This class covers the fundamentals of modeling weather and climate for undergraduate students of Mathematics and Data Science. It has the following main themes:
Introduction; The Lorenz equations as a paradigm (selected topics from Kaper/Engler, Chapter 7
Some basic notions from dynamical systems: flow, attractor (cf. Kaper/Engler, Chapter 4), Lyapunov exponents (see, e.g., these lecture notes from BYU)
No class due to MIDS event
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
Discussion of Problem Sheet 1
No class due to conference
Discussion and review
Introduction to linear response theory
A simple energy balance model for the earth (Kaper/Engler, Chapter 2)
Discussion of implementation issues for the linear response calculation on Problem Sheet 2
2-box model for the thermohaline circulation (Kaper/Engler, Chapter 3)
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)
Introduction to the Euler and Navier-Stokes equations (ctd.)
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)
Derivation of the shallow water equations (e.g. Section 2.6 in Franzke et al. with some simplifications and additional details)
Geostrophic balance, Potential vorticity, the quasi-geostrophic equations (Section 2.7 in Franzke et al., reduced to the quasi-geostrophic limit, with additional details)
The quasi-geostrophic equations (ctd.)
Inertia gravity waves and Rossby waves in shallow water, dispersion relation (also see video of gravity wave simulation)
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)
Energy cycle, geostrophic turbulence (see simple Python code here)
Geostrophic turbulence ctd.; energy and enstrophy cascade (see, e.g., selected topics from this set of slides
Review