### Mathematics and Climate

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When all else fails in making "small talk," one can always fall back on the weather. For those of us used to living where there is a discernible change in seasons, one can chat about it being too cold, too hot, too wet, or, despite how lovely the snow is, how much of a pain it has become to shovel so much of it recently. The changeability of the weather is one constant fact of life. However, what about climate?

## Mathematics of Climate

I started as a graduate student in mathematics at the University of Wisconsin in the fall of , and for the winter holidays, I headed for the airport for my first airplane trip I had taken a bus to Madison from my home in New York, to carry my gear for the year out there.

When I arrived at the airport I remember seeing a sign which reported that the temperature was degrees. I had a moment of fright as to whether or not planes could really fly in degree weather. Happily, Bernoulli's Principle and Newton's Laws apply in this range!

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Madison friends tell me that though Madison winters are still cold, they are not as cold as those of the middle 's. In a single person's lifetime 40 years is a large period of time, but not on a geological scale. With news about melting ice caps and glaciers, the possibility of shore erosion and flooding due to raised ocean levels, changes in the amount of fresh water entering the Arctic Ocean disrupting the ocean current system, Earth's current and medium range climate is of great concern.

There is much talk of global warming and there are some who deny that it is happening. By its very nature climate, as opposed to weather, is concerned with issues over relatively long time scales, and, thus, what has been true for the last few weeks, days, or even years does not necessarily give a picture of the issues. Although mathematics grows because it attempts to solve problems that arise outside of mathematics e.

Newton developed calculus in part to help understand gravity and the motion of the planets, Euler wrote a paper concerning where to optimally position the masts of a sailing ship , it also grows because mathematicians enjoy looking for patterns of all kinds for their own sake. Though, in part, the theory of games was developed to understand questions arising in economics and political science, once it was born it took off on its own, developing methods and ideas that were independent of applications settings.

It might not seem that game theory has much to say about Mathematics and Climate, but the tools of game theory are being employed to try to help the world deal better with earthly issues that affect the climate. Because of its ability to abstract and generalize, mathematics is a unique tool for getting insight into the increasingly large number of domains that it is being applied to. Climate issues are just one of these. When an issue is raised in an area outside of the mathematics community, people who have mathematical skills in these other areas often begin by constructing mathematical models of this new area.

A mathematical model is a simplified representation of a more complex reality. The value of constructing a model is that one can explicitly see the simplifying assumptions being made, look at the "predictions" or "insights" offered by the model, and use this as a basis for either refining the model or doing experiments which help provide the feedback in seeing whether the model is providing one with the insights that are being sought.

A somewhat different approach is to take the area one is trying to get insight into and apply existing tools to study what are emerging as the key issues in this area. What are the major tools mathematics uses to probe climate issues? Change is at the heart of a lot of phenomena in nature. The great mathematician and physicist Isaac Newton needed insight into the way one quantity might change with another in order to get the profound insights he did into the nature of gravity and the motion of the planets.

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To do this he "invented" calculus, building on the ideas of those who came before him. Calculus among other things is a mathematical tool to help understand how one quantity changes with respect to another. Thus, if d is distance and t is time, you can compute an average speed. For example if:. Since in 4 units of time the distance covered was 12, the average or mean speed for this "trip" was 3. Newton saw his way from how to go from an average rate of change to an instantaneous rate of change.

When you tell a friend that for your mile trip your average velocity was 30 miles per hour, it can be concluded that you took 5 hours to make the trip, and during that trip your velocity at some of the times you traveled may have been 60 mph, 0 miles per hour as you waited for traffic to clear behind an accident or stopped for a cup of coffee.

The "instantaneous" velocity at any time of your trip was available by looking at the "speedometer' on your car, which told you how fast you were traveling at that instant. Newton knew that if one has a polynomial function for simplicity, think of n as a positive integer. Thus, the derivative of a function which has been multiplied by a constant merely multiplies the derivative by a constant. Furthermore, the derivative of a constant function is 0.

This last fact makes sense because if a variable is not changing, then its instantaneous rate of change should be 0. It turns out that the derivative of the sum of two or more generally a finite sum of functions is the sum of the derivatives of these functions. However, the derivative of a product does not generally obey the rule of being the product of the derivatives of the functions, though beginning calculus students master how finding the derivative of the product and quotient of functions does work.

If we apply these ideas to the result to equation 1 above, we get that the derivative is t.

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## Mathematics of Planet Earth | EPSRC Centre for Doctoral Training

We can conclude that at time 4 the instantaneous velocity is 4. If we take a derivative of the velocity function, we get the second derivative which can be thought of as the instantaneous rate of change of the velocity with respect to time, and, thus, is what physicists call the acceleration. It turned out that Newton was able to formulate many of nature's laws with this new derivative tool.

Thus, the famous force law of is formulated as a differential equation. In this equation a stands for the second derivative of the distance. Similarly, Newton's Law of Cooling, which also has a connection to climate issues, was formulated as a differential equation. Newton's Law of Cooling is used to model the temperature change of an object placed into a "new" environment of a different temperature.

Perhaps a small sphere at room temperature is inserted into ice water. The law states that. What this law says in words is that the rate of change of temperature of the object is proportional to the difference between the starting temperature of the object and that of the surrounding environment. It is easy to verify to do this one needs to know that the derivative of e kt is k e kt that the "general solution" of this differential equation is:.

Based on the predictions made in solving differential equations such as the one associated with Newton's Law of Cooling, one can do experiments or make observations which can confirm the accuracy with which the differential equation serves as a model for the phenomenon to be explored. What happens when a quantity changes or varies with more than one other variable?

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This is much more common because it is rare that changes in one variable only depend on the change in one other variable. How does one deal with rates of change now? One approach is to fix all of the variables except one, and compute the derivative with respect to the one remaining variable while treating all of the other variables as constants.

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This is the idea of a partial derivative. Here is a simple example: If. For the first line above we treat t as a constant and find the "ordinary" derivative with respect to s , and for the second line above we treat s as a constant and find the "ordinary" derivative with respect to t. It is not difficult to give these partial derivatives geometric interpretations using three-dimensional geometry, in this case. Many important laws in the physical sciences can be formulated in terms of partial differential equations.

Examples of these are the heat equation, the wave equation, and the Laplace equation. Work on these equations, born of interest in mathematical physics for a period of many years, is associated with many of the most famous practitioners of mathematics: Euler, Laplace, Cauchy, and Fourier. Somewhat less well known are the engineer and physicist Claude-Louis Navier. These scholars became interested in understanding the motion of fluids. Newton had also been interested in fluids, and he initiated what today are known as Newtonian fluids.

A Newtonian fluid is one which can be described in terms of its viscosity. Intuitively, viscosity measures the fluid's resistance to flow. Honey and water are fluids but they display very different viscosity. However, with mathematical tools that were developed after Newton, more progress could be made. The more complex Navier-Stokes equations they are partial differential equations also apply to Newtonian fluids but carry the modeling to a more detailed level. These equations can be used in a variety of different settings.

The fluid flow work that was done by Euler and involves what today are known as the Euler equations, can be viewed as a "limiting case" of the Navier-Stokes equations. However, though the Euler and Navier-Stokes equations have led to very fruitful insights into fluid flow and have been of use to engineers concerned with fluids, mathematically the situation is not all that satisfactory.

Part of the issue, as noted by Charles Fefferman Princeton , an expert in the field, is "Solutions of the Euler equation behave very differently from the solutions of the Navier-Stokes equation. Definitive answers supported by rigorous proofs seem a long way off. The mathematical modeling issues that arise in trying to handle the various circumstances has increased both our practical and theoretical insights.

The Navier-Stokes equations are used in part as the basis for the growing field of computational fluid dynamics. Many laboratories devoted to this area have sprung up around the country.