EART30351 Lecture 7

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

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

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

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

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

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

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

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)

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

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

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

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

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

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