1. Concepts, Definitions, and the DiffusionEquationEnvironmental fluid mechanics is the study of fluid mechanical processes that affect the fate andtransport of substances through the hydrosphere and atmosphere at the local or regional scale1(up to 100 km). In more layman’s terms, environmental fluid mechanics studies how fluidsmove substances through the natural environment as they are also transformed. In general,the substances we may be interested in are mass, momentum or heat. More specifically, masscan represent any of a wide variety of passive and reactive tracers, such as dissolved oxygen,salinity, heavy metals, nutrients, and many others. This course and textbook discusses passiveprocesses affecting the transport of species in a homogeneous natural environment. That is, asthe substance is transported, its presence does not cause a change in the dynamics of the fluidmotion. The book for part 2 of this course, “Stratified Flow and Buoyant Mixing,” incorporatesthe effects of buoyancy and stratification to deal with active mixing problems where the fluiddynamics change in response to the transported substance.This chapter introduces the concept of mass transfer (transport) and focuses on the physicsof diffusion. Because the concept of diffusion is fundamental to this part of the course, wesingle it out here and derive its mathematical representation from first principles through to animportant solution of the governing partial differential equation. The mathematical rigor of thissection is deemed necessary so that the student gains a fundamental and complete understandingof diffusion and the diffusion equation. This foundation will make the complicated processesdiscussed in the remaining chapters trac table and will start to build the engineering intuitionneeded to solve problems in environmental fluid mechanics.1.1 Concepts, Significance and DefinitionsStated simply, environmental fluid mechanics is the study of natural processes that changeconcentrations.2These processes can be categorized into two broad groups: transport and transformation.Transport refers to those processes which move substances through the hydrosphere and atmo-sphere by physical means. As an analogy to the postal service, transport is the process by whicha letter goes from one location to another. The postal truck is the analogy for our fluid, and1At larger scales we must account for the Ear th’s rotation through the Coriolis effect, and this is the subject ofgeophysical fluid dynamics.2A glossary at the end of this text provides a list of important terms and their definitions in environmental fluidmechanics (with the associated German term) to help orient the reader to a wealth of new terminology.Copyrightc 2004 by Scott A. Socol ofsky and Gerhard H. Jirka. All rights reserved.2 1. Concepts, Definitions, and the Diffusion Equationthe letter itself is the analogy for our chemical species. The two primary modes of transport inenvironmental fluid mechanics are advection (transport associated with the flow of a fluid) anddiffusion (transport associated with random motions within a fluid). The second process, trans-formation, refers to those processes that change a substance of interest into another substance.Keeping with our analogy, transformation is the paper recycling factory that turns our letterinto a shoe box. The two primary modes of transformation are physical (transformations causedby physical laws, such as radioactive decay) and chemical (transformations caused by chemicalor biological reactions, such as dissolution and respiration).In engineering practice, environmental fluid mechanics provides the tools to (1) assess theflow of nutrients and chemicals vital to life through the ecosystem, (2) limit toxicity, and (3)minimize man’s impact on global climate.1. Ecosystem Dynamics. Nutrients are food sources used by organisms to generate energy.Engineers need to know the levels of nutrients and their transformation pathways in order topredict species populations in freshwater ecosystems, such as the growth and decay of algalblooms in response to phytoplankton and zooplankton dynamics. Some common nutrientsand vital chemicals are oxygen, carbon dioxide, phosphorus, nitrogen, and an array of heavymetals, among others.2. Toxicity. For toxic chemicals, engineers need to understand natural transport and trans-formation processes to design projects that minimize the probability of occurrence of toxicconcentrations while maintaining an affordable budget. Some common toxic chemicals areheavy metals (such as iron, zinc, and cadmium), radioactive substances (such as uraniumand plutonium), and poisons and carcinogenic substances (such as PCBs, MTBE, carbonmonoxide, arsenic, and strong acids).3. Global Climate Change. Some chemical species are also of interest due to their ef-fects on the global climate system. Some notable substances are the chlorofluorocarbons(CFCs) which deplete the ozone layer, the greenhouse gases, in particular, carbon dioxideand methane, which maintain a warm planet, and other substances, such as sulfate aerosolsthat affect the Earth’s reflectivity through cloud formation.It is important to remember that all chemicals are necessary at appropriate levels to sustainlife and that anthropogenic input of chemicals into the hydrosphere is a necessary characteristicof industrialization. Engineers use environmental fluid mechanics, therefore, to avoid adverseimpacts and optimize the designs of engineering projects and to mitigate the effects of accidents.1.1.1 Example ProblemsAlthough environmental fluid mechanics addresses basic processes that we are all familiar withthrough our natural interaction with the environment (e.g. sensing smoke in a crowded bar),its application in engineering is not frequently taught and students may find a steep learningcurve in mastering its concepts, terminology, and significance. This problem is compounded bythe fact that a whole new set of equations must be mastered before meaningful design problemscan be addressed. Here, we pause to introduce several typical problems and their relationship1.1 Concepts, Significance and Definitions 3TurbulenceDiffusionAir-flow inAir-flow outGas or aerosol releaseFig. 1.1. Schematic of the mixing processes in an enclosed space. A point source of substance is released in thelower left corner of a r oom. Mixing is caused by flow into and out or the room through the ventilation system andby random motions in the