Slide 1

The mathematics of weather and climate Dr Emily Shuckburgh Slide 2 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 Slide 3 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) Slide 4 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) Slide 5 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 Slide 6 Effects of sphere Coriolis: vertical comp negligable cf g, and w small, so left with where Resulting eqns: [geo.mpg] Slide 7 Rotating fluids have strange properties [nonrot.mpg, taylor.mpg] Slide 8 Physical effects of rotation geostrophic wind If constant density then Taylor columns If not constant then thermal wind [hadley.mpg] Slide 9 Slide 10 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. Slide 11 Understanding weather/climate 1.Winter storms - midlatitudes 2.El Nino - equatorial region Key difference: 70 17 7% 41045 Coriolis small in tropics Beta-effect largest at equator Different energy sources for forcing Slide 12 Winter Storms Slide 13 Midlatitude dynamics Emitted Longwave Absorbed solar Net radiation temperature wind equator pole ~15km ground Slide 14 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] Slide 15 Winter (North Atlantic) [nao.webarchive] or http://sciencebulletins.amnh.org/earth/f/nao.20040910/ Slide 16 Slide 17 North Atlantic Oscillation Slide 18 +ve-ve Slide 19 Understanding weather/climate 1.Winter storms - midlatitudes 2.El Nino - equatorial region Key difference: 70 17 7% 41045 Coriolis small in tropics Beta-effect largest at equator Different energy sources for forcing Slide 20 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. Slide 21 Equatorial waves Eastward and westward propagating waves with different speeds Slide 22 How waves excited? Slide 23 El Nino Slide 24 Equatorial waves [pacific92-05.mpg] Slide 25 El Nino Slide 26 La Nina Slide 27 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]