8.1: A Useful and Simple Model
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Before we explore the comings and goings of salt in the ocean, it’s helpful to formalize a conceptual model introduced in Chapter 6, what I call the reservoir model. It helps us understand beach sand budgets, salinity, the water cycle, and a whole lot more.
The reservoir model has three features:
- a reservoir, a place where something is stored
- a source, a process that increases the mass or volume of a reservoir
- a sink, a process that decreases the mass or volume of a reservoir
A fish tank, a swimming pool, and the ocean are all places where water is stored; they are reservoirs. The atmosphere is a reservoir for various gases. The solid Earth contains reservoirs of fossil fuels (oil, coal, and natural gas), diamonds, and other kinds of rocks and minerals. Energy reservoirs exist too. Chemical energy is stored in food and wood. Heat energy is stored in the ocean, the atmosphere, and land surfaces. Even human systems feature reservoirs. Your refrigerator is a reservoir of food. Your bank account is a reservoir of money.
The money metaphor illustrates an important property: reservoirs are not static. Practically speaking, a single number represents your money reservoir, but this reservoir may be subdivided into smaller reservoirs, such as checking, savings, loans, or retirement funds. If your checking account is anything like mine, the balance changes almost daily. Reservoirs represent a snapshot of a quantity at one place in a given moment of time.
Reservoirs may grow in volume, shrink, or remain the same. And that’s where the various sources and sinks come into play. Sources, also known as inputs, add to a reservoir. Sinks, also known as outputs, subtract from a reservoir. The volume of material, energy, or some other quantity in a reservoir at any moment in time may be described by the following simple mathematical formula:
Vtreservoir = Vtsources – Vtsinks
(Eq. 11.1)
where Vtreservoir is the volume of the reservoir at time, t, Vtsources is the sum of all additions to the reservoir at time, t, and Vtsinks is the sum of all subtractions from the reservoir at time, t.
Consider the implications of this formula. What happens to the reservoir volume when sources are greater than sinks? What happens when sinks are greater than sources? What happens when sources and sinks are equal?
When sources exceed sinks, the volume of a reservoir may grow. More is added than is taken away. When sinks exceed sources, the volume of a reservoir may contract. More is being removed than is being added. But when the rate of addition is the same as the rate of removal, then the size of the reservoir remains the same.
Note that even though a reservoir may not be changing in size, it can still have active sources and sinks. Imagine you are adding water to a bathtub from a faucet but you’ve left the drain partially open. If the source (water in) exceeds the sink (water out), the bathtub will fill. If the water drains faster than it’s being added by the faucet, the volume of water will fall. What happens when the faucet and drain flow at equal speeds? The volume in the bathtub will not change. Just because a reservoir is not visibly changing, it doesn’t mean nothing is happening. The sources and sinks of a reservoir may be quite active and maintain a steady volume.
Let’s use this model to think about exchanges of water in the ocean, the main reservoir for water on Earth’s surface. If we combine sources and sinks, we get a simple equation that looks like this:
- Volume of ocean water = sources – sinks
- (Eq. 11.2)
Now this model may seem obvious and not very useful. But when we start talking about phenomena such as sea level rise, this simple model proves useful. The most profound scientific ideas often start with simple models.