1. Vorticity equation in z-coords

2. Physical meaning

3. Examples

Vorticity equations

From Wikipedia

Governing equations

Primitive equations in z-coords

ut + uux + vuy + wuz − fv = −1ρ

px

vt + uvx + vvy + wvz + fu = −1ρ

py

Governing equations

Primitive equations in z-coords

Vorticity equation (in z-coords)

ut + uux + vuy + wuz − fv = −1ρ

px

vt + uvx + vvy + wvz + fu = −1ρ

py

DDt (ζ + f) = − (ζ + f) ( ∂u∂x + ∂v∂y )

−( ∂w∂x ∂v∂z − ∂w∂y ∂u∂z )+

1ρ2 ( ∂ρ∂x ∂p∂y − ∂ρ∂y ∂p∂x )

Stretching termDDt (ζ + f) = − (ζ + f) ( ∂u∂x + ∂v∂y )

u

v

Tilting term

xy

z

DDt (ζ + f) = − ( ∂w∂x ∂v∂z − ∂w∂y ∂u∂z )

u

u

w

w

Solenoidal termDDt (ζ + f) =

1ρ2 ( ∂ρ∂x ∂p∂y − ∂ρ∂y ∂p∂x )

1010 hPa

1000 hPa

990 hPa

ρ

Governing equations

Primitive equations in z-coords

Vorticity equation (in z-coords) (where, and )

ut + uux + vuy + wuz − fv = −1ρ

px

vt + uvx + vvy + wvz + fu = −1ρ

py

∇ = ∇2 ⃗u = iu + jvDDt (ζ + f) = − (ζ + f)∇ ⋅ ⃗u − ∇w ×

∂ ⃗u∂z

+1ρ2

∇ρ × ∇p

StretchingTerm

TiltingTerm

SolenoidalTerm

Example (stretching)

Youtube NSF Fluid Mechanics Series (9.Vorticity 2/2 11분30초)

Example (tilting)

From Holton, originally from Klemp (1987)

Youtube: Vorticity 4K (Mike Olbinski) (Wind shear 0:59, Rotation 4:21, Tornado: 5:07)

Example (tilting)

Youtube NSF Fluid Mechanics Series 
(9.Vorticity 2/2 14:15)

곡류천의 예
(https://educalingo.com/)

Example (solenoid)

From Essentials of Meteorology (Ahrens)

Example (solenoid)

From Holton