1. EART30351
Lecture 7

2. Mass continuity equation 1
Geometric coordinates
Mass of element ΔM = ρΔxΔyΔz 𝑑∆𝑀 𝑑𝑡 = 𝑑𝜌 𝑑𝑡 ∆𝑥∆𝑦∆𝑧+ 𝜌 𝑑∆𝑥 𝑑𝑡 ∆𝑦∆𝑧+𝜌 𝑑∆𝑦 𝑑𝑡 ∆𝑥∆𝑧 + 𝜌 𝑑∆𝑧 𝑑𝑡 ∆𝑦∆𝑥 But 𝑑 𝑑𝑡 ∆𝑥 =𝑢 𝑥+∆𝑥 −𝑢 𝑥 = 𝜕𝑢 𝜕𝑥 ∆𝑥 Δz
x+Δx x Δy Δx

3. Mass continuity equation 1
Geometric coordinates
Mass of element ΔM = ρΔxΔyΔz 𝑑∆𝑀 𝑑𝑡 = 𝑑𝜌 𝑑𝑡 ∆𝑥∆𝑦∆𝑧+ 𝜌 𝑑∆𝑥 𝑑𝑡 ∆𝑦∆𝑧+𝜌 𝑑∆𝑦 𝑑𝑡 ∆𝑥∆𝑧 + 𝜌 𝑑∆𝑧 𝑑𝑡 ∆𝑦∆𝑥 But 𝑑 𝑑𝑡 ∆𝑥 =𝑢 𝑥+∆𝑥 −𝑢 𝑥 = 𝜕𝑢 𝜕𝑥 ∆𝑥
𝑑∆𝑀 𝑑𝑡 = 𝑑𝜌 𝑑𝑡 ∆𝑥∆𝑦∆𝑧+ 𝜌 𝜕𝑢 𝑑𝑥 ∆𝑥∆𝑦∆𝑧+…
1 ∆𝑀 𝑑∆𝑀 𝑑𝑡 = 1 𝜌 𝑑𝜌 𝑑𝑡 +𝛻.𝑽=0
by conservation of mass. So:
1 𝜌 𝑑𝜌 𝑑𝑡 +𝛻.𝑽 = 0 Δz
x+Δx x Δy Δx

4. Mass continuity equation 2
Pressure coordinates
Mass of element
ΔM = g-1ΔxΔyΔp
since
𝜕𝑝 𝜕𝑧 =−𝜌𝑔
𝑑∆𝑀 𝑑𝑡 = 1 𝑔 𝑑∆𝑥 𝑑𝑡 ∆𝑦∆𝑝+ 1 𝑔 𝑑∆𝑦 𝑑𝑡 ∆𝑥∆𝑝 + 1 𝑔 𝑑∆𝑝 𝑑𝑡 ∆𝑦∆𝑥
But
𝑑 𝑑𝑡 ∆𝑥 =𝑢 𝑥+∆𝑥 −𝑢 𝑥 = 𝜕𝑢 𝜕𝑥 ∆𝑥 Δp
x+Δx x Δy Δx

5. Mass continuity equation 2
Pressure coordinates
Mass of element
ΔM = g-1ΔxΔyΔp
since
𝜕𝑝 𝜕𝑧 =−𝜌𝑔
𝑑∆𝑀 𝑑𝑡 = 1 𝑔 𝑑∆𝑥 𝑑𝑡 ∆𝑦∆𝑝+ 1 𝑔 𝑑∆𝑦 𝑑𝑡 ∆𝑥∆𝑝 + 1 𝑔 𝑑∆𝑝 𝑑𝑡 ∆𝑦∆𝑥
But
𝑑 𝑑𝑡 ∆𝑥 =𝑢 𝑥+∆𝑥 −𝑢 𝑥 = 𝜕𝑢 𝜕𝑥 ∆𝑥
𝑑∆𝑀 𝑑𝑡 = 1 𝑔 𝜕𝑢 𝑑𝑥 ∆𝑥∆𝑦∆𝑝+…
1 ∆𝑀 𝑑∆𝑀 𝑑𝑡 =𝛻.𝑽=0
by conservation of mass. So:
𝛻.𝑽 = 0
Under the hydrostatic assumption, air behaves like an incompressible fluid!
(compressibility of air only important on small scales – e.g. sound waves) Δp
x+Δx x Δy Δx

6. Convergence and Divergence
Since .V  0, the term ‘divergence’ is used for the horizontal components only:
𝛻 𝐻 .𝑼=− 𝜕𝜔 𝜕𝑝
= dp/dt is the vertical velocity in pressure coordinates. Roughly:
≈ -gρw
Patterns of (horizontal) convergence and divergence determine vertical motion

7. Convergence and Divergence
Since .V  0, the term ‘divergence’ is used for the horizontal components only:
𝛻 𝐻 .𝑼=− 𝜕𝜔 𝜕𝑝
= dp/dt is the vertical velocity in pressure coordinates. Roughly:
≈ -gρw
Patterns of (horizontal) convergence and divergence determine vertical motion
Weather systems are vertically coherent (300 and 700 m charts qualitatively similar). So, the sign of w and  doesn’t change in the vertical At the ground, w, = 0 At the tropopause w, ≈ 0 since stratosphere restricts vertical motion

8. Convergence-Divergence Dipoles
Positive  (downward motion)
Negative  (upward motion)
Trop
Trop
∂/∂p < 0
Divergence
Convergence
∂/∂p > 0
Height
Pressure
Height
Pressure
∂/∂p > 0
Convergence
Divergence
∂/∂p < 0  
Convergence aloft means divergence below and vice versa

9. Divergence of the Geostrophic wind
𝑼 𝐺 = 1 𝑓 𝒌×𝛁Φ= 1 𝑓 − 𝜕Φ 𝜕𝑦 , 𝜕Φ 𝜕𝑥 ,0
Neglecting variation of f with latitude:
𝛻. 𝑼 𝐺 = 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 =0
(𝛻. 𝑈 𝐺 = 1 𝑓 𝛻. 𝒌×𝛁Φ − 1 𝑓 2 𝜕𝑓 𝜕𝑦 𝒌×𝛁Φ
=− 𝑈 𝐺 𝑓 𝜕𝑓 𝜕𝑦 =- 𝑈 𝐺 𝑓𝐴 𝜕𝑓 𝜕𝜆 =− 𝑈 𝐺 𝐴 𝑐𝑜𝑡𝜆, ~5x10-6 s-1.
Actual values around jet stream ~ 3 x 10-5 s-1)

10. Ageostrophic wind
Decompose U into: U = UG + UA Geostrophic and ageostrophic wind. Divergence/convergence therefore depend mainly on departures from geostrophy – i.e. UA

11. Ageostrophic wind
Decompose U into: U = UG + UA Geostrophic and ageostrophic wind. Divergence/convergence therefore depend mainly on departures from geostrophy – i.e. UA
From the momentum equation: 𝑑𝑼 𝑑𝑡 =−𝛁Φ−𝑓𝒌×𝑼 By the definition of UG : 𝑑𝑼 𝑑𝑡 =−𝑓𝒌× 𝑼 𝑨 𝑼 𝑨 = 1 𝑓 𝒌× 𝑑𝑼 𝑑𝑡 Ageostrophic wind is proportional to acceleration

12. Ageostrophic wind around jet stream
Acceleration is greatest where wind is greatest, i.e. jet stream UA is perpendicular to the acceleration (x product) UA
Jet stream accelerating from left to right (jet entrance)
accn

13. Convergence around jet streak
Undisturbed flow, dU/dt = 0 C UA D
dU/dt
Jet
dU/dt UA D C
Undisturbed flow, dU/dt = 0
Ageostrophic wind at jet entrance (LHS) points poleward. This means air ‘piles up’ on the poleward side – convergence. As this is near the tropopause it tends to force downward motion – and by symmetry we can see a quadrupole pattern

14. Dines Compensation
Tropopause C D C D C D
Ground
Schematic of synoptic-scale atmospheric circulation – overturning cells (showing w rather than  vertical velocity)
Areas of vertical motion related to vertical dipoles of convergence and divergence

15. Relation to flow
Tropopause J J J H L H
Ground
Convergence and divergence is greatest at jet stream level: jet stream essentially drives the circulation cells.
Convergence aloft => More mass in the column of air => High pressure at surface
Divergence aloft => Less mass in the column of air => Low pressure at surface