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Atmospheric Dynamics: lecture 2

Topics Some aspects of advection and the Coriolis-effect (1.7) Composition of the atmosphere (figure 1.6) Equation of state (1.8&1.9) Water vapour in the atmosphere (1.10) Potential temperature and Exner function (1.13) Buoyancy (1.4) Vertical accelerations and instability > convection (1.15) Latent heat release and conditional instability (1.16)

(a.j.vandelden@uu.nl) (http://www.phys.uu.nl/~nvdelden/dynmeteorology.htm)

Problem 1.2 (p.16); problem 1.6 (p. 44); problem 1.7 (p.45)

Topics for “BOX” (about 1000 words)

Moist adiabat and moist convective adjustment How to calculate the temperature of the lifting condensation level Convective inhibition Conservation of potential vorticity (Ertel’s theorem) Difference between confluence and convergence Storm tracks Cold front, warm front, occluded front, backbent front Polar stratospheric vortex Breaking planetary waves Sudden stratospheric warming North Atlantic Oscillation and Arctic Oscillation Seasonal cycle of the Hadley circulation The Inter-Tropical Convergence Zone (ITCZ) Brewer-Dobson circulation ENSO and Walker circulation Difference between a tornado and a tropical cyclone

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Examples of a “BOX”

Data for case study

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20092011-13

21092011-03

Tropical cyclone

21092011-00

COLD FRONT

☐ ☐

☐

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warm sector

below cloud band

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cold sector

The equations*

€

d v dt

= −α ∇ p − g ˆ k − 2

Ω × v + Fr

€

dρdt

= −ρ ∇ ⋅ v

€

Jdt = cvdT + pdα

€

pα = RT

momentum

mass

energy

state

*see geophysical fluid dynamics or “Holton” and sections 1.7 and 1.8 of lecture notes

Pressure gradient (1.5) Gravity (1.4) Coriolis (1.6&1.7) Friction (1.3)

Unknowns are:

€

v ,ρ,T , p

€

α ≡1ρ

eqs. 1.4a,b,c

eq. 1.7b

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Material derivative of a scalar

€

ddt

=∂∂t

+ v ⋅ ∇ =

∂∂t

+ u ∂∂x

+ v ∂∂y

+ w ∂∂z

Material derivative

Local derivative

“Advection” Non-linear!!!

Section 1.7

advection

eq. 1.6

Scalar is a function of x, y, z and t

€

∂∂t(...) = 0

cloud advection & stationary gravity waves

€

ddt(...) = 0

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cloud advection & stationary gravity waves

€

∂∂t(...) = 0

€

ddt(...) = 0

Material derivative of a vector

Additional terms due to curved coordinate system!!

Section 1.7

€

d v dt≡

dudt−

uvtanφa

+uwa

⎛ ⎝ ⎜

⎞ ⎠ ⎟ i + dv

dt+

u2tanφa

+vwa

⎛

⎝ ⎜

⎞

⎠ ⎟ j +

dwdt

−u2 +v2

a⎛

⎝ ⎜

⎞

⎠ ⎟ k

These terms are frequently neglected in theoretical analysis

eq. 1.5

(see geophysical fluid dynamics)

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Coriolis effect

€

Ω × v = wΩcosφ − vΩsinφ( )ˆ i + uΩsinφ( )ˆ j - uΩcosφ( ) ˆ k

€

2 Ω × v ≈ − 2Ωvsinφ( )ˆ i + 2Ωusinφ( )ˆ j ≡ − fvˆ i + fu j

From “scale analysis”:*

*w<<v and w<<u , see Holton, chapter 2 or geophysical fluid dynamics

f is the Coriolis Parameter

Section 1.7

Composition of the atmosphere

Annual and global average concentration of various constituents in the atmosphere of Earth, as function of height above the Earth’s surface. The concentration is expressed as a fraction of the total molecule number density. This fraction is proportional to the mixing ratio. F11 and F12 denote the chlorinated fluorocarbons Freon-11 and Freon-12. Note that the concentration of carbon dioxide is constant up to a height of 100 km, while the concentrtaion of water vapour decreases by several orders of magnitude in the lowest 20 km.

Figure 1.6

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Equation of state Here k is Boltzman’s constant (=1.381 10-23 J K-1). If the air is dry, R is the specific gas constant for dry air (=287 J K-1kg-1), while n is the molecular number density (in numbers per m3). If air is a mixture of dry air and water vapour, R is the specific gas "constant" for this mixture €

p = nkT

€

p = ρRT

€

q ≡ ρvρ

The water vapour concentration in the atmosphere is expressed in terms of either the fraction of the total number of molecules, or as the fraction of the mass density of air (specific humidity),

Section 1.8

Clausius Clapeyron

Equilibrium water vapour pressure as a function of temperature, according to the Clausius Clapeyron equation assuming Lv is constant (=2.5×106 J K-1)

€

∂pe∂T

=peLvRvT

2

Clausius-Clapeyron equation for the water vapour pressure, pe, which is in equilibrium with the liquid phase:

Lv and Rv are, respectively, the so-called latent heat of evaporation (2.5×106 J kg-1) and the gas constant for water vapour (461.5 J K-1 kg-1).

es=pe

Section 1.9

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Water cycle Precipitable water

http://www.ecmwf.int/research/era/ERA-40_Atlas/

Section 1.11

Hadley circulation The average meridional circulation in the tropics, called the Hadley circulation, is thought to be driven by latent heat release in large convective clouds in the ITCZ. The subsidence in the subtropics leads to warming of the air and a concomitant reduction of the relative humidity

ITCZ

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Zonal mean heating

http://www.ecmwf.int/research/era/ERA-40_Atlas/

J<0 J>0

Potential temperature, θ

€

θ ≡Tprefp

⎛

⎝ ⎜

⎞

⎠ ⎟

κ

€

Jdt = cvdT + pdα

€

pα = RT €

κ ≡ R /cp

€

dθdt

=JΠ

€

}

€

Π≡ cpppref

⎛

⎝ ⎜

⎞

⎠ ⎟

κ

Exner-function

If J=0 (adiabatic) θ is materially conserved!!

eq. 1.54

eq. 1.55

Section 1.13

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Equations in terms of potential temperature and Exner-function

€

d v dt

= −θ ∇ Π− g ˆ k − 2

Ω × v + Fr

€

dθdt

=JΠ

€

dΠdt

= −RΠcv

∇⋅ v + RJ

cvθ

problem 1.6

Three differential equations with three unknowns!

eq. 1.56

eq. 1.58

eq. 1.55

Archimedes principle & buoyancy

*see Holton, chapter 2

An element immersed in a fluid at rest experiences an upward thrust which is equal to the weight of the fluid displaced. If ρ0 is the density of the fluid and V1 is the volume of the object, the upward thrust is therefore equal to gρ0V1 . The net upward force, F (the so-called buoyancy force), on the object is equal to (gρ0V1-gρ1V1), where ρ1 is the density of the object. With the equation of state and some additional approximations we can derive that

€

F ≈ mgT0 −T1T0

Gravity is dynamically important if there are temperature differences

Section 1.4

problem 1.2 (15 minutes to do this problem)

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Acceleration under buoyancy

€

θ0 = θ *+dθ0dz

δz€

F = md2zdt2

≈ mgT1 −T0T0

≈ mgθ1 −θ0θ0

Potential temperature environment of air parcel:

Force on air parcel:

Air parcel has temperature

€

θ =θ *

Acceleration under buoyancy

€

θ0 = θ *+dθ0dz

δz€

F = md2zdt2

≈ mgT1 −T0T0

≈ mgθ1 −θ0θ0

Potential temperature environment of air parcel:

Force on air parcel:

€

θ1 −θ0θ0

=θ1 −θ *−

dθ0dz

δz

θ0=−dθ0dz

δz

θ0

Air parcel has temperature

Buoyant force is proportional to

€

d2δzdt2

= −gθ0

dθ0dz

δzTherefore

€

θ =θ *

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Stability of hydrostatic balance

€

d2δzdt2

= −gθ0

dθ0dz

δz ≡ −N 2δz

The solution:

€

N 2 ≡ gθ0

dθ0dz

€

δz = exp ±iNt( )

€

N 2 =gθ0

dθ0dz

< 0If Exponential growth instability

Brunt Väisälä-frequency, N

€

N 2 =gθ0

dθ0dz

> 0If oscillation stability

Brunt Väisälä frequency

€

N 2 ≡ gθ0

dθ0dz

Extra problem:

Demonstrate that the Brunt-Väisälä frequency is constant in an isothermal atmosphere.

What is the typical time-period of a buoyancy oscillation in the atmosphere?