Summer Semester 2025

Summary

This class covers the fundamentals of modeling weather and climate for undergraduate students of Mathematics and Data Science. It has the following main themes:

  • Simple models that illustrate the difference between weather and climate from a dynamical systems viewpoint (Lorenz model) and simple models of climate (energy balance models),
  • Fundamental equations of geophysical fluid dynamics and basic phenomenology,
  • Depending on time and interest: focus more in-depth on selected aspects, and/or do simple practical experiments with a toy climate model.

Textbooks

  • Shen, Somerville, Climate Mathematics: Theory and Applications, Cambridge University Press, 2019
  • Engler, Kaper, Mathematics of Climate, SIAM, 2013
  • Warner, Numerical Weather and Climate Prediction, Cambridge University Press, 2010
  • Vallis, Atmospheric and Oceanic Fluid Dynamics, 2017

Grading

  • Regular exercise assignments
  • The grade for this class is determined by a final oral exam
  • Successful submission of exercises will add a grade bonus of 2/3 of a grade step to your final grade. Note that the pass/fail decision is not affected by the bonus, and the top grade can be achieved without the bonus.

Topics

Apr 24, 2025

Introduction; The Lorenz equations as a paradigm (selected topics from Kaper/Engler, Chapter 7

Apr 28, 2025

Some basic notions from dynamical systems: flow, attractor (cf. Kaper/Engler, Chapter 4), Lyapunov exponents (see, e.g., these lecture notes from BYU)

May 5, 2025

No class due to MIDS event

May 8, 2025

Background knowledge for completing Problem Sheet 1: writing a determinant as a Laplace expansion, change of variables for multiple variables, Computing the leading Lyapunov exponent in practice; Tangent propagator

May 12, 2025

Discussion of Problem Sheet 1

May 15, 2025

No class due to conference

May 19, 2025

Discussion and review

May 22, 2025

Introduction to linear response theory

May 26, 2025

A simple energy balance model for the earth (Kaper/Engler, Chapter 2)

May 30, 2025

Discussion of implementation issues for the linear response calculation on Problem Sheet 2

June 3, 2025

2-box model for the thermohaline circulation (Kaper/Engler, Chapter 3)

June 12, 2025

2-box model for the thermohaline circulation (finish); introduction to the Euler and Navier-Stokes equations (see, e.g., the introductory paper by Kerr and Oliver, Section 2; as supplementary material, reading Appendix A for a big-picture review of Vector Calculus and Appendix B.1 for an example calculation of the total energy budget is recommended)

June 16, 2025

Introduction to the Euler and Navier-Stokes equations (ctd.)

June 30, 2025

Forces in geophysical fluid flow: Coriolis force, buoyancy force, and viscous forces; the Boussinesq equations in three spatial dimensions (see, e.g., Franzke et al., Section 2.1); mid-latitude scalings (see Section 2.3 in the same review paper - in class, I have simplified the presentation straight away to the case of a homogeneous fluid, i.e., a fluid with constant density, so that equation (21d) drops out and the buoyancy term also simplifies)

July 2, 2025

Derivation of the shallow water equations (e.g. Section 2.6 in Franzke et al. with some simplifications and additional details)

July 3, 2025

Geostrophic balance, Potential vorticity, the quasi-geostrophic equations (Section 2.7 in Franzke et al., reduced to the quasi-geostrophic limit, with additional details)

July 7, 2025

The quasi-geostrophic equations (ctd.)

July 10, 2025

Inertia gravity waves and Rossby waves in shallow water, dispersion relation (also see video of gravity wave simulation)

July 17, 2025

Baroclinic instability (see, e.g., the book by Vallis, Section 9.5 “The Eady problem”, without some of the computational details; see video of laboratory experiment, a DIY version, and video of computer simulation)

July 14, 2025

Energy cycle, geostrophic turbulence (see simple Python code here)

July 21, 2025

Geostrophic turbulence ctd.; energy and enstrophy cascade (see, e.g., selected topics from this set of slides

July 24, 2025

Review