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Physics of the Atmosphere Physik der Atmosphäre

Physics of the Atmosphere Physik der Atmosphäre. SS 2010 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de. Last Week. The Navier-Stokes Equation describes the conservation of momentum in fluids

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Physics of the Atmosphere Physik der Atmosphäre

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  1. Physics of the Atmosphere Physik der Atmosphäre SS 2010 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de

  2. Last Week • The Navier-Stokes Equation describes the conservation of momentum in fluids • Frontal zones and fronts are an important phenomenon in the Earth‘s atmosphere • Fronts are strongly tilted  Formula of Margules • High- and low pressure systems form in the descending and ascending branch of baroclinc waves

  3. Contents

  4. Characterisation of the Planetary Boundary Layer (PBL) • The PBL is the part of the atmosphere which is in direct contact to the Earth‘s (or ocean) surface • Here the exchange of scalar tracers (heat, momentum, gases) between surface and atmosphere occurs • The lowermost layer (thickness in the order of mm) is governed by molecular diffusion  Molecular- viscous layer • Above the molecular-viscous layer, turbulence is the dominant transport process • Size of eddies increases with altitude • What is the flux from the surface to the free atmosphere above the PBL? How does it depend on: • Shear stress • Temperature profile and vertical stability • Surface roughness • General definition of the PBL: The layer which is influenced by surface frictionIn this layer the shear stress τis nearly constant with altitude

  5. Boundary Layer in a Wind TunnelSeeds Dispersed in Olive Oil http://efd.safl.umn.edu/research/wind_tunnel/

  6. Boundary Layer on the Wing of an Aircraft

  7. Boundary Layer on a Ship Shaded areas: Vorticity Contour lines: Axial velocity http://www.iihr.uiowa.edu/~shiphydro/cfd_IG_5512_forward_speed_diffraction.htm

  8. Structure of the PBL Classification by shear stress: • Molecular-viscous layer (z ~ mm): • v(z=0) = 0 • τ = -ρυ ∂vx/ ∂z ≡ const • Prandtl layer (z ~ 10-100 m): • τ = -ρ K ∂vx/ ∂z = -ρ u*2≡ const; K = K(z) • Ekman layer (z ~ 1 km): • τlinearly decreasing with altitude, change in wind direction • Free atmosphere (z > 1 km): • τ ≈ 0

  9.  0 Free atmosphere Ekman - layer z [m] Planetary Boundary Layer Prandtl - layer  const  0 Molecular – viscous layer 104 103 102 101 100 10-1 10-2 10-3 The Atmospheric Boundary Layer

  10. low p Fp vg FC high p low p Fp vr FR vg FC high p Wind Profile in the PBL (1) • In the free atmosphere (free of friction), the wind is geostrophic (i.e., parallel to isobars due to the balance between pressure gradient and Coriolis force) • Close to the surface, friction will cause a deviation of the wind direction from geostrophic solution (flow from high to low pressure) • Consequences: • wind speed increases with altitude • wind direction canges with altitude in form of a spiral, the so-called Ekman Spiral

  11. B C A Wind Profile in the PBL (2) Close to the surface friction reduces the wind speed to levels well below the geostrophic speed vg. Since (Fc v) the influence of the Coriolis force is reduced. The direction of the friction force is opposite to the direction of the wind the, therefore close to the ground the wind will turn into the direction of the pressure gradient. A) Close to the ground the friction force is relatively large, v points approximately in the direction of pressure gradient force. B) In intermediate altitudes there is already a considerable angle between FP and v. C) In the geostrophic case (at several 100 m altitude) the friction force can be neglected and FC is anti parallel to FP. The air parcel moves at right angle to the pressure gradient force.

  12. Vertical Wind Profile in the Boundary Layer- Neutral Conditions - Very close to the surface the wind velocity is determined by molecular friction (kinematic viscosity υ), the velocity profile is linear: Inspecting the dimension of the expression we find that  (Velocity)2 Calling this velocity „Friction Velocity“ u* we may write:

  13. Turbulence near the Surface From z  few mm turbulence sets in: In the turbulent regime we set: The “Turbulent Diffusion Constant” Kz=Kz(z) will certainly increase with height, since close to the surface only small eddies can exist (c.f. Kolmogorow – theory). We thus assume: with  0.4   von Kármán constant

  14. Vertical Wind Profile in the Boundary Layer - 2 Vertical wind velocity – profile: After integration we obtain vx(z) under the assumption that τ = ρ u*2 = const: with the Roughness Parameter z0depending on the surface properties.For aerodynamically smooth surfaces, z0 is given by z0 ≈ υ/9u*

  15. Vertical Wind Profile in the Boundary Layer - 3 • Usually the surface wind is driven by the wind in the free atmosphere • Assume that the velocity vr(zr)at a refererence altitude zr is known (e.g., geostrophic wind) • Thuswe have: • With u* = const this yields:

  16. Rough surfaces: • Earth surface no longer reference height  Zero Point displacement d • Interpret integration constant as „Roughness Parameter“ z0 z0=10-2 mmz0=1 mm z0=100 mm Wind Profiles for Different Surface Roughness

  17. rough smooth Transition of Wind Profiles Change of vertical wind profile at the boundary rough  smooth surface • An ‚inner boundary layer‘ forms as a transition between both wind profiles • The upward propagation of this inner boundary can be described by turbulent diffusion • Height of boundary given by implicit equation (see Roedel):

  18. The non-neutral PBL • For the neutral PBL, measurement of wind profile is sufficient for a complete description of dynamics • This is not valid anymore if the PBL is • Unstable: increased vertical exchange, larger diffusion coefficients, smaller gradients • Stable: reduced vertical exchange, stronger gradients, eventually (during strong inversions) complete surpression of turbulent mixing • In these cases, the buoancy of air parcels in relation to shear forces needs to be considered

  19. Influence of Water Vapour on Vertical Stability • So far, we have only considered the release of latent heat on vertical stability • Even without condensation water vapour also influences vertical stability because moist air is less dense than dry air • Ratio of molar masses of air and water vapour: • Density fluctuations under consideration of water vapour with density ρw:

  20. Influence of Water Vapour on Vertical Stability • The density fluctuations of moist air lead to energy production due to buoyancy forces: • The first term in brackets describes the flux of sensible heat, for which we had already inferred • The last term in brackets describes the turbulent flux of latent heat with the evaporation heat L: • Thus the turbulent energy production rate becomes • Over land:Hl≈ H, contribution of water vapour to production of turbulent energy only several percent • Over ocean:Hl≈ 9H, contribution of water vapour to production of turbulent energy similar to contribution of thermal convection

  21. Transport of Trace Species in the Atmospheric Boundary Layer Trace Gas Flux JC: Integration and Division by JC yields: where R12 denotes the transfer resistance for trace gas transport between the altitude levels z2, z1. Its reciprocal is the transfer velocity: 1/R12 = v12(or „piston velocity“) Transfer resistances are additive:

  22. The Trace Gas Profile The trace species –Vertical profile at a given (height independent) vertical flux of the trace species JC : At sufficient distance from the ground (at neutral layering) we have: K  = u*z + D  u*z thus c(z)  ln(z) The transfer resistance R12 between two altitude layers (z1, z2):

  23. R10, 100 R1, 10 R0.1, 1 R0.01, 0.1 R0.001, 0.01 Rlaminar RG The Transfer Resistance Each decade in z (0.1 – 1m, 1m – 10m, ...) represents the same resistance for the trace species. Thus Rges =  ?

  24. Vertical Flux – Example: NO2 C. Volpe Horii, J.W. Munger, and S.C. Wofsy, M. Zahniser, D. Nelson, and J. B. McManus (2004), Fluxes of nitrogen oxides over a temperate deciduous forest, J. Geophys. Res. 109, D08305, doi:10.1029/2003JD004326.

  25. Gas Exchange Atmosphere - OceanBasic Gas Flux Equation Gas Flux (outward is positive): F = kL (Cl -  Cg) kL: Gas transfer velocity, also called piston velocity, gas exchange coefficient or deposition velocity. kL = 1/RL Cl: Concentration in water near the surface Cg: Concentration in air near the surface : Solubility of the gas in water (Cl/Cg)equilibrium • Time Scale considerations: • - Characteristic time scale of gas transfer ( = h/k) is on order of weeks • - Forcing function change on order of hours. • In order to quantify gas fluxes on a regional or global scale • we must have synoptic and co-located estimates of gas concentrations • and forcing function. Conceptual view of air-sea gas exchange of inert gases Rick Wanninkhof

  26. Gas Phase: F= kg(Csg-Cg) Cg Csg Csl Csg=Csl Cl Water Phase: F=kl(Cl-Csl) Basic Conceptual Model Conceptual view of air-sea gas exchange of inert gases Rick Wanninkhof

  27. Water side resistance Air and water side resistance of importance Air side resistance Air/water Resistance Magnitude of typical Ostwald solubility coefficients: He ≈ 0.01 O2 ≈ 0.03 CO2 ≈ 0.7 DMS ≈ 10 CH3Br ≈ 10 PCB's ≈ 100-1000 H2O ≈ ∞ Conceptual view of air-sea gas exchange of inert gases Rick Wanninkhof

  28. Global CO2 Budget 1990-2000 (PgC a-1) Positive values: flux into atmosphere IPCC-TAR, Prentice et al., 2001

  29. Oceanic Carbon Cycle Transport Mechanisms: • Advection and mixing through ocean currents (“Solubility Pump”) Marine biological “pumps”: • Organic carbon • Carbonates

  30. ‘AEOLOTRON’ – The Heidelberg Wind-Wave Facility • circular wind-wave flumeInstitute of Environmental Physics, • University of Heidelberg: • Diameter: 10 m (Perimeter: 29.2 m) • Width: 0.6 m • Height: 2.4 m • Water depth: 1.2 m • Surface area: 18.4 m2 • Water volume: 21000 l • Wind speed up to 14 m/s B. Jähne, M. Schmidt. R. Rocholz (2005), • Thermal imaging: passive und active, spectroscopy • Fourier-Transform-Spectroscopy ( FTIR ) • gas Chromatography ( He, H2 ) • mass balance methods ( CO2, CH4, F12, N2O ) • wind waves (slope) • water- and wind current, temperature, humidity

  31. Imaging Slope / Height Gauge digital image processing wave state: slope  refraction at the surface height  absorption in the water body area extended light source Setup for the Wave State Measurement at the AELOTRON B. Jähne, M. Schmidt and R. Rocholz (2005), Combined optical slope/height measure-ments of short wind waves: principle and calibration, Meas. Sci. Technol. 16,1937-1944.

  32. y x S= y/x water surface reconstruction • slopesaturationspectra: slope in x slope in y plus height information • surface roughness • <s2> as a better parameter for gas transfer velocities B. Jähne, M. Schmidt. R. Rocholz (2005), Measures of the Wave State

  33. wind turbulence windwaves surfactants bubbles Parameters influencing air sea gas exchange B. Jähne, M. Schmidt. R. Rocholz (2005), in order to improve the parameterizations and the models of gas exchange, the different transport mechanisms have to be understood in detail and quantitatively measured.

  34. flux scale vertical space scale time scale Scaling Parameters of Transport ProcessesAcross the Sea-Surface Microlayer Typical microlayer thickness: ~ 20 – 200 mm for diffusive sublayer (gas) ~ 400 mm – 2 mm thermal sublayer (heat) ~ 0.5 – 5 mm viscous sublayer (momentum) experimentally extreme difficult (e.g. wavy surface) typical time scale: 0.1 – 10 sec All concepts (transport models, scaling, parameters) apply for transport of momentum, heat, and mass due to similarity of transport equations B. Jähne, M. Schmidt. R. Rocholz (2005),

  35. Parameterizations of the gas transfer velocity Wanninkhof, [1992]: Quadratic fit of natural 14C disequilibrium and bomb 14C inventory methods. Wanninkhof & McGillis, [1999]: Cubic fit for transfer rates GASEX 1998 CO2 covariance methods. Nightingale, [2000]: Best fit (quadratic) to North Sea 92, 93, Georges Banks 97, 98 data, 3He/SF6 deliberate tracer studies.

  36. Parameterizations versus measurements • Wind speed is not the only parameter influencing air-sea gas transfer: • Transfer rate is correlated with mean square slope of short wind waves, e.g. in [Jähne, 1980] • Surface films lead to strong decrease in transfer rate, e.g. in [Frew et al., 1990] • Fetch conditions have to be taken into account to infer from the wind speed to the sea state e.g. in [Woolf, 2005] • Bubble mediated transport: air entrainment due to wave breaking, e.g. in [Woolf, 1987]

  37. Global Air – Sea Flux of CO2

  38. Estimation of the global exchange rate betweenocean and atmosphere utilizing radar backscatter Cooperation with D. M. Glover, N. M. Frew, and S. J. McCue, Woods Hole Oceanographic Institution, Woods Hole, MA, USA: “Estimating regional and global air-sea gas exchange rates using the dual-frequency TOPEX and JASON-1 altimeters” Estimation of the Global gas transfer velocity based on parameterization with mean square slope of short wind waves.

  39. Summary • The planetary boundary layer is the layer where surface friction has an impact (τ ≠ 0). It can be subdivided into different regimes: • Molecular-viscous layer governed by molecular diffusion • Prandl- layer, where shear stress is constant with altitude • Ekman- layer, where shear stress decreases with altitude (until it is zero in the free atmosphere) • Basic assumption: Turbulent diffusion coefficient is proportional to altitude  Logarithmic wind profile • Water vapour has an impact on vertical stability not only due to the release of latent heat, but also due to its lower density • The transport of scalar tracers in the boundary layer can be parameterised with the transfer resistance R or the piston velocity v12: • In the turbulent regime, the transfer resistance is proportional to the logarithmic ratio of the altitude difference • Air/sea gas exchange is a very important issue in the chemistry and climate of the atmosphere (how much anthropogenic CO2 is taken up by the oceans?) • It can be investigated using wind-wave facilities, such as the Aelotron at the IUP

  40. The non-neutral PBLBouyant forces • Buoyant forces Fb due to turbulence are caused by density fluctuations: • Thus the turbulent power density (per volume) due to buoancy is: • Density fluctuations are caused by: • Temperature fluctuations • Fluctuations in water vapour content (due to the smaller density of moist air, not due to the release of latent heat!)

  41. The non-neutral PBLBouyant forces • Express density fluctuations as fluctuations of the potential temperature • Since ρ = const/θ , we have • Thus the turbulent power density becomes: with the turbulent heat flux:

  42. The non-neutral PBLShear stress • Work done due to shear stress per unit area is dW = τ dx • Thus the power per unit area (= Energy flux) is • The negative divergence of the energy flux yields the power (energy production) per volume due to shear stress:

  43. The non-neutral PBLShear stress • For neutral conditions, we had (logarithmic wind profile) • For the general case of non-neutral conditions, a correction function Φ(H, u*, z) is introduced:with • Thus the energy production due to shear stress becomes:

  44. The non-neutral PBLRichardson Number and Monin-Obuchow Length • The Flux-Richardson number is defined as the negative ratio of energy production rates due to thermal forces and due to shear stress: • Sign of Rf: • Rf > 0 for stable conditions • Rf = 0 for neutral conditions • Rf < 0 for labile conditions • It has been shown empirically that Rfand Φonly depend on altitiude z and a scale length called Monin-Obuchow Length L*: • L* is (in first approximation) independentfrom altitude Monin-Obuchow length as a function of 10 m wind speed and turbulent heat flux

  45. The non-neutral PBLRichardson Number • The Flux-Richardson number is (or rather was) difficult to measure (simultaneous measurement of heat flux and shear stress). • A quantity more easy to measure (only temperature and wind speed profile necessary) is the Richardson-Number, given by • The Richardson-Number is related to the Flux-Richardson number vial the ratio of turbulent diffusion coefficients for heat and momentum, KH and K, respectively: Labile Stable

  46. The non-neutral PBLRichardson Number as a Funciton of z/L*

  47. The non-neutral PBLTurbulent Diffusion Coefficient • The diffusion coefficient for momentum can be obtained from the definition of shear stressand the vertical wind profile:yielding: Labile Stable

  48. Characterisation of Vertical Exchange

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