Atmospheric Thermodynamics Outline - Universität … · N. K ampfer Atmospheric Thermodynamics Aim...

N. Kampfer

AtmosphericThermodynamics

Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure

Hydrostaticequilibrium

Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity

Saturation vaporpressure

Moist adiabatic lapserate

Clouds

Atmospheric Thermodynamics

N. Kampfer

Institute of Applied PhysicsUniversity of Bern

28. Feb. and 2. March 2012

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

OutlineAimGas law

Ideal gasvan der Waals gasPhase transitions

PressureHydrostatic equilibriumScale heightMixingColumn density

TemperatureLapse rateStability

CondensationHumiditySaturation vapor pressureMoist adiabatic lapse rateClouds

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Aim

A planetary atmosphere consists of different gases hold tothe planet by gravityThe laws of thermodynamics apply

I pressure structureI pressure as vertical coordinate→ some planets have no solid surface

I hydrostatic equilibrium for compressible gasI scale heightI column densityI mean free path

I temperature structureI lapse rateI stabilityI latent heat and condensation → cloudsI wet lapse rate

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Ideal gas lawThe ideal gas law is an equation of state, an equationrelating measurable thermodynamic quantities

pV = NkT

N = amount of particlesk = 1.381 · 10−23 J/K is Boltzmann’s constantn = N/V is the number density, particles per Volume

p = nkT

one mole contains NA = 6.022 · 1023 particleswith q moles of a substance N = qNA and the gas law gets

pV = qNAkT = qRT

where R = kNA

R = 8.314 J mol−1 K−1 resp.R = 8314 J kmol−1 K−1 is the universal gas constant

The mass of one mole of substance is called molar weight:Mwater = 18.016 kg/kmol Mair = 28.97 kg/kmol

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Ideal gas lawmass of q moles is m = qMdensity ρ can be expressed as

ρ =m

V=

qM

V=

Mp

RT

very often gas law is expressed as

pV =m

MRT = m

R

MT = mRGT

orp = ρRGT

RG is the gas constant for the gas under discussion!For the Earth:

I for dry air Rd = 287 JK−1 kg−1

I for water vapor Rv = 461 JK−1 kg−1

Don’t mix up RG and R !! RG = R/MIn the literature often R is written as R∗ and RG as R!

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Ideal gas law

Basic properties of planetary atmospheres

gases satisfiesPp!P ~total pressure!, we can use Dalton’sand ideal gas equations to obtainmC5«Pp /P, where «5mV /m is the ratio between the vapor molecular weight andthe mean atmospheric molecular weight~in this contextmCis also called the specific humidity!. If saturation occurs, thenthe saturated mass mixing ratio is given by

mS~T,P!5«PV~T!

P, ~8!

where PV(T) is the saturated vapor pressure. The relativehumidity is defined as f (T,P)5100PV /PV(T)5100(mC /mS). Another way to measure the abundance of agas mixture is by means of the volume mixing ratio~alsocalled the molar mixing ratio or mole fraction!

XC5VC

V5

Pp

P5

mC

«. ~9!

Note that in general for a mixture ofi gases with volumemixing ratiosXi , we have

m5(i

m iXi . ~10!

The slope of the vapor pressure curvePV(T) in Eq. ~8!marks the phase transition where two phases are in equilib-rium and is known as the Clausius–Clapeyron equation,6

dPV

dT5

L

T~V22V1!, ~11!

whereL is the latent heat of the phase transition~in J g21!and Vi51/r i is the specific volume~phase 15vapor phase,25liquid or ice phase!. In Fig. 2 we show the pressure-temperature phase diagram for water, which is a commonliquid in planetary atmospheres.

A useful approximation to Eq.~11! can be obtained underthe following assumptions: the temperature variations in thelatent heat can be neglected, the vapor is an ideal gas, and thespecific volume of the liquid or solid phases is neglected

compared to that of the vapor, that is,V22V1;Vvapor

;1/rV5RVT/P, with RV the specific gas constant for thevapor. Then Eq.~11! can be rewritten as

dPV

dT5

LPV

RVT2, ~12!

which is the approximate form of the Clausius–Clapeyronequation employed in introductory meteorologicaltextbooks.5,7 If we integrate Eq.~12! with L5const, we ob-tain

PV~T!5PV0 expFLS 1

RVT02

1

RVTD G , ~13!

where PV0 is the saturation vapor pressure at temperatureT0 . A more accurate expression for the latent heat can beobtained by integrating the relation8

S ]L

]TDP

5DCP , ~14!

whereDCP is the change of the specific heat between thetwo phases. In the followingD denotes an increment. If weexpand the specific heat for each phase asCP(T)5a1bT1¯, we obtain

L5L01DaT1Db

2T21O~T3!, ~15!

whereL0 is an integration constant,a andb are empiricallydetermined constants for each phase, andDa and Db indi-cate the change of the constantsa and b between the twophases. We combine Eqs.~12! and ~15! to find the generalform for the saturation vapor pressure curve:

ln~PV!5 ln~C!

11

RVF2

L0

T1Da ln T1

Db

2T1O~T2!G . ~16!

Table I. Basic properties of planetary atmospheres. The quantities are defined in Sec. II.

Planet orsatellite

Maincomponents~%!

m~g mol21!

R*~J g21 K21!

Cp

~J g21 K21!g

~m s22!Ga

~K Km21!T0

~K!P0

~bar!

Venus CO2 ~0.96! 44.01 0.19 0.85 8.89 10.50 731 92N2 ~0.035!

Earth N2 ~0.78! 28.97 0.29 1.00 9.80 9.80 288 1.013O2 ~0.21!

Mars CO2 ~0.953! 44.01 0.19 0.83 3.74 4.50 214 0.07N2 ~0.027!

Jupiter H2 ~0.864! 2.22 3.75 12.36 24.25 2.00 165 1.00He ~0.136!

Saturn H2 ~0.85! 2.14 3.89 14.01 10.00 0.70 134 1.00He ~0.14!

Titan N2 ~0.65–0.98! 28.67 0.29 1.04 1.35 1.30 94 1.50Ar ~0.25–0!

Uranus H2 ~0.85! 2.30 3.61 13.01 8.80 0.70 76 1.00He ~0.15!

Neptune H2 ~0.79! 2.30 3.61 13.01 11.10 0.85 76 1.00He ~0.21!

HD 209458 H2 ~1.0! 2.00 4.16 14.00 8.00 0.60 1300 1.00

769 769Am. J. Phys., Vol. 72, No. 6, June 2004 Sańchez-Lavega, Pe´rez-Hoyos, and Hueso

copied from Sanchez: Clouds in planetary atmospheres: A useful application of the Clausius-Clapeyron equation, DOI:10.1119/1.164527

Note: P0 is surface pressure for Venus, Earth, Mars, Titan.For the giant planets a surface cannot be defined. For these T0 = T0(P0)

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Ideal gas lawWhen is the ideal gas law not valid?

Ideal gas law was obtained under the condition that:I density of gas is low enough that individual molecules

do not feel significant attractive forcesI space occupied by molecules does not represent

significant fraction of the total volume

More precise formulation given by van der Waals equation[p + a

q2

V 2

](V − qb) = qRT

I a characterizes the relative importance of attractiveforces between molecules

I b is a measure of the effective volume occupied by themolecules

I a and b depend on the gas

For the Earth the ideal gas law is valid under mostconditions, BUT not necessarily for other planets!

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

van der Waals gas

reduced p is relative to pc

I Isotherms of a van derWaals gas can have amaximum and a minimum

I Temperature above whichisotherms have noextremum is called criticaltemperature, Tc

I If some part of isothermhas a positive slope →phase transition

In the case of Venus and its CO2 atmosphere the ideal gaslaw is not strictly valid any moreCritical point of CO2 is at 304K and 73.9atm!

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

van der Waals gas and phase transitions

phase diagram for water

I Isotherms of a van derWaals gas can have amaximum and a minimum

I Temperature above whichisotherms have noextremum is called criticaltemperature, Tc

I If some part of isothermhas a positive slope →phase transition

In the case of Venus and its CO2 atmosphere the ideal gaslaw is not strictly valid any moreCritical point of CO2 is at 304K and 73.9atm!

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Phase transitionsChange of form or phase of a substance leads to an energyrelease or absorption.The amount of energy involved in a phase transition is calledlatent heat, L ⇒ Transport of energy in the atmosphere

Condensable substances play a central role in theatmospheres of many planets and satellites:

I Earth: Water condenses in liquid and ice. Ice floats onocean ⇒ effect on albedo

I Mars: CO2 condenses into dry ice in clouds and frost atthe surface. Dry ice sinks in a CO2-ocean

I Jupiter and Saturn: Water and NH3 condense

I Venus: clouds from condensed sulfuric acid

I Titan: Methane condenses

I Triton: Nitrogen condenses!

All gases are condensable at low enough temperatures orhigh enough pressures → saturation vapor pressure

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

General form of phase diagram

copied from R.Pierrehumbert: Principles of planetary climate

At the triple point all three phases coexist

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Phase diagram for water and ice


N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Phase diagram for water in relation to planets

copied from Sanchez: Clouds in planetary atmospheres: A useful application of the Clausius-Clapeyronequation, DOI: 10.1119/1.164527

V Venus surface, E Earth surface and cloud level, M Mars surface

and cloud level, J Jupiter, S Saturn, U Uranus, N Neptune. Details

are shown in the inset. Note the inset’s vertical scale is linear.

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Partial pressureAn atmosphere is a mixture of gasesDalton’s law: The total pressure p is the sum of the partialpressures of each component pj

p = p1 + p2 + p3 + ... =∑

pj

The partial pressure of water vapor is denoted by e and iscalled vapor pressure

For relative amounts of gases it follows

Nj

N=

pjp

This is the volume mixing ratio, or VMR often expressed inppm or ppb or even ppt → trace gases

The mass mixing ratio is defined as

MMR =ρiρ

=mi

min gkg−1

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Most abundant gases in planetary atmospheres

copied from Y.Yung: Photochemistry of planetary atmospheres

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

VMR of gases in Earth atmosphere

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Mean molecular weight versus height for Earth

copied from C.Bohren: Atmospheric Thermodynamics

Why this shape of the curve?→ we have to look in more detail at the pressure behavior

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Hydrostatic equilibriumWhy and how does pressure decrease with altitude?7 a q

öo

>a

;Ä

6<

;?z H

=Q

-r-T =

>-i

.' :

7:-

? =

.--=

- =

==

---

-7.^

..:=

::

-.=

_:=

?1

=-=

?z

i;

;==

=i

==

7

.2

=..

i

Ei*

iF€

7E

t?i

j=?

ä[ä

.? E

li+

:s

if

31

72

-* rF

;i;

u

s3

e;=

i E

?4

.Vt

€

a6

=1

#i

.:i!

3ä

\ =

':."

a

T=

2

.7-ö

6':

" F

ß_

€*i

s

äa

iii,

.ä

rE äI a

ii

,E*:

l*F

+A

a

*z'e

-za

+

!,

A

;A

- :

igiF

:ä

äjr

ii''

=l+

=

€ = ?

t=z

?Il

r;l!

E?

t+Z

$I;

älo

-i

:_il

ä 1

ää

s ä

a

56

'=

--N

ol+

-

"-

---l

- I

I I \ P @

C)

c z

figure from Wallace&Hobbs

With no vertical accelerartion→ slab in equilibrium→ sum of forces is zero:weight equals pressure force→ hydrostatic equilibrium

Mg = ρgA∆z = p(z)A− p(z + ∆z)Adp = −ρgdz

ρ =Mp

RT⇒ dp

dz= − pg

RGT

p(z) = p(z0) exp

(−∫ z

z0

Mg

RT (z ′)dz ′)

Vertical variation of p is much larger than horizontal ortemporal variation! What is p(z0)?

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Scale heightM, g ,T depend on the planet and on heightAssume T does not vary much and take an average Tav

p(z) = p0 exp

(− Mg

RTavz

)= p0 exp

(−mgz

kTav

)The quantity RTav

Mg has dimensions of a length

→ scale height (Skalenhohe) H

H =RTav

Mg=

RGTav

g=

kTav

mg≈ 8400m for Earth

Hydrostatic law expressed with H

p ≈ p0 exp(− z

H

)ρ ≈ ρ0 exp

(− z

H

)If T 6= const then Hp < Hρ → 1

Hρ= 1

Hp+ 1

TdTdz

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Scale height for different planets

from Y.Yung

Physical properties of planetary atmospheres at 1 bar

Assume density ρ(z) would be constant→ H would be height of atmosphere

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Discussion of scale height

Discussion:

I pressure decreases with height faster for lower T

I as T 6= const also H will change

I H depends on mass → each constituent would have itsown scale height → own pressure distribution → VMRof unreactive gases would depend on altitude

but this is not observed!

at least the lower part of the atmosphere behaves as if itwere built up of a single species with a mean molar massEarth: 28.8, Venus and Mars: 44, Jupiter 2.2

Homogeneity of lower atmosphere is a consequence ofmixing due to fluid motions

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Homosphere - Turbosphere

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Homosphere - TurbosphereIn the absence of sinks and sources ratios of various gaseousconstituents at any level in the atmosphere are determinedby two competing processes: molecular diffusion and mixingdue to fluid motions.

Mixing on a macroscale by convection, turbulence and smalleddies does not discriminate according molecular mass

Relative importance of molecular and bulk motions dependson relative distances moved between transport eventsFor bulk fluid motions → mixing lengthFor molecular motion → mean free path: λm

λm ≈1

nσ≈ 1

σ

kT

p

Collision cross section σ of air molecule: ≈ 3 · 10−15 cm−2

At sea level number density n ≈ 3 · 1019 cm−3

Average separation between molecules d = n−1/3 ≈ 3.4nmMean free path λm ≈ 0.1µm, i.e. ≈ 30d

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Homosphere - Turbosphere

Transition region in anatmosphere from turbulentmixing to diffusion is knownas the turbopause orhomopause

For the Earth mixing lengthand mean free path lengthsare approx. equal at100-120 km

Well mixed region below turbopause: homosphereGravitationally separated region above: heterospehre

Above ≈ 500km λm very big → individual molecules followballistic trajectories → escape to space

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Column density

The total content in a column of unit cross section of anatmosphere with a constant scale height is given by thecolumn density (Saulenhohe)

Nc =

∫ ∞0

ndz = n0 exp(− z

H

)dz = n0H =

p0

mg0

Column density in its general form is also used for particledistributions that do not obey the exponential law

Total mass of a planetary atmosphere can be expressed by

Matm =

(p

g

)s

4πR20

where s is at the surface (whatever this is /)

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Temperature profile of Earth

from Jacobson: Atmospheric modeling

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Thermal structureThe thermal structure of an atmosphere is governed by theefficiency of energy transportThere are three principal mechanisms to transport energy:

I conduction

I radiation

I convection

particular mechanism usually dominant in a certain altituderangeconduction: transfer of energy by collisions between particlesPrimary mechanism for heat transfer near a planet’s surface

radiation: dominates in regions where the optical depth ofthe gas is not too large nor too smallUsually dominant in upper troposphere and stratosphereRadiation efficiency critically depends upon emission andabsorption properties of the gas⇒ Radiative transfer equation

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Thermal structure ctd.convection: motion of fluids caused by density gradientswhich result from temperature differences.Most efficient in dense atmospheres, i.e. usually in thetroposphereWhen convection dominates temperature profile will beadiabatic ⇒ dry adiabatic lapse rate

Equation that governs the thermal structure (without proof)

ρcpdT

dt+

dΦc

dz+

dΦk

dz= q

q = net heating rate = rate of heating - rate of coolingΦc = conduction heat fluxΦk= convection heat flux

Φc = −K dT

dzand Φk = −KHρcp

(dT

dz+

g

cp

)K=thermal conductivity and KH=eddy diffusivity

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Temperature profile of inner planets

copied from Yung and DeMore, Photochemsitry of planetray atmospheres

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Temperature profile of outer planets

copied from Yung and DeMore, Photochemsitry of planetray atmospheres

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Lapse rate

Radiative transfer tends to produce highest temperatures atthe lowest altitudes→ hot, lighter air lies under cold, heavier air→ one would guess that convection would arise, BUT

gases are compressible and pressure decreases with height→ rising air parcel will expand, will do work on theenvironment→ air is cooled

Consequence:Temperature drop from expansion can exceed decrease intemperature of surrounding atmosphere→ in that case convection will not occur!

What is the decrease in temperature with altitude?What is the lapse rate?

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Lapse rateConsider air parcel thermally insulated from environmentAir parcel can move up and down under adiabatic conditions

First law of Th.D. dU = dq + dW = dU − pdVEnthalpy dH = dU + pdV + Vdp

For our case → dH = VdpHeat capacity at constant pressure Cp = (dH/dT )p

CpdT = Vdp

dp = −ρgdz from hydrostatic equilibrium

CpdT = −V ρgdz

For a unit mass of gas (cp) we get

−dT

dz=

g

cp= Γd

Γd is called the dry adiabatic lapse rate

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Hydrostatic law with lapse rate

For an adiabatic and dry atmosphere temperature in thetroposphere decreases with altitude

T (z) = T0 − Γd(z − z0)

Together with hydrostatic law

p(z) = p0

[1− Γd(z − z0)

T0

]g/ΓdRG

or expressed as function of temperature by the adiabatic law

p(T ) = p0

[T

T0

]g/ΓdRG

⇔ T (p) = T (p0)

[p0

p

]g/ΓdRG

T0 is close to the radiative equilibrium temperature. Howeverfor the terrestrial planets it is modified by the greenhouseeffect and for the giant planets by internal energy sources.

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Lapse rate for different planetsΓd only depends on gravity and average heat capacity perunit mass

from Y.Yung

Physical properties of planetary atmospheres at 1 bar

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Tropospheres on planets: measurements


N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Stability

Actual temperature gradient of atmosphere: Γ = −dTdz

I Γ < Γd

→ any attempt of an air parcel to rise is counteractedby cooling → parcel gets colder and denser, it sinks→ any attempt of an air parcel to sink is counteractedby warming → parcel gets warmer and lighter, it rises→ atmosphere is stable

I Γ > Γd

→ any attempt of an air parcel to rise is enforced bywarming → parcel gets warmer and lighter, it continuesto rise→ any attempt of an air parcel to sink is enforced bycooling → parcel gets colder and denser, it continues tosink→ convection is working → atmosphere is unstable

Actual Γ rarely exceed Γd by more than a very small amount

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

StabilityN

J15 |F 90 r+ (n

F

f\ \ ö (n rF s s r+

lr.

ts.

n\

f{H

JJ

-.

oa 0

a '5

-U)<

1l- X

H

. rJ

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copied from G.Petty, Atmospheric Thermodynamics

Γ < Γd stable

Γ ≈ Γd neutral

Γ > Γd unstable

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Stability

DALR=dry adiabatic lapse rate Γd

- - - = actual (measured) lapse rate Γ

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

StabilityBuoyancy

z

P,T d, T,

Surface

Air parcel: density ρ′ and temp T ′

Ambient: density ρ and temp TWhat is the acceleration on theparcel of unit volume?

Acting forces: gravity and buoyancy(Auftrieb)

Net force: F = (ρ− ρ′)g

Acceleration:

d2z

dt2=

F

ρ′=

(ρ− ρ′

ρ′

)g = g

(T ′ − T

T

)What happens if parcel is displaced upwards fromequilibrium position z0?

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

StabilityOscillations → Brunt-Vaisala frequency

Equilibrium level at z0 → T of environment be T0

Vertical displacement be ∆z

Temperature of air parcel will become T ′(z) = T0 − Γd∆zTemperature of environment will be T (z) = T0 − Γ∆z

Acceleration:

d2z

dt2= g

(T ′ − T

T

)=

g

T(Γ− Γd)dz

This is the equation of an harmonic oscillator

ω2 = N2 =g

T(Γd − Γ)

N is called the Brunt-Vaisala frequency

ν =1

2π

√g

T(Γd − Γ)

⇒ N is a measure for static stability. If N2 > 0 → stable

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

StabilityOscillations → an example

Gravity Waves Ripple over Marine Stratocumulus Clouds

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Condensation

However: Presence of condensable vapors in atmosphericgases complicates matters!

I Condensation to liquid or solid releases latent heat tothe air parcel

I For a saturated vapor, every decrease in temperature isaccompanied by additional condensation

I Saturated adiabatic lapse rate, Γs , must be smallerthan Γd

I Clouds can formI Clouds are mainly made of H2O for the Earth, but not

alone, e.g. PSC are HNO3

I Clouds on giant planets made from NH3, H2S, CH4I Clouds on Mars from CO2 and on Venus from H2SO4

For the derivation of Γs we need Clausius -Clapeyronequation

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

HumidityDifferent ways to express humidity in the atmosphere

Mixing ratio g/kg

w ≡ mv

md=ρvρd

=Mv

Md

e

p − e

where e is the partial pressure of water vapor

As p � e and with MvMd

= ε = 0.622:

w ≈ 0.622e

p

As long there is no condensation or evaporation the mixingratio is conserved!

Specific humidity is defined as

s =ρvρ

=ρv

ρd + ρv=

εe

p − (1− ε)e

In most cases s ≈ w

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Saturation vapor pressure

Equilibrium between condensation and evaporation→ saturation vapor pressure es→ is valid for other gases than water vapor

Relation between saturation pressure and temperature isgiven by equation of Clausius and Clapeyron

desdT

=1

T

Lvαv − αl

≈ 1

T

Lvαv≈ es

LvRvT 2

Lv = enthalpy of vaporization, for H2O: Lv = 2.5 · 106 J/kgαv resp. αl specific density of vapor and liquid phases

es(T ) = es(T0)e− Lv

Rv

(1T− 1

T0

)= Ce−

LvRv T = Ce−

mv LvkT

• numerator: energy required to break a water molecule freefrom its neighbors• denominator: average molecular kinetic energy available⇒ NOTE: es only depends on T .

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Saturation vapor pressureMeasures of humidity expressed with saturation pressure

Useful approximation for water vapor:

lnes

6.11hPa=

LMv

R

(1

273− 1

T

)= 19.83− 5417

T [K]

Saturation mixing ratio ws(T , p) ≈ 0.622 es(T )p

Relative humidity, RH in % RH = 100 wws

= 100 ees

Dew point, Td , is the temperature where RH = 100%

es(T = Td) = e

Boiling point, Tb, is the temperature where es is equal to p

es(Tb) = p

Frost point: counterpart of the dew point at T < 0C

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Saturation vapor pressure for water vapor

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

Temperature [C]

Sa

tura

tio

n v

ap

or

pre

ss

ure

[h

Pa

]

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

Temperature [C]

Sa

tura

tio

n v

ap

or

pre

su

re [

hP

a]

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Moist adiabatic lapse rate

A rising air parcel will cool according Γd

→ condensation will set in at the lifting condensation levelWhen condensation occurs, the parcel becomes a mixture of

I dry air

I water vapor (at saturation)

I liquid water droplets

→ Lapse rate will change from Γd to Γs

Moist adiabatic lapse rate, Γs , can be shown to be

Γs = −dT

dz=

g

cp

1 + Lvws/RT

1 + L2vws/cpRvT 2

< Γd

� �� Sorry, no further simplifications!

In case of Earth: Γs ≈ 5K/km in contrast to Γd ≈ 10K/km

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Clouds, a few facts

I Clouds can form on all planets with condensable gases

I Temperature must drop below the condensation orfreezing temperature of such gases

I Cloud condensation nuclei must be present

I Most terrestrial clouds consist of water droplets and icecrystals but other cloud particles are possible, eg.HNO3·2H2O or H2SO4/H2O in PSCs

I On ♀ exist H2SO4 clouds

I On ♂ exist water ice clouds

I On titan clouds of CH4 are expected

I NH3- ice may form on X and YI H2S-ice may form on Z and [ and also CH4-ice

I Clouds are often related to precipitation

I Clouds are extremely important for radiation budget→ often little is known

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Condensation levels for some planets

copied from Sanchez: Clouds in planetary atmospheres: A useful application of the Clausius-Clapeyron equation, DOI:10.1119/1.164527

solid line: Temperature profile

dashed line: saturation vapor pressure

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Polar stratospheric clouds

photo from H.Berg, Karlsruhe

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Polar mesospheric clouds

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Clouds on Mars

photo from NASA

N. Kampfer


Aim

Gas law

Ideal gas

van der Waals gas

Phase transitions

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity



Clouds

Clouds on Venus

photo from NASA

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