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Understanding Geophysical Fluid Dynamics: Key Concepts and Techniques

This comprehensive summary covers essential topics in Geophysical Fluid Dynamics, including eigenfunctions of the Laplacian, differential equations, and the separation of variables. It highlights techniques for differentiating scalar and vector functions, the principles of equilibrium, perturbations, and oscillations. Key concepts such as Taylor series, eigenvalues, and the gradient are discussed, alongside practical applications like tidal analysis and the impact of Coriolis and viscosity. This resource serves as a fundamental guide for studying the dynamics of fluid motions within geophysical contexts.

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Understanding Geophysical Fluid Dynamics: Key Concepts and Techniques

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Presentation Transcript

  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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