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The General Circulation of the Atmosphere Background and Theory

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

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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

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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

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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.

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

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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

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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 )

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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

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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)

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

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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

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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)

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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)

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