1. Don’t forget Keys homework

2. Sunday Sept 20th: Summary Main threads Equations of geophysical fluid dynamics Eigenfunctions of the Laplacian

3. Threads Techniques o Eigenvalues/vectors/functions o Differentiating scalar and vector functions o Differential equations Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

4. Threads Techniques o Eigenvalues/vectors/functions o Differentiating scalar and vector functions o Differential equations Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

5. Recall: Finding eigvals and eigvecs

6. The derivative as a limit

7. Taylor series constant derivative at x 0 power of h=x-x 0

8. Rules of differentiation

9. Properties of the exponential function Sum rule: Power rule: Taylor series: Derivative Indefinite integral

10. The Gradient

11. The Laplacian

12. Divergence of a vector field

13. Curl

14. Partial differential equations Algebraic equation: involves functions; solutions are numbers. Ordinary differential equation (ODE): involves total derivatives; solutions are univariate functions. Partial differential equation (PDE): involves partial derivatives; solutions are multivariate functions.

15. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

16. Separation of Variables

17. Separation/superposition in linear systems superposition separation

18. Wave superposition, Beats wave beats

19. Fri 18Wed16Mon 14Sat 12Sun 20Sun 13Tue 15Thu 17Sat 19 Tides at Newport, Sept 12-20 2009 Spring tides

20. Superposition (linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.

21. Superposition in PDEs

22. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

23. Perturbations about equilibrium Equilibrium: F=0 ~0 (LINEARIZATION)

24. Lunar tides

25. Interpreting σ

26. b. repellora. attractor c. saddle d. limit cycle e. unstable spiral f. stable spiral Interpreting two σ’s

27. 3D system: chaos

28. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

29. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

30. The equations of geophysical fluid dynamics flow velocity diffusion density

31. Pressure gradient CoriolisViscosity (velocity diffusion) Buoyancy

32. Motionless equilibrium

33. Motionless (hydrostatic) equilibrium

34. Geostrophic equilibrium Pressure gradient Coriolis Neglect viscosity Assume velocity is small, so that products are negligible Set time derivatives to zero.

35. Geostrophic equilibrium Flow is along isopycnals.

36. Geostrophic equilibrium Flow is along isopycnals.

38. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

39. Eigenfunctions of the Laplacian: diffusion

42. Eigenfunctions of the Laplacian:waves

43. Cylindrical coordinates cylindrical Chain rule:

44. Waves on a circular basin

46. Analyze just like exp, sin, cos. Bessel’s equationBessel functions

47. Y singular at r=0. To fit initial conditions:(superposition)

48. Spherical coordinates

49. Threads Techniques o Differentiating scalar and vector functions o Differential equations o Eigenvalues/vectors/functions Principles o Separation o Superposition Objectives o Equilibrium, o Perturbations – oscillations, instabilities

50. Don’t forget Keys Homework Mart!n’s coat, stapler