivision of a larger spill.

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.