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Heat storage

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- 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:
- ΔQS / Δz = CsΔT/ Δt.
- where the term on the lhs denotes the heat flux density change in layer Δz, 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 Δz = 0.5 m of dry clay, we then obtain:
- ΔT/ Δt = 90 J m-2 s-1 / {(0.5 m)(1.42 × 106 J m-3 K-1)}
- then, ΔT/ Δt = 1.27 × 106 K s-1 = 0.46 K h-1

Oke (1987)

- 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.

Oke (1987)

- 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 CaΔT/ Δz (= -k ΔT/ Δz) .

- 100 W m-2 = (20.5 × 10-6 m2 s-1) ×
(0.0012 × 106 J m-3 K-1) ×ΔT/0.003 m

- solving for ΔT yields:
- ΔT = {(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.

- Equations for water vapour and momentum transfer are similar:
- E = -ρaKva∂ρv/∂z and
- τ = ρaKma∂u/∂z
- Since molecular diffusivities (“K”-values) are small, gradients are large in the laminar boundary 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.

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 11:00 – 12:30

- 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.

- 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.

Oke (1987)

- 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].

- 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.

- 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.

- 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” (θ) instead of temperature.

- 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 Γ (DALR) and effectively rotates T curves by Γ.
- If θ is used rather than T, analysis of stability is simplified
θ = T + Γ z

Oke (1987)