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ATS150:GlobalClimateChange 4:TheLayerModel What is a Model? The Layer Model of the Greenhouse Effect ATS 150 Lecture 4 Please read Chapter 3 in Archer Textbook Empirical Models CSU Deterministic Models • Formulated as “cause and effect” • Common in physics and chemistry • Usually take the form of differential equations • Initial & boundaryvalue problems • May still have adjustable coefficients • Generalized mathematical formulation with adjustable coefficients • Combinations of – polynomials – exponential growth & decay – Periodic sines and cosines • Coefficients fit to data (e.g., least squares) • Interpolation or extrapolation Sco=Denning • Models represent reality in a simplified or idealized way • Used to understand or predict • Doesn’t have to be realistic to be useful CMMAP ATS 1 ATS150:GlobalClimateChange 4:TheLayerModel Energy Balance Energy Balance Fin = Fout Energy In = Energy Out • Assume both Sun and Earth are • Let the rate of energy flow from the Sun to the Earth be called Fin • Let the rate of energy flow from the Earth to outer space be called Fout Energy Balance blackbodies (ε=1), so F = σ T4 • Fin = absorbed sunlight x daylight area • Fout = σ Tearth4 x total area Solar Radiation Fin = Fout Fin = Solar brightness x (1 – albedo) x (area of Earth’s shadow) Fout = σ (Tearth)4 x (area of Earth’s surface) Sco=Denning CSU CMMAP ATS • 30% reflected by clouds, air, dust, and surface • 19% absorbed by the atmosphere (mostly clouds) • 51% absorbed at the surface 2 ATS150:GlobalClimateChange 4:TheLayerModel Energy Balance Energy Balance Fin = Fout Fin = Fout Fin = Solar brightness x (1 – α) x π S(1− α )π r 2 = 4π r 2σ T 4 r2 S(1− α ) = 4σ T 4 Fout = σ (Te)4 x 4π r2 S(1− α ) =T4 4σ Energy Balance Thermal Emission Both are almost perfect blackbodies! SUN Fin = Fout 1/4 1/4 $ (1360Wm−2 )(1− 0.3) ' $ S(1− α ) ' T =& ) = 255K ) =& −8 −2 −4 % 4σ ( % 4(5.67x10 Wm K ) ( But the observed surface temperature is about 288 K Sco=Denning CSU CMMAP ATS EARTH 255 K The hot sun radiates at shorter (visible) wavelengths that carry more energy Energy absorbed by the cooler earth is then reradiated at longer (thermal infrared) wavelengths 3 ATS150:GlobalClimateChange 4:TheLayerModel Works for Other Planets Too Bare Rock Model ht g Sunli Fin = Fout Fsurface Surface S(1− α ) =T4 4σ Fin = Fout S(1− α ) =σT4 4 Experiment!h=p://Lnyurl.com/planetary-balance Two-Layer Model ght Sunli Glass Fglass Fglass Fsurface Surface Sco=Denning CSU • Now imagine a layer of glass above the solid rock • Glass is transparent to sunlight • Glass is opaque to “Earthlight” (thermal IR) CMMAP ATS S(1− α ) 4 • If sunlight were the only source of energy, the surface of the Earth would be way too cold for life! • 255 °K = -18 °C = -1 °F Two-Layer Model • Sun shines right through transparent Glass glass layer σ Tglass 4 • Glass layer emits σ Tsfc 4 both upward and Surface downward Fin = Fout • Glass layer “feels” S(1− α ) = σ Tglass 4 both space and Boundary 4 σ Tsfc 4 = 2σ Tglass 4 surface Glass • Surface “feels” both S(1− α ) + σ Tglass 4 = σ Tsfc 4 Surface Sun and glass 4 boundary σ Tglass 4 4 ATS150:GlobalClimateChange 4:TheLayerModel Seen from Above S(1− α ) 4 boundary g Sunli g Sunli Glass Fglass ht ht Surface Both models: • Absorb the same sunshine • Reflect the same amount of sunlight back out • Emit the same amount of Earthlight S(1− α ) 4 Two-Layer Model • Recall from σ Tglass boundary boundary balance glass that σ Tglass 4 S(1-α)/4 = σTglass4 σ Tsfc 4 • Substitute this into surface surface balance Fin = Fout • Find that surface is S(1− α ) + σ Tglass 4 = σ Tsfc 4 Surface much warmer 4 σ Tglass 4 + σ Tglass 4 = σ Tsfc 4 under glass! 2Tglass 4 = Tsfc 4 • 303 °K = 30 °C 4 2Tglass = Tsfc = 1.189 Tglass = 303 K = 86 °F Sco=Denning σ Tglass 4 boundary glass σ Tglass 4 σ Tsfc 4 Fglass Fsurface Fsurface Two-Layer Model surface Fin = Fout S(1− α ) = σ Tglass 4 Boundary 4 1/4 $ S(1− α ) ' Glass & ) = Tglass % 4σ ( 255 K = Tglass • Use balance at boundary to space to find temperature of the glass • Since energy in from Sun and energy out from glass are same as bare rock model, temperature is too! Summary 4 CSU CMMAP ATS • Layer models are not meant to be accurate or predictive, just to help us understand how the world works • Energy balance of “bare rock” model is way too cold to support life! • Adding a layer of glass makes the surface nice and toasty • Where did the heat come from? 5 ATS150:GlobalClimateChange 4:TheLayerModel Bathtub Analogy • If faucet runs faster than drain, level rises • And vice versa • Drain runs faster when water is deep • Adding glass to layer model acts like a clog in the drain Faucet~Sun • Water rises until Drain~Thermalemission drainage = inflow Waterlevel~temperature again Three Layers • Top boundary (“skin”) temperature is always the same • As we add layers, the surface gets hotter • (bathtub drain gets slower) Sco=Denning CSU CMMAP ATS 6 ATS150:GlobalClimateChange 4:TheLayerModel Ten Layers • Top boundary (“skin”) temperature is always the same • As we add layers, the surface gets hotter • (bathtub drain gets slower) Sco=Denning CSU 100 Layers! • Top boundary (“skin”) temperature is always the same • As we add layers, the surface gets hotter • (bathtub drain gets slower) CMMAP ATS 7