A Rotational View of the Atmoshere Chapter 4
Paul A. Ullrich [email protected]
Part 2: The Vorticity Equation Vorticity: Key Questions
Question: How is positive / negative vorticity generated?
Question: How do we describe the time rate of change of vorticity?
Question: How do we describe conservation of vorticity? Is vorticity actually conserved following the fow?
Question: What is the role of the Earth’s rotation?
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation Start from the scaled horizontal momentum equation in z coordinates, no viscosity:
Du 1 @p = + fv Dt ⇢ @x Dv 1 @p = fu Dt ⇢ @y
Take derivatives on both sides and compute: @ Dv @ @v = + u v @ Dv @ Du @x Dt @x @t · r ✓ ◆ ✓ ◆ @x Dt @y Dt @ Du @ @u ✓ ◆ ✓ ◆ = + u u @y Dt @y @t · r ✓ ◆ ✓ ◆
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
@ @u @ @ 1 @p @ (u u)=+ (fv) @y @t @y · r @y ⇢ @x @y ✓ ◆ @ @v @ @ 1 @p @ + + (u v)= (fu) @x @t @x · r @x ⇢ @y @x ✓ ◆
A rather long deriva on
@⇣ @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 ✓ ◆ ✓ ◆ ✓ ◆
Paul Ullrich Introduction to Atmospheric Dynamics 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 ✓ ◆ ✓ ◆ ✓ ◆
Vorticity Equation
D @w @v @w @u (⇣ + f)= (⇣ + f)( u) ( ↵ p) k Dt rh · @x @z @y @z r ⇥r · ✓ ◆ 1 With specifc volume ↵ = ⇢
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
What are these terms?
D @w @v @w @u (⇣ + f)= (⇣ + f)( u) ( ↵ p) k Dt rh · @x @z @y @z r ⇥r · ✓ ◆
Tilting or twisting Divergence term Solenoidal term term
Rate of change of the absolute vorticity following the motion
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
Take a closer look at the divergence term…
(⇣ + f)( u) rh ·
Horizontal Absolute vorticity divergence
• Remember that divergence is related to vertical motion (column integrated divergence gives the local vertical velocity)
• We now see that it is also related to changes in vorticity…
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vertical Velocity Obtained from horizontal divergence Continuity Equation in @u @v @! + + =0 Pressure Coordinates @x @y @p ✓ ◆p ⇥u ⇥v d = + dp ⇥x ⇥y ✓ ◆p @ p=0 p=0 ⇥u ⇥v d = + dp ⇥x ⇥y Z @ ps Zps ✓ ◆p p=0 ⇥u ⇥v (p = 0) (p )= + dp s ⇥x ⇥y Zps ✓ ◆p ps (p )= ( u)dp s ⇥p · Zp=0 Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vorticity and Divergence
Vertical wind is related to the divergence of the horizontal wind.
(which requires an ageostrophic component to the wind)
Changes in vorticity are partially driven by divergence of the horizontal wind.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vorticity and Divergence
This fow feld is purely vortical (zero divergence)
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vorticity and Divergence
This fow feld is purely divergent (zero vorticity)
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vorticity and Divergence
Question: What is the effect of divergence on vorticity?
We can think about this like angular momentum
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 A Spinning Skater
Motion is in the (x,y) plane
Axis of rotation is in the vertical plane
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 A Spinning Skater
Motion is in the (x,y) plane Question: A skater that is spinning with a given radius then brings in her arms. What happens to her angular velocity?
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 A Spinning Skater
Recall: Angular momentum Motion is in the is a conserved quantity. Its (x,y) plane magnitude is given by L = rm! | |
Answer: Since her radius decreases, conservation of angular momentum says that her angular velocity must increase.
Question: How is this like divergence?
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Vorticity and Divergence
Question: What is the effect of divergence on vorticity?
Answer: A divergent fow must lead to a decrease in angular velocity. A convergent fow must lead to an increase in angular velocity.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation Divergence term only
D (⇣ + f)= (⇣ + f)( u) Dt rh ·
Horizontal Absolute vorticity divergence
Question: What happens to absolute vorticity when the fow is
(a) Divergent? ( h u) > 0 r · (b) Convergent? ( u) < 0 rh ·
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
What are these terms?
D @w @v @w @u (⇣ + f)= (⇣ + f)( u) ( ↵ p) k Dt rh · @x @z @y @z r ⇥r · ✓ ◆
Tilting or twisting Divergence term Solenoidal term term
Rate of change of the absolute vorticity following the motion
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting and Twisting
Recall how relative vorticity is obtained:
Relative velocity: u =(u, v, w)
@w @v @u @w @v @u 3D vorticity vector: u = , , r⇥ @y @z @z @x @x @y ✓ ◆ @v @u Relative vorticity: ⇣ = k ( u)= · r⇥ @x @y ✓ ◆
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting and Twisting
Tilting or twisting term @w @v @w @u @x @z @y @z ✓ ◆
Tilting represents the change in the vertical component of vorticity by tilting horizontal vorticity into the vertical.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting or twisting term @w @v @w @u @x @z @y @z ✓ ◆
z
Consider a long, thin cylinder of fuid aligned with the y-axis. y
x
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting or twisting term @w @v @w @u @x @z @y @z ✓ ◆ =0 > 0 z Initial axis of rotation Consider a long, thin cylinder of fuid aligned with the y-axis. y
The fow satisfes v=0 and is sheared in the vertical so that the vorticity vector is along the y axis. x
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting or twisting term @w @v @w @u @x @z @y @z ✓ ◆ =0 > 0 > 0 z
Consider a long, thin The fow also satisfes cylinder of fuid dw/dy>0 leading to an aligned with the y-axis. upward tilting of the y cylindrical tube The fow satisfes v=0 and is sheared in the vertical so that the vorticity vector is along the y axis. x
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting and Twisting
z
Tilting or twisting term y @w @v @w @u @x @z @y @z ✓ ◆ =0 > 0 > 0
x
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Tilting and Twisting
Rotation in the (y,z) plane. Vorticity vector points along the x axis.
As the wheel is turned there is a component of vorticity in the z plane.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
What are these terms?
D @w @v @w @u (⇣ + f)= (⇣ + f)( u) ( ↵ p) k Dt rh · @x @z @y @z r ⇥r · ✓ ◆
Tilting or twisting Divergence term Solenoidal term term
Rate of change of the absolute vorticity following the motion
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Solenoidal / Baroclinic Terms
@ 1 @p @ 1 @p @↵ @p @↵ @p + = + @x ⇢ @y @y ⇢ @x @x @y @y @x ✓ ◆ ✓ ◆
The solenoidal / baroclinic terms capture changes in vorticity due to the alignment of surfaces of constant density and pressure.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Solenoidal / Baroclinic Terms
Defnition: In a barotropic fuid density depends only on pressure.
By the ideal gas law, this implies that surfaces of constant density are surfaces of constant pressure are surfaces of constant temperature.
Defnition: In a baroclinic fuid density depends on pressure and temperature.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Solenoidal / Baroclinic Terms
Barotropic: surfaces of constant density parallel to surfaces of constant pressure
y ↵ p =0 p-2Δp r ⇥r
ρ-2Δρ p-Δp
ρ-Δρ
ρ p
A barotropic fuid does not x induce changes in vorticity
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Solenoidal / Baroclinic Terms
Baroclinic: surfaces of constant density intersect surfaces of constant pressure
y ↵ p =0 p-2Δp r ⇥r 6 ρ-2Δρ
p-Δp ρ-Δρ
ρ p
A baroclinic fuid induces x a change in vorticity.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Solenoidal / Baroclinic Terms
In the real atmosphere, the solenoidal terms are a primary driver of the development of low-level low pressure systems in the middle latitudes.
This term is also important in understanding circulation around fronts.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Aside: Baroclinic Fronts
Note that this case has to do with the development of horizontal vorticity (perpendicular to x-z plane) z p-2Δp ρ-2Δρ
p-Δp ρ-Δρ
ρ p
Cold air sinking Warm air rising x behind front ahead of front
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
What are these terms?
D @w @v @w @u (⇣ + f)= (⇣ + f)( u) ( ↵ p) k Dt rh · @x @z @y @z r ⇥r · ✓ ◆
Tilting or twisting Divergence term Solenoidal term term
Rate of change of the absolute vorticity following the motion
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Advection of Vorticity
D ⇥ ⇥ ⇥ ⇥ ( + f)= ( + f)+u ( + f)+v ( + f)+w ( + f) Dt ⇥t ⇥x ⇥y ⇥z
@f @f @f =0 =0 =0 @t @x @z
D @⇣ @f (⇣ + f)= + u ⇣ + v Dt @t · r @y Advection of Advection of relative vorticity planetary vorticity
Paul Ullrich Introduction to Atmospheric Dynamics 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 Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation Scale Analysis
Changes in relative vorticity are caused by:
– Divergence – Tilting – Gradients in density – Advection
Question: Which of these terms are the most important for large-scale fows?
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 Scale Analysis Typical scales associated with large-scale mid-latitude storm systems: U 10 m s 1 P 10 hPa = 1000 Pa ⇡ ⇡ 3 W 0.01 m s 1 ⇢ 1kgm ⇡ ⇡ 2 L 106 m ⇢/⇢ 10 ⇡ ⇡ 4 4 1 H 10 m f0 10 s ⇡ ⇡ 11 1 L/U 105 s = @f/@y 10 s ⇡ ⇡ a 107 m (Radius of Earth) ⇡ 2 g 10 m s (Gravity) ⇡ 5 2 1 ⌫ 10 m s (Kinematic Viscosity) ⇡
Paul Ullrich Analysis of the Dynamical Equations March 2014 Scale Analysis
@v @u U 5 1 Relative Vorticity: ⇣ = 10 s @x @y ⇡ L ⇡
4 1 Planetary Vorticity: f 10 s 0 ⇡
Defnition: The Rossby number of a fow is a ⇣ U dimensionless quantity which represents the Ro ratio of inertia to Coriolis force. f0 ⇡ f0L ⌘
⇣ 1 In the mid-latitudes planetary vorticity is 10 f0 ⇡ generally larger than relative vorticity.
Paul Ullrich Analysis of the Dynamical Equations March 2014 Scale Analysis
Time rate of change and horizontal advection of relative vorticity:
2 @⇣ @⇣ @⇣ U 10 2 ,u ,v 10 s @t @x @y ⇡ L2 ⇡
Vertical advection of relative vorticity:
@⇣ WU 11 2 w 10 s @z ⇡ HL ⇡
Paul Ullrich Analysis of the Dynamical Equations March 2014 Scale Analysis Remaining Terms Recall this is smaller than U/L
@u @v 10 2 Divergence term: (⇣ + f) + f ( u) 10 s @x @y ⇡ 0 rh · ⇡ ✓ ◆
@w @v @w @u WU 11 2 Tilting term: 10 s @x @z @y @z ⇡ HL ⇡ ✓ ◆ @f 10 2 Planetary vorticity advection: v U 10 s @y ⇡ ⇡
⇢ p 11 2 Solenoidal term: k ( ↵ p) 10 s · r ⇥r ⇡ ⇢2L2 ⇡
Paul Ullrich Analysis of the Dynamical Equations March 2014 The Vorticity Equation
“Large” terms (10-10 s-2)
@⇣ @⇣ @⇣ @f @u @v + u + v + v +(⇣ + f) + @t @x @y @y @x @y ✓ ◆ @⇣ @w @v @w @u @↵ @p @↵ @p = w + @z @x @z @y @z @x @y @y @x ✓ ◆
“Small” terms (10-11 s-2)
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation Scale Analysis
Divergence term dominates along with horizontal advec on and the local me rate of change of rela ve vor city.
Til ng term important where there is a large shear and strong horizontal gradient in the ver cal velocity (boundary layer, smaller scales).
Solenoidal term important where there are strong density (temperature) gradients that intersect lines of constant pressure (sea breeze, fronts).
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation Relative Vorticity, Planetary Vorticity and Divergence
f ⇣ f 0 10 10 0 100 ⇣ ⇡ h u ⇡ u ⇡ r · rh ·
The rota on of the Earth is about 10 mes larger than rela ve vor city and 100 mes larger than divergence.
Paul Ullrich Introduction to Atmospheric Dynamics March 2014 The Vorticity Equation
Retaining leading order terms…
@⇣ @⇣ @⇣ @f @u @v + u + v + v +(⇣ + f) + =0 @t @x @y @y @x @y ✓ ◆
Defnition: The horizontal material D @ @ @ h = + u + v derivative is defned by the equation Dt @t @x @y
D @u @v h (⇣ + f)= (⇣ + f) + Dt @x @y ✓ ◆
Paul Ullrich Introduction to Atmospheric Dynamics March 2014