Dynamics I

• 23. Nov.: circulation, thermal wind, vorticity

• 30. Nov.: shallow water theory, waves

• 7. December: numerics: diffusion-advection

• 14. December: computer simulations

Fluid dynamics

properties of mass, momentum and energy

in a control volume

Substatial or material derivative

Fluid dynamics

properties of mass, momentum and energy

in a control volume

Substatial or material derivative

To Add: Vorticity !

Scale Analysis

Scale Analysis

Scale Analysis

dominant terms: geostrophy

Scale Analysis

Validity of the geostrophic approximation:

Geostrophic approximation:

Geostrophic balance:

Parallelism between wind velocities

and pressure contours (isobars)

Taylor–Proudman theorem (Taylor, 1923; Proudman, 1953)

gConst.

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.

Non-Geostrophic Flows

g still suppose that the fluid is homogeneous and frictionless, no vertical structure

Additional terms

Non-Geostrophic FlowsContinuity equation:

Non-Geostrophic Flows

Surface elevation η = b + h − H:

Continuity equation:

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

Shallow Water ModelCase b=0

This is a formulation that we will encounter in layered models !

Vorticity

Vorticity:

Elimination of pressure terms:

Continuity:

d/dt [(f+ zeta)/h] = 0Conservation of potential vorticity

ambient vorticity (f) plus relative vorticity zeta

vorticity vector is strictly vertical

(Volume conservation)

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.

Kelvin’s theorem

(f + zeta)/h : the potential vorticity, is also conserved.

Kelvin’s theorem

(f + zeta)/h : the potential vorticity, is also conserved.

This product canbe 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

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 !

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.

Shallow Water ModelCase b=0

This is a formulation that we will encounter in layered models !

Shallow Water ModelCase b=0

Schrödinger 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:

where x is the position operator, and p is the momentum operator

we must solve the time-independent Schrödinger equation:

Solution

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:

Graph

Graph

Rossby

Gravity

Kelvin

Yanai, mixed G-R

Homework