The General Circulation of the Atmosphere

Background and Theory

Overview

• 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

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.

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

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 planetΩ = angular rotation rateФ = latitudeu = zonal velocity

• Conserved, since tidal forces negligible

• “Coriolis force deflects to the right in NH” = conservation of angular momentum

Vorticity

= x u• 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 / fLf = 2 Ω cos(Ф) (Coriolis parameter)U = velocity scaleL = 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 large-scale 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

boundarieszonal winds weaker than that at surface

Surface wind determined by boundary conditionsM <= Ω a2

u <= um = Ωa sin2 (Ф)/cos(Ф)