The General Circulation of the
Atmosphere
Tapio Schneider
Overview
• Aims
• Axisymmetric features of Earth’s
atmosphere
• Tropical Hadley Circulation
– Hide’s theorem
• Extratropical Circulation
• Atmospheric Macroturbulence
http://www.gps.caltech.edu/~tapio/papers/annrev06_supp.html
Aims
• Require a theory of general circulation of the atmosphere
to produce models of the Earth’s atmosphere, both past,
future and for atmospheric models of other planets.
• A general circulation theory for idealised atmospheres
with axisymmetric rotations is a prerequisite for any
future more complete, general circulation theory, which
must be reducible to this canonical case.
• To draw attention to unresolved, fundamental questions
about the general circulation of dry atmospheres,
questions whose resolution is a prerequisite for any
general circulation theory, moist or dry.
Axisymmetric Circulation
Temporal and Zonal Circulations
Axisymmetric Flow
• Proposed by Hadley
• Axisymmetric circulation  baroclinically
unstable
• Eddies transport heat
polewards
Macroturbulence
• Mactroturbulence – large scale eddies,
+1000km.
• Eddies produced by baroclinic instability.
• Transport angular momentum into
latitude zones in which they are created.
• Angular momentum flux into zone compensated by surface drag 
surface westerlies appear in baroclinic zones into which angular
momentum is being transported.
• Vertical structure of winds and strength of upper level jets linked to
surface winds by thermal/gradient wind balance.
Thermal Wind
• Relates vertical shear of the zonal wind to
meridional temperature.
• Not actually a wind, but the difference in the
geostrophic wind between two pressure levels
p1 and p0, with p1 < p0.
• Only present in an atmosphere with horizontal
gradients of temperature i.e. baroclinic.
• Flows around areas of high and low temperature
as the geostrophic wind flows around areas of
high and low pressure.
Axisymmetric Circulation Vs.
Macroturbulence
Explanation of Figure 3
• Bottom row fig 3 temporal and zonal means of mass flux stream
function and angular momentum in steady states of macroturbulent
circulation that correspond to the axisymmetric circulation in top row.
• Macroturbulent Hadley cells extend further poleward than
axisymmetric simulations.
• Streamlines in upper parts of Hadley cell cut angular momentum
contours.
• Local Rossby numbers reduced relative to axisymmetric circulation.
• Eddies strengthen the equinoctial Hadley cells (3a and 3b) and
weaken the winter cell (3b and 3e)
• Mass flux in Hadley cells in macroturbulent model same order of
magnitude as in Earth’s Hadley cells.
• When max heating moved to 6 degrees latitude, winter cell 1.5 times
bigger and summer cell 1.5 times smaller (3d and e).
Implications of Hide’s Theorem
• u <= um = Ωa sin2 ()/cos()
• Assume gradient-wind balance, then from meridional momentum
equation:
•  Ф <= 2 Ω2 a2 3
Ф=gz, (assuming small latitude = tropics)
• Use ideal-gas result p = p0 exp(-Ф/RT)
(T is vertically averaged)
• => constraints on meridional decrease in temperature
• Assume T ~ h cos2
• Then Hadley circulation extends to
• m ~ sqrt (gz* h) / (Ω2 a2 T0)
h = pole-equator T difference
Meridional Extent of Hadley Cells
Potential Vorticity & Entropy
Entropy – measures amount of disorder in a system
• For an ideal gas: s = cp ln (T p –R/c )
•   = 0 exp(-s/cp)
•   constant  s constant
Potential vorticity – measure of vorticity, normalized by entropy
• P = (planetary vorticity + relative vorticity) / (width of entropy contour)
= (f + )/H
• Conserved quantity
for adiabatic processes
Isentropic Mass Circulation
Extratropical flow ~ large-scale eddies ~
adiabatic  convenient to use
isentropic coordinates
Entropy transported poleward
Eddy entropy flux >> mean entropy flux
Eulerian mass flux 
Isentropic mass flux 
Isentropic, meridional
mass flux
Isentropic eddy flux of
potential vorticity P
Eddy flux of  at sfc
(boundary term)
Assume eddies mix P
downgradient & P>0 in
interior  southward P flux
…
i
Ekman mass flux
b
Turbulence as a diffusive process
• Assume eddies mix potential vorticity & potential temp. diffusively
• Assume there is e so that above e, atmos. is in radiativeconvective equilibrium. Integrate previous eqn.  LHS vanishes,
ignore Ekman flux
•
up to which entropy fluxes are significant – this
level must be lower than the tropopause  pe >= pt 
 find
1
bulk stability
supercriticality – measure of vertical extent
of eddy entropy fluxes
Supercriticality constraint
x-axis – negative  gradient
~ entropy gradient
y-axis – bulk stability
Sc<1 regime – eddy entropy
fluxes weak, tropopause set
by radiation/convection
Sc~1 regime – eddy entropy
fluxes large & stabilize the
thermal stratification 
tropopause height adjusted
 A state with strong
nonlinear eddy-eddy
interactions (Sc>>1) adjusts
thermal stratification so that
Sc<~1 (and has weak eddyeddy interactions)
Summary
• Differential heating causes Hadley circulation in tropics, Polar cell near
poles
• In midlatitudes, differential heating causes baroclinic instability
• Hide’s Theorem imposes upper limit to Hadley circulation extent
• Extratropical circulation associated with (adiabatic) eddy fluxes of P, 
• If eddies act diffusively, supercriticality <=1
– thermal stratification / tropopause height linked to eddy strength