dynamics i 23. nov.: circulation, thermal wind, vorticity 30. nov.: shallow water theory, waves 7....
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Dynamics I
• 23. Nov.: circulation, thermal wind, vorticity
• 30. Nov.: shallow water theory, waves
• 7. December: numerics: diffusion-advection
• 14. December: computer simulations
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Fluid dynamics
properties of mass, momentum and energy
in a control volume
Substatial or material derivative
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Fluid dynamics
properties of mass, momentum and energy
in a control volume
Substatial or material derivative
To Add: Vorticity !
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Scale Analysis
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Scale Analysis
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Scale Analysis
dominant terms: geostrophy
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Scale Analysis
Validity of the geostrophic approximation:
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Geostrophic approximation:
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Geostrophic balance:
Parallelism between wind velocities
and pressure contours (isobars)
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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.
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Non-Geostrophic Flows
g still suppose that the fluid is homogeneous and frictionless, no vertical structure
Additional terms
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Non-Geostrophic FlowsContinuity equation:
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Non-Geostrophic Flows
Surface elevation η = b + h − H:
Continuity equation:
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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
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Shallow Water ModelCase b=0
This is a formulation that we will encounter in layered models !
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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)
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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.
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Kelvin’s theorem
(f + zeta)/h : the potential vorticity, is also conserved.
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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
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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 !
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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.
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Shallow Water ModelCase b=0
This is a formulation that we will encounter in layered models !
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Shallow Water ModelCase b=0
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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:
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Solution
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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:
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Graph
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Graph
Rossby
Gravity
Kelvin
Yanai, mixed G-R
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Homework
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