91 In the previous lectures, we learned the most commonly-used tools for derivatives. While doing so, we introduced the exponential function as a very special function whose derivative is itself. In this Section, we will learn more about the derivative of exponential functions, and understand why the exponential is really the most remarkable function of all.

3.3 3.3.1

Derivatives of exponential functions Case Study: The Carbon Cycle and the danger of ocean acidification

Over the past 200 years, the pH of the ocean has decreased significantly (from about 8.25 to 8.15), with a corresponding increase in “acidity” by about 25%. This has dramatic consequences for the long-term survival of a variety of corals and zooplankton (which have difficulty growing under these acidic conditions), and therefore poses enormous risks to the entire oceanic ecosystem. This case study proposes a very simplified model of the problem of ocean acidification, which nevertheless explains and clarifies the relevant physical processes and possible future outcomes. The problem of ocean acidification is closely tied with the problem of the increase of the atmospheric CO2 content, through what is commonly referred to as the “Global Carbon Cycle”. At any point in time, a steady exchange of CO2 exists between the ocean and the atmosphere through degassing and absorption. If the atmospheric CO2 abundance is larger, some of it gets absorbed into the ocean, while if the oceanic CO2 abundance is larger, some of it is released back into the atmosphere. The timescale for this exchange has been estimated to be about 300 years. When CO2 is absorbed in the seawater, a fraction of the molecules dissolve according to the following reaction: + CO2 + H2 O ↔ HCO− 3 +H Since the pH of a solution depends on the concentration cH of Hydrogen ions H+ , following the formula: pH = − log10 (cH ) dissolving CO2 decreases the pH of seawater. This can be illustrated simply in the following plot:

In order to model the expected evolution of the oceanic pH, we therefore need to model the exchange of CO2 between the ocean and the atmosphere, and determine how the oceanic CO2 concentration changes as a result of anthropogenic emissions. Let’s define a few variables:

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The atmospheric CO2 content changes as a result of many factors. Here, we will for simplicity only consider two of them: natural and anthropogenic emissions, and absorption by the ocean. The rate of change of atmospheric CO2 is therefore given by:

Any molecule that dissappears from the atmosphere necessarily goes into the ocean, so the oceanic and atmospheric rates of change of CO2 are related to one another. In addition, the CO2 that is absorbed into the ocean dissolves, as described earlier, and the HCO− 3 ions created further combine into Calcium Carbonate (CaCO3 ) which is the main ingredient necessary for coral growth, and for the growth of the shells of zooplankton. When the zooplankton dies, it sinks to the bottom of the ocean (if it gets eaten by fish instead, then the fish sinks to the bottom of the ocean when it dies), and accumulates in Carbon rich sedimentary layers. This process, which has been called a “Biological Carbon Pump”, leads to a steady removal of Carbon from the seawater. Here, we will assume that this Carbon pump removes oceanic CO2 at a constant “sink” rate S.

As a result, the rate of change of the the oceanic CO2 content is:

How do we use these equations to find out what the evolution of the oceanic pH is expected to be?

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Let’s first consider the global Carbon evolution prior to the industrial revolution. At this point, the anthropogenic Carbon emissions were still negligible compared with the natural ones. As a result, the set of equations describing the system can be simplified to:

The first thing to notice is these equations are differential equations, and furthermore, the differential equation for fsea (t) depends on fair (t), and vice-versa: they are called COUPLED differential equations. Unfortunately, the method we learned to solve differential equations graphically does not work “as is” for coupled differential equations. Fortunately, however, we can use an interesting trick here. Note that the total Carbon content (i.e. the sum of the atmospheric and oceanic content) is

Because of that, we have:

So in fact, by taking the sum of the two equations above, we constructed a third much simpler one! If the emission rate is assumed to be constant, then f (t) is a function with a constant derivative. This implies:

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Over billions of years, the Earth’s ecosystem has adjusted itself to stabilize the natural Carbon cycle, so Carbon emissions and natural Carbon sink exactly compensate one-another. In other words:

To find out what this implies for the actual atmospheric and oceanic abundances fsea (t) and fair (t), we need to solve for each of them individually. In order to do this, a similar trick can be used to create another simple equation starting from the original two:

Let us now “rename” the function fair (t) − fsea (t) as g(t). Then

This, on the other hand, is a differential eqation that we know how to solve graphically:

We see that g(t) tends to a constant after a long time, implying that

We now have two equations for the two unknown quantities fair (t) and fsea (t) (for large-time t). We can solve for each of them separately by substitution:

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This simplified approach gives a good “first impression” of the long-term balance in the system under the assumptions we made, prior to human intevention. During that time, one of the only serious upsets to the balance were occasional supervolcanoes going off, which would release a great amount of CO2 in the atmosphere in a very short time. An interesting question is, how long would it take for the system to adjust? In order to determine this, we need to solve for fair (t) and fsea (t) more quantitatively (i.e. not just graphically). It is now time to learn about derivatives of exponential functions, in order to find a real analytical solution of the equations.

3.3.2

Mathematical corner: Derivative of exponential functions

Derivative of the natural exponential function:

Derivative of exponentials of functions:

Proof:

Examples: • Deduce the derivative of h(x) = e3x

• Deduce the derivative of h(t) = eat where a is constant

• Deduce the derivative of h(x) = 3ex

2

+1

.

• Deduce the derivative of h(y) = 2esin(y) .

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CHAPTER 3. TOOLS FOR DERIVATIVES • Deduce the derivative of h(x) = ax .

The exponential as the solution of simple differential equations: Based on what we just saw, what is a possible solution of: • The differential equation h0 (x) = h(x) ?

• What other solution is there?

• The differential equation h0 (x) = 4h(x) ?

• What other solution is there?

• The differential equation h0 (t) = −2h(t)?

• What other solution is there?

In short, we find that exponential functions ubiquitously appear as solutions of differential equations where the first derivative is proportional to the function itself!

3.3.3

Case Study: The Carbon Cycle and the danger of ocean acidification (part II)

As it turns out, the differential equation

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has the solution :

How do we check this? In the case of algebraic equations, we can easily check that a number is indeed a solution by plugging it into the equation and verifying that it does satisfy it. We do the same for differential equations....

Note that the number “g0 ” in the formula for g(t) is the initial difference between the atmospheric and oceanic Carbon abundances (i.e. it is equal to g(t = 0)). Indeed,

The following plot shows g(t) (in units of 1015 grams, or equivalently, 1 Ptg) for different initial conditions (different values of g0 ), for τ = 300 years. Suppose that the atmoshere and ocean were at equilibrium for billions of years, and that a volcano goes off at t = 0, releasing 500 Ptg of CO2 in the atmosphere. How long does it take for the system to go back to normal, roughly? 1000

g(t) (in Petagrams)

800

600

400

200

0 0

200 400 600 800 Time since perturbation to CO2 balance (years)

1000

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One of the main differences between the case of a volcano going off, and the rise of anthropogenic emissions, is that the volcano releases a great deal of CO2 in one go, but the emission rate goes back to “normal” very quickly, so overall, we still have the balance Enatural = S. By contrast, since the beginning of the industrial revolution, anthropogenic emissions have risen steadily. As of now, Ehuman ' 6 Ptg / year. We will (somewhat naively) assume that governmental policies will rapidly be adjusted to keep the emissions at that level from here on). Let us see what implications this has for the global CO2 cycle. Let’s simply re-trace our steps so far, but with the new term added to the equations. First, let’s reconstruct and solve the equation for the total Carbon content. We’ll start “counting” time (ie. set t = 0) at the beginning of the industrial revolution. At that time, the total content in the atmosphere+ocean was 1000 Ptg.

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To find out how much of the increase in the total Carbon content goes into the ocean, and how much goes in the atmosphere, we now have to solve for fair (t) and fsea (t) separately. Again, we proceed as before and construct the function g(t) = fair (t) − fsea (t). We will assume that at t = 0, the two were at equilibrium for the “natural” state studied earlier, so that g(t = 0) = 300Ptg.

This time, the equation describing the evolution of g(t) is:

g(t) post industrial revolution (in Ptg)

We can check that this function is indeed a solution to the equation, and that it is indeed equal to 300 at t = 0.

1400 1200 1000 800 600 400 200 0 0

200 400 600 800 Time since industrial revolution (years)

1000

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CHAPTER 3. TOOLS FOR DERIVATIVES Now that we know g(t), we can calculate fair (t) and fsea (t) separately.

fair and fsea post industrial revolution (in Ptg)

The functions fair (t) and fsea (t) are shown in the plot below. 2500

2000

1500

1000

500

0 0

100 200 300 400 Time since industrial revolution (years)

500

We see that the amount of CO2 in the ocean will begin to increase more rapidly within the next 100 years. As this CO2 dissolves into seawater, H+ ions will be produced, and the system will become progressible more acidic. Why is ocean acidification such a problem? Unfortunately, it is much more difficult for organisms to absorb Calcium Carbonate molecules into their shells in an acidic environment. In fact, if the environment is too acidic, the solution actually dissolves the exisiting shells! Coral growth for instance has notably slowed down in the past 20 years, and there are fears that corals will soon die completely if ocean acidification continues at the present rate. An even more dangerous outcome (at least, from a human point of view) is that the biological Carbon pump (the sink term) depends on zooplanktons being able to grow significant shells before they die. In an acidic environment, however, they cannot do so, and the sink rate drops. If S = 0, then the rate of increase of total CO2 will be even higher. What are our options for the future? Even if we are able to maintain anthropogenic emissions at the same level as today’s we are still pumping more CO2 into the world than the biological Carbon pump is able to remove. One of the possible resolutions of the problems proposed by scientists is to increase the zoo-

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plankton growth rate to increase the sink term. Methods currently investigated include “seeding” the ocean surface with more nutrients (and in particular iron) to increase the carrying capacity for phytoplancton, which are the main food source of zooplankton.

3.3.4

Mathematical corner: More examples of equations which support exponential functions as solutions.

The exponential function crops up everywhere, principally because it is a very common solution of wellknown differential equations that describe real-world systems. Example 1: Bank interest rates.

Example 2: Radioactive decay.

Example 3: The logistic equation.

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CHAPTER 3. TOOLS FOR DERIVATIVES (continued)