atmospheric thermodynamics outline - iap · n. k ampfer atmospheric thermodynamics aim gas law...

N. Kampfer

AtmosphericThermodynamics

Aim

Gas law

Pressure

Hydrostaticequilibrium

Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity

Saturation vaporpressure

Clouds

Atmospheric Thermodynamics

N. Kampfer

Institute of Applied PhysicsUniversity of Bern

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Outline

Aim

Gas law

PressureHydrostatic equilibriumScale heightMixingColumn density

TemperatureLapse rateStability

CondensationHumiditySaturation vapor pressureClouds

N. Kampfer


Aim

Gas law

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 hold

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

I hydrostatic equilibriumI 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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Ideal gas law

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

a mole contains NA = 6.022 · 1023 particlesa kmole contains NA = 6.022 · 1026 particleswith q moles of a substance N = qNA and the gas law gets

pV = qNAkT = nRT

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

N. Kampfer


Aim

Gas law

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 dry air Rd = 287 JK−1 kg−1

for water vapor Rv = 461 JK−1 kg−1

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

N. Kampfer


Aim

Gas law

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=

Vj

V=

pj

p

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 =ρi

ρ=

mi

min gkg−1

N. Kampfer


Aim

Gas law

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 atmospheress

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

VMR of gases in Earth atmosphere

N. Kampfer


Aim

Gas law

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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Hydrostatic equilibrium

As a gas is compressible → density falls with altitude

Vertical pressure profile can be predicted by consideringchange in overhead force, dF , for a change in altitude dz ina column of gas with density ρ and area A

dF = −ρgAdz

Pressure and altitude are related by hydrostatic equilibrium

dp = −ρgdz

For an ideal gas at temperature T → ρ = MpRT

p(z) = p(z0) exp

(−∫ z

z0

Mg

RTdz

)M, g ,T depend on the planet and on height

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Scale heightAssume T does not vary much and take an average Tav

p(z) = p0 exp

(− Mg

RTavz

)The quantity RTav

Mg has dimensions of a length

→ scale height (Skalenhohe) H

H =RTav

Mg=

RGTav

g=

kTav

mg

Hydrostatic law expressed with H

p = p0 exp(− z

H

)n = n0 exp

(− z

H

)ρ = ρ0 exp

(− z

H

)

N. Kampfer


Aim

Gas law

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

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Discussion of hydrostatic lawHow well do these expressions fit with reality?

from Y.Yung

N. Kampfer


Aim

Gas law

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 parts of atmospheres behave as theywere built up of a single species with a mean molar massEarth: 28.8, Venus and Mars: 44, Jupiter 2.2

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

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Homosphere - Turbosphere

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Homosphere - TurbosphereMixing on a macroscale by

I convection

I turbulence

I small eddies

does not discriminate according molecular mass

Relative importance of molecular and bulk motions dependson relative distances moved between transport eventsFor bulk 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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Homosphere - Turbosphere

Transition region in an atmospherefrom turbulent mixing to diffusion isknown as the turbopause orhomopause

For the Earth both lengths are approx. equal at 100-120 km

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

N. Kampfer


Aim

Gas law

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

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

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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Thermal structureThe thermal structure of an atmosphere is the result of aninteraction between radiation, composition and dynamics

Equation that governs the thermal structure (without proof)

ρcpdT

dt+

dΦc

dz+

dΦk

dz= q

Cp = heat capacity per unit mass at constant pressureq = net heating rate = rate of heating - rate of coolingΦc = conduction heat fluxΦk = convection heat flux

Φc = −KdT

dz

Φk = −KHρcp

(dT

dz− g

cp

)K=thermal conductivity and KH=eddy diffusivity

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Thermal structure

ρcpdT

dt+

dΦc

dz+

dΦk

dz= q

I First term only important for modeling diurnal variations

I Third term (convection) dominates in the troposphere

I Fourth term dominates in the middle atmosphere

I Second term (conduction) balances the fourth term inthe thermosphere

Thermal structure of a planetary atmosphere depends on thechemical composition

Chemical composition may be affected by

I temperature through temperature dependent reactions

I condensation of chemical species

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Temperature profile of inner planets

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Temperature profile of outer planets

N. Kampfer


Aim

Gas law

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

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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Lapse rate for different planets

from Y.Yung

Physical properties of planetary atmospheres at 1 bar

N. Kampfer


Aim

Gas law

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

I Γ > Γd

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

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

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Stability

DALR=dry adiabatic lapse rate

N. Kampfer


Aim

Gas law

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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Humidity

Different ways to express humidity in the atmosphere:

Mixing ratio g/kg w ≡ mv

md=ρv

ρd=

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 =ρv

ρ=

ρv

ρd + ρv=

eε

p − (1− ε)e

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Saturation vapor pressureEquilibrium 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

des

dT=

1

T

Lv

Vv − Vl=

1

T

lv1ρv− 1

ρl

where: Lv = enthalpy of vaporizationVv resp. Vl are volumina of vapor and liquid phasesfor H2O: lv = 2.5 · 106 J/kg

es ≈ Ce

“− lv

Rv T

”= Ce(−mv lv

kT )

numerator: energy required to break a water molecule freefrom its neighborsdenominator: average molecular kinetic energy available

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Saturation vapor pressure

Useful approximation for water vapor:

lnes

6.11mb=

LMv

R

(1

273− 1

T

)= 19.83− 5417

T

Saturation mixing ratio ws ≈ 0.622 esp

Relative humidity, RH RH = 100 wws

= 100 ees

Dew point is the temperature where RH = 100%

Lapse rate for saturated conditions, Γs , can be shown to be

Γs = −dT

dz=

g

cp

1 + lvws/RT

1 + l2v ws/cpRvT 2

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

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Saturation vapor pressure for water vapor

N. Kampfer


Aim

Gas law

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

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

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Polar stratospheric clouds

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Clouds on Mars

photo from NASA

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Clouds on Venus

photo from NASA

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Level of cloud formationThe lifting condensation level, LCL, is the level to which aparcel of air would have to be lifted dry adiabatically toreach a RH of 100% → base of clouds

Height of LCL is a function of T and humidity resp.condensable matter

If a parcel with T0 is lifted from z0 to height z then

T (z) = T0 − Γd(z − z0)

For the dew point at any z

Td(z) = Td0 − Γdew (z − z0)

zLCL is reached when both are equal

zLCL = z0 +T0 − Tdo

Γd − Γdewwhere Γdew = −dTd

dz=

g

εlv

T 2d

T

→ Rule of thumb: zLCL − z0 = (T0 − Td0)/8 in km-units

N. Kampfer


Aim

Gas law

Pressure


Scale height

Mixing

Column density

Temperature

Lapse rate

Stability

Condensation

Humidity


Clouds

Ceilometer at IAP for cloud base measurements

Laser-ceilometerfrom M.Schneebeli

Cloud base as determined with a ceilometer

http://www.iapmw.unibe.ch/staff/schneeb/ceilometer

atmospheric thermodynamics outline - iap · n. k ampfer atmospheric thermodynamics aim gas law...

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