Understanding Atmospheric Circulation: Key Concepts and Theoretical Framework

1. The General Circulation of the Atmosphere Background and Theory

2. Overview • Definitions • Potential Temperature • Stream function • Vorticity • Angular Momentum • Rossby number • Geostrophic wind • Gradient wind • Baroclinic Instability • Turbulence & Eddies • Hide’s Theorem

3. 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

4. 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

5. 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.

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

7. 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

8. 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 )

9. 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

10. 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)

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

12. 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

13. 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)

14. 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)