# The General Circulation of the Atmosphere Background and Theory.

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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 p 0. 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  =  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 / 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 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 boundaries  zonal winds weaker than that at surface Surface wind determined by boundary conditions  M <= Ω a 2  u <= u m = Ωa sin 2 (Ф)/cos(Ф)