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Vertical stability - Institut für Umweltphysik

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Physik der Atmosphäre Atmosph re I<br />

WS 2008/09<br />

Ulrich Platt<br />

<strong>Institut</strong> f. <strong>Umweltphysik</strong><br />

R. 424<br />

Ulrich.Platt@iup.uni-heidelberg.de<br />

Ulrich.Platt@iup.uni heidelberg.de<br />

Physik der Atmosphäre I


Last week<br />

• Atmosphere crucial for live<br />

• Atmosphere is a complex and non-linear<br />

system, interacting with the other geophysical<br />

compartments (ocean, land, ice sheet…)<br />

• The primary source of energy is solar radiation<br />

• Strongest variability (T, p, …) is in the vertical<br />

• Large range of spatial and temporal scales<br />

• Hydrostatic equation:<br />

p<br />

= p0e<br />

Physik der Atmosphäre I<br />

−<br />

z<br />

H


Contents<br />

7.10.08 Introduction – Literature - <strong>Vertical</strong> structure of the atmosphere<br />

14.10.08 Adiabatic processes - <strong>Vertical</strong> <strong>stability</strong><br />

21.10.08 Atmospheric radiation: Absorption, scattering, emission<br />

28.10.08 Atmospheric radiation: The energy budget of the atmosphere<br />

4.11.08 Atmospheric dynamics: Navier-Stokes equation, continuity equation<br />

Atmospheric dynamics: Pressure gradients and wind fields, thermal<br />

11.11.08<br />

wind<br />

18.11.08 Atmospheric dynamics: Vorticity<br />

25.11.08 Atmospheric dynamics: The planetary boundary layer<br />

2.12.08 Atmospheric circulation: Global circulation patterns, planetary waves<br />

9.12.08 Atmospheric circulation: Pressure systems, Hadley cell<br />

13.12.08 Diffusion and turbulence: Molecular diffusion, basics of turbulence<br />

13.01.09 Diffusion and turbulence: Theorem of Taylor, correlated fluctuations<br />

20.01.09 Diffusion and turbulence: Diffusion of scalar tracers<br />

27.01.09 Near-surface dynamics: Wind profile, influence of surface friction<br />

Physik der Atmosphäre I


Outline for Today<br />

1. Basic thermodynamic definitions<br />

2. Dry-adiabatic lapse rate<br />

3. Moist-adiabatic lapse rate<br />

4. The potential temperature<br />

5. The equivalent temperature<br />

6. <strong>Vertical</strong> <strong>stability</strong><br />

7. Buoyancy oscillations<br />

Physik der Atmosphäre I


The vertical structure of the atmosphere<br />

Physik der Atmosphäre I


How to determine the<br />

tropospheric temperature profile<br />

1. Atmosphere is heated by the Earth‘s surface<br />

2. Cooling/heating rates of the air by emission/absorption of<br />

radiation are much smaller than typical transport times<br />

3. Transport processes are (nearly) adiabatic<br />

4. The condensation of water vapour is an additional source<br />

of heat<br />

5. To determine the tropospheric temperature profile, we<br />

need<br />

a. the ideal gas law<br />

b. the first law of thermodynamics<br />

c. the hydrostatic equation<br />

Physik der Atmosphäre I


Ideal gas equation<br />

• Ideal gas:<br />

– molecules have no volume<br />

– no collisions between gas molecules<br />

– error for assuming that atmospheric gases are ideal gases is small<br />

• Ideal gas equation<br />

– p: pressure<br />

– V: volume<br />

– n: amount of gas in moles<br />

– T: absolute temperature<br />

– R: gas constant (8.314 J/mol K)<br />

pV= nRT<br />

n<br />

p= RT= ρ RT<br />

V<br />

• Extensive properties:<br />

– proportional to size of system, e.g. n, V<br />

– „2x size 2x property“<br />

• Intensive property:<br />

– independent of mass: p,T, ρ<br />

– „intensify“ extensive property by dividing by mass or volume “specific property”, e.g. ρ = m/V<br />

Physik der Atmosphäre I


First law of thermodynamics<br />

• First law (1. Hauptsatz):<br />

dU = dQ +<br />

dW<br />

• dQ – heat added to the system<br />

• dW – work done ON the system<br />

• dU – change in internal energy<br />

• Work done on system: compression of air<br />

dW =−p⋅dV • Internal energy of an ideal gas is independent of<br />

volume and proportional to temperature<br />

dU = Cv<br />

dT<br />

• Cv – heat capacity (at constant volume)<br />

⇒ dQ = dU − dW = pdV + CvdT Physik der Atmosphäre I


First law of thermodynamics (contd contd.) .)<br />

1. First law (1. Hauptsatz):<br />

dQ= pdV+ C dT⇒ pdV= dQ−CdT 2. Ideal gas law:<br />

p V = n RT<br />

⇒ pdV<br />

+ V dp =<br />

⇒<br />

dQ V dp nR<br />

Cv<br />

dT ) ( + + − =<br />

• Convert to intensive<br />

quantities by dividing by n:<br />

V<br />

= − dp + ( R + cv<br />

) dT<br />

n<br />

– q = Q/n: heat per mole<br />

– c v = C v /n: specific heat capacity<br />

– c p = c v + R: specific heat capacity at<br />

constant pressure<br />

v v<br />

RT<br />

p<br />

nRdT<br />

dp<br />

dq p<br />

=<br />

−<br />

Physik der Atmosphäre I<br />

+<br />

c<br />

dT


Dry-adiabatic<br />

Dry adiabatic lapse rate<br />

1. First law (1. Hauptsatz):<br />

2. Hydrostatic equation:<br />

⇒<br />

=<br />

−<br />

RT<br />

p<br />

dp<br />

+<br />

c<br />

dq p<br />

p<br />

=<br />

p<br />

0<br />

e<br />

M g<br />

− z<br />

RT<br />

⇔<br />

dp<br />

p<br />

dT<br />

=<br />

−<br />

dq = M gdz<br />

+ c pdT<br />

M g<br />

RT<br />

• Movement of air parcel is assumed to be adiabatic, i.e. no<br />

heat is exchanged with the environment: dq = 0. This is<br />

justified since cooling rates in the troposphere are much<br />

smaller than typical transport times.<br />

⇒<br />

dT<br />

c p<br />

=<br />

−M<br />

gdz<br />

dz<br />

Physik der Atmosphäre I


Dry-adiabatic<br />

Dry adiabatic lapse rate (contd ( contd.) .)<br />

• Temperature gradient in the atmosphere:<br />

dT<br />

c p<br />

= −M<br />

gdz<br />

⇒<br />

Values for air:<br />

– c p = 28.97 J/(K mole)<br />

– M = 28.97 g/mole<br />

– g = 9.81 m/s 2<br />

• Dry-adiabatic lapse rate:<br />

⎛ dT ⎞<br />

Γ ≡ −⎜<br />

⎟<br />

⎝ dz ⎠<br />

adiabatic<br />

K<br />

= 0. 00981 ≈ 1K<br />

m<br />

– The temperature gradient is given by Γ only if no additional heat<br />

sources (condensation, absorption of radiation) are present (dq=0)<br />

• Actual temperature gradient present in the atmosphere:<br />

γ ≡<br />

⎛<br />

−⎜<br />

⎝<br />

dT ⎞<br />

⎟⎠<br />

dz<br />

actual<br />

dT<br />

dz<br />

=<br />

−<br />

M g<br />

c p<br />

Physik der Atmosphäre I<br />

100m


Atmospheric Energy<br />

• Kinetic energy (per volume):<br />

– E=½ ρ air v 2 , ρ air = 1.293 kg m -3 , v = 10 ms -1<br />

E = 65 J m -3<br />

• Sensible heat (per volume):<br />

– Q s = ρ air c p ∆T, c p = 10 3 J kg -1 K -1 , ∆T = 10K<br />

Q s = 1.3x10 4 J m -3<br />

• Latent heat (per volume):<br />

– Q L = ρ air L q, L = 2.256x10 6 J kg -1 , q = 10 g kg -1<br />

Q L = 2.9x10 4 J m -3<br />

Phase transfer processes (e.g. convection, cloud<br />

formation) play major roles in energy balance<br />

Physik der Atmosphäre I


• Vapour pressure [hPa]:<br />

Moisture parameters<br />

– partial pressure of H 2 O vapour: e<br />

– p=Σ i p i =p dry +e<br />

– assume H 2 O vapour to be an ideal gas: e = ρ v R v T<br />

• Saturation vapour pressure [hPa]: e*<br />

• Absolute humidity [g m -3 ]:<br />

a = ρ v = m v / V<br />

• Specific humidity [kg/kg=1]:<br />

q = ρ v / ρ air , with ρ air density of moist air<br />

• Relative humidity [1]:<br />

f = e / e*<br />

• Dew point: T @ e = e* (p=const)<br />

Physik der Atmosphäre I


The Moist-Adiabatic<br />

Moist Adiabatic Lapse Rate<br />

• Condensation of water vapour releases heat, i.e. dQ ≠ 0:<br />

dQ = −Ldmw<br />

– L ≈ 2250 J/g: specific evaporation heat<br />

– dm w : change in mass of gaseous water vapour<br />

• Convert to intensive quantities by dividing by n:<br />

dq<br />

= −<br />

L<br />

n<br />

dm<br />

w<br />

V RT<br />

= −L<br />

dρ<br />

w = −L<br />

dρw<br />

= −L<br />

n p<br />

– ρ w = m w /V is the saturation water vapour concentration<br />

• Re-ordering yields the moist-adiabatic lapse rate:<br />

w<br />

p<br />

Physik der pAtmosphäre dT I<br />

RT<br />

p<br />

dρw<br />

dT<br />

RT<br />

dρw<br />

dq = M gdz<br />

+ c p dT ⇒ −L<br />

dT = c p dT + M gdz<br />

p dT<br />

⎛ dρ<br />

⎞<br />

R T<br />

⇒ ⎜− − ⎟ =<br />

⎝ p dT ⎠<br />

w L cpdT Mgdz dT Mg<br />

=−<br />

dz RLT dρ<br />

c + ⋅<br />

dT


Saturation vapour pressure<br />

Magnus formula for e* [hPa]:<br />

e water<br />

e ice<br />

* =<br />

* =<br />

6.<br />

11hPa<br />

6.<br />

11hPa<br />

⎛ 17.<br />

1(<br />

T − 273K)<br />

⋅exp⎜<br />

⎝ 235 + ( T − 273K)<br />

⎛ 22.<br />

4(<br />

T − 273K)<br />

⋅exp⎜<br />

⎝ 272 + ( T − 273K)<br />

rule of thumb:<br />

∆T=10K ∆e ≈ 2 e<br />

Bergeron-Findeisen process<br />

⎞<br />

⎟<br />

⎠<br />

⎞<br />

⎟<br />

⎠<br />

Physik der Atmosphäre I<br />

Wallace and Hobbs, 2006


Water vapour saturation curve<br />

Water vapour partial pressure E (right axis) and water vapour density ρ w (left axis) over<br />

liquid water as a function of temperature. Adapted from Roedel (1992).<br />

Physik der Atmosphäre I


Moist-adiabatic<br />

Moist adiabatic lapse rate<br />

Moist-adiabatic lapse rate as a function of air temperature, with pressure<br />

as parameter (Roedel, 1994).<br />

Physik der Atmosphäre I


The (Alpine) Föhn F hn<br />

Inflow of moist air cooling, precipitation heating outflow of warm, dry air<br />

Source: Roedel<br />

Physik der Atmosphäre I


Potential Temperature<br />

Potential temperature Θ = temperature of an air parcel<br />

if it would be adiabatically compressed to a pressure of 1013 mBar<br />

• 1st law of thermodynamics:<br />

RT<br />

( c p − cv<br />

) T<br />

= − dp + c p dT = − dp + c<br />

p<br />

p<br />

• dq = 0 (adiabatic process) yields:<br />

dp dT<br />

( c p − cv<br />

) = c p<br />

p T<br />

• Integration yields:<br />

dq p<br />

T<br />

p<br />

T<br />

0<br />

− 1 −1<br />

0<br />

= κ k<br />

κ κ<br />

p<br />

with κ = c p /c v and the surface pressure p 0 = 1013 mBar.<br />

• Θ ≡ T 0 is the temperature of the air parcel after compression from p to p 0 :<br />

Θ<br />

=<br />

⎛<br />

T<br />

⎜<br />

⎝<br />

p<br />

p<br />

0<br />

⎞<br />

⎟<br />

⎠<br />

κ −1<br />

κ<br />

dT<br />

with (κ-1)/κ ≈ 0.286 in air<br />

Physik der Atmosphäre I


Höhe (Km)<br />

Strahlungsheizung durch O 2 (UV)<br />

Strahlungskühlung durch H 2 O & CO 2<br />

Strahlungsheizung durch O 3 (UV)<br />

Θ=800K<br />

Strahlungskühlung (IR) durch H2O & CO2 Konvektion<br />

Θ=1900K<br />

Temperatur (K)<br />

Θ=340K<br />

Physik der Atmosphäre I<br />

Druck (mBar)<br />

Homosphäre


Equivalent Temperature<br />

Equivalent temperature T eq = temperature of a moist air parcel<br />

if all water vapour would condense<br />

• Release of latent heat per volume...<br />

∆<br />

q v<br />

= ρ L<br />

• .. leads to an increase in internal energy:<br />

u air c p T ∆ = ∆ ρ<br />

• ∆u = ∆q Temperature increase:<br />

∆<br />

ρ L<br />

ρ c<br />

v T = =<br />

air<br />

p<br />

q<br />

L<br />

c<br />

p<br />

with water vapour mixing ratio q = ρ v /ρ air<br />

• Equivalent and equivalent-potential temperature:<br />

T = T + q<br />

eq<br />

L<br />

c<br />

p<br />

Θ<br />

eq<br />

= Θ + q<br />

L<br />

c<br />

p<br />

Physik der Atmosphäre I<br />

Example:<br />

• L ≈ 2250 J/g<br />

• c p ≈ 1 J/(g K)<br />

• T = 15°C<br />

• Specific humidity: 70%,<br />

corresponding to q ≈ 12.4⋅10 -3<br />

∆T ≈ 31 K<br />

Equivalent temperature T eq ≈ 46°C


Potential temperature and vertical <strong>stability</strong><br />

• From the hydrostatic equation, we have<br />

p<br />

=<br />

p<br />

• Total differential of the potential temperature:<br />

dΘ<br />

=<br />

Θ<br />

0<br />

e<br />

dT<br />

T<br />

M g<br />

− z<br />

RT<br />

• Thus, with<br />

−<br />

R<br />

c<br />

⇔<br />

p<br />

dp<br />

p<br />

dp<br />

p<br />

=<br />

= −<br />

dT<br />

T<br />

dΘ<br />

Θ ⎛ dT ⎞ Θ<br />

= ⎜ + Γ⎟<br />

=<br />

dz T ⎝ dz ⎠ T<br />

M g<br />

RT<br />

M g<br />

+ dz<br />

c T<br />

Γ = ( g)<br />

( c T )<br />

M p<br />

p<br />

dz<br />

( Γ −γ<br />

)<br />

where γ = − dT dz is the actual temperature gradient.<br />

Physik der Atmosphäre I


<strong>Vertical</strong> <strong>stability</strong><br />

• γ = Γ, dΘ/dz = 0: neutral<br />

• γ < Γ, dΘ/dz > 0: stable<br />

• γ > Γ, dΘ/dz < 0: unstable<br />

Physik der Atmosphäre I


z<br />

z 2<br />

z 1<br />

z<br />

z 2<br />

z 1<br />

dΘ<br />

><br />

dz<br />

T(z)<br />

Potential Temperature and <strong>Vertical</strong> Stability<br />

0 Stabile Schichtung<br />

F<br />

F<br />

Θ P<br />

Θ P<br />

F<br />

F<br />

Θ(z)<br />

∆z<br />

Θ<br />

∆z<br />

Physik der Atmosphäre I<br />

Θ<br />

Auslenkung eines Luftpakets nach<br />

oben (von z 2 nach z 3 ):<br />

Θ P =const.<br />

Θ P < Θ(z 3 )<br />

Dichte des Luftpaketes größer<br />

als Dichte der umgebenden Luft<br />

Rücktreibende Kraft<br />

dΘ<br />

<<br />

dz<br />

0 Labile Schichtung<br />

Auslenkung eines Luftpakets nach oben<br />

(von z2 nach z3 ):<br />

ΘP =const.<br />

ΘP > Θ(z3 )<br />

Dichte des Luftpaketes geringer als<br />

Dichte der umgebenden Luft<br />

Resultierende Kraft nach oben, treibt<br />

Luftpaket weiter von der<br />

ursprünglichen Lage weg


Dry and moist <strong>stability</strong><br />

Physik der Atmosphäre I


Stüve St ve diagram (1)<br />

http://www.csun.edu/~hmc60533/CSUN_103/weather_exercises/soundings/smog_and_inversions/Understanding%20Stuve_v3.htm<br />

Physik der Atmosphäre I


Isobars<br />

Stüve St ve diagram (2)<br />

Air temperature<br />

Inversion<br />

Wind speed<br />

and direction<br />

Temperature [K]<br />

Pressure Altitude Temperature [°C]<br />

Isotherms<br />

Physik der Atmosphäre I


Stüve St ve diagram (3)<br />

Dew point temperature<br />

Saturation water vapour mixing ratio [g water/kg air]<br />

Physik der Atmosphäre I


Dry<br />

lapse rate<br />

Stüve St ve diagram (4)<br />

Condensation<br />

starts<br />

Physik der Atmosphäre I<br />

Yellow line:<br />

Temperature of an<br />

air parcel lifted up<br />

adiabatically from<br />

950 mBar.<br />

Initial water vapour<br />

content: 8g/kg<br />

(inferred from dew<br />

point temperature).<br />

Air parcel is<br />

undersaturated and<br />

thus follows the dry<br />

adiabatic<br />

temperature gradient<br />

(~1K/100m)<br />

Condensation starts<br />

at 800mBar, where<br />

saturation water<br />

vapour mixing ratio<br />

equals water vapour<br />

content.


Dry<br />

lapse rate<br />

Stüve St ve diagram (5)<br />

Moist<br />

lapse rate<br />

Temperature of rising air parcel is always below actual temperature<br />

Stable conditions<br />

Physik der Atmosphäre I<br />

Yellow line:<br />

Temperature of an<br />

air parcel lifted up<br />

adiabatically from<br />

950 mBar.<br />

Initial water vapour<br />

content: 8g/kg<br />

(inferred from dew<br />

point temperature).<br />

Air parcel is<br />

undersaturated and<br />

thus follows the dry<br />

adiabatic<br />

temperature gradient<br />

(~1K/100m)<br />

Condensation starts<br />

at 800mBar, where<br />

saturation water<br />

vapour mixing ratio<br />

equals water vapour<br />

content.<br />

Now the air parcel<br />

follows the moist<br />

adiabat due to<br />

release of latent<br />

heat.


Stüve St ve diagram (6)<br />

Temperature of rising air parcel is above actual temperature<br />

Unstable conditions, convection<br />

Physik der Atmosphäre I<br />

Yellow line:<br />

Temperature of an<br />

air parcel lifted up<br />

adiabatically from<br />

950 mBar.<br />

Dew point<br />

temperature is equal<br />

to actual temperature<br />

up to 750 mBar <br />

condensation,<br />

formation of clouds.


Brunt-Väis Brunt isälä or buoyancy oscillations<br />

How can atmospheric <strong>stability</strong> be defined quantitatively?<br />

• Equation of motion for the buoyancy of an air parcel:<br />

dv z<br />

( )<br />

ρ −<br />

dt<br />

= g ρ * ρ<br />

ρ: density of air parcel; ρ*: density of surrounding air<br />

• ρ proportional to 1/T:<br />

dvz dt<br />

*<br />

1 T −1<br />

T T −T<br />

= g = g *<br />

1 T T<br />

• Adiabatic and actual lapse rate:<br />

*<br />

T ( z)<br />

= T0<br />

− Γ∆z<br />

T ( z)<br />

= T0<br />

− γ∆z<br />

• It follows:<br />

dvz γ − Γ<br />

= g ∆z<br />

*<br />

dt T<br />

• Potential temperature:<br />

dΘ Θ dvz<br />

g dΘ<br />

= ( Γ − γ ) ⇒ = − ∆z<br />

*<br />

dz T<br />

dt Θ dz<br />

*<br />

Physik der Atmosphäre I


Brunt-Väis Brunt isälä or buoyancy oscillations<br />

How can atmospheric <strong>stability</strong> be defined quantitatively?<br />

• Equation of motion for the buoyancy of an air parcel:<br />

dvz 2<br />

ρ = −ρ<br />

B ∆z<br />

dt<br />

with<br />

2 g dΘ<br />

B =<br />

Θ dz<br />

• This is the equation of an harmonic oscillator with the<br />

Brunt-Väisälä frequency<br />

f<br />

B 1<br />

= =<br />

2π<br />

2π<br />

g dΘ<br />

Θ dz<br />

• For dΘ/dz > 0 (stable conditions), this leads to buoyancy oscillations.<br />

• Typical periodic times:<br />

– 15 minutes for γ = 0.8 (stable conditions)<br />

– 5 minutes for γ = 0 (isotherm stratification)<br />

Physik der Atmosphäre I


Lee waves<br />

Buoyancy oscillations downwind of mountains<br />

„Lenticular clouds“<br />

Physik der Atmosphäre I<br />

Formation of lee waves


Lee waves<br />

… can propagate up to the stratosphere<br />

Polar stratospheric clouds<br />

observed in Kiruna, northern<br />

Sweden.<br />

These clouds are formed in the<br />

ascending branch of the lee<br />

waves occuring downwind of<br />

the Scandinavian mountains.<br />

(Photo courtesy of C.F. Enell)<br />

Physik der Atmosphäre I


Summary<br />

• Rising air cools down due to adiabatic expansion<br />

– Dry-adiabatic lapse rate: Γ ≈ 1K/100m<br />

• Condensation of water vapour leads to the release of latent heat<br />

– Moist-adiabatic lapse rate: (0.5 – 1)K/100m<br />

• Potential temperature Θ = temperature of air parcel, if compressed<br />

adiabatically to 1013 mBar<br />

• <strong>Vertical</strong> <strong>stability</strong> of the atmosphere is determined by the actual<br />

vertical temperature gradient compared to the dry/moist lapse rate:<br />

– dΘ/dz = 0: neutral<br />

– dΘ/dz > 0: stable<br />

– dΘ/dz < 0: unstable<br />

• Buoyancy oscillations (Brunt-Väisälä oscillations) can occur for stable<br />

conditions (dΘ/dz > 0)<br />

Formation of lee waves downwind of mountains<br />

Physik der Atmosphäre I

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