Vorticity The vector measure of rotation around a

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:

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 Relative Vorticity Coriolis Parameter = 2Ωsin(Φ) where Φ = latitude

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

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

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

Directional Divergence

Speed Divergence Convergence

The Vorticity Equation • Relates the local rate of change of vorticity to several forcing mechanisms: 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

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: Advection of absolute vorticity Divergence term

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