The General Circulation of the
Atmosphere
Background and Theory
Overview
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Definitions
Potential Temperature
Stream function
Vorticity
Angular Momentum
Rossby number
Geostrophic wind
Gradient wind
Baroclinic Instability
Turbulence & Eddies
Hide’s Theorem
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
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
Potential Temperature (θ)
• 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.
Stream Function
• A function whose contours are stream lines
• Helpful for visualization (i.e. plots)
• In 2D:
Angular Momentum
• For an air parcel in the atmosphere on a rotating planet:
M = (Ω a cos(Ф) + u ) a cos(Ф)
a = radius of planet
Ω = angular rotation rate
Ф = latitude
u = zonal velocity
• Conserved, since tidal forces negligible
• “Coriolis force deflects to the right in NH” = conservation
of angular momentum
Vorticity
• =xu
• Measures amount of rotation in a flow
• Can separate into 2 components:
– planetary vorticity = f = 2 Ω cos()
– relative vorticity =  = -((u cos )) / (a cos )
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 Ω cos(Ф) (Coriolis parameter)
U = velocity scale
L = length scale
• Ro << 1 – Rotation dominant
• Ro ~ 1 – Rotation and advection important
• Ro >> 1 – Advection dominant
Geostrophic Wind
• If Ro <<1 and friction can be
neglected =>
• 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)
Gradient Wind
•
Gradient-wind: geostrophy + centrifugal force
–
adds a correction to geostrophic velocities, depending on orientation of feature rotation relative to
planetary rotation
Baroclinic Instability
• Important for flows
with Ro <<1
• How does differential
heating of poles vs.
equator affect
atmospheric flow?
http://www.gps.caltech.edu/~tapio/papers/annrev06_supp.html
Turbulence & Eddies
• Turbulence as a
diffusive process
• Generally, turbulence
occurs at all scales
• Often expressed as
rotating structures
(eddies)
• Cyclones an
example of largescale eddies
• can transfer energy
from small to large
scale (inverse energy
cascade)
Hide’s 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 <= Ω a2
u <= um = Ωa sin2 (Ф)/cos(Ф)