1. Vorticity equation in z-coords 2. Physical meaning
3. Examples
Vorticity equations
From Wikipedia
Governing equations
Primitive equations in z-coords u
t+ uu
x+ vu
y+ wu
z− fv = − 1
ρ p
xv
t+ uv
x+ vv
y+ wv
z+ fu = − 1
ρ p
yGoverning equations
Primitive equations in z-coords
Vorticity equation
(in z-coords)
u
t+ uu
x+ vu
y+ wu
z− fv = − 1 ρ p
xv
t+ uv
x+ vv
y+ wv
z+ fu = − 1
ρ p
yD
Dt ( ζ + f ) = − ( ζ + f ) ( ∂u
∂x + ∂v
∂y )
− ( ∂w
∂x ∂v
∂z − ∂ w
∂y ∂u
∂z ) + 1 ρ
2( ∂ρ
∂x ∂p
∂y − ∂ ρ
∂y ∂p
∂x )
Stretching term
D
Dt ( ζ + f ) = − ( ζ + f ) ( ∂u
∂x + ∂v
∂y )
u
v
Tilting term
y x
z D
Dt ( ζ + f ) = − ( ∂w
∂x ∂v
∂z − ∂ w
∂y ∂u
∂z )
u u w
w
Solenoidal term
D
Dt ( ζ + f ) = 1
ρ
2( ∂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 )
u
t+ uu
x+ vu
y+ wu
z− fv = − 1 ρ p
xv
t+ uv
x+ vv
y+ wv
z+ fu = − 1
ρ p
y∇ = ∇
2u ⃗ = iu + jv D
Dt ( ζ + f ) = − ( ζ + f ) ∇ ⋅ u ⃗ − ∇w × ∂ u ⃗
∂z + 1
ρ
2∇ ρ × ∇ p Stretching
Term
Tilting Term
Solenoidal
Term
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
