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The mathematics of weather and climate Dr Emily Shuckburgh

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Equations of fluid flow I Cloud over a hill C(x,y,z,t). In steady state cloud doesnt change in time. Following a particle Lagrangian derivative Advection: ability of fluid to carry properties with it as it moves

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Equations of fluid flow II 5 key variables (u,v,w), p, T 5 eqns: Newtons 2nd (3 eqns), conserve mass (1 eqn), thermodynamics (1 eqn) Hydrostatic balance Ideal gas: (R gas constant, T temperature)

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Equations of fluid flow III Conserve mass - mass changes if flux into volume: First law of thermodynamics gives: DQ/Dt comes from latent/radiative heating (note T, p dependency)

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Effects of rotation Live on rotating planet & observe winds/currents in rotating frame (N2 in inertial frame) [coriolis.mpg] Angular momentum from rotation gives rotating fluids unusual properties For equations of motion in rotating frame need to translate D/Dt acting on a vector into rotating frame Coriolis acceleration Centrifugal acceleration

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Effects of sphere Coriolis: vertical comp negligable cf g, and w small, so left with where Resulting eqns: [geo.mpg]

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Rotating fluids have strange properties [nonrot.mpg, taylor.mpg]

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Physical effects of rotation geostrophic wind If constant density then Taylor columns If not constant then thermal wind [hadley.mpg]

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Summary: fluid flow on the Earth Newtons second law, conservation of mass, thermodynamics Rotating sphere Atmosphere & ocean are fluids Numerically solve equations on a grid, parameterise any sub-grid physics, e.g. clouds.

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Understanding weather/climate 1.Winter storms - midlatitudes 2.El Nino - equatorial region Key difference: 70 17 7% 4º10º45º Coriolis small in tropics Beta-effect largest at equator Different energy sources for forcing

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Winter Storms

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Midlatitude dynamics Emitted Longwave Absorbed solar Net radiation temperature wind equator pole ~15km ground

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Barotropic instability: wave instability associated with horizontal shear. Extracts KE from mean flow. Baroclinic instability: wave instability associated with vertical shear. Converts PE associated with mean horizontal temp gradient that must exist to provide thermal wind balance for vertical shear. [eddies.mpg]

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Winter (North Atlantic) [nao.webarchive] or http://sciencebulletins.amnh.org/earth/f/nao.20040910/

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North Atlantic Oscillation

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+ve-ve

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Understanding weather/climate 1.Winter storms - midlatitudes 2.El Nino - equatorial region Key difference: 70 17 7% 4º10º45º Coriolis small in tropics Beta-effect largest at equator Different energy sources for forcing

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Tropical dynamics Equatorial waves trapped near equator. Coriolis force changes sign at the equator. Take fluid eqns, make approximations for equator. Consider wave-like disturbances. Eqns of same form as Schrodinger equation, solutions are Hermite polynomials.

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Equatorial waves Eastward and westward propagating waves with different speeds

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How waves excited?

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El Nino

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Equatorial waves [pacific92-05.mpg]

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El Nino

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La Nina

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Summary Behaviour of rotating fluid is unusual Midlatitude dynamics very different to tropical dynamics Many aspects of weather/climate can be understood in terms of waves/instabilities of fluid flow. [more movies at http://paoc.mit.edu/labweb]

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