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 h = pole-equator T difference

• Then Hadley circulation extends to

m ~ sqrt (gz* h) / (Ω2 a2 T0)

Meridional Extent of Hadley Cells

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

Potential Vorticity & Entropy

Isentropic Mass Circulation

Eulerian mass flux

Isentropic mass flux

Extratropical flow ~ large-scale eddies ~ adiabatic convenient to use isentropic coordinates

Entropy transported poleward

Eddy entropy flux >> mean entropy flux

i

b

Isentropic, meridional mass flux

Isentropic eddy flux of potential vorticity P

Eddy flux of at sfc (boundary term)

Ekman mass fluxAssume eddies mix P downgradient & P>0 in

interior southward P flux …

• Assume eddies mix potential vorticity & potential temp. diffusively

• Assume there is e so that above e, atmos. is in radiative-convective equilibrium. Integrate previous eqn. LHS vanishes, ignore Ekman flux

find up to which entropy fluxes are significant – this level must be lower than the tropopause pe >= pt

Turbulence as a diffusive process

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 eddy-eddy 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