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 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)