HEAT BUDGET AND RADIATION A Heat Budget 1 Black body radiation Definition. A perfect black body is defined as a body that absorbs all radiation that falls on it. The intensity of radiation emitted by a black body depends only on its temperature. The radiated flux, Stefan-Boltzmann law, where σ =5.67 10 8 Wm 2 K 4. The wavelength of maximum energy, Wien s law, Emissivity ɛ (< 1) used if not a perfect black body, F = σt 4, (1) λ m T = k = const. (2) F = ɛσt 4. The Sun (T 6000 K effective temperature) and Earth (T 300 K effective temperature) to a good approximation behave as black bodies. Wavelength of maximum radiation intensity: λ m (Sun) 0.5µm (visible), (3) λ m (Earth) 10.0µm (infra-red). (4) Figure 1 shows the normalized radiation of the Sun and Earth. E λ /E max λ Sun Earth 10 1 10 0 10 1 10 2 Wavelength, λ (µm) Figure 1: Normalized black body radiation of the Earth and Sun. There is almost no overlap of the two radiation curves. Sun 99% of energy at 0.15-4 µm Earth 99% of energy at 4-120 µm Throstur Thorsteinsson - 1 - research.turdus.net
2 Earth s Surface Heat Budget All heat received on earth surface from the sun. Energy falling outside the earth s atmosphere Solar constant 1368 Wm 2. Figure 2 shows the annual energy balance for the earth-atmosphere system. Less energy is received at the surface because of absorption, scattering, reflection. The surface of Earth emits long-wave radiation. Gases and clouds emit long-wave radiation also. SPACE Incoming solar radiation Reflected solar radiationoutgoing infrared radiation 100 6 20 4 6 38 26 ATMOSPHERE Absorbed by H2O dust, O3 16 3 Absorbed in clouds Backscattered by air 51 Reflected by clouds OCEAN / LAND Reflected by surface Absorption by H2O, CO2 Net emission by H2O, CO2 Emission by clouds 15 Sensible heat flux Latent heat flux 21 7 23 Net emission of infrared radionation from surface Figure 2: The annual global energy balance for the earth-atmosphere system (Wallace and Hobbs, 1977, modified from their figure 7.1). Energy emitted by the Earth s climate system tends to maintain a balance with solar energy coming into the system. This balance, known as the radiation budget, allows the Earth to maintain the moderate temperature range essential for life as we know it. There is positive radiation balance between 35 S and 40 N, which drives the weather systems. Ocean currents even out the difference (Wallace and Hobbs, 1977). When incoming short-wave solar radiation (Figure 3), known as insolation, enters the Earth s climate system, a portion of it is absorbed at the Earth s surface, causing the surface to heat up. Some of the absorbed energy is then radiated outward in the form of long-wave infrared radiation. Cloud layers trap some of the radiation from the Earth s surface, and then emit long-wave radiation, both outward and back to the surface. The temperature of the Earth s surface is about 33 C higher due to long-wave radiation contribution from the atmosphere (Wallace and Hobbs, 1977). The amount of radiation emitted by the Earth s surface that makes it back to space is the result of many interrelated influences, such as the amount of cloud cover, cloud heights, characteristics of cloud droplets, amount and distribution of water vapor and other greenhouse gases, land features, surface temperature, and the transparency of the atmosphere. In the warm tropical areas, low values of outgoing longwave radiation (OLR) correspond to large amounts of high, cold clouds while high values of OLR correspond to relatively clear areas or cloudy areas with low, warm clouds. In the extra-tropics OLR values typically decrease with decreasing temperature. Figure 4 shows average long-wave radiation. Throstur Thorsteinsson - 2 - research.turdus.net
January 350 300 250 200 150 100 50 0 June 350 300 250 200 150 100 50 0 Figure 3: Surface short-wave radiation in January and June. Mean of 5 years of ERBE measurements. Throstur Thorsteinsson - 3 - research.turdus.net
Figure 4: This image represents the average amount and distribution of heat, in the form of long-wave electromagnetic radiation, that was radiated from Earth to space by the Earth s climate system in the period, 1985-1986, as measured by the NASA Earth Radiation Budget Experiment (ERBE) instruments on the Earth Radiation Budget Satellite (ERBS) and the NOAA-9 satellite. Areas in the purple-to-blue range indicate smaller amounts of outgoing long-wave radiation (OLR); areas in the light-blue-to-red range indicate greater amounts of long-wave radiation leaving the Earth. Clouds play an important role in the Earth s climate system by affecting the amount of heat in the form of electromagnetic radiation that is allowed to pass into or out of the system. The balance between radiation coming to the Earth from the sun and the radiation emitted and reflected from the Earth system is known as the Earth s radiation budget. Generally speaking, low, thick clouds tend to cool the Earth by reflecting the sun s radiation and preventing it from reaching the Earth s surface. In contrast, high, thin clouds tend to warm the planet by allowing solar radiation to pass easily through to the Earth s surface while, at the same time, trapping some of the Earth s infrared radiation and radiating it back to the surface. Whether a given cloud will cause heating or cooling depends on several factors, such as the cloud s height, its size, and the make-up of the particles that form the cloud. The balance between the cooling and warming actions of global cloud cover is very close although, overall, cloud cover produces cooling on a global basis. Figure 5 shows average cloud cover. Two other processes transfer heat between surface and atmosphere: Sensible heat: Eddies warm surface if T air >T surf Latent heat: Gain if vapor condensates. Throstur Thorsteinsson - 4 - research.turdus.net
Figure 5: This image from the High Resolution Infrared Radiation Sounder (HIRS)-2/Mirowave Sounding Unit (MSU) on the NOAA-10 satellite represents the average amount and global distribution of cloud cover during May 1987. Clear-to-mostly-sunny skies are shown in purple and dark blue, increasing cloud cover in lighter shades of blue and green, and mostly-cloudy-to-completely-cloudy skies in yellow, orange, and red. 2.1 Sensible heat Warm air moves over the glacier and cools. Releases sensible heat, where the energy transfer is mostly by eddies near the surface. The equation for sensible heat, T Q H = K H z, (5) where the eddy heat flux coefficient K H is a function of v wind speed,z 0 surface roughness, dt/dz,.... Figure 6 shows warm air above glacier surface. v warm air T 1 Q H glacier T0 z 0 Figure 6: Sensible heat. 2.2 Latent heat Moist air cools and loses its ability to carry moisture. The moisture condenses and releases latent heat. At 0 C and 1 atm, 1 m 3 of air can hold (max) 4.8 g of water vapor m s saturation. At 10 C and 1 atm m s = 9.4 g m 3 water vapor. Throstur Thorsteinsson - 5 - research.turdus.net
Latent heat of: Sublimation, ice to vapour L s 2833 kj kg 1 Vaporization, liquid to vapour L v 2500 kj kg 1 Fusion, liquid to ice L f 333 kj kg 1 When 1 g of water vapor condenses it releases enough heat to melt 7.5 g of ice. The heat gained by the surface is L v E, where L v = 2500 10 3 Jkg 1 is specific latent heat of vaporization, and E is the vertical flux of water vapour, m Q E = L v E = K E z, (6) where m = moisture, mass of water vapor per unit volume, [m] =kgm 3. Figure 7 shows moist air above glacier surface. moist air glacier Q E Figure 7: Latent heat. Similarity theory (Monin and Obukhov, 1954) assumes that the profiles of wind, temperature, and moisture content are similar, and can be described by the momentum transfer, that is K H K E K, where K is the momentum calculated from the wind profile. We can therefore estimate these from a measured wind profile. Relative moisture is given by, r = m m s (T ). (7) 2.3 Energy balance When there is no horizontal movement, where M heat used to melt snow and ice G rate of gain of heat of a vertical column of the glacier R net radiation M + G = R + Q H + Q E + Q P, (8) Throstur Thorsteinsson - 6 - research.turdus.net
Q H rate of transfer of sensible heat from the air to the surface by turbulence Q E = L v E heat of vaporization Q P where Q P = L f P, L f is latent heat of fusion and P precipitation rate of rain. 2.4 Net radiation The net radiation is, where Q α I i I o 2.5 Radiation rate of incoming solar radiation albedo incoming long-wave radiation outgoing long-wave radiation. R = Q(1 α)+i i I o, (9) Measured, can compare long-wave measured with black-body radiation Albedo, α, Q out = αq in. (10) Table below shows albedo values (per cent) (Paterson, 1994). Range Mean Dry snow 80-97 84 Melting snow 66-88 74 Firn 43-69 53 Clean ice 34-51 40 Slightly dirty ice 26-33 29 Dirty ice 15-25 21 Debris-covered ice 10-15 12 The radiation that reaches the glacier surface, It decreases with depth in the glacier, Q 0 = Q in Q out. Q d = Q 0 e kλ. (11) in snow in ice λ 1m λ 10 m 2.6 Heat from Precipitation Q p = ρc p h T, (12) where h is rate of precipitation per unit time, ρ = 1000 kg m 3 is water density, and c p = 2097 J kg 1 K 1. Throstur Thorsteinsson - 7 - research.turdus.net
2.7 Note: A glacier ending in a lake, like Breida in Jökulsárlón, gets a lot of energy from the lake, which absorbs most of the incoming solar radiation. 2.8 Melting Warming of surface layer and then melting. At the surface, G + M = Q i (1 α)+i i I o + Q H + Q E + Q P. First energy G goes into warming the ice to the melting point. Then, when G =0,M goes into melting snow and ice. G = dg dt = c p ρ dt dz, dt (13) where c p is heat capacity. Summer melt at Baegisarjökull (1968), Radiation Q i,q o,i i,i o 55% Warm air Q H 25% Condensation Q E 15% 2.9 Estimating melt H = M. (14) ρ ice L f Difficult to measure all the components contributing M. Heat used for melting, M = L f a, where a is ablation. Then, parameterize H = a + bt +... Degree days: mean temperature T over 24 hours (1 C over 24 hrs = 1 degree day). Degree days model of melting, 2.10 Melting at the base Geotherm outside of geothermal area geothermal area Frictional heat S = i α i T i, α i = { 0, if T i < 0, 1, if T i > 0. 0.13 W m 2, about 1 cm ice per year 50Wm 2, about 5 m ice per year. (15) Q N = u τ, (16) where τ is shear stress, about 1 bar = 10 5 Nm 2, and u velocity. With u = 5 m per hour, and τ = 1 bar, we get Q N = 138 W m 2, which may apply during a surge. Throstur Thorsteinsson - 8 - research.turdus.net
Radiation Problems 1 a) Calculate the wavelength of maximum energy from a human, body temperature 37 C. b) What is the temperature of a body that radiates at λ m =2µm? 2 Q N = u τ, where u = 5m per hour, and τ = 1 bar, calculate Q N. 3 Relative moisture. If at T =10 C m =4.8g m 3, what is the relative moisture in %? 4 Q P = ρc p h T, where c p =4.2 Jg 1 K 1 =1calg 1 C 1 and ρ =1gcm 3.Ifh = 10 mm per hour and T =10 C, calculate Q P. 5 Would the temperature on Earth be lower/higher if no atmosphere? 6 Show that when 1 g of vapor condenses it releshes enough heat to melt 7.5 g of ice. Here L v = 2500 kj kg 1 (liquid to/from vapor), and L f = 333 kj kg 1 (liquid to/from ice). Paterson, W. S. B. 1994. The physics of Glaciers. Pergamon, 3rd edition. Wallace, J. M. and P. V. Hobbs. 1977. Atmospheric Science (An introductory survey). Academic Press, San Diego, 1st edition. Throstur Thorsteinsson - 9 - research.turdus.net