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Physical Basis of Boundary Layer

From Boundary Layer Climates, T.R. OKE

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Physical Basis of Boundary Layer

  1. 1. Pabitra Gurung CHAPTER 2 PHYSICAL BASIS OF BOUNDARY LAYER
  2. 2. CONTENTS 1. Exchanges and climatic response near surfaces (a) The ‘active’ surface (b) Exchange in a volume (c) Exchange in boundary layers 2. Sub-surface climates (a) Soil heat flux and soil temperature (b) Soil water flow and soil moisture 3. Surface layer climates (a) Lapse rates and stability (b) Momentum flux and air temperature (c) Sensible heat flux and air temperature (d) Water vapour and latent heat fluxes and atmospheric humidity (e) Further remarks on convective exchange 4. Outer layer climates
  3. 3. 1. EXCHANGES AND CLIMATIC RESPONSE NEAR SURFACES  A plane that separate two different media; it contains no energy & mass; important for energy & mass exchange and conversion  The surface may be in motion, semi-transparent, undulation  Principal plane of climatic activities in a system, where omajority of the radiant energy is absorbed, reflected, and emitted; omain transformation of energy and mass occur; oprecipitation is intercepted; othe major portion of drag on airflow is exerted  Energy absorption by day and depletion by night (a) The ‘active’ surface
  4. 4. 1. EXCHANGES AND CLIMATIC RESPONSE NEAR SURFACES (b) Exchange in a volume Convergence Divergence Advection: Convergence and divergence of horizontal fluxes (warming and cooling)  Radiation budget and surface energy balance, 𝑄∗ = 𝑄 𝐻 + 𝑄 𝐸 + 𝑄 𝐺  Energy balance in a volume or layer, 𝑄∗ = 𝑄 𝐻 + 𝑄 𝐸 + 𝑄 𝐺 + ∆𝑸 𝑺, Where, ∆𝑄 𝑆 is net energy storage  In a soil layer (∆𝑧): ∆𝑄 𝑠 ∆𝑧 = 𝐶𝑠 ∆ 𝑇𝑠 ∆𝑡  𝑄𝑖𝑛 = 𝑄 𝑜𝑢𝑡 ± ∆𝑄 𝑆  Flux convergence (warming) and Flux divergence (cooling); and advection  In a time period ∆𝑡, Temperature change (∆ 𝑇) ∝ Change of heat flux (∆𝑄 𝑆)  Specific heat (c) (J kg-1 K-1) and heat capacity (C) (J m-3 K-1)
  5. 5. 1. EXCHANGES AND CLIMATIC RESPONSE NEAR SURFACES  Division of the soil-atmosphere system (parallel to the active surface) o Sub-surface layer o Laminar boundary layer o Roughness layer o Turbulent surface layer o Outer layer (c) Exchange in boundary layers
  6. 6.  Sub-surface layer o Soil heat fluxes, 𝑄 𝐺 = −𝑘 𝐻 𝑠 𝐶𝑠 𝜕 𝑇 𝜕𝑧 Where, 𝑘 𝐻 𝑠 = Soil thermal diffusivity (m2 s-1) [It measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy] o The energy transfer occurs due to molecular collisions transferring kinetic energy (molecular exchange) Exchange in boundary layers 𝐹𝑙𝑢𝑥 𝑜𝑓 𝑎𝑛 𝐸𝑛𝑡𝑖𝑡𝑦 = 𝐴𝑏𝑖𝑙𝑖𝑡𝑦 𝑡𝑜 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 × 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑅𝑒𝑙𝑒𝑣𝑒𝑛𝑡 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 Idealized mean profiles of air and soil temperature near the soil-atmosphere interface in fine weather 1. SUB-SURFACE CLIMATES & EXCHANGES
  7. 7. Soil heat flux (𝑸 𝑮) and soil temperature (𝑻 𝑺) HEAT HEAT Idealized mean profiles of air and soil temperature near the soil-atmosphere interface in fine weather  Soil Thermal Properties: o Thermal conductivity ( 𝓀 𝑠), [W m-1 K-1] o Heat capacity (𝐶𝑠), [J m-3 K-1] o Thermal diffusivity (𝑘 𝐻 𝑠 ), [m2 s-1] o Thermal admittance (𝜇 𝑠), [J m-2 s-1/2 K-1] (a measure of the ability of a surface to accept or release heat)  𝑄 𝐺 = −𝓀 𝑠 𝜕 𝑇𝑠 𝜕𝑧 ≈ −𝓀 𝑠 ( 𝑇2− 𝑇1) (𝑧2−𝑧1)  𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶 𝑠  𝜇 𝑠 = 𝐶𝑠 𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶𝑠  𝜇 𝑠 𝜇 𝑎 = 𝑄 𝐺 𝑄 𝑎 1. SUB-SURFACE CLIMATES & EXCHANGES
  8. 8. Soil heat flux (𝑸 𝑮) and soil temperature (𝑻 𝑺)  Soil Thermal Properties: o Thermal conductivity ( 𝓀 𝑠), [W m-1 K-1] o Heat capacity (𝐶𝑠), [J m-3 K-1] o Thermal diffusivity (𝑘 𝐻 𝑠 ), [m2 s-1] o Thermal admittance (𝜇 𝑠), [J m-2 s-1/2 K-1] (a measure of the ability of a surface to accept or release heat)  𝑄 𝐺 = −𝓀 𝑠 𝜕 𝑇𝑠 𝜕𝑧 ≈ −𝓀 𝑠 ( 𝑇2− 𝑇1) (𝑧2−𝑧1)  𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶 𝑠  𝜇 𝑠 = 𝐶𝑠 𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶𝑠  𝜇 𝑠 𝜇 𝑎 = 𝑄 𝐺 𝑄 𝑎 Relationship between soil moisture content: (a) thermal conductivity, (b) heat capacity, (c) thermal diffusivity and (d) thermal admittance 1. SUB-SURFACE CLIMATES & EXCHANGES
  9. 9. 1. SUB-SURFACE CLIMATES Soil heat flux (𝑸 𝑮) and soil temperature (𝑻 𝑺)  Soil Thermal Properties: o Thermal conductivity ( 𝓀 𝑠), [W m-1 K-1] o Heat capacity (𝐶𝑠), [J m-3 K-1] o Thermal diffusivity (𝑘 𝐻 𝑠 ), [m2 s-1] o Thermal admittance (𝜇 𝑠), [J m-2 s-1/2 K-1] (a measure of the ability of a surface to accept or release heat)  𝑄 𝐺 = −𝓀 𝑠 𝜕 𝑇𝑠 𝜕𝑧 ≈ −𝓀 𝑠 ( 𝑇2− 𝑇1) (𝑧2−𝑧1)  𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶 𝑠  𝜇 𝑠 = 𝐶𝑠 𝑘 𝐻 𝑠 = 𝓀 𝑠 𝐶𝑠  𝜇 𝑠 𝜇 𝑎 = 𝑄 𝐺 𝑄 𝐻 Generalized cycles of soil temperature at different depths for (a) daily and (b) annual periods
  10. 10.  Soil moisture potential (Ψ): Energy required to extract water from the soil matrix (displaced water head).  Ψ ∝ 1 S , and Ψ ∝ 𝐸𝑣𝑎𝑝𝑜𝑡𝑟𝑎𝑛𝑠𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛  Darcy’s Law for the vertical flux of liquid in the soil matrix: 𝐽𝑙 = −𝐾𝑓 𝜕Ψ 𝜕𝑧 , where 𝐾𝑓 is hydraulic conductivity  The flux-gradient relationship for the vertical flux of vapour in the soil matrix: 𝐽 𝑉 = −𝑘 𝑉𝑎 𝜕 𝜌 𝑉 𝜕𝑧  The saturation vapour concentration is directly related to temperature  Vapour flow: Day (↓) and Night (↑) Soil water flow (𝐉) and Soil moisture (𝐒) Particularly in this site, soil-air exchange is occurred to a depth of nearly 0.8 m (in a sandy loam during a drying phase) 1. SUB-SURFACE CLIMATES & EXCHANGES
  11. 11.  Laminar boundary layer o Speed of flow, distance and viscosity creates turbulence flow o Thickness of Laminar sub-layer depend on the surface roughness and the external wind speed o No convection: All non-radiative transfer is by molecular diffusion o Sensible heat flux: 𝑄 𝐻 = − 𝜌𝑐 𝑝 𝑘 𝐻 𝑎 𝜕 𝑇 𝜕𝑧 = −𝐶 𝑎 𝑘 𝐻 𝑎 𝜕 𝑇 𝜕𝑧 o For water vapour: 𝐸 = −𝑘 𝑉𝑎 𝜕 𝜌 𝑉 𝜕𝑧 o For momentum: 𝜏 = 𝜌𝑘 𝑀 𝑎 𝜕 𝑢 𝜕𝑧 Exchange in boundary layers (a) Development of a laminar boundary layer over a flat plate and its transition to turbulent flow, (b) The vertical variation of the flux of any entity, the associated diffusion coefficients and the concentration of its property. 2. SURFACE LAYER CLIMATES & EXCHANGES
  12. 12.  Roughness layer o Complex flows; and formed eddies and vertices (Due to surface roughness elements) o 3-D effects: Depends on characteristics of elements; and their shape, plan density, flexibility o Difficult to express: Exchange of heat, mass and momentum Exchange in boundary layers  Turbulent surface layer o Small-scale turbulence above the surface o Constant flux layer o Depth about 10% of planetary boundary layer o Day: Lapse profile & -ve gradient o Night: Inversion profile & +ve gradient 2. SURFACE LAYER CLIMATES & EXCHANGES
  13. 13.  Dry adiabatic lapse rate (Γ): Constant (9.8 ℃ 𝑘𝑚−1) for dry/unsaturated air  Environmental lapse rate (ELR): Based on actual observed temperature structure above a given location  (a) Unstable (ELR > Γ), (b) Stable (ELR < Γ), & (c) Neutral (ELR = Γ) Lapse rates and stability Warmer Colder Warmer Colder  With fine weather: Unstable by day and Stable by night  Over high latitude snow surfaces in winter: Stable boundary layer for longer period  Over tropical ocean surfaces: Unstable boundary layer for longer period Height vs temperature (a) unstable atmosphere on sunny days and (b) stable atmosphere at night 2. SURFACE LAYER CLIMATES & EXCHANGES
  14. 14.  Wind field is largely controlled by the frictional drag imposed on the flow by the underlying rigid surface  Above Zg, wind speed is approximately constant due to negligible frictional drag  Depth of Zg will increase with strong surface heating and will decrease with cooling  Surface shearing stress (τ): The force exerted on the surface by the air  𝜏 = 𝜌𝐾 𝑀 𝜕 𝑢 𝜕𝑧 , where 𝐾 𝑀 is eddy viscosity (m2 s-1)  𝑢 𝑧 = 𝑢∗ 𝓀 𝑙𝑛 𝑧 𝑧0 , where 𝑢 𝑧 is mean wind speed (m s-1) at the height 𝑧, 𝑢∗ is friction velocity (m s-1),𝓀 is von Karman’s constant (≈ 0.40), 𝑧0 is roughness length (m)  Surface sharing stress, 𝜏 = 𝜌 𝑢∗ 2 (the shearing stress is proportional to the square of the wind velocity at some arbitrary reference height) Momentum flux (𝛕) and wind speed (𝐮) The wind speed profile near the ground. (a) the effect of terrain roughness; and (b, c, d, e) the effect of stability on the profile shape and eddy structure 2. SURFACE LAYER CLIMATES & EXCHANGES
  15. 15.  In the turbulent surface layer, 𝑄 𝐻 = −𝐶 𝑎 𝐾 𝐻 𝜕 𝑇𝑠 𝜕𝑧 + Γ = −𝐶 𝑎 𝐾 𝐻 𝜕 𝜃 𝜕𝑧 Where, 𝐾 𝐻 is eddy conductivity (m2 s-1)  Vertical turbulent transfer Sensible heat flux (𝑸 𝑯) and air temperature (𝑻 𝒂) Results from fast response instruments at a height of 23 m over grass in day time unstable conditions (at Edithvale, Australia) updraft downdraft Generalized daily cycle of air temperature in the atmosphere on a cloudless day 2. SURFACE LAYER CLIMATES & EXCHANGES and 𝜃 is a potential temperature (temperature of an air parcel at the arbitrary pressure value of 100 kPa)
  16. 16.  Idealized weather conditions o Profile 1: Before sunrise o Profile 2: Soon after sunrise o Profile 3: Mid-day o Profile 4: Near sunset Sensible heat flux (𝑸 𝑯) and air temperature (𝑻 𝒂) Generalized form of the air temperature profile in the lowest 150 m of the atmosphere at different times on a day with fine weather. [Where, h* is the depth of the mixed layer] Potential Temperature Environmental Temperature Profile SRB ST 𝑸 𝑯 h* 1 − ↓ ↓ ~ 0 2 + ↑ ↑ shallow 3 + ↑ ↑ deep 4 − ↓ ↓ ~ 0 SRB = Surface radiation budget ST = Surface temperature  This idealized conditions will effect by cloud cover and wind speed due to impact on radiation and turbulence respectively.  Due to cloud and wind, the daily range of temperature (lower maxima and higher minima) and extremes of stability (more neutral) will reduce. 2. SURFACE LAYER CLIMATES & EXCHANGES
  17. 17.  Vapour content (humidity)  Vapour density (𝜌 𝑉) [kg m-3], and Vapour pressure (𝑒) [Pa]  The Ideal Gas Law: 𝑒 = 𝜌 𝑉 𝑅 𝑉 𝑇  Saturation vapour density (𝜌 𝑉 ∗ ) and Saturation vapour pressure (𝑒∗ ) [Water molecules escaping to the air = Water molecules returning to the liquid]  Vapour pressure (or density) deficit: 𝑣𝑝𝑑 = (𝑒∗ − 𝑒), 𝑎𝑛𝑑 𝑣𝑑𝑑 = (𝜌 𝑉 ∗ − 𝜌 𝑉)  Greater the vpd/vdd → Greater the evaporation at the surface (at same temperature)  Dew-point (or frost-point) [Useful to consider condensation of fog or dewfall due to cooling] Water vapour and latent heat fluxes (𝑬, 𝑸 𝑬) and atmospheric humidity (𝝆 𝑽 𝒐𝒓 𝒆) 17 157 Relationship between saturation vapour pressure and temperature over a plane surface of pure water 2. SURFACE LAYER CLIMATES & EXCHANGES
  18. 18.  In the turbulent surface layer, Evaporation mass flux, 𝐸 = −𝐾 𝑉 𝜕 𝜌 𝑉 𝜕𝑧 and the flux of latent heat, 𝑄 𝐸 = 𝐿 𝑉 𝐸 = −𝐿 𝑉 𝐾 𝑉 𝜕 𝜌 𝑉 𝜕𝑧  Evaporation process depends on; o The availability of water and energy o The existence of a vapour concentration gradient o A turbulent atmosphere to carry the vapour away Water vapour and latent heat fluxes (𝑬, 𝑸 𝑬) and atmospheric humidity (𝝆 𝑽 𝒐𝒓 𝒆) (a) Idealized mean profiles of water vapour concentration near the ground’s surface, and (b) the diurnal variation of vapour pressure on cloudless days in May (Quickborn, Germany) 2. SURFACE LAYER CLIMATES & EXCHANGES
  19. 19.  Convection is the principal means of transporting the daytime energy surplus of the surface away from the interface  The relative importance of sensible versus latent heat is mainly governed by the availability of water for evaporation  Bowen’s ratio, 𝛽 = 𝑄 𝐻 𝑄 𝐸 ; If 𝛽 > 1: relatively warming of the lower atmosphere, and if 𝛽 < 1: may increase the humidity of the lower atmosphere, and negative 𝛽 indicates that the two fluxes (𝑄 𝐻 & 𝑄 𝐸) have different signs (especially at night due to evaporation) Further remarks on convective exchange 2. SURFACE LAYER CLIMATES & EXCHANGES
  20. 20. o Mixed layer: Day time convective layer (mixing of airborne materials: dust, pollutants, spores, etc.) o Heat flux ≈ 0 (near the inversion base) o Convective entrainment: Heat transport downwards o Potential temperature, θ = T P0 P R cp [Temperature of an air parcel at the arbitrary pressure value of 100 kPa (P0). Where, R is gas constant] (c) Exchange in boundary layers  Outer layer o Turbulent layer to the top of the planetary boundary layer (≈ 90% of its depth) Schematic representation of airflows in the outer layer 3. OUTER LAYER CLIMATES AND EXCHANGES
  21. 21. 3. OUTER LAYER CLIMATES AND EXCHANGES (a) Daily variation of the boundary layer on an ‘ideal’ day. (b) Idealized mean profiles of potential temperature ( 𝜃), wind speed ( 𝑢) and vapour density ( 𝜌 𝑉) for the daytime convective boundary layer (c) same as (b) for nocturnal stable layer  The depth of mixed layer (ℎ∗ ) starts to rise when the surface sensible heat flux density becomes positive  When sunrise, ℎ∗ rapidly increases by eliminating the previous night’s inversion and reaches maximum in mid-afternoon and the complete layer is convectively unstable The temporal dynamics of the boundary layer under ‘ideal’ weather conditions  The mixing equalizes temperature, wind speed, humidity and other properties throughout the layer Day Night
  22. 22. (a) Stages in the temporal development of a thermal (b) Initiation of a thermal by a hill and cumulus clouds (c) Formation of cloud streets Convective structures associated with instability Little or no horizontal wind Surface wind and hills/islands Less convection and high surface wind speed 3. OUTER LAYER CLIMATES AND EXCHANGES
  23. 23. THANK YOU

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