Advanced Graduate Lecture Course:
Consisting of a Lecture Course of 2 hours including a 1 hour Problem Class per week:
Lecture Course (250 059 VU): Tuesday, 9:30 - 11.30 hrs
Start: October 11, 2016
End: December 13, 2016
There will be an extra lecture on Monday, October 31, 2016, 9:30 - 11:30 hrs.
Problem Class: Thursday, 9:30 - 10.30 hrs
Start: October 6, 2016
End: Friday, December 9, 2016
There will be an extra and final Problem Class on Friday, December 9, 2016, 9:30 - 10:30 hrs.
Note there will be no activities in the period November 1 - 22, 2016.
The study of the flow of fluids has a long and illustrious history, with strong links to both pure and applied mathematics, and to physics and engineering. At the theoretical level, we are beginning to understand, in depth, some of the processes involved in the motions of fluids in many different contexts: pipes and arteries, rivers, oceans, the atmosphere, and much more. However, in order to undertake an investigation in any one of these areas requires a significant grounding in the mathematics of fluid flows. With this in place, a study of a more specific nature is possible. So, for example, we can begin the process of examining the mathematical issues that pertain to ocean flows. This is one application, which only in recent years has begun to show some successes, that has been made possible, to a large extent, by the explosion in the available data. Even so, a comprehensive mathematical approach is quite beyond us: oceanographers often have to resort, by necessity, to much ad hoc modelling.
The flows associated with our oceans are obviously complex; they are affected by the winds at the surface, large- and small-scale currents, considerable areas of upwelling or downwelling, thermal and saline variations, variable bottom topography, and more. However, in some regions of some oceans, the observed flows follow a more well-ordered pattern; one such is the flow in the neighbourhood of, and along, the Pacific Equator, and another is the only fully circular flow on Earth: the Antarctic Circumpolar Current. The first example, which includes a complicated vertical structure, is highly nonlinear, three-dimensional and contains a thermocline, but is amenable to a mathematical approach based on classical theoretical fluid dynamics; for the second type of flow, we have only recently found that a mathematical description (in spherical coordinates on a rotating Earth) can be attempted. This course will introduce all the basic and relevant ideas of theoretical fluid dynamics, define the classical water-wave problem, and then describe and analyse some oceanic flows.
The material to be covered is provided in the syllabus (see below); primarily, the ideas will be presented via a set of lectures, but additional calculations, and associated examples, will be described in the problem classes. Handouts will be available at the start of each lecture and problem class, and a full text of all the material, in extended form, will be available at the completion of the course.
1. Basic ideas and governing equations
Definition of a fluid; the continuum hypothesis; variables and notation; streamlines and particle paths; material derivative; acceleration; Eulerian & Lagrangian formulations; equation of mass conservation; incompressibility; hydrostatic equilibrium; Euler’s equation (mentioning Navier-Stokes in passing); vorticity and rotation; Helmholtz’s equation; irrotational flow; circulation; Bernoulli’s equation; pressure equation; stream function.
2. The classical water-wave problem I
Boundary conditions for water waves; rȏle and importance of non-dimensionalisation & scaling.
3. Asymptotic expansions
Short introduction to asymptotic expansions with a parameter: basic definitions and elementary techniques.
4. The classical water-wave problem II
Rȏles of the parameters to define different classes of problem; introduction to some classical results.
5. Oceanic flows
Introduction to the nature of these flows: complexity, geometry, rotation; simple approximations used by oceanographers; aims of a mathematical approach.
Formulation of the Pacific Equatorial Undercurrent wave problem, and the nature of the Antarctic Circumpolar Current; presentation of some results.
See also the course directory of the University of Vienna (link).