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Mathematical Representation • Vorticity is the curl of the velocity vector • For 3-D vorticity in Cartesian coordinates:  i   V  x u  j  y v  k    w v    u w   v u    j   i      k   z  y z   z x   x y  w The horizontal relative vorticity is found by eliminating terms with vertical (w) components:  i    k  V   x u  j  y v  k   v u     z  x y  w
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Mathematical Representation • The absolute vorticity (η) is the sum of the relative vorticity (spin of the fluid) and the planetary vorticity (rotation due to the spin of the earth):    f Absolute Vorticity Relative Vorticity Coriolis Parameter = 2Ωsin(Φ) where Φ = latitude
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Mathematical Representation • In natural coordinates, the relative vorticity is: V V    Rn n “Curvature Term” R = Radius of Curvature Vorticity due to the spin around a point “Shear Term” Vorticity due to shear of the wind
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Divergence A scalar field (no direction) that measures the tendency of a vector field to spread out
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Mathematical Representation • In Cartesian coordinates (2-Dimensional): u v  V   x y In natural coordinates:   V  V V  n s How the wind speed changes in the t direction Note: Convergence  is just    V How the wind direction changes in the n direction
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The Vorticity Equation • Relates the local rate of change of vorticity to several forcing mechanisms:  u v  D   f     f     Dt  x y   w v w u  1    2   x z y z     p  p      x y y x  Divergence Term Tilting Term Solenoidal Term Convergence = Increase in vorticity Divergence = Decrease in vorticity Represents effects on vorticity due to changes in the vertical velocity (w) in the horizontal direction Effects of pressure gradient force on changes in vorticity
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Simplified Vorticity Equation • A scale analysis shows that the tilting and solenoidal terms are much smaller than the other terms • Furthermore, break the total derivative into its local change and advective components to yield:  u v  V  (  f )  (  f )(  ) t x y Advection of absolute vorticity Divergence term
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Vorticity Advection • Positive Absolute Vorticity Advection (PAVA) – – – – Approaching trough Rising motion Surface convergence Possible intensification of surface low • Negative Absolute Vorticity Advection (NAVA) – – – – Approaching ridge Sinking motion Surface divergence Intensification of anti-cyclones
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