Mathematics of Planet Earth


Dr Shane Keating, Senior Lecturer, The University of New South Wales School of Mathematics and Statistics.
Associate Professor Lisa Alexander, Associate Professor, The University of New South Wales Climate Change Research Centre and ARC Centre of Excellence in Climate Extremes (ARCCLEX).


This course aims to introduce key concepts and mathematical methods for understanding our planet and the dynamical processes that shape it. We will introduce the fundamental equations of motion and conservation laws that govern the fluid dynamics of the atmosphere and ocean. These equations are systematically simplified to develop the elegant and powerful theory of geophysical fluid dynamics, which is then used to quantatively model a rich variety of atmospheric and oceanic phenomena and their interaction with Earth’s climate.

Students will also develop an understanding of how climate models are built, executed and evaluated. We will examine statistical methods for interrogating large climate datasets, including model evaluation metrics, uncertainty quantification, and extreme value theory. Theoretical understanding gained in lectures will be explored in practical sessions and hands-on activities using a coupled climate model (CSIRO Mk3L) to simulate a range of climate scenarios such as double-CO2, Snowball Earth, or the Last Glacial Maximum.

Course Overview

1: Fundamental concepts

Eulerian and Lagrangian frames, continuity, Navier-Stokes equation, Vorticty, frozen-in law, Kelvin’s theorem, equation of state, energy conservation

2: Geophysical fluid dynamics

Stratification, Boussinesq approximation, internal gravity waves, rotating coordinate systems, Coriolis force, geostrophic balance, f-plane and beta-plane approximations, Rossby waves

3. Atmospheric circulation

Overview of atmospheric circulation, thermal wind balance, baroclinic instability, wave-mean interaction, zonal jets.

4. Ocean circulation

Overview of ocean circulation, planetary geostrophic equations, wind-driven circulation, Ekman layer, Sverdrup balance, abyssal flow, Stommel-Arons model, buoyancy-driven circulation.

5. Climate models

Model hierarchies from conceptual to comprehensive, chaos and initial conditions, climate models, energy budgets, fluxes

6. Climate data science and statistics

Model evaluation metrics, uncertainty estimation, interpreting ensembles, bias correction, weighting, statistical methods for climate date, extreme value theory.

Contact hours

28 hours


multivariate calculus, some PDEs. Computing experience with Unix, Python, Matlab recommended but not required.


Mid-School assignment: 40%
Final examination: 60%


Vallis, G. K., 2006. Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press
– Excellent and comprehensive text on geophysical fluid dynamics and large-scale circulation of the atmosphere and ocean.

Marshall, J., & Plumb, R. A. (2016). Atmosphere, ocean and climate dynamics: an introductory text (Vol. 21). Academic Press.
– Good mid-level text based on undergraduate course taught at MIT.

Mathematics of Planet Earth
Mathematics of Planet Earth


Dr Shane Keating, Senior Lecturer, The University of New South Wales School of Mathematics and Statistics.

Shane is a leading expert in geophysical fluid dynamics, ocean turbulence, and satellite remote sensing. His research at the interface of applied mathematics and earth science has resulted in collaborations and visiting positions at the US National Center for Atmospheric Research, NASA’s Jet Propulsion Laboratory, and the NZ Bureau of Meteorology. He is also a member of the NASA Science Team designing the next generation of Earth observing satellites. Shane is an award-winning educator and has been an invited lecturer at the ARC Centre of Excellence in Climate System Science winter school and the Australian Bureau of Meteorology graduate summer school.



Associate Professor Lisa Alexander, The University of New South Wales Climate Change Research Centre and ARC Centre of Excellence in Climate Extremes (ARCCLEX).

Lisa is an internationally recognised expert on climate variability and extremes and has contributed significantly to the Intergovernmental Panel on Climate Change (IPCC) assessments. She was awarded the 2011 Priestley Medal by the Australian Meteorological and Oceanographic Society, the 2013 Australian Academy of Science Dorothy Hill award, and the 2017/18 World Meteorological Organisation (WMO) Commission for Climatology Outstanding Service award. Lisa has taught and led several student workshops, including the International Centre for Theoretical Physics (ICTP) summer school (Trieste) and the ARCCLEX graduate winter school.

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