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Don’t forget

Don’t forget. Keys homework. Sunday Sept 20th: Summary. Main threads Equations of geophysical fluid dynamics Eigenfunctions of the Laplacian. Threads. Techniques Eigenvalues /vectors/functions Differentiating scalar and vector functions Differential equations Principles Separation

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Don’t forget

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  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 • Eigenvalues/vectors/functions • Differentiating scalar and vector functions • Differential equations • Principles • Separation • Superposition • Objectives • Equilibrium, • Perturbations – oscillations, instabilities

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

  5. Recall:Finding eigvals and eigvecs

  6. The derivative as a limit

  7. Taylor series power of h=x-x0 constant derivative at x0

  8. Rules of differentiation

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

  10. The Gradient

  11. The Laplacian

  12. Divergence of a vector field

  13. Curl

  14. Partial differential equations Algebraicequation: 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 • Differentiating scalar and vector functions • Differential equations • Eigenvalues/vectors/functions • Principles • Separation • Superposition • Objectives • Equilibrium, • Perturbations – oscillations, instabilities

  16. Separation of Variables

  17. Separation/superposition in linear systems separation superposition

  18. Wave superposition, Beats wave beats

  19. Tides at Newport, Sept 12-20 2009 Sat 12 Sun 13 Mon 14 Tue 15 Wed16 Thu 17 Fri 18 Sat 19 Sun 20 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 • Differentiating scalar and vector functions • Differential equations • Eigenvalues/vectors/functions • Principles • Separation • Superposition • Objectives • Equilibrium, • Perturbations – oscillations, instabilities

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

  24. Lunar tides

  25. Interpreting σ

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

  27. 3D system: chaos

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

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

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

  31. Pressure gradient Coriolis Viscosity (velocity diffusion) Buoyancy

  32. Motionless equilibrium

  33. Motionless (hydrostatic) equilibrium

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

  35. Geostrophic equilibrium Flow is along isopycnals.

  36. Geostrophic equilibrium Flow is along isopycnals.

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

  38. Eigenfunctions of the Laplacian:diffusion

  39. Eigenfunctions of the Laplacian: diffusion

  40. Eigenfunctions of the Laplacian: diffusion

  41. Eigenfunctions of the Laplacian:waves

  42. Cylindrical coordinates cylindrical Chain rule:

  43. Waves on a circular basin

  44. Waves on a circular basin

  45. Waves on a circular basin Bessel’s equation Bessel functions Analyze just like exp, sin, cos.

  46. Bessel functions Y singular at r=0. To fit initial conditions: (superposition)

  47. Spherical coordinates

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

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

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