As part of our Math Awareness Month celebrations, we posed our series of questions about mathematics and climate study to Tapio Schneider, a Professor of Environmental Science and Engineering at Caltech. Dr. Schneider conducts research on the dynamics of the Earth’s climate changes, turbulence, and turbulent transport in the atmosphere and oceans. He is also co-editor with Adam H. Sobel of the PUP book The Global Circulation of the Atmosphere.

**PUP: **What are you currently working on?

**Tapio Schneider: **I am working on theories of how large-scale (>1000 km) atmospheric turbulence influences the global climate. For example, we study how turbulent transport affects tropical circulations and how it controls the distribution of atmospheric water vapor and rainfall.

**PUP: **How did you become interested in this field?

**TS: **I am fascinated by how nature works. I was trained as a physicist and loved how physics helped explain the inanimate world around me, from refrigerators to cell phones to the blue color of the sky and the red color of sunsets. I particularly like the physics of everyday phenomena—phenomena that occur roughly at the energy of sunlight (for example, many quantum phenomena occur at the energy of sunlight, and in part because of that, quantum devices such as the transistor revolutionized our life). When I was looking for a research area for graduate studies, I was looking for a young field with open questions to which young scientists can make lasting and fundamental contributions. Atmospheric dynamics is such a field—and the phenomena certainly occur at the energy of sunlight!

**PUP: **How do you use mathematics in your work?

**TS: **Mathematics to me is a means to an end. It gives succinct descriptions of complex relations among natural phenomena. From these relations, we can draw inferences (explanations, predictions) about phenomena through mathematical reasoning. For example, Newton’s laws of mechanics are succinctly expressed in terms of differential equations, and from (numerical) solutions of these equations, we can predict how properties of atmospheric turbulence change with climate. Mathematics is an extraordinary efficient and effective language for expressing the laws of nature, and of climate in particular. In Wigner’s words, “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve” (from E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Communications in Pure and Applied Mathematics, vol. 13, 1960).

**PUP: **How is mathematics helping us to understand climate change and how the earth works?

**TS: **Mathematical relations expressing the laws of nature are at the core of everything we do, from building models of climate, to analyzing simulations conducted with such models, to making predictions. Climate models are based on Newton’s laws of mechanics, the laws of thermodynamics, and conservation laws for atmospheric and water mass, expressed in mathematical form for a continuum of microscopic volumes.

Tools of modern applied mathematics, developed in the middle and late 20th century, are used to find numerical solutions of the complex equations with the help of computers. We practice what may be called computational science and what some call experimental mathematics–we use computer simulations the way a laboratory scientist would use experiments, for example, to study how changes in the concentration of atmospheric greenhouse gases affects rainfall patterns. We also use mathematics to analyze data, for example, to condense complex information in high-dimensional datasets into more manageable and more easily interpretable low-dimensional information.

**PUP: **What are the top 3 biggest problems in climate science that still

need to be solved? How will mathematics help solve these problems?

**TS: **How do clouds and their radiative properties respond to climate change? This question is central to understanding how much surface temperatures change for a given change in insolation or in the concentration of greenhouse gases (the “climate sensitivity”). We need coarse-grained (>10-100 km scale) mathematical descriptions of the dynamics and thermodynamics that govern the small scales of rain drops in clouds. We need these coarse-grained descriptions because, with current computers and for the foreseeable future, it is impossible to simulate the small-scale dynamics explicitly in climate models.

How do the large-scale turbulent fluxes of heat, momentum, and water vapor in the atmosphere depend on the mean climate state? This question affects almost all physical questions in climate science, from how the pole-equator surface temperature contrast is controlled, to how the distribution of water vapor, Earth’s most important greenhouse gas, varies with climate. We need macroscopic mathematical descriptions of turbulent fluxes to resolve this question (though we can simulate the microscopic dynamics giving rise to the turbulent fluxes reasonably well on computers).

And, how do ice sheets and glaciers respond to climate change? This question is central to understanding how sea level may change with climate and to understanding long-term climate changes such as the cycle of ice ages. We need to know more precisely the equations that describe glacier and ice sheet dynamics, and even where we do know the equations, we need better mathematical tools to solve them numerically on a computer (for example, understanding where an ice stream with rapid ice flow forms involves difficult free-boundary problems).

**PUP: **Why should students who are good in math consider research in climate or earth science?

**TS: **It is a young field in need of talented mathematicians and physicists to work on some of the most interesting, relevant, and challenging scientific problems!

**PUP: **What books, Princeton or otherwise, would you recommend to people who want to learn more about how mathematics and physics are helping us to understand how climate works?

**TS: **Alley’s The Two-Mile Time Machine: Ice Cores, Abrupt Climate Change, and Our Future is an engaging introduction to how ice ages come about, what we know about past climate changes, and what they may imply for the future.

Hartmann’s Global Physical Climatology is a more formal introduction to the laws governing climate on an undergraduate level.

There are many more advanced texts, such as, Peixoto and Oort’s Physics of Climate (a summary of observations and basic laws) and Holton’s An Introduction to Dynamic Meteorology. The book Adam Sobel and I edited, The Global Circulation of the Atmosphere, gives a summary of theories for the atmospheric circulation on the graduate level.