The General Circulation of the Atmosphere

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The General Circulation of the Atmosphere

• Background and Theory

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Overview

• Definitions
• Potential Temperature
• Stream function
• Vorticity
• Angular Momentum
• Rossby number
• Geostrophic wind
• Gradient wind
• Baroclinic Instability
• Turbulence Eddies
• Hides Theorem

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Definitions
Inviscid Flow A fluid flow where viscous
(friction) forces are small in comparison to
inertial forces.
Meridional Along a meridian (N-S).
Zonal Along a latitude circle (E-W).
Axisymmetric Symmetrical about the axis of
planetary rotation that is, zonally symmetric
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Definitions
Reversible Process A processe which can be
reversed by means of infinitesimal changes in
some property of the system without loss or
dissipation of energy
Isentropic Process A process in which the
entropy of the system remains constant. It is
both adiabatic and reversible.
Macroturbulence Totality of irregular motions
of large scale eddies, characterised by a small
Rossby number.
Advection The horizontal movement of air or
atmospheric properties, solely by the motion of
the atmosphere
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Potential Temperature (?)

• The temperature an air parcel will have if
  adiabatically and reversibly moved to a reference
  pressure level p0.
• For an ideal gas
• A conserved property for all dry adiabatic
  processes.

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Stream Function

• A function whose contours are stream lines
• Helpful for visualization (i.e. plots)
• In 2D

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Angular Momentum

• For an air parcel in the atmosphere on a rotating
  planet
• M (O a cos(?) u ) a cos(?)
• a radius of planet
• O angular rotation rate
• ? latitude
• u zonal velocity
• Conserved, since tidal forces negligible
• Coriolis force deflects to the right in NH
  conservation of angular momentum

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Vorticity

• ? ? x u
• Measures amount of rotation in a flow
• Can separate into 2 components
• planetary vorticity f 2 O cos(?)
• relative vorticity ? -(??(u cos ?)) / (a cos
  ?)

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Rossby number

• Measure of the relative importance of rotation
  and advection
• -or- of the importance of planetary vorticity vs.
  relative vorticity
• Ro U / fL
• f 2 O cos(?) (Coriolis parameter)
• U velocity scale
• L length scale
• Ro ltlt 1 Rotation dominant
• Ro 1 Rotation and advection important
• Ro gtgt 1 Advection dominant

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Geostrophic Wind

• If Ro ltlt1 and friction can be neglected gt
• Geostrophy Pressure gradient force balances
  Coriolis force
• Atmosphere is geostrophic to first approximation
• Wind is along pressure contours (pressure is
  essentially the stream function for velocity)

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Gradient Wind

• Gradient-wind geostrophy centrifugal force
• adds a correction to geostrophic velocities,
  depending on orientation of feature rotation
  relative to planetary rotation

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Baroclinic Instability

• Important for flows with Ro ltlt1
• How does differential heating of poles vs.
  equator affect atmospheric flow?

http//www.gps.caltech.edu/tapio/papers/annrev06_
supp.html
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Turbulence Eddies

• Turbulence as a diffusive process
• Generally, turbulence occurs at all scales
• Often expressed as rotating structures (eddies)
• Cyclones an example of large-scale eddies
• can transfer energy from small to large scale
  (inverse energy cascade)

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Hides Theorem

• Axisymmetry Diffusion of angular momentum (eg.
  from small scale turbulence)
• No extremum of angular momentum away from
  boundaries
• zonal winds weaker than that at surface
• Surface wind determined by boundary conditions
• M lt O a2
• u lt um Oa sin2 (?)/cos(?)

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