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Physik der Atmosphäre I Physik der Atmosphäre I Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424

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Physik der Atmosphäre I Last week Atmosphere crucial for live Atmosphere is a complex and non-linear system, interacting with the other geophysical compartments (ocean, land, ice sheet…) The primary source of energy is solar radiation Strongest variability (T, p, …) is in the vertical Large range of spatial and temporal scales Hydrostatic equation:

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Physik der Atmosphäre I Contents

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Outline for Today 1.Basic thermodynamic definitions 2.Dry-adiabatic lapse rate 3.Moist-adiabatic lapse rate 4.The potential temperature 5.The equivalent temperature 6.Vertical stability 7.Buoyancy oscillations

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Physik der Atmosphäre I The vertical structure of the atmosphere

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Physik der Atmosphäre I How to determine the tropospheric temperature profile 1.Atmosphere is heated by the Earth‘s surface 2.Cooling/heating rates of the air by emission/absorption of radiation are much smaller than typical transport times 3. Transport processes are (nearly) adiabatic 4.The condensation of water vapour is an additional source of heat 5.To determine the tropospheric temperature profile, we need a.the ideal gas law b.the first law of thermodynamics c.the hydrostatic equation

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Physik der Atmosphäre I Ideal gas equation Ideal gas: –molecules have no volume –no collisions between gas molecules –error for assuming that atmospheric gases are ideal gases is small Ideal gas equation –p: pressure –V: volume –n: amount of gas in moles –T: absolute temperature –R: gas constant (8.314 J/mol K) Extensive properties: –proportional to size of system, e.g. n, V –„2x size 2x property“ Intensive property: –independent of mass: p,T, –„intensify“ extensive property by dividing by mass or volume “ specific property ”, e.g. = m/V

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Physik der Atmosphäre I First law of thermodynamics First law (1. Hauptsatz): dQ – heat added to the system dW – work done ON the system dU – change in internal energy Work done on system: compression of air Internal energy of an ideal gas is independent of volume and proportional to temperature C v – heat capacity (at constant volume)

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Physik der Atmosphäre I First law of thermodynamics (contd.) 1.First law (1. Hauptsatz): 2.Ideal gas law: Convert to intensive quantities by dividing by n: –q = Q/n: heat per mole –c v = C v /n: specific heat capacity –c p = c v + R: specific heat capacity at constant pressure

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Physik der Atmosphäre I Dry-adiabatic lapse rate 1.First law (1. Hauptsatz): 2.Hydrostatic equation: Movement of air parcel is assumed to be adiabatic, i.e. no heat is exchanged with the environment: dq = 0. This is justified since cooling rates in the troposphere are much smaller than typical transport times.

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Physik der Atmosphäre I Dry-adiabatic lapse rate (contd.) Temperature gradient in the atmosphere: Values for air: –c p = J/(K mole) –M = g/mole –g = 9.81 m/s 2 Dry-adiabatic lapse rate: –The temperature gradient is given by only if no additional heat sources (condensation, absorption of radiation) are present (dq=0) Actual temperature gradient present in the atmosphere:

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Physik der Atmosphäre I Atmospheric Energy Kinetic energy (per volume): –E=½ ρ air v 2, ρ air = kg m -3, v = 10 ms -1 E = 65 J m -3 Sensible heat (per volume): –Q s = ρ air c p ΔT, c p = 10 3 J kg -1 K -1, ΔT = 10K Q s = 1.3x10 4 J m -3 Latent heat (per volume): –Q L = ρ air L q, L = 2.256x10 6 J kg -1, q = 10 g kg -1 Q L = 2.9x10 4 J m -3 Phase transfer processes (e.g. convection, cloud formation) play major roles in energy balance

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Physik der Atmosphäre I Moisture parameters Vapour pressure [hPa]: –partial pressure of H 2 O vapour: e –p=Σ i p i =p dry +e –assume H 2 O vapour to be an ideal gas:e = ρ v R v T Saturation vapour pressure [hPa]: e* Absolute humidity [g m -3 ]: a = ρ v = m v / V Specific humidity [kg/kg=1]: q = ρ v / ρ air, with ρ air density of moist air Relative humidity [1]: f = e / e* Dew point: e = e* (p=const)

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Physik der Atmosphäre I The Moist-Adiabatic Lapse Rate Condensation of water vapour releases heat, i.e. dQ ≠ 0: –L ≈ 2250 J/g: specific evaporation heat –dm w : change in mass of gaseous water vapour Convert to intensive quantities by dividing by n: – w = m w /V is the saturation water vapour concentration Re-ordering yields the moist-adiabatic lapse rate:

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Physik der Atmosphäre I Saturation vapour pressure Wallace and Hobbs, 2006 Magnus formula for e* [hPa]: rule of thumb: ΔT=10K Δe ≈ 2 e Bergeron-Findeisen process

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Physik der Atmosphäre I Water vapour saturation curve Water vapour partial pressure E (right axis) and water vapour density w (left axis) over liquid water as a function of temperature. Adapted from Roedel (1992).

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Physik der Atmosphäre I Moist-adiabatic lapse rate Moist-adiabatic lapse rate as a function of air temperature, with pressure as parameter (Roedel, 1994).

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Physik der Atmosphäre I The (Alpine) Föhn Source: Roedel Inflow of moist air cooling, precipitation heating outflow of warm, dry air

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Physik der Atmosphäre I Potential Temperature Potential temperature = temperature of an air parcel if it would be adiabatically compressed to a pressure of 1013 mBar 1 st law of thermodynamics: dq = 0 (adiabatic process) yields: Integration yields: with = c p /c v and the surface pressure p 0 = 1013 mBar. ≡ T 0 is the temperature of the air parcel after compression from p to p 0 : with ( -1)/ ≈ in air

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Physik der Atmosphäre I Homosphäre Konvektion Strahlungsheizung durch O 3 (UV) Strahlungskühlung durch H 2 O & CO 2 Strahlungsheizung durch O 2 (UV) Strahlungskühlung (IR) durch H 2 O & CO 2 Temperatur (K) Höhe (Km) Druck (mBar) =340K =800K =1900K

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Physik der Atmosphäre I Equivalent Temperature Equivalent temperature T eq = temperature of a moist air parcel if all water vapour would condense Release of latent heat per volume..... leads to an increase in internal energy: Δu = Δq Temperature increase: with water vapour mixing ratio q = v / air Equivalent and equivalent-potential temperature: Example: L ≈ 2250 J/g c p ≈ 1 J/(g K) T = 15°C Specific humidity: 70%, corresponding to q ≈ 12.4 ΔT ≈ 31 K Equivalent temperature T eq ≈ 46°C

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Physik der Atmosphäre I Potential temperature and vertical stability From the hydrostatic equation, we have Total differential of the potential temperature: Thus, with where is the actual temperature gradient.

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Physik der Atmosphäre I Vertical stability , d /dz = 0: neutral , d /dz > 0: stable , d /dz < 0: unstable

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Physik der Atmosphäre I Potential Temperature and Vertical Stability z zz z1z1 z2z2 F F PP Auslenkung eines Luftpakets nach oben (von z 2 nach z 3 ): Θ P =const. Θ P > Θ(z 3 ) Dichte des Luftpaketes geringer als Dichte der umgebenden Luft Resultierende Kraft nach oben, treibt Luftpaket weiter von der ursprünglichen Lage weg Auslenkung eines Luftpakets nach oben (von z 2 nach z 3 ): Θ P =const. Θ P < Θ(z 3 ) Dichte des Luftpaketes größer als Dichte der umgebenden Luft Rücktreibende Kraft z zz z1z1 z2z2 F F PP (z) T(z)

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Physik der Atmosphäre I Dry and moist stability

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Physik der Atmosphäre I Stüve diagram (1)

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Physik der Atmosphäre I Stüve diagram (2) PressureAltitude Temperature [°C] Temperature [K] Wind speed and direction Air temperature Isobars Isotherms Inversion

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Physik der Atmosphäre I Stüve diagram (3) Saturation water vapour mixing ratio [g water/kg air] Dew point temperature

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Physik der Atmosphäre I Stüve diagram (4) Dry lapse rate Condensation starts Yellow line: Temperature of an air parcel lifted up adiabatically from 950 mBar. Initial water vapour content: 8g/kg (inferred from dew point temperature). Air parcel is undersaturated and thus follows the dry adiabatic temperature gradient (~1K/100m) Condensation starts at 800mBar, where saturation water vapour mixing ratio equals water vapour content.

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Physik der Atmosphäre I Stüve diagram (5) Dry lapse rate Moist lapse rate Yellow line: Temperature of an air parcel lifted up adiabatically from 950 mBar. Initial water vapour content: 8g/kg (inferred from dew point temperature). Air parcel is undersaturated and thus follows the dry adiabatic temperature gradient (~1K/100m) Condensation starts at 800mBar, where saturation water vapour mixing ratio equals water vapour content. Now the air parcel follows the moist adiabat due to release of latent heat. Temperature of rising air parcel is always below actual temperature Stable conditions

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Physik der Atmosphäre I Stüve diagram (6) Yellow line: Temperature of an air parcel lifted up adiabatically from 950 mBar. Dew point temperature is equal to actual temperature up to 750 mBar condensation, formation of clouds. Temperature of rising air parcel is above actual temperature Unstable conditions, convection

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Physik der Atmosphäre I Brunt-Väisälä or buoyancy oscillations Equation of motion for the buoyancy of an air parcel: : density of air parcel; *: density of surrounding air proportional to 1/T: Adiabatic and actual lapse rate: It follows: Potential temperature: How can atmospheric stability be defined quantitatively?

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Physik der Atmosphäre I Brunt-Väisälä or buoyancy oscillations Equation of motion for the buoyancy of an air parcel: with This is the equation of an harmonic oscillator with the Brunt-Väisälä frequency For d /dz > 0 (stable conditions), this leads to buoyancy oscillations. Typical periodic times: –15 minutes for = 0.8 (stable conditions) –5 minutes for = 0 (isotherm stratification) How can atmospheric stability be defined quantitatively?

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Physik der Atmosphäre I Lee waves Buoyancy oscillations downwind of mountains „Lenticular clouds“ Formation of lee waves

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Physik der Atmosphäre I Lee waves … can propagate up to the stratosphere Polar stratospheric clouds observed in Kiruna, northern Sweden. These clouds are formed in the ascending branch of the lee waves occuring downwind of the Scandinavian mountains. (Photo courtesy of C.F. Enell)

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Physik der Atmosphäre I Summary Rising air cools down due to adiabatic expansion –Dry-adiabatic lapse rate: ≈ 1K/100m Condensation of water vapour leads to the release of latent heat –Moist-adiabatic lapse rate: (0.5 – 1)K/100m Potential temperature = temperature of air parcel, if compressed adiabatically to 1013 mBar Vertical stability of the atmosphere is determined by the actual vertical temperature gradient compared to the dry/moist lapse rate: –d /dz = 0: neutral –d /dz > 0: stable –d /dz < 0: unstable Buoyancy oscillations (Brunt-Väisälä oscillations) can occur for stable conditions (d /dz > 0) Formation of lee waves downwind of mountains

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