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Vorticity

Vorticity. The vector measure of rotation around a point. Mathematical Representation. Vorticity is the curl of the velocity vector For 3-D vorticity in Cartesian coordinates:. The horizontal relative vorticity is found by eliminating terms with vertical (w) components:.

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Vorticity

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  1. Vorticity The vector measure of rotation around a point

  2. Mathematical Representation • Vorticity is the curl of the velocity vector • For 3-D vorticity in Cartesian coordinates: The horizontal relative vorticity is found by eliminating terms with vertical (w) components:

  3. 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): Absolute Vorticity Coriolis Parameter = 2Ωsin(Φ) where Φ = latitude Relative Vorticity

  4. Mathematical Representation • In natural coordinates, the relative vorticity is: “Shear Term” Vorticity due to shear of the wind “Curvature Term” R = Radius of Curvature Vorticity due to the spin around a point

  5. Divergence A scalar field (no direction) that measures the tendency of a vector field to spread out

  6. Mathematical Representation • In Cartesian coordinates (2-Dimensional): In natural coordinates: How the wind speed changes in the t direction Note: Convergence is just How the wind direction changes in the n direction

  7. Directional Divergence

  8. Speed Divergence Divergence Convergence

  9. The Vorticity Equation • Relates the local rate of change of vorticity to several forcing mechanisms: Divergence Term Convergence = Increase in vorticity Divergence = Decrease in vorticity Tilting Term Represents effects on vorticity due to changes in the vertical velocity (w) in the horizontal direction Solenoidal Term Effects of pressure gradient force on changes in vorticity

  10. 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: Divergence term Advection of absolute vorticity

  11. 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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