# The mathematics of weather and climate Dr Emily Shuckburgh.

## Presentation on theme: "The mathematics of weather and climate Dr Emily Shuckburgh."— Presentation transcript:

The mathematics of weather and climate Dr Emily Shuckburgh

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

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)

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)

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

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

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

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

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.

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

Winter Storms

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

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]

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

North Atlantic Oscillation

+ve-ve

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

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.

Equatorial waves Eastward and westward propagating waves with different speeds

How waves excited?

El Nino

Equatorial waves [pacific92-05.mpg]

El Nino

La Nina

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]