Mathematics and the environment


The aim of the course is to illustrate the techniques of mathematical modelling in their particular application to environmental problems. The mathematical techniques used are drawn from the theory of ordinary differential equations and partial differential equations. However, the course does require the willingness to become familiar with a range of different scientific disciplines. In particular, familiarity with the concepts of fluid mechanics will be useful.


Applications of mathematics to environmental or geophysical problems involving the use of models with ordinary and partial differential equations. Examples to be considered are:
Climate dynamics. River flow and sediment transport. Glacier dynamics.

Reading list

A. C. Fowler 2004 Mathematics and the environment. Mathematical Institute lecture notes. (Revised edition, September 2006.)
K. Richards 1982 Rivers. Methuen.
G. B. Whitham 1974 Linear and nonlinear waves. Wiley, New York.
W. S. B. Paterson 1994 The physics of glaciers, 3rd edition. Pergamon Press.
J. T. Houghton 2002 The physics of atmospheres, 3rd ed. C.U.P., Cambridge.

Lecture synopsis

(Page numbers refer to the printed lecture notes.) (Updated 7/10/06.)

    Climate dynamics.

  1. Radiative heat transfer (pp. 3-17).
  2. Convection (pp. 17-19).
  3. Energy balance models (pp. 20-29).
  4. Ice ages (pp. 29-42).


  5. Chézy and Manning flow laws (pp. 66-70).
  6. Slowly varying flow; flood hydrograph (pp. 70-72).
  7. Rapidly varying flow, St. Venant equations (pp. 72-76).
  8. Waves and instability (pp. 77-88).

    Sediment transport

  9. Patterns and bedforms (pp. 103-107).
  10. Bedload transport; the Exner equation. Suspended sediment. Potential flow model. (pp. 108-113).
  11. St. Venant type models (pp. 113-118).
  12. Anti-dunes and dunes (pp. 119-129).

    Glacier dynamics

  13. Waves and surges (pp. 156-164, 186-188).
  14. Surface waves. Subglacial sliding (pp. 164-168, 180-184).
  15. Subglacial drainage (pp. 184-185).
  16. Linked cavity systems and surges (pp. 188-194).

Course materials

These can be downloaded as pdf files; currently available are