Atmospheric_Dynamics_II-v2

# Atmospheric_Dynamics_II-v2 - Vorticity The Law of...

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Unformatted text preview: Vorticity The Law of conservation of Vorticity • Relative vorticity is defined as • The time rate of change of vorticity can be written by taking the equations of horizontal momentum and deriving the x component with respect to y and the y component with respect to x • By doing that, the relative vorticity and Coriolis parameter can be isolated (try to show this is true –Problem 7.32) and we can rewrite the equation in the vorticity equation in the form: y u x v ∂ ∂- ∂ ∂ = ζ In Cartesian coordinate form ∂ ∂- =- ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ x p fv z u w y u v x u u t u y ρ 1 ∂ ∂- = + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ y p fu z v w y v v x v u t v x ρ 1 Momentum Equations: ( 29 = + ∂ ∂ ∂ ∂- ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ + + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ dy df v z u y w z v x w y v x u f z w y v x u t ζ ζ ζ ζ ζ ∂ ∂ ∂ ∂- ∂ ∂ ∂ ∂ x p y y p x ρ ρ ρ 2 1 dt d ζ V ⋅ ∇ ∂ ∂ ∂ ∂- ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂- ∂ ∂ ∂ ∂- ∂ ∂ + ∂ ∂ +- = + x p y y p x z u y w z v x w y v x u f dt f d ρ ρ ρ ζ ζ 2 1 ) ( ) ( Divergence Term: Very important in synoptic-scale movements Tilting Term: important for mesocale processes (e.g. formation of tornadoes Solenoidal term important only in particular small- scale circulations x x y y z z Suppose v increasin g with height δ / δ v z Suppose w varying with x δ / δ w x • For synoptic purposes, lets neglect the tilting and solenoidal terms. In Lagrangian form the vorticity variation equation is: ∂ ∂ + ∂ ∂ +- = + y v x u f dt f...
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## This note was uploaded on 12/28/2011 for the course GEOG 226 taught by Professor Leila during the Fall '09 term at UCSB.

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Atmospheric_Dynamics_II-v2 - Vorticity The Law of...

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