2. Energy Balance. 1. All substances radiate unless their temperature is at absolute zero (0 K). Gases radiate at specific frequencies, while solids

Transcription

1 I. Radiation 2. Energy Balance 1. All substances radiate unless their temperature is at absolute zero (0 K). Gases radiate at specific frequencies, while solids radiate at many Click frequencies, to edit producing Master a continuous text spectrum. styles 2. An object that absorbs all radiation incident on it is called a Second level blackbody. Maximum radiation at a given temperature is called Third level blackbody radiation. 3. Plank s Law describes the radiation of Fourth a blackbody level as a function of temperature and frequency. It is Fifth derived level from «the probable molecular energy distribution at a given temperature. B ν (T) = 2ħ ν 3 c 2 e ħν/kt -1 Where: ħ = Plank s const.=6.624x10-34 Js c = speed of light=3x10 8 ms -1 k = Boltzman s const.=1.37x10-23 JK -1 T = temperature (K) ν = frequency (s -1 )

2 4. Emissivity: the actual emitted radiant energy/radiant energy emitted by a blackbody at the same temperature. 5. Absorptivity: the amount of energy absorbed/total amount of energy incident. 6. Stefan-Boltzman law is the integral of the Plank function over all frequencies. B(T) = B ν (T) d ν = σt 4..and can be expressed in terms of a flux F = σt 4 2

3 7. Wien s Displacement Law: wavelength of maximum emission is inversely proportional to temperature. The hotter the object, the higher the frequency and the shorter the wavelength of emitted radiation. When db/dλ = 0 λ max = 2897 T [λ in µm] For T earth ~ 288K => λ max = 10 µm T sun ~6000K => λ max = 0.48 µm

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8 8. Emission if a substance is given by: E ν = ε ν B ν (T) Where ε=emissivity; for a black body ε ν =1 at all frequencies; grey bodies have a constant ε ν but less than 1 at all frequencies. 9. Kirchoff s Law: In thermodynamic equilibrium, the emissivity of a substance must equal its absorptivity (absorptivity = emissivity). A good absorber implies a good emitter.

9 II. Solar Radiation 1. Radiation from the sun (photosphere) is essentially continuous similar to a blackbody at ~6000 K, although a bit less, particularly around 0.3 µm and less.

10 2. From Wien s Law λ max =2897/6000 = 0.48 µm Actually, λ max = 0.47 µm, so the sun s colour temperature is 6110 K. 3. Solar Constant (S o ): the flux of solar radiation received on a surface, held normal to the sun s direction, at the mean distance between the sun and the Earth. S o = 1367 Wm -2 From F = σt 4 = S o x 4πd 2 (d=earth-sun dist) T sun = 5796 K (partial absorption of solar beam in the sun s outer layers)

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13 4. Electromagnetic spectrum: Most of the radiation from the sun (99%) is between the wavelengths of 0.15 µm and 4 µm, and is called shortwave radiation. Of this 9% is in the UV (λ < 0.4 µm), 45% is in the visible ( µm), and 46% in the infrared (λ > 0.74 µm).

14 III. Temperature of the Earth 1. The amount of solar radiation intercepted by the earth (πr e2 S o ) is spread over the earth s surface (4 πr e2 ) so that the earth receives on average 1367/4 = 342 Wm -2.

15 2. Albedo: percentage of solar radiation reflected back to space (reflectivity is used for a single wavelength). For the earth, the average albedo is 30%. Clouds reflect 50% (20-70%); snow varies between 80% (fresh snow) and 50%. For water the albedo varies with the angle of the sun thus with latitude: 2% at equator, 2.1% at 20 o, 2.5% at 40 o, 6% at 60 o and 35% at 80 o.

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18 3. At equilibrium (constant surface temperature), the radiation getting in = radiation going out. So/4 x (1-A) = σ T W/m 2 = T e T e = 255 K = -18 C With Aledo (A) = 0.3; σ = 5.67x10-8 Wm -2 K -4

19 4. Measured surface temperate (T s ) is actually 288 K. The difference is due to the natural greenhouse effect. The transmissivity τ of the atmosphere, the amount of radiation transmitted out to space is: τ (σ T s4 ) = σ T e 4 = So/4 x (1-A) If τ=1, T s =T e and all transmitted radiation gets through the atmosphere, and out to space. In our case: τ = T e4 /T s 4 = 0.62 Absorptivity = 1- τ, so 38% of the radiation emitted from the ground is absorbed by the atmosphere (mainly by water vapor). 5. For a temperature of 288 K, λ max =2897/288 ~ 10µm Earth s peak radiation.

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21 T sun = 6000K T e = 287K

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23 IV. Absorption of Solar Radiation in Earth s Atmosphere 1. Quantum theory: an atom or molecule has only discrete values of energy, and absorption or emission of electromagnetic radiation (of frequency ν) is associated with a change of energy from one discrete level to another. If a quantum of energy is h ν, and the allowed energy states are E and E, then E -E = h ν; or ν = E h 2. Rotational Energy: energy levels are associated with the rotation of molecules as a whole about its center of mass. There are only small differences between energy levels.

24 3. Vibrational Energy: potential energy varies with the distance in the molecule, relative to the equilibrium position r o. repulsion attraction r o Atoms/molecules oscillate about this mean position with an energy E = h ν (V+1/2), where V=0, 1, 2,. is the vibrational quantum number. At atmospheric temperatures, most molecules are in the V=0 state. Large energy differences exist between these levels. When they are excited they generally are accompanied by excitation of the easily excited rotational levels as well (called vibrational-rotational bands)

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26 4. Electronic Transitions: an atom/molecule absorbs electromagnetic radiation and electrons jump to a higher energy level, however, only when the frequency of the radiation corresponds to the difference between the initial energy and that of some permitted higher energy level (emissions result when the molecule/atom reverts spontaneously back to the lower level).

27 5. Ionization Energy: after an electron jumps out of the atom/molecule, we are left with an ionized particle. This molecule can now absorb more radiation than otherwise allowed, and it carries this excess energy with it as kinetic energy. There is therefore an ionization continuum (region of continuous absorption) on the high frequency (high energy) side of the ionization frequency. 6. Dissociation Energy: when the vibrational energy within molecules gets too large, attraction can t hold the molecule together, and dissociation occurs

28 7. Low energy is associated with low frequency so absorption in the far infrared (tens of µm) excites rotational bands; in the near infrared (7 µm and less) vibrational-rotational states are excited; visible and ultraviolet radiation excites molecular electronic transitions; and extreme ultraviolet (EUV) causes ionization and dissociation. 8. Molecules don t show vibrational-rotational absorption unless they have electronic dipole moments. Molecules with pure covalent bonding such as those made up of two similar atoms (N 2, O 2, H 2 ) have no electric dipole moment, and thus do not absorb infrared radiation.

29 9. There is a certain width to the allowed frequencies of absorption and emission for a specific molecule. Collisions between molecules in the atmosphere adds a little extra energy which can also be lost when reverting back to a lower energy level. Collisions with like molecules are more effective in this regard (the energy is more equally shared). So increasing the atmospheric pressure increases the range of frequencies molecules can absorb and emit called pressure broadening. Low Pressure High Pressure 10. Dense bodies (solids) have atoms so close together (high pressure) that their mutual interactions give multitudes of adjacent quantum states, so absorption and emission is essentially continuous with ν

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31 V. Absorption in the Earth s Atmosphere 1. Ionization extreme UV: N 2 => N + 2 (700 A) O => O + (850 A) O 2 => O 2+ (1110 A) All occur, in general, above 100 km altitude [1 A = 10-4 µm ] For this reason the ionosphere occurs above 100 km. 2. Dissociation extreme UV, UV: O 2 + hν => O + O (λ<0.24 µm) O 3 + hν => O 2 + O ( µm) In general this occurs above 12 km, and mostly between 30-50km defining the ozone layer. 3. Electronic transitions near UV and visible: O 3 absorption ( µm, 0.6 µm) at km H 2 O absorption (0.3-1 µm) near surface. Both of these are weak effects.

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33 4. Vibrational-rotational transitions: -H 2 O moderate absorption in the 1-25 µm range in bands -Strong H 2 O absorption band in the µm band -CO 2 strongly absorbs at 2.7 µm, 4.3 µm and µm -O 3 in the stratosphere has moderate absorption at 4.5 µm, 9.6 µm and 15 µm (heating the stratosphere) 5. Rotational bands far infrared: H 2 O at > 25 µm

34 6. Effect on terrestrial radiation (radiation emitted from the Earth s surface) also known as longwave radiation. The Earth is forced (from radiation balance) to have its maximum radiation out to space at 10 µm, since very little absorption occurs there. The atmosphere: -completely absorbs radiation from µm and 29 µm due to H 2 O -is semi-transparent from µm (CO 2 and H 2 O) -is semi-transparent from µm (H 2 O) -is semi-transparent from µm (CO 2 and H 2 O) -is transparent in the window region from µm [If N 2 and O 2 did absorb in the window region the atmospheric temperature, and hence surface temperature, would increase, resulting in the maximum radiation shifting to a different wavelength to reduce this absorption. Therefore, the fact that the trace gases absorb at wavelengths around 10 µm is NOT a coincidence.]

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38 VI. Earth s Energy Balance

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40 VII. Seasonal Variations of Energy Balance 1. Albedo is maximum in winter months (snow and ice), while longwave radiation (from Earth) maximum in summer months. Globally, the northern hemisphere dominates in both cases due to the larger are of continents in the NH. The seasonal cycle of global albedo and IR emission are out of phase by 180 o Month of Year

41 2. Absorbed solar radiation is in phase with the incoming radiation, which varies with the earth s distance from the sun (and has a greater seasonal variation than the albedo). Presently the incoming solar radiation us greatest in January (closer to the sun). So the incoming and outgoing radiation peaks are also 180 o out of phase. Jan 3 Jul 4

42 32 Albedo OLR Absorbed Solar Radiation Month of Year Heating Heating -5 Cooling Month of Year 3. Global net radiation is thus presently positive (radiative heating) in the NH winter, and negative (radiative cooling) during the NH summer.

43 VIII. 0-D Climate Models (Radiative Equilibrium Models) 1. These models balance the incoming and outgoing energy by putting everything in terms of one dependent variable. Generally, the surface temperature T is the dependent variable we wish to predict. The change in temperature with time is caused by the difference between the incoming and outgoing radiation: dt c dt = R -R where c=heat capacity, T=temperature and R= radiative fluxes. The heat capacity is mostly associated with energy absorption in the oceans. Assume this energy is absorbed in the first 70m of the ocean, and the ocean areal coverage is 70.8%. c = ρ w c w hα h=mixed layer depth, α=areal coverage, ρ w =density of water

44 The incoming radiation depends on the solar constant µs o, where µ=1 is for the present day value, and the planetary albedo is A. R = µs o (1-A) 4 The planetary albedo will be greater when the temperature is Colder, as there will be more snow and ice. A = a bt where a, b are constants to be determined The outgoing radiation varies with the temperature and with the emissivity: R = ε σt 4 where ε is the emissivity.

45 The emissivity of the system is a function of the emissivity of the surface (ε s ), and the transmissivity of the atmosphere (τ s ). ε = ε s τ s So the full equation becomes: dt= 1{ S o µ b T + S o µ (1-a) - εσt 4 } dt c 4 4 To produce the current climate, with dt/dt = 0, we need a=2.8, b=0.009, ε=0.69, µ =1. This gives T s = K

46 2. If one assumes equilibrium conditions (dt/dt=0), one can solve for the equilibrium temperature T as a function of anything else in particular as a function of solar constant. As the equation is non-linear (due to the outgoing radiation formulation) there will be more than one solution. One should keep only the physically reasonable ones (not say, -150K). dt= 1{ S o µ b T + S o µ (1-a) - εσt 4 } dt c 4 4 Then plotting the values of T against the independent Variable (here, µ ): T T s + T s - µ c µ

47 3. This determines the external stability of the system how the equilibrium solution varies with a change in the independent variable, which is external to the system. Here, as the solar constant is reduced to some critical value (µ c ), the number of solutions are reduced from 2 to 1. Below µ c no solution is possible. This point is the bifurcation point. With lower temperatures, the albedo gets too high for energy balance to be restored, and the temperature approaches minus infinity. If some limit is put on how high the albedo may get, say, A=0.75, the solution ends up as an ice-age earth. T T s + T s - µ c µ

48 4. Internal Stability: are both branches stable in the sense that if T deviates from the equilibrium solution for some reason, it will revert back to that equilibrium solution? To determine this, in the time dependent problem, use as an initial guess for the solution some value very close to the equilibrium solution. This will produce a value for dt/dt. Update the new guess accordingly, and see if the answer converges back to the equilibrium result. If it does, the solution is stable. In this case only top branch is stable. T T s + T s - µ c µ

49 5. Internal sensitivity: How do changes to the values of the constants that are internal to the system influence the independent variable T. The sensitivity is generally expressed as: B x = dt s + d(ln x) where x = a, b, c, or ε. For this system: B a = -1380K implies that a 1% increase in the value of a will result in a 13.8K decrease in Ts. When a increases, albedo becomes less sensitive to changes in T. The feedback is negative because as a increases, T decreases. B b = 1280K which means a 1% increase in b would produce a 12.8 K increase in T; as b increases A becomes more sensitive to changes in T. So as T increases, A gets lower, which produces an even higher T (positive feedback).

50 B ε = -393K so that as ε increases by 1%, T decreases by 3.93K. An increase in emissivity could be brought about by a change in the absorptivity of the atmosphere (e.g. a decrease in CO 2 concentrations in the atmosphere). 6. The external sensitivity test generally used is B µ. Here B µ =393K (just the opposite to the emissivity change). To put this is context, the global temperature during the last ice age is estimated to have been no more than 3 or 4 K colder than today; with this model, such an effect is produced by a 1% decrease in the solar constant. 7. The value of B may change with the value of T. As one approaches the bifurcation point, the value of B increases. These 0-D models are very sensitive to small changes in model parameters (a problem with these simple models).

51 Homework: M. H. Hart, 1978: The evolution of the atmosphere of the Earth, ICARUS, 33,