Institute of Applied Physics University of Bern Outline
A planetary atmosphere consists of different gases hold to the planet by gravity The laws of thermodynamics hold structure as vertical coordinate some planets have no solid surface hydrostatic scale height column density mean free path temperature structure lapse rate stability latent heat and condensation clouds wet lapse rate Ideal gas law pv = NkT N amount of particles k = 1.381 10 23 J/K is Boltzmann s constant n = N/V is the number density, particles per Volume a mole contains N A = 6.022 10 23 particles a kmole contains N A = 6.022 10 26 particles with q moles of a substance N = qn A and the gas law gets pv = qn A kt = nrt where R = kn A 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: M water = 18.016 kg/kmol M air = 28.97 kg/kmol
Ideal gas law mass of q moles is m = qm density ρ can be expressed as ρ = m V = qm V very often gas law is expressed as or = Mp RT pv = m M RT = m R M T = mr G T p = ρr G T R G is the gas constant for the gas under discussion! for dry air R d = 287 JK 1 kg 1 for water vapor R v = 461 JK 1 kg 1 Don t mix up R G and R!! In the literature often R is written as R and R G as R! Partial An atmosphere is a mixture of gases Dalton s law: The total p is the sum of the partial s of each component p j p = p 1 + p 2 + p 3 +... = p j The partial of water vapor is denoted by e and is called vapor For relative amounts of gases it follows N j N = V j V = p j p This is the volume mixing ratio, or VMR often expressed in ppm or ppb or even ppt trace gases The mass mixing ratio is defined as MMR = ρ i ρ = m i m in gkg 1
Most abundant gases in planetary atmospheres copied from Y.Yung: Photochemistry of planetary atmospheress VMR of gases in Earth atmosphere
Mean molecular weight versus height for Earth copied from C.Bohren: Why this shape of the curve? we have to look in more detail at the behavior As a gas is compressible density falls with altitude Vertical profile can be predicted by considering change in overhead force, df, for a change in altitude dz in a column of gas with density ρ and area A df = ρgadz and altitude are related by hydrostatic dp = ρgdz For an ideal gas at temperature T ρ = Mp RT p(z) = p(z 0 ) exp ( z z 0 Mg RT dz ) M, g, T depend on the planet and on height
Assume T does not vary much and take an average T av ( p(z) = p 0 exp Mg ) z RT av The quantity RT av Mg scale height (Skalenhöhe) H has dimensions of a length H = RT av Mg = R G T av g law expressed with H ( p = p 0 exp z ) ( H n = n 0 exp z ) ( H ρ = ρ 0 exp z ) H = kt av mg for different planets from Y.Yung Physical properties of planetary atmospheres at 1 bar
Discussion of hydrostatic law How well do these expressions fit with reality? from Y.Yung Discussion of scale height Discussion: decreases with height faster for lower T as T const also H will change H depends on mass each constituent would have its own scale height own distribution VMR of unreactive gases would depend on altitude but this is not observed! at least the lower parts of atmospheres behave as they were built up of a single species with a mean molar mass Earth: 28.8, Venus and Mars: 44, Jupiter 2.2 Homogeneity of lower atmospheres is a consequence of mixing due to fluid motions
Homosphere - Turbosphere Homosphere - Turbosphere on a macroscale by convection turbulence small eddies does not discriminate according molecular mass Relative importance of molecular and bulk motions depends on relative distances moved between transport events For bulk motions mixing length For molecular motion mean free path: λ m λ m 1 nσ 1 kt σ p Collision cross section σ of air molecule: 3 10 15 cm 2 At sea level number density n 3 10 19 cm 3 Average separation between molecules d = n 1/3 3.4nm Mean free path λ m 0.1µm, i.e. 30d
Homosphere - Turbosphere Transition region in an atmosphere from turbulent mixing to diffusion is known as the turbopause or homopause For the Earth both lengths are approx. equal at 100-120 km Well mixed region below turbopause: homosphere Gravitationally separated region above: heterospehre The total content in a column of unit cross section of an atmosphere with a constant scale height is given by the column density N c = 0 ndz = n 0 exp ( z ) dz = n 0 H = p 0 H mg 0 in its general form is also used for particle distributions that do not obey the exponential law Total mass of a planetary atmosphere can be expressed by ( ) p M atm = 4πR0 2 g s where s is at the surface (whatever this is )
profile of Earth from Jacobson: modeling Thermal structure The thermal structure of an atmosphere is the result of an interaction between radiation, composition and dynamics Equation that governs the thermal structure (without proof) dt ρc p dt + dφ c dz + dφ k dz = q C p = heat capacity per unit mass at constant q = net heating rate = rate of heating - rate of cooling Φ c = conduction heat flux Φ k = convection heat flux Φ c = K dt dz ( dt Φ k = K H ρc p dz g ) c p K=thermal conductivity and K H =eddy diffusivity
Thermal structure dt ρc p dt + dφ c dz + dφ k dz = q First term only important for modeling diurnal variations Third term (convection) dominates in the troposphere Fourth term dominates in the middle atmosphere Second term (conduction) balances the fourth term in the thermosphere Thermal structure of a planetary atmosphere depends on the chemical composition Chemical composition may be affected by temperature through temperature dependent reactions condensation of chemical species profile of inner planets
profile of outer planets Radiative transfer tends to produce highest temperatures at the lowest altitudes hot, lighter air lies under cold, heavier air one would guess that convection would arise, BUT gases are compressible and decreases with height rising air parcel will expand, will do work on the environment air is cooled Consequence: drop from expansion can exceed decrease in temperature of surrounding atmosphere in that case convection will not occur! What is the decrease in temperature with altitude? What is the lapse rate?
Consider air parcel thermally insulated from environment Air parcel can move up and down under adiabatic conditions First law of Th.D. Enthalpy du = dq + dw = du pdv dh = du + pdv + Vdp For our case dh = Vdp Heat capacity at constant C p = (dh/dt ) p C p dt = Vdp dp = ρgdz from hydrostatic C p dt = V ρgdz For a unit mass of gas (c p ) we get dt dz = g c p = Γ d Γ d is called the dry adiabatic lapse rate for different planets from Y.Yung Physical properties of planetary atmospheres at 1 bar
Actual temperature gradient of atmosphere: Γ = dt dz Γ < Γ d any attempt of an air packet to rise is counteracted by cooling packet gets colder and denser, it sinks any attempt of an air packet to sink is counteracted by warming packet gets warmer and lighter, it rises atmosphere is stable Γ > Γ d any attempt of an air packet to rise is enforced by warming packet gets warmer and lighter, it continues to rise any attempt of an air packet to sink is enforced by cooling packet gets colder and denser, it continues to sink convection is working atmosphere is unstable Actual Γ rarely exceed Γ d by more than a very small amount DALR=dry adiabatic lapse rate
However: Presence of condensable vapors in atmospheric gases complicates matters! to liquid or solid releases latent heat to the air parcel For a saturated vapor, every decrease in temperature is accompanied by additional condensation Saturated adiabatic lapse rate, Γ s, must be smaller than Γ d can form are mainly made of H 2 O for the Earth, but not alone, e.g. PSC are HNO 3 on giant planets made from NH3, H 2 S, CH 4 on Mars from CO2 and on Venus from H 2 SO 4 For the derivation of Γ s we need Clausius -Clapeyron equation Different ways to express humidity in the atmosphere: ratio g/kg w m v m d = ρ v ρ d = M v M d e p e where e is the partial of water vapor As p e and with M v M d = ε = 0.622: w 0.622 e p As long there is no condensation or evaporation the mixing ratio is conserved! Specific humidity is defined as s = ρ v ρ = ρ v ρ d + ρ v = eε p (1 ε)e
Equilibrium between condensation and evaporation saturation vapor e s is valid for other gases than water vapor Relation between saturation and temperature is given by equation of Clausius and Clapeyron de s dt = 1 T L v V v V l = 1 T 1 ρ v l v 1 ρ l where: L v = enthalpy of vaporization V v resp. V l are volumina of vapor and liquid phases for H 2 O: l v = 2.5 10 6 J/kg e s Ce l v Rv T = Ce ( m v lv kt ) numerator: energy required to break a water molecule free from its neighbors denominator: average molecular kinetic energy available Useful approximation for water vapor: e s ln 6.11mb = LM ( v 1 R 273 1 ) T Saturation mixing ratio w s 0.622 e s p = 19.83 5417 T Relative humidity, RH RH = 100 w w s = 100 e e s Dew point is the temperature where RH = 100% for saturated conditions, Γ s, can be shown to be Γ s = dt dz = g c p 1 + l v w s /RT 1 + l 2 v w s /c p R v T 2 In case of Earth: Γ s 5K/km in contrast to Γ d 10K/km
for water vapor, a few facts can form on all planets with condensable gases must drop below the condensation or freezing temperature of such gases Cloud condensation nuclei must be present Most terrestrial clouds consist of water droplets and ice crystals but other cloud particles are possible, eg. HNO 3 2H 2 O or H 2 SO 4 /H 2 O in PSCs On exist H 2 SO 4 clouds On exist water ice clouds On titan clouds of CH 4 are expected NH 3 - ice may form on and H 2 S-ice may form on and and also CH 4 -ice are often related to precipitation are extremely important for radiation budget often little is known
Polar stratospheric clouds photo from H.Berg, Karlsruhe Polar stratospheric clouds
on Mars photo from NASA on Venus photo from NASA
Level of cloud formation The lifting condensation level, LCL, is the level to which a parcel of air would have to be lifted dry adiabatically to reach a RH of 100% base of clouds Height of LCL is a function of T and humidity resp. condensable matter If a parcel with T 0 is lifted from z 0 to height z then For the dew point at any z T (z) = T 0 Γ d (z z 0 ) T d (z) = T d0 Γ dew (z z 0 ) z LCL is reached when both are equal z LCL = z 0 + T 0 T do where Γ dew = dt d Γ d Γ dew dz = g Td 2 ɛl v T Rule of thumb: z LCL z 0 = (T 0 T d0 )/8 in km-units Ceilometer at IAP for cloud base measurements Laser-ceilometer from M.Schneebeli Cloud base as determined with a ceilometer