Solar radiation / radiative transfer

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1 Solar radiation / radiative transfer The sun as a source of energy The sun is the main source of energy for the climate system, exceeding the next importat source (geothermal energy) by 4 orders of magnitude! Sources of energy for the climate systems solar radiation geothermal energy anthropogenic energy generation infrared emission by the full moon solar radiation reflected by the full moon radiation by the stars W W W W W W 1

2 Correlation and causation 2

3 Solar zenith angle Essentially the position of the sun with respect to an observer at the surface is determined by the solar zenith angle θ o, that is the angle between the vertical above the observer (the normal) and the sun. The solar zenith angle can be expressed in terms of the solar declination δ (the angle subtended by the sun with respect to the equatorial plane), the hour angle ω (< before true solar noon, at noon, > after solar noon), and the geographic latitude ϕ of the observer O. N θ o O ϕ δ ω 3

4 Solar zenith angle (2) The equation reads: cosθ o sin ϕsin δ + cosϕcosδcosω The hour angle is equal to zero at true solar noon, increasing by.26 radians or 15 pro hour. Therefore ω -π/2 at 6: and ω π/2 at 18:. Since θ o π/2 at sunrise and sunset (except at the Poles), the hour angles -Ω at sunrise and Ω at sunset (atronomical values) can be found from: cosω tan ϕ tan δ provided that -1 cosω +1. If cosω > +1 we have polar night, and if cosω < -1 polar day with midnight sun. According to Iqbal (1983) a useful formula to express solar declination as a function of the running date (day of the year, D [1,365]) is: δ 2π.493 sin D [radians] Iqbal, M. 1983: An Introduction to Solar Radiation. Academic Press, Toronto, 39 pp 4

5 Solar path Source: 5

6 Solar insolation at TOA The distribution of solar insolation at the top of the atmosphere is given by: 2 a S So cosθo r where S o is the solar constant, a the mean distance between the sun and the earth, and r the current distance at a particular day of the year D. Following Iqbal (1983), the square of the ratio (a/r) can be calculated as: a 2 r 2π cos D 365 Note: more accurate formulas for the solar declination and the relative distance to the sun can be found in Iqbal (1983) or Liou (22). The daily insolation is found by integrating the above equation between sunrise and sunset: S D Ω S Ω 2S o o a r 2 a r 2 sin δsin ϕ + cosδcosϕcosωdω [ Ωsin δsin ϕ + cosδcosϕsin Ω] 6

7 Distribution of daily insolation at TOA After Liou (22). 7

8 Annual mean insolation at TOA As for the latitudinal distribution of the annual mean insolation we have: 8

9 Zenith angle and air mass When considering absorption and scattering of solar radiation it is necessary to know the total mass of absorbing or scattering substance. Recall the Beer- Bouguer-Lambert law: s N (s) N exp k ds ρ N exp ( k u) where u is the optical path. The distance covered by a beam in the atmosphere depends on the solar zenith angle: θ ο θ ο 9

10 Zenith angle and air mass (2) To account for the effect of the zenith angle, we make use of the so-called relative optical air mass defined as (Paltridge and Platt, 1976): m ρdz ρds As seen in the above figure, to the extent that the atmosphere can be considered as a non-refreacting, plane-parallel medium: m 1 cosθ o secθ o In practice, due to the curvature of the earth s atmosphere and the density dependence of the refractive index, this equation holds true only for θ o < 6 o. A more accurate formula is due to Kasten (1966). It reads: m { sin γ +.15 ( γ + 3. ) } 1 o o 885 where γ o 9 ο θ o is the observed solar altitude (in degrees). z θ s 1

11 Radiation at ground: direct and indirect irradiance Atmospheric scatter ensures that the downcoming flux density has a direct as well as a diffuse component. At any solar zenith angle θ o the total vertical flux density, the so-called global radiation, is given by (Paltridge and Platt, 1976): Gl F F dir cosθ o + N dω diff where F dir is the irradiance of the direct beam on a surface perpendicular to the beam and N diff is the radiance of diffuse radiation and the integral is over the all solid angles of the upper hemisphere. In terms of the total optical depth [(τ R, + τ oz, + τ wv, + τ D, ) m] relative to the extinction of solar radiation by Rayleigh scattering, ozone and water vapor absorption and extinction by aerosols, the direct beam can be expressed as: [ ( τ + τ + τ + τ ) m] d F dir S exp R, oz, wv, D, where S is the spectral radiance at the top of the atmosphere. 11

12 Direct radiation The formula for the direct radiation can be simplified by introducing the transmissivity, given (see class on Radiative Transfer ) as T which allows one to write: F dir exp ( ) τ S R, oz, wv, D, m ( T T T T ) d By defining the transmittance (transmission function) q as the average transmissivity of the atmosphere, such that: F dir q m S d the formula for the direct radiation reduces to: F dir Sq m S o a r 2 q m cosθ Note that the global average of q for a cloudless atmosphere has been estimated in ~.7. o 12

13 Distribution of selected fluxes of solar radiation All of the following picture refer to annual mean fluxes and were taken from Raschke and Ohmura (25). First we show a map of the net solar radiation at the top of the atmosphere. Raschke, E. and A. Ohmura, 25: Radiation budget of the climate system. In Hantel (ed.), Landolt-Börnstein, Group V, Vol. 6, Observed Global Climate, Springer, Berlin. 13

14 Distribution of the selected fluxes of solar radiation (2) Map of the global radiation at the earth s surface. Raschke, E. and A. Ohmura, 25: Radiation budget of the climate system. In Hantel (ed.), Landolt-Börnstein, Group V, Vol. 6, Observed Global Climate, Springer, Berlin. 14

15 Distribution of the selected fluxes of solar radiation (3) Map of the mean transmittance. Raschke, E. and A. Ohmura, 25: Radiation budget of the climate system. In Hantel (ed.), Landolt-Börnstein, Group V, Vol. 6, Observed Global Climate, Springer, Berlin. 15

16 Geometrical considerations 16

17 Radiance and irradiance Radiation falling on a surface at an angle θ from the normal to the surface gives at the surface an irradiance confined to the solid angle dω equal to N cosθ dω. The total irradiance at the surface is therefore the integral over the half-sphere of which the surface is the diametral plane: F N cosθdω In terms of polar co-ordinates θ and φ: F 2π dφ π/ 2 N cosθsin θdθ For isotropic radiation, that is when N is independent of direction, the integral reduces to: F N 2π dφ π/ 2 cosθsin θdθ π N 17

18 Blackbody radiation: Planck s law In 191 Planck was able to derive an analytical expression for the (unpolarized) radiant energy emitted by an enclosure in thermal equilibrium at an absolute temperature T (a blackbody) per unit volume per unit wavelength interval, the so-called Planck function: B where c h k Τ (T) 2 h c h c exp 1 k T 1 [W m m s -1, speed of light in vacuum J s, Planck constant J K -1, Boltzmann constant absolute temperature [K] Note that B has units of a spectral radiance. Spectral quantities are related to their total counterparts through: m sr ] N dn d 18

19 Spectrum of blackbody radiation The Planck function implies that the spectrum of blackbody radiation has a strong dependence on absolute temperature, as seen by contrasting the spectra for the solar (578 K ~ temperature of the photosphere) and terrestrial (255 K) emission. 19

20 Wien s and Stefan-Boltzmann s laws Given a spectrum of blackbody radiation, the wavelength max were the maximum emission takes place is found by setting db /d : max T K m This is Wien s displacement law. The total radiance is found by integrating B over all wavelengths: B(T) where σ B d σt π W m -2 K -4, Stefan-Boltzmann constant. Since the black-body emission is isotropic, the corresponding irradiance is: F(T) 4 2π π/ 2 σt dφ π cosθsin θdθ σt 4 This is Stefan-Boltzmann s law. 2

21 Kirchhoff s law Contrary to a blackbody, which absorbs all of the incident radiation, a socalled grey body reflects part of the incident radiation (with a reflectivity R ). Hence, in thermal equilibrium with a blackbody a grey body can only emit a radiant energy e B since (see figure) B e + r. The emissivity ε of a grey body is defined as: e ε 1 R A 1 B where R reflectivity absorptivity A Hence for a grey body ε A. This is Kirchhoff s law. An average emission coefficient ε can be defined by requiring that the total radiant energy emitted by a gray body follows Stefan-Boltzmann s law as: F π e d π ε B d π ~ ε B d εσ T 4 21

22 Emission coefficient Most natural surfaces have an emission coefficient ε ~ 1; for vegetation, for instance, ε.9 (see Oke, 1987 or Garratt, 1992). Snow, too, behaves in the longwave range almost like a blackbody, with ε.99 for fresh snow. Metals, on the other hand, are poor emitters: ε.3 for aluminium, ε.2 for iron, ε.2 for silver. The effective emissivity of the atmosphere, ε a, depends on the concentrations of the greenhouse gases and the presence of clouds. Note that the total content u of a constituent is usually expressed in terms of the total mass per unit cross-sectional area (in units of [kg m -2 ] but more typically [g cm -2 ]), which is the path integral of the density: u ρdz 22

23 Absorption spectra Note, however, that molecular compounds such as H 2 O and CO 2 do not emit or absorb continuously over the whole spectral range. The implications for the atmospheric absorption (emission) are shown in the following figure. Absorption spectra for various atmospheric gases between the top of the atmosphere and the surface. Peixoto, J.P. and A.H. Oort, 1992: Physics of Climate. American Institute of Physics, New York, 52 pp. 23

24 The atmospheric window This differential absorption/emission gives rise to the so-called atmospheric window. Courtesy of Rolf Philippona, MeteoSwiss, Payerne. 24

25 Short- and longwave radiation Solar radiation mainly stems from the sun s photosphere, which has an effective temperature of 58 K, more than an order of magnitude larger than the mean temperature at the earth s surface, 288 K. This contrast gives rise to a considerable difference in the Planck function. Solar radiation received at the Earth s surface has been weakened by its interaction with the atmosphere (see Solar Radiation ). The spectra of the incoming solar radiation (direct beam) and of the radiant energy emitted by the atmosphere are nevertheless distinct (see figure on the previous page), with only little overlap at approximately 4 µm. This limit is used to discriminate between short- and longwave radiation. In summary: < 4 µm shortwave or solar radiation > 4 µm longwave or terrestrial (incl. atmosphere) radiation. 25

26 Radiative transfer Radiation traversing a medium will be weakened by its interaction with matter. This interaction is called extinction or attenuation, an overall designation for the processes of absorption and scattering. We assume that the medium has a density ρ and is characterized by a mass extinction coefficient of k [m 2 kg -1 ]. ds N N + dn ρ, k According to the above figure and to first order: dn ρ k N ds 26

27 Beer-Bouguer-Lambert law If scattering and emission can be neglected: dn ρk ds With N (s ) N, the equation can be integrated to yield: N (s) N s N exp ρk ds If k is independent of s, then s N (s) N exp k ds ρ N exp where the optical path u has been defined as ( k u) This is Beer s or Beer-Bouguer-Lambert law. u s ρ ds. 27

28 Optical depth, transmissivity, absorptivity and reflectivity If k depends on s, then it is more convenient to define the so-called optical depth τ and the transmissivity (spectral transmittance) T as: τ (,s) s N k ρds and T exp( τ ) N For a non-scattering medium, the fraction of radiation absorbed by the medium is: A 1 T where A is the absorptivity. Note that all of the above are monochromatic or spectral quantities. If scattering takes place, a certain portion of the incident radiation can be reflected back into the incident direction. The ratio of the reflected (backscattered) to the incident intensity is called monochromatic reflectivity, R. In this case: T A + R

29 Clouds radiative properties Consider for instance the properties of water clouds as discussed by Stephens (1978). After: Stephens, G.L., 1978: Radiation profiles in extended water clouds. J. Atmos. Sciences, 35,

30 Scattering Scattering is the physical process by which a particle (or molecule) in the path of an electromagnetic wave continuously extracts energy from the incident wave and re-radiates that energy in all directions. In the atmosphere, the particles responsible for scattering range from gas molecules (~ 1-4 µm) to large raindrops and hail particles (~ 1 4 µm). We can broadly distinguish the following categories: solid aerosols (.1 to 1 µm), irregular shape, variable refractive index; haze water drops (.1 to 1 µm), spherical, known refractive index; cloud water drops (1 to 1 µm), spherical, known refractive index; cloud ice particles (1 to 1 µm), irregular shape, known refractive index. Based on the size of the scattering particles, we distinguish between Rayleigh scattering, particle diameter << wavelength of the incident beam Mie scattering*, particles diameter ~ wavelength of the incident beam * Mie theory provide a framework for describing scattering caused by spherical particles. 3

31 Directional dependence of scattering Rayleigh scattering is characterized by symmetry between forward and backward scattering. As the particles become larger, an increasing proportion of the incident radiation is scattered in the forward direction (Mie scattering). Sketches of the angular pattern of the scattered intensity from particles of various sizes are shown here. After: Liou, K.N., 22: An Introduction to Atmospheric Radiation (2nd Ed.). Academic Press, Amsterdam 31

32 Wavelength dependence of scattering Rayleigh scattering decreases with increasing wavelength of the incident beam according to: N N sc 4 where N and N sc are the total intensities of the incident and scattered radiation. This relation explains why the sky appear blue under cloudless conditions. Rayleigh scattering is also responsible for the albedo of the clear-sky atmosphere. It can be shown that the planetary albedo of a purely Rayleigh atmosphere is ~.2 (assuming a surface albedo of.16), which is somewhat higher than the observed clear-sky planetary albedo of.17. The dependence of Mie scatter on the wavelength of the incident beam is more complex. As a rule of thumb: N N sc ~

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