Vorticity Vertical component of vorticity: i.e., the rotation about the local vertical v u ς k V or in cartesian coordinate s, ς x y There are three types of vorticity used in geophysical fluid dynamics Relative vorticity = ς Planetary vorticity = f Absolute vorticity = ςa = ς + f If the wind is geostrophic, the geostrophic relative vorticity is given by g 2 1 2 ςg = ∇ Φ = ∇ z, where Φ = gz is geopotenti al f f Vorticity y + ∂v ∂u ς= ∂x ∂y + u 0 y v 0 x x ς 0 for cyclonic circulation ς 0 for anticyclon ic circulation Divergence ∇ • VH = ∂u ∂v + ∂x ∂y Divergence: From the continuity equation - u v w 1 dρ V ρ dt x y z For incompressible flow: dρ 0 dt u v w VH x y z Most meteorologists simply use divergence when they mean the horizontal divergence VH Divergence u v VH x y ∂u ? ∂x u 0 x v 0 y ∂u ∂v ∇ • VH = + > 0, divergence ∂x ∂y ∂u ∂v ∇ • VH = + < 0, convergenc e ∂x ∂y Vorticity and Divergence Why they are important? What’s the unit? s-1 What’s the order of the magnitude? Relative Vorticity ~ 10 -5 s -1 Scale Analysis (Synoptic Scales) ∂v 10 ms -1 10 ms -1 = = = 10 6 ∂x 1000km 10 m -5 s -1 For severe weather , vorticity can go up to ~ 10 -3 s-1 . Divergence , ~ 10 -6 s -1 Why? For severe weather , divergence can go up to ~ 10 -3 s-1 . Vorticity and Divergence (Synoptic Scale) Relative Vorticity ~ 10 -5 s -1 ∂v ∂u ς= ∂x ∂y ∂v when >0 ∂x Divergence , ~ 10 -6 s -1 u v VH x y ∂u => >0 ∂y Same signs for both terms ∂u when >0 ∂x ∂v => <0 ∂y Divergence smaller because ∂u ∂v and are opposite signs! ! ! ∂x ∂y Vorticity in Nature Coordinates v ∂v ς= R ∂n v R Curvature effect n V S Shear effect The smaller the R (i.e., curved more, shorter waves), the stronger the curvature term. R is positive for cyclonic circulation and negative for anticyclonic circulation. Vorticity ? curvature Shear Vorticity Advection Ridge Trough Trough V wind speed L dςa 0, dt 0 Relative vorticity advection Planetary vorticity advection ∂ςa ∂ς ∂f ∂ς df = + = = -V • ∇ςa = -V • ∇ς - v , ∂t ∂t ∂t ∂t dy df >0 dy Vorticity Advection (shorter Rossby waves) For short waves, relative vorticity dominates Ridge Trough Trough ς min V ς max ς max L Wavelength ∂ς ∂ς => = - V • ∇ς = - V (nature coordinate s) ∂t ∂s Downstream of a trough or upsteam of a ridge ∂ς ∂ς ∂ς V > 0, < 0, => ≈ -V > 0 (PVA) ∂s ∂t ∂s Vorticity Advection (shorter Rossby waves) ∂ς >0 ∂t PVA If one uses geostrophi c wind to approximat e real wind, then g 1 ςg = ∇2Φ = ∇2z, where Φ = gz is the geopotenti al. f f (assume wave solution) g 2 ∂ςg f ∂(∇ z) g ∂z ∂z = ∝> 0, => < 0, => z decreases with time. ∂t ∂t f ∂t ∂t PVA Adiabatic cooling z decreases with time divergence aloft ascent convergenc e Vorticity Advection (shorter Rossby waves) For short waves, relative vorticity dominates Ridge Trough Trough ς min V ς max ς max L ς ς V ς V (nature coordinate s) t s Down stream of a trough or upsteam of a ridge V 0, ς ς ς 0, -V 0 s t s Positive Vorticity Advection, PVA. Vorticity Advection (shorter Rossby waves) For short waves, relative vorticity dominates Ridge Trough Trough ς min V ς max ς max L Waves propagate eastward! ς ς V ς V (nature coordinate s) t s Down stream of a trough or upsteam of a ridge V 0, ς ς ς 0, -V 0 s t s Negative Vorticity Advection, NVA. Vorticity Advection (shorter Rossby waves) NVA ∂ς <0 ∂t If one uses geostrophi c wind to approximat e, then (assume wave solution) g 2 ∂ςg ∂ f ∇ z g ∂z = ∝< 0, ∂t ∂t f ∂t NVA Adiabatic warming z increases with time ∂z => > 0, => z inceases with time. ∂t convergence aloft decent divergence Vorticity Advection (shorter Rossby waves) 564 βL2 C = U- 2 , 4π Trough df where β = , dy Wave propagation speed 570 576 U C wave speed Tropopause L2 z C 4 0 speed 2 level of non-divergence 500-600 mb Vertical Coupling 564 Trough 570 576 U C wave speed Tropopause L2 z C 4 level of non-divergence 500-600 mb 2 0 speed Will grow Will grow Low-level cyclone and 500mb heights 5520 5460 996 cold air mass 992 ** . cold front 32 31 988 .. 39 37 direction of propagation 49 46 56 52 warm air mass warm front Upper and low-level coupled together – upper level trough enhances the low-level cyclone development. Low-level cyclone and 500mb heights 5520 5460 5520 5460 upper level trough is about to catch low-level system. upper level trough catches low-level system – low level system is decaying.