Introduction to Environmental Modeling

Introduction

What we need to do in modeling is identify regions we wish to analyze. We will define a volume for study and then look at what goes into and comes out of that volume. These flows across the boundary, called fluxes, have units of quantity of interest per unit area per unit time, and are contributors to change within the volume being studied. Without fluxes, and other sources, a system will relax toward equilibrium where there is no flow and no variables are changing with time. Fluxes of energy, momentum, and mass into and out of a system effect change in that system. A complicating issue is that we need to describe these fluxes, and the system, at scales consistent with the questions we intend to address by modeling. For example, one might wish to assess a lake as a whole if interested in its fish population. Or if one is trying to monitor an invasive fish population, such as Asian carp into Lake Michigan, it may be important to know if and where the species is distributed in the lake. It may also be useful to have information about the currents within the lake and how they are distributed. We want to use some basic equations that describe important phenomena, and we also want to be able to apply those equations at different scales.

In the last chapter, we developed expressions for the general time derivative, material derivative, partial time derivative, gradient operator, and divergence operator. The time derivative forms are indicative of properties of a region being studied and how this region changes or deforms in time. Here, we seek some general mathematical tools that facilitate the transfer of derivative expressions for the observation of systems between scales. The primary tools that allow such transformations are two theorems: the divergence theorem and the transport theorem. Note that these theorems are mathematical relations that apply based on the properties of the functions and regions being studied.