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Unformatted text preview: 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 doesn’t change in time. • Following a particle • Lagrangian derivative ∂ C ¶ t ae è ç ö ø ÷ fixed point in space = 0 δ C = dt ¶ C ¶ t + dx ¶ C ¶ x + dy ¶ C ¶ y + dz ¶ C ¶ z DC Dt = ∂Χ ∂τ + υγ∇Χ 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: Newton’s 2nd (3 eqns), conserve mass (1 eqn), thermodynamics (1 eqn) • • Hydrostatic balance • Ideal gas: ( R gas constant, T temperature) ρ d xd ydz D u Dt = F fric + F gravity + F pressure 1 r Ñ p g ö z p / ρ = ΡΤ g ρ = γπ ΡΤ =  ∂π ∂ζ p = π εγζ/ΡΤ 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) DQ Dt = χ π ∆Τ ∆τ 1 ρ ∆π ∆τ D ρ ∆τ + ρ∇γυ = 0 Effects of rotation...
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 Fall '11
 Kearn
 Fluid Dynamics, Equations, Rotation, Equator

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