1. Dynamics I 23. Nov.: circulation, thermal wind, vorticity 30. Nov.: shallow water theory, waves 7. December: numerics: diffusion-advection 14. December: computer simulations

2. Fluid dynamics properties of mass, momentum and energy in a control volume Substatial or material derivative

3. Fluid dynamics properties of mass, momentum and energy in a control volume Substatial or material derivative To Add: Vorticity !

4. Scale Analysis

6. dominant terms: geostrophy

7. Scale Analysis Validity of the geostrophic approximation:

8. Geostrophic approximation:

9. Geostrophic balance: Parallelism between wind velocities and pressure contours (isobars)

10. Taylor–Proudman theorem (Taylor, 1923; Proudman, 1953) g Const. vertical derivative of the horizontal velocity is zero: Physically, it means that the horizontal velocity field has no vertical shear and that all particles on the same vertical move in concert. Such vertical rigidity is a fundamental property of rotating homogeneous fluids.

11. Non-Geostrophic Flows g still suppose that the fluid is homogeneous and frictionless, no vertical structure Additional terms

12. Non-Geostrophic Flows Continuity equation:

13. Non-Geostrophic Flows Surface elevation η = b + h − H: Continuity equation:

14. Non-Geostrophic Flows In the absence of a pressure variation above the fluid surface (e.g., uniform atmospheric pressure over the ocean), this dynamic pressure is Continuity equation

15. Shallow Water Model Case b=0 This is a formulation that we will encounter in layered models !

16. Vorticity Vorticity: Elimination of pressure terms: Continuity: d/dt [(f+ zeta)/h] = 0 Conservation of potential vorticity ambient vorticity (f) plus relative vorticity zeta vorticity vector is strictly vertical (Volume conservation)

17. Conservation of volume in an incompressible fluid This implies that if the parcel is squeezed vertically (decreasing h), it stretches horizontally (increasing ds), and vice versa Vorticity horizontal divergence (∂u/∂x + ∂v/∂y > 0) causes widening of the cross-sectional area ds convergence (∂u/∂x + ∂v/∂y < 0) narrowing of the crosssection.

18. Kelvin’s theorem (f + zeta)/h : the potential vorticity, is also conserved.

19. Kelvin’s theorem (f + zeta)/h : the potential vorticity, is also conserved. This product can be interpreted as the vorticity flux (vorticity integrated over the cross- section) and is therefore the circulation of the parcel. Two-dimensional flows: Kelvin’s theorem conservation of circulation in inviscid fluids

20. Circulation of a parcel (f + zeta)/h : the potential vorticity, is also conserved. This product can be interpreted as the vorticity flux (vorticity integrated over the cross-section) and is therefore the circulation of the parcel. Two-dimensional flows: Kelvin’s theorem guarantees conservation of circulation in inviscid fluids Exercise !

21. Potential Vorticity Rapidly rotating flows, in which the Coriolis force dominates. In this case, the Rossby number is much less than unity (Ro = U/L << 1), which implies that the relative vorticity (ζ = ∂v/∂x − ∂u/∂y, scaling as U/L) is negligible in front of the ambient vorticity f. The potential vorticity reduces to q =f/h ! if f is constant – such as in a rotating laboratory tank or for geophysical patterns of modest meridional extent – implies that each fluid column must conserve its height h.

22. Shallow Water Model Case b=0 This is a formulation that we will encounter in layered models !

23. Shallow Water Model Case b=0

28. Schrödinger equationSchrödinger equation: harmonic oscillator In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V(x) = (1/2)mω^2 x^2. The Hamiltonian of the particle is:Hamiltonian where x is the position operator, and p is the momentum operatorpositionmomentum we must solve the time-independent Schrödinger equation:Schrödinger equation

29. Solution

30. Ladder operator method a acts on an eigenstate of energy E to produceanother eigenstate of energy a† acts on an eigenstate of energy E to produce an eigenstate of energy a "lowering operator", a† "raising operator„ The two operators together are called "ladder operators". In quantum field theory, a and a† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy. Operator a and a† have properties:

34. Graph

35. Rossby Gravity Kelvin Yanai, mixed G-R

36. Homework