Heat storage

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## Heat storage

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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: • 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**Layers in the Lower Atmosphere**• Laminar boundary layer**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.**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.**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.**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**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.**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.**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].**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.**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