Atmospheric Turbulence Surface Fluxes, and the Planetary Boundary Layer

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Atmospheric TurbulenceSurface Fluxes,
and thePlanetary Boundary Layer
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Turbulent Surface Fluxes

• Recall turbulent fluxes of sensible and latent
  heat can be written as

• Weve learned about physiological resistances rs,
  but how about aerodynamic resistance ra? How can
  we estimate this?
• Think about momentum flux, or aerodynamic drag.
  Also called surface stress

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Surface-Layer Mixing

• Turbulent eddies near the surface act to mix
  atmospheric properties (T, q, u) and reduce
  vertical gradients
• Assume a characteristic length scale l for eddy
  mixing, then

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Surface Layer Stress

• Momentum flux (surface stress) is proportional to
  the square of the product of the wind speed
  gradient (shear) and the turbulent length scale
• Define an eddy viscosity or eddy diffusivity
  Km which is analogous to molecular diffusivity
• Define a velocity scale u for the turbulent
  eddies near the surface, called friction velocity

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Surface Layer (contd)

• Near the surface, eddies are limited in size by
  the proximity of the ground, so l in Km is l(z)
• Assume l kz, where k 0.4 is an empirical
  coefficient known as von Karmans constant
• Leads to a characteristic relationship for
  variation of mean wind speed with height the
  log-wind profile

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Log-wind Profile

• Mean wind speed in the surface layer is
  decelerated by friction whose influence is felt
  aloft through eddy momentum flux
• Varies logarithmically with height
• Y-intercept of log-linear plot of SL wind vs z is
  z0, which we define as the roughness length

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Roughness Lengths

• Roughness length is the intercept of the SL log
  wind profile
• Related to underlying surface elements
• Varies over many orders of magnitude over common
  surfaces!
• Over water, depends sensitively on waves sea
  state (Cd u2)

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Surface Layer Over Vegetation

• Same equations, but consider a displacement
  height due to elevated vegetation surfaces
  (typically about 2/3 to ¾ of height of individual
  veg elements)
• Aerodynamic resistance (for momentum) is simply
  the vertical integral of 1/Km

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Turbulence Kinetic Energy

• Decompose momentum equation into mean and
  turbulent perturbation u
• Multiply equation for by u
• Same for v and w
• Rearrange and obtain predictive equations for
  u2, v2, and w2
• Define turbulence kinetic energy TKE as (u2v
  2w 2)/2

terms advection and other redistribution
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Buoyancy vs Shear and TKE

• Consider the ratio of buoyancy to shear forcing
  of turbulence kinetic energy
• Let
• And define a gradient Richardson number

flux Richardsonnumber
definitions of KH, KM K-theory
Why should Km be different from KH As z
increases, shear forcing decreases faster than
bouyancy forcing
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Non-Neutral Conditions

• Unstable Case
• If H is positive z/L is negative, because then L
  is negative. This refers to unstable conditions.
  In that case buoyancy generates turbulence, and j
  are less than 1,.i.e. the Ks are greater than
  their "neutral" value.
• Stable Case
• Contrarily, if H is negative (z/L is positive),
  which refers to stable conditions, and j are
  greater 1 than, and the Ks becomes smaller than
  their "neutral" values. In that case buoyancy
  suppresses turbulence.

For non-neutral SL, adjust Km
Define another length scale
L Obukhov length height at which
buoyancy and shear forcing become equal
Adjusted non-neutral profiles
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Non-neutral SL Similarity

• Fluxes vary in complex ways depending on height
  and stability
• Using z/L collapses variability SL similarity
• Empirical functions adjust fluxes for stability
  in similar ways

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Dimensionless Gradients
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Adjusted Wind Profiles

• Surface heating mixes momentum (decelerates wind
  aloft)
• Surface cooling decouples surface

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PBL Wind SpeedsAnnual Mean Diurnal Cycles

• Surface winds are maximum at midday
• Winds aloft are maximum at night (decoupling)
• Momentum mixing during daytime allows surface
  friction to be felt throughout ML

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PBL TemperaturesDiurnal Cycle

• Morning inversion broken by surface heating
• Shallow ML by 10 AM under RL from yesterday
• Superadiabatic surface layer at 2 PM
• New inversion forms near surface by 6 PM
• Nocturnal BL grows from the bottom up

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Structure of a Thermal

• Updrafts tend to be more organized and cover a
  smaller area than downdrafts

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Boundary Layer Thermal

• Surface heat flux creates superadiabatic lapse
  rate
• Air parcel accelerates upward due to positive
  buoyancy throughout ML
• Overshoot at capping inversion can entrain air
  from FA into turbulent ML

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Daytime PBL Profiles

• Surface is a source of heat and moisture, and a
  sink for momentum
• Heat flux reverses sign near top of PBL due to
  entrainment

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Typical Diurnal Cycle of PBL Over Land
(Stull, 1988)
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Boundary Layer Clouds

• Big thermals that reach lifting condensation
  level are often capped by shallow cumulus clouds
• If these clouds are forced to the level of free
  convection, they grow on their own by
  condensation heating
• PBL-top clouds are an important means for venting
  PBL air into the free troposphere

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Encroachment Model
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Jump or Slab PBL Model
well-mixed slab with sharp jump in (q, u, q) at
top lapse rate g in FA heating from bottom (H)
and top (entrainment)
5 equations, 5 unknowns Predicts evolution of
BL during day
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Stable (Nocturnal) Boundary Layer

• Nocturnal BL is typically quite shallow (few
  hundred m)
• Strong cooling at surface can produce a moisture
  sink (dew or frost)
• Decoupling from surface friction can allow
  nocturnal jet to form
• Shear under nocturnal jet sometimes produces
  bursts of mixing