Heat storage

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(No Transcript)
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Heat storage

• Conservation of energy requires that incoming
  energy balances outgoing energy plus a change in
  storage.
• To relate changes in heat content with
  temperature, we use
• ?QS / ?z Cs ?T/ ?t.
• where the term on the lhs denotes the heat flux
  density change in layer ?z, and the term on the
  rhs represents the heat capacity times the
  heating rate.
• If we use as an example Qin 100 W m-2 and Qout
  10 W m-2, and a layer thickness ?z 0.5 m of
  dry clay, we then obtain
• ?T/ ?t 90 J m-2 s-1 / (0.5 m)(1.42 106 J
  m-3 K-1)
• then, ?T/ ?t 1.27 106 K s-1 0.46 K h-1

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Oke (1987)
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Layers in the Lower Atmosphere

• Laminar boundary layer

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Laminar Boundary Layer

• This skin is only a few mm thick and adheres to
  all surfaces.
• In this layer, the motion is laminar, i.e.
  streamlines are continuously parallel to the
  surface.
• Thus adjacent layers of the fluid remain distinct
  and do not intermix.
• In addition, there is no convection such that
  transfers of heat, water, etc. are by conduction.

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Oke (1987)
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• As an example, take a laminar boundary layer that
  is 3 mm thick, a sensible heat flux QH 100 W
  m-2, and an air temperature Ta 10oC.
• Then what is the gradient in temperature between
  the surface and the top of the laminar boundary
  layer?
• Use QH -Ka Ca ?T/ ?z ( -k ?T/ ?z) .

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• 100 W m-2 (20.5 10-6 m2 s-1)
• (0.0012 106 J m-3 K-1) ?T/0.003 m
• solving for ?T yields
• ?T (100 W m-2)(0.003 m )/ (20.5
• m2 s-1)(0.0012 J m-3 K-1) 0.3/0.0246 K 12.2
  K
• Thus very large temperature gradients exist in
  the laminar boundary layer.

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• Equations for water vapour and momentum transfer
  are similar
• E -?a Kva ??v/?z and
• t ?a Kma ?u/?z
• Since molecular diffusivities (K-values) are
  small, gradients are large in the laminar
  boundary layer.

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Roughness Layer

• The surface roughness causes complex 3D flows,
  including eddies and vortices, that are dependent
  on the details of the surface.
• Exchanges of heat, mass and momentum and related
  climatic characteristics are difficult to express
  in this zone, but generalized features can be
  established.

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Join us for the Environmental Science
Engineering Welcome Back Lunch

• Why? free pizza, meet colleagues faculty, hear
  about program changes, etc
• Where? Bentley Garden
• When? Thursday,
• September 25th
• from 1100 1230

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Turbulent Surface Layer

• The TSL is above the roughness layer where small
  scale turbulence dominates and vertical fluxes
  are approximately constant (constant flux
  layer) - about 10 of the PBL depth.
• Processes of transfer are turbulent, not
  molecular, in this layer.
• However, we can write a flux gradient transfer
  equation that is analogous to conduction, by
  replacing the K's with eddy diffusivities.
• These are not simple constants, but vary with
  time and space (if they were constant, turbulent
  would be a solved problem and weather forecasting
  would be nearly perfect!).
• The eddy diffusivities vary with the size of the
  eddies, that tend to increase with height above
  the surface.

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• Values of K increase from about 10-5 m2
• s-1 near the laminar boundary layer to as
  large as 102 m2 s-1 higher up in the PBL (that
  equates to 7 orders of magnitude!).
• Since the flux is approximately constant but that
  the diffusivities increase with height, the
  related climatic property (wind, temperature,
  humidity) has a curved (logarithmic) shape with a
  decreasing gradient away from the surface.

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Oke (1987)
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• In an analogy with the soil, the greatest
  temperature range is near the surface and
  decreases away from it and there is a time lag
  between surface and air temperatures.
• However it is less than for soil because
  turbulent transfers are more efficient than
  conduction at moving heat around.
• see Oke, p. 51.

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Stability

• A dominant process in the lower atmosphere is
  convection, and a major control on the amount of
  convection is the vertical temperature structure
  (stability).
• To look at stability, consider a discrete
  parcel of air that does not exchange any heat
  with the air around it as it moves (adiabatic
  motion).
• If you move the parcel up it will encounter lower
  pressure because the mass of air above it becomes
  progressively less dense.
• As it encounters lower pressure it will tend to
  expand to make its internal pressure match that
  of its environment, but the expansion requires
  both work and energy.

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• Since the only available energy is in the form of
  heat, the rising parcel will cool.
• In unsaturated air the parcel cools at the
  constant rate of 9.8 10-3 oC m-1 called the
  Dry Adiabatic Lapse Rate (DALR).
• On the other hand, a parcel moving downward will
  warm at the DALR.
• If a parcel is saturated, some water vapour will
  condense as it rises, thus releasing latent heat
  and reducing the rate of cooling.

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• In this case, the parcel of air will cool at the
  Saturated Adiabatic Lapse Rate (SALR) that has
  an approximate value of 6.0 10-3 oC m-1.
• The actual temperature profile of the atmosphere
  (not the DALR!) is called the Environmental Lapse
  Rate (ELR).
• When considering stability, it is useful to use
  potential temperature (?) instead of
  temperature.

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• Potential temperature is the temperature that a
  parcel would have if it were moved adiabatically
  to 1000 hPa.
• This is like correcting the observed temperature
  to allow for G (DALR) and effectively rotates T
  curves by G.
• If ? is used rather than T, analysis of stability
  is simplified
• ? T G z

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Oke (1987)