The Vorticity Equation

A Rotational View of the Atmoshere Chapter 4

Part 3: Potential Vorticity Barotropic Vorticity Equation Assume a homogeneous, incompressible fuid (constant density with ρ = 0).

Continuity Equation @⇢ = (⇢u) @t r ·

@u @v @w u =0 + = r · @x @y @z

Paul Ullrich A Rotational View of the Atmosphere March 2014 Barotropic Vorticity Equation

Vorticity Equation (Divergence term only) D @u @v h (⇣ + f)= (⇣ + f) + Dt @x @y ✓ ◆ @u @v @w + = @x @y @z D (⇣ + f) @w h =(⇣ + f) Dt @z

D (⇣ + f) For purely horizontal fow (w=0): h =0 Dt

Paul Ullrich A Rotational View of the Atmosphere March 2014 Barotropic Vorticity Equation

The barotropic vorticity equation states that when there is no vertical velocity, the absolute vorticity is conserved following the horizontal motion.

D (⇣ + f) h =0 Dt

Let’s try a more general derivation…

Paul Ullrich A Rotational View of the Atmosphere March 2014 Potential Vorticity In a Barotropic Fluid

Assume a homogeneous, incompressible fuid (constant density with ρ = 0).

D (⇣ + f) @w h =(⇣ + f) Dt @z Approximate vorcity by geostrophic vorcity,

integrate vercally from z1 to z2

D (⇣ + f) w(z ) w(z ) h g =(⇣ + f) 2 1 Dt g z z 2 1 Dhh Dh(⇣g + f) Dhh Use = w(z ) w(z ) h =(⇣ + f) Dt 2 1 Dt g Dt

Paul Ullrich A Rotational View of the Atmosphere March 2014 Potential Vorticity In a Barotropic Fluid

Dh(⇣g + f) w(z2) w(z1) =(⇣g + f) Dt z2 z1 Defne layer thickness D h h h = z 2 z1 Use = w(z2) w(z1) Dt

D (⇣ + f) D h h h g =(⇣ + f) h Dt g Dt

Rearrange

1 D (⇣ + f) 1 D h h g = h (⇣g + f) Dt h Dt

Paul Ullrich A Rotational View of the Atmosphere March 2014 Potential Vorticity In a Barotropic Fluid

1 D (⇣ + f) 1 D h h g = h (⇣g + f) Dt h Dt

Calculus of logarithms D D h log(⇣ + f)= h log h Dt g Dt

Properes of logarithms

D (⇣ + f) D ⇣ + f h log g =0 h g =0 Dt h Dt h 

Paul Ullrich A Rotational View of the Atmosphere March 2014 Potential Vorticity In a Barotropic Fluid

For a barotropic, D ⇣ + f incompressible and h g =0 Dt h homogeneous fuid: 

Defnition: The barotropic potential ⇣ + f PV = g vorticity of a fuid column is defned as h

Barotropic potential vorticity is conserved following a fuid column and measures the absolute vorticity relative to the depth of the vortex.

Paul Ullrich A Rotational View of the Atmosphere March 2014 PV Conservation In a Barotropic Fluid

Question: What happens when the vortex meets a hill?

Surface with a Hill

Paul Ullrich A Rotational View of the Atmosphere March 2014 PV Conservation In a Barotropic Fluid

⇣g + f PV = = Constant h

If a fuid column remains at a constant latitude, relative vorticity must change with the depth of the fuid.

Paul Ullrich A Rotational View of the Atmosphere March 2014 Vorticity and Depth

Potential vorticity provides a link between depth and vorticity. As the depth of the vortex changes, the ⇣g + f relative vorticity has to change in PV = = Constant order to conserve potential vorticity. h

This result links: • The rotational component of the wind, which is responsible for rotation in the horizontal plane. • The irrotational component, which is responsible for divergence / convergence and hence changes in the height of the fuid column.

Paul Ullrich A Rotational View of the Atmosphere March 2014 Vorticity and Depth

Potential vorticity indicates an inerplay between relative and planetary vorticity through conservation of absolute angular momentum.

⇣g + f PV = = Constant h

Paul Ullrich A Rotational View of the Atmosphere March 2014 Vorticity

Relative Vorticity

Planetary Vorticity

Paul Ullrich A Rotational View of the Atmosphere March 2014 The Vorticity Equation

@⇣ @u @v + u ⇣ +(⇣ + f) + @t · r @x @y ✓ ◆ @w @v @w @u @f @ 1 @p @ 1 @p + + v = + @x @z @y @z @y @x ⇢ @y @y ⇢ @x ✓ ◆ ✓ ◆ ✓ ◆

Changes in absolute Advection Divergence vorticity are caused by: Tilting Baroclinicity

Paul Ullrich A Rotational View of the Atmosphere March 2014 Scale Analysis

@v @u U 5 1 Relative Vorticity: ⇣ = 10 s @x @y ⇡ L ⇡

4 1 Planetary Vorticity: f 10 s 0 ⇡

⇣ 1 In the mid-latitudes planetary vorticity is 10 f0 ⇡ generally larger than relative vorticity.

Paul Ullrich A Rotational View of the Atmosphere March 2014 Relative / Planetary Vorticity

Observation 1: Planetary vorticity is cyclonic / positive vorticity in the Northern Hemisphere.

Observation 2: Planetary vorticity, in the mid-latitudes, is usually larger than relative vorticity.

A growing cyclone “adds to” planetary vorticity, leading to intense lows.

A growing anti-cyclone “opposes” planetary vorticity, leading to weak highs.

Paul Ullrich A Rotational View of the Atmosphere March 2014 PV Conservation In a Barotropic Fluid

⇣g + f PV = = Constant h

Figure: Stretching and shrinking of a column will change the relative vorticity.

Recall: Constant density implies a fuid does not change in volume.

Paul Ullrich A Rotational View of the Atmosphere March 2014