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Simple notion about radiation and radiation geometry.

Simple notion about radiation and radiation geometry.

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13 solar radiation Presentation Transcript

  • 1. Solar Radiation Physics and Geometry for hydrologists Il Sole, F. Lelong, 2008, Val di Sella Riccardo RigonMonday, December 10, 12
  • 2. When you see the Sun rise, do you not see a round disc of fire somewhat like a guinea? Oh no, no! I see an innumerable company of heavenly host crying “Glory, glory, glory is the Lord God Almighty.” W. Blake 2R. RigonMonday, December 10, 12
  • 3. Introduction Educational Goals • To recognise that the water cycle is powered by solar energy • To gain knowledge of the spatial and temporal variation of the radiation distribution on the Earth • To present the ways in which radiation is produced, received by the Earth, transmitted by the atmosphere, reflected, absorbed, and reemitted by the Earth’s surface • To introduce the concepts necessary to better understand the elements of the energy balance needed in remote-sensing applications, the snow balance, and evapotranspiration 3R. Rigon 1Monday, December 10, 12
  • 4. The Sun The Sun is the origin of the water cycle 4R. Rigon 2Monday, December 10, 12
  • 5. The Sun Composition of the Sun The Sun is mainly composed of hydrogen. The rest is prevalently He4. Hydrogen is the fuel for the nuclear fusion that takes place inside the Sun and produces helium. However, the He4 contained in the Sun for the most part originates from previous stellar lives. 5R. Rigon 3Monday, December 10, 12
  • 6. The Sun Sun Fact Sheet The Sun is a G2 type star, one of the hundred billion stars of this type in our galaxy (one of the hundred billion galaxies in the known universe). Diameter: 1,390,000 km (the Earth: 12,742 km or 100 times smaller) Mass: 1.1989 x 1030 kg (333,000 times the mass of the Earth) Temperature: 5800 K (at the surface) 15,600,000 K (at the core) The Sun contains 99.8% of the total mass of the Solar System (Jupiter contains nearly all the rest). Chemical composition: Hydrogen 92.1% Helium 7.8% 6 Other elements: 0.1%R. Rigon 4Monday, December 10, 12
  • 7. The Sun The Sun and the planets to scale 7R. Rigon 5Monday, December 10, 12
  • 8. The Sun The internal structure of the Sun The Sun’s energy is created in the core by fusing hydrogen into helium. This energy is irradiated through the radiative layer, then transmitted by convection through the convective layer, and, finally, radiated through the photosphere, which is the part of the Sun that we see. 8R. Rigon 6Monday, December 10, 12
  • 9. The Sun Provide a relatively constant rate of radiation energy that in few minutes from the cromosphere arrives to the Earth. Detail of a Pellizza da Volpedo Painting 9R. RigonMonday, December 10, 12
  • 10. The Sun Solar Spots Radiation flux is regular up to a point. In reality it manifests variations. Solar spots appear as dark spots on the surface of the Sun and they have a temperature of 3,700 K (to be compared to the 5,800 K of the surrounding photosphere). A solar spot can last for may days, the most persistent lasting for many weeks. 10R. Rigon 7Monday, December 10, 12
  • 11. The Sun Variability of the Emissions An image of the sun in X-ray band, taken by the Yohkoh solar observatory satellite, which shows changes in emissions of the solar corona   from a maximum in 1991 (left) to a minimum in 1995 (right). 11R. Rigon 8Monday, December 10, 12
  • 12. The Sun Variability of the Emissions Solar radiation is subject to fluctuations, some of which are localised in restricted areas, while others are more global and follow an 11-year cycle. Every 11 years the sun goes from a limited number of solar spots and flares to a maximum, and vice versa. During this cycle the Sun’s magnetic poles switch orientation. The last solar minimum was in 2006. 12R. Rigon 8Monday, December 10, 12
  • 13. The Sun Variability of the Emissions The graph shows the solar spot cycle over the last 400 years. It should be noted that before 1700 there was a period in which very few solar spots were observed. This period coincides with the Little Ice Age, which is why there are suggestions that there is a connection between solar spot activity and the climate on Earth. The most evident cycle has a period of 11 years. But there is a second cycle which seems to have a period of 55-57 years. 13R. Rigon 9Monday, December 10, 12
  • 14. The Sun The Stefan-Boltzmann law Every body with a temperature different than T=0 K emits radiation as a function of its temperature according to the Stefan-Boltzmann law R=✏ T4 14R. Rigon 10Monday, December 10, 12
  • 15. The Sun The Stefan-Boltzmann law Every body with a temperature different than T=0 K emits radiation as a function of its temperature according to the Stefan-Boltzmann law R=✏ T4 Radiation emitted 14R. Rigon 10Monday, December 10, 12
  • 16. The Sun The Stefan-Boltzmann law Every body with a temperature different than T=0 K emits radiation as a function of its temperature according to the Stefan-Boltzmann law R=✏ T4 emissivity Radiation emitted 14R. Rigon 10Monday, December 10, 12
  • 17. The Sun The Stefan-Boltzmann law Every body with a temperature different than T=0 K emits radiation as a function of its temperature according to the Stefan-Boltzmann law R=✏ T4 Stefan-Boltzmann constant emissivity Radiation emitted 14R. Rigon 10Monday, December 10, 12
  • 18. The Sun The Stefan-Boltzmann law Every body with a temperature different than T=0 K emits radiation as a function of its temperature according to the Stefan-Boltzmann law R=✏ T4 absolute temperature Stefan-Boltzmann constant emissivity Radiation emitted 14R. Rigon 10Monday, December 10, 12
  • 19. The Sun The physics of Radiation On the basis of the temperature of the Sun photosphere (~6000 K), and the Stephan-Boltzmann law, the total energy emitted by the Sun is RSun = ✏ T 4 = 1 ⇤ 5.67 ⇤ 10 8 ⇤ 60004 ⇡ 25.12 ⇤ 109 J m 2 s 1 15R. Rigon 11Monday, December 10, 12
  • 20. The Sun The Sun is nearly a “blackbody”! The Sun is practically a blackbody. The difference between a true blackbody and the Sun is due to the fact that the corona and the chromosphere selectively absorb certain wavelengths. 16R. Rigon 12Monday, December 10, 12
  • 21. The Sun The Sun is nearly a “blackbody”! The area below the curves is given by the Stefan-Boltzmann law. The curves themselves are given by Planck’s law. 17R. Rigon 13Monday, December 10, 12
  • 22. The Sun The complete electromagnetic spectrum Figure 2.9C.B. Agee The spectrum of solar radiation stretches far beyond the visible band where, however, nearly half the available energy is concentrated 18 R. Rigon 16 Monday, December 10, 12
  • 23. The Sun Planck’s Law •Planck’s law is the general law for electromagnetic emission from the surface of a blackbody*: 2⇡c2 h 5 W = ch [W m 2 µm 1 ] e KT 1 * Stefan-Boltzmann law is just the integration of Plank’s law over wavelengths 19R. Rigon 14Monday, December 10, 12
  • 24. From Sun To Earth From Sun to Earth The energy irradiated by the Sun passes through an imaginary disc with diameter the same as the Earth’s. The energy flow is maximum at that point on the Earth where the radiation is perpendicular. 20R. Rigon 18Monday, December 10, 12
  • 25. From Sun To Earth Solar radiation The Sun irradiates approximately at the solar constant rate, which is, on the average, on the top of the atmosphere, Frolich, 1985 http://en.wikipedia.org/wiki/Solar_constant 21R. Rigon 19Monday, December 10, 12
  • 26. Copying with Earth surface Astronomical variability of radiation In its orbit around the Sun, the Earth keeps its north-south rotational axis unvaried, causing a different angle between the Sun’s rays and the surface of the Earth. 22R. RigonMonday, December 10, 12
  • 27. From Sun To Earth Seasons Figure 3.1 The Earth is 5 million kilometers closer to the Sun during the northern winter: a clear indication that temperature is controlled more by orientation than by distance. 23R. RigonMonday, December 10, 12
  • 28. From Sun To Earth Corrections to the solar constant The Earth’s orbit around the Sun is an ellipse. The shape of the ellipse is determined by its eccentricity, which varies in time, changing the distances of the aphelion and perihelion 24 http://www.ascensionrecta.com/R. Rigon 20Monday, December 10, 12
  • 29. From Sun To Earth Precession of the polar axis The axis of rotation moves with a slow period, executing a complete precession every 26,000 years. Polar stars behave like this for only a very short period 25R. RigonMonday, December 10, 12
  • 30. From Sun To Earth Astronomical influences Orbit shape Orbit change Orbit angleR. Rigon 26Monday, December 10, 12
  • 31. From Sun To Earth Solar radiation in hydrological models Therefore the solar contant must be corrected S (e.g. Corripio, 2002): 27R. RigonMonday, December 10, 12
  • 32. From Sun To Earth Solar radiation in hydrological models Therefore the solar contant must be corrected S (e.g. Corripio, 2002): where: N is the day of the year (in 1, ..., 365) 28R. RigonMonday, December 10, 12
  • 33. Copying with Earth surface Radiation intensity Solar intensity governs seasonal climatic changes and the local climatic niches which are linked to the apparent height of the Sun. 29R. RigonMonday, December 10, 12
  • 34. Copying with Earth surface Insolation and latitude Figure 3.7 Incoming solar radiation is not evenly distributed across all lines of latitude, creating a heating imbalance. 30R. RigonMonday, December 10, 12
  • 35. Copying with Earth surface Radiative imbalance 31R. RigonMonday, December 10, 12
  • 36. Copying with Earth surface Radiation received from the Sun decreases towards the poles and it is reduced in areas where clouds form frequently For example, the complete energy balance is greater at the equator but the greatest amount of insolation is in the subtropical deserts Average annual radiation is < 80 W/m2 in the cloudy parts of the arctic and the antarctic >280 W/m2 in the subtropical deserts 50R. RigonMonday, December 10, 12
  • 37. Copying with Earth surface The geometry of radiation From a subjective point of view, the Sun varies its height in the sky seasonally. This is the subject of interest in the study of the geometry of radiation. 33R. RigonMonday, December 10, 12
  • 38. Copying with Earth surface To sum up Calculations of the incident radiation onto the surface of the Earth need to take account of the geometry of the interaction between the Sun’s rays and the surface of the Earth, which is curved and therefore variably exposed with respect to the direction of the Sun in function of latitude, time of day (longitude) and, naturally, day of the year. Moreover the Earth rotation is inclined with respect to its orbit around the Sun , and this causes seasons to happen. 34R. RigonMonday, December 10, 12
  • 39. Copying with Earth surface The geometry of radiation To calculate the aforementioned quantities it is usual to use a topocentric coordinate system, Nautic Almanac Office, 1974 that is, with the origin in the geographic position of the observer, which is right-handed and positioned on the plane tangent to the Earth’s surface in the considered point. N.B. - A coordinate system located at the centre of the Earth id called geocentric. 35R. RigonMonday, December 10, 12
  • 40. Copying with Earth surface The geometry of radiation The X-axis is, therefore, tangent to the earth and positive in a West-East direction. The Y-axis Nautic Almanac Office, 1974 is tangent in the North-South direction and is directed towards the South. The Z-axis lies on the segment joining the centre of the Earth with the point being considered on the surface. It is assumed that the Sun lies in the ZY plane at the solar noon. 36R. RigonMonday, December 10, 12
  • 41. Copying with Earth surface Solar Vector The solar vector can be expressed as a function of the angles that have been defined. The resulting trigonometric expression is: Z ⇥ sin ⇥ cos ⌥ = ⇤ sin ⇤ cos ⇥ cos s cos ⇤ cos ⌅ cos⇤ cos ⇥ cos + sin ⇤ sin Y X Therefore, to determine the position of the Sun one needs to know the latitude, the hour angle, and the solar declination. 37R. RigonMonday, December 10, 12
  • 42. Copying with Earth surface Hour angle The hour angle can be easily calculated as: ⇥ t ⇥= 1 12 if t is the solar hour 38R. RigonMonday, December 10, 12
  • 43. Copying with Earth surface Solar declination Is the angular height of Sun from the horizon at equator at noon* The solar declination is a function of the day of the year (and the era). It requires complex calculations that take account of the precession movements of the Earth. There are, however, various approximations. The one that is presented here is due to Bourges, 1985: where is the day of the year *http://en.wikipedia.org/wiki/Declination 39R. RigonMonday, December 10, 12
  • 44. Copying with Earth surface Projection on a plane at a certain latitude If is the vertical unit row-vector corresponding to the Z axis: Z and ⇥ sin ⇥ cos Y ⌥ = ⇤ sin ⇤ cos ⇥ cos s cos ⇤ cos ⌅ cos⇤ cos ⇥ cos + sin ⇤ sin X is the solar vector 40R. RigonMonday, December 10, 12
  • 45. Copying with Earth surface Projection on a plane at a certain latitude Then the projection of the solar irradiation on the plane YX is reduced by the factor where: Z or: Y X with the symbols explained above 41R. RigonMonday, December 10, 12
  • 46. Copying with Earth surface To sum up: The solar constant can be modified as follows. Was: Is now: 42R. RigonMonday, December 10, 12
  • 47. Absorption and transmission of short wave radiation Atmosphere is a gray body • The blackbody is an ideal object that absorb all the radiative energy it receives • Real objects (bodies, “gray bodies”) are not capable of absorbing all radiation. • To understand the difference between a blackbody and a gray body we need to analyse the interactions between a surface and the electromagnetic radiation incident onto it. 43R. RigonMonday, December 10, 12
  • 48. Absorption and transmission of short wave radiation Atmospheric absorption Radiation passes quite freely through the Earth’s atmosphere and it warms the surfaces of seas and oceans. A portion of between 45% and 50% of the incident radiation onto the Earth reaches the ground 44R. RigonMonday, December 10, 12
  • 49. Absorption and transmission of short wave radiation Shortwave Radiation budget The solar radiation penetrates the atmosphere, and it is transferred towards the ground, after being reflected and scattered. 45R. RigonMonday, December 10, 12
  • 50. Absorption and transmission of short wave radiation Shortwave Radiation budget Radiation reflected The solar radiation penetrates the atmosphere, and it is transferred towards the ground, after being reflected and scattered. 45R. RigonMonday, December 10, 12
  • 51. Absorption and transmission of short wave radiation Shortwave Radiation budget Radiation reflected The solar radiation penetrates the atmosphere, and it is transferred towards the ground, after being reflected and scattered. Radiation transmitted 45R. RigonMonday, December 10, 12
  • 52. Absorption and transmission of short wave radiation Shortwave Radiation budget S It should not be forgot that the radiation budget is an energy budget, for which the incoming radiation equals the reflected one plus the absorbed plus the transmitted 46R. RigonMonday, December 10, 12
  • 53. Absorption and transmission of short wave radiation Shortwave Radiation budget S It should not be forgot that the radiation budget is an energy budget, for which the incoming radiation equals the reflected one plus the absorbed plus Radiation the transmitted absorbed 46R. RigonMonday, December 10, 12
  • 54. Absorption and transmission of short wave radiation Shortwave Radiation budget S This budget can be apply to any slice of the atmosphere 47R. RigonMonday, December 10, 12
  • 55. Absorption and transmission of short wave radiation Shortwave Radiation budget S Corrected Solar constant This budget can be apply to any slice of the atmosphere 47R. RigonMonday, December 10, 12
  • 56. Absorption and transmission of short wave radiation Shortwave Radiation budget S Solar radiation reflected back to space Corrected Solar constant This budget can be apply to any slice of the atmosphere 47R. RigonMonday, December 10, 12
  • 57. Absorption and transmission of short wave radiation Shortwave Radiation budget S Transmitted radiation Solar radiation reflected back to space Corrected Solar constant This budget can be apply to any slice of the atmosphere 47R. RigonMonday, December 10, 12
  • 58. Absorption and transmission of short wave radiation Shortwave Radiation budget S Transmitted radiation Solar radiation reflected back to space Energy absorbed by atmosphere Corrected Solar constant This budget can be apply to any slice of the atmosphere 47R. RigonMonday, December 10, 12
  • 59. Absorption and transmission of short wave radiation Coefficients The following coefficients can also be defined • is the transmission coefficient, said atmospheric transmissivity • is the reflection coefficient, said atmospheric reflectivity (albedo) • is the absorption coefficient, said atmospheric absorptivity 48R. RigonMonday, December 10, 12
  • 60. Absorption and transmission of short wave radiation Shortwave Radiation budget Energy conservation: implies that reflectivity, transmissivity and absorptivity sum to one: Which is, indeed, valid for reflectivity, transmissivity and absorptivity of any other body 49R. RigonMonday, December 10, 12
  • 61. Absorption and transmission of short wave radiation Shortwave Radiation budget S 50R. RigonMonday, December 10, 12
  • 62. Absorption and transmission of short wave radiation Shortwave Radiation budget S We just forget for a moment this. It will be splitted into two parts: one depending on diffuse radiation and another on cloud cover 50R. RigonMonday, December 10, 12
  • 63. Absorption and transmission of short wave radiation Shortwave Radiation budget S 51R. RigonMonday, December 10, 12
  • 64. Absorption and transmission of short wave radiation Shortwave Radiation budget S Atmosphere is pretty transparent: which means that we can, as a first approximation, neglect it (atmosphere is heated from below) 51R. RigonMonday, December 10, 12
  • 65. Absorption and transmission of short wave radiation Shortwave Radiation budget S In any case let’s concentrate on the transmitted radiation This can be decomposed into two parts: direct and diffuse solar radiation 52R. RigonMonday, December 10, 12
  • 66. Absorption and transmission of short wave radiation Shortwave Radiation budget S Evidently, for simmetry is also composed by reflected and diffuse solar radiation 53R. RigonMonday, December 10, 12
  • 67. Absorption and transmission of short wave radiation Diffuse radiation comes from scattering Incident solar radiation strikes gas molecules, dust particles, and pollutants, ice, cloud drops and the radiation is scattered. Scattering causes diffused radiation. Two types of light diffusion can be distinguished: Mie scattering Rayleigh scattering 5R. RigonMonday, December 10, 12
  • 68. Absorption and transmission of short wave radiation Rayleigh Scattering •The impact of radiation with air molecules smaller than λ/π causes scattering (Rayleigh scattering) the entity of which depends on the frequency of the incident wave according to a λ-4 type relation. •In the atmosphere, the wavelengths corresponding to blue are scattered more readily than others. incident radiation diffuse radiation transmitted radiation 55R. RigonMonday, December 10, 12
  • 69. Absorption and transmission of short wave radiation Mie Scattering •When in the atmosphere there are particles with dimensions greater than 2 λ/π (gases, smoke particles, aerosols, etc.) there is a scattering phenomenon that does not depend on the wavelength, λ, of the incident wave (Mie scattering). incident radiation diffuse radiation transmitted radiation •This phenomenon can be observed, for example, in the presence of clouds. 56R. RigonMonday, December 10, 12
  • 70. Absorption and transmission of short wave radiation Diffused Light Scattering selectively eliminates the shorter visible wavelengths, leaving the longer wavelengths to pass. When the Sun is on the horizon, the distance travelled by a ray within the atmosphere is five or six times greater than when the Sun is at the Zenith and the blue light has practically been completely eliminated. 57R. RigonMonday, December 10, 12
  • 71. Absorption and transmission of short wave radiation Tilt of the Earth’s axis and atmospheric effects The tilt of the earth’s axis and atmospheric effects together affect the amount of radiation that reaches the ground. 58R. RigonMonday, December 10, 12
  • 72. Absorption and transmission of short wave radiation One way to take into account of absorption Would be to run a full model of atmospheric transmission (e.g. Liou, 2002). However hydrologists prefer to use parameterizations, and the concept of atmospheric transmissivity. 59R. RigonMonday, December 10, 12
  • 73. Absorption and transmission of short wave radiation Solar radiation transmitted to the ground under clear sky conditions S Finally: Corripio, 2002 60R. RigonMonday, December 10, 12
  • 74. Absorption and transmission of short wave radiation Solar radiation transmitted to the ground under clear sky conditions S Finally: Corripio, 2002 Fraction of direct solar radiation included between the considered 60 wavelengthsR. RigonMonday, December 10, 12
  • 75. Absorption and transmission of short wave radiation Solar radiation transmitted to the ground under clear sky conditions S Finally: Corripio, 2002 Transmittance of the atmosphere Fraction of direct solar radiation included between the considered 60 wavelengthsR. RigonMonday, December 10, 12
  • 76. Absorption and transmission of short wave radiation Solar radiation transmitted to the ground under clear sky conditions Correction due to S elevation of the site Finally: Corripio, 2002 Transmittance of the atmosphere Fraction of direct solar radiation included between the considered 60 wavelengthsR. RigonMonday, December 10, 12
  • 77. Absorption and transmission of short wave radiation Solar radiation transmitted to the ground under clear sky conditions S We do not enter in the details of how and are determined. Please look, for instance, at Formetta et al., 2012 61R. RigonMonday, December 10, 12
  • 78. Considering Clouds Hydrologists (and not only them) treat the influence of clouds separately It is assumed that the effects of clouds is an attenuation of the transmitted solar radiation Transmitted direct radiation at the surface after clouds correction 62R. RigonMonday, December 10, 12
  • 79. Considering Clouds Hydrologists (and not only them) treat the influence of clouds separately It is assumed that the effects of clouds is an attenuation of the transmitted solar radiation Transmitted direct Transmitted direct radiation at the surface radiation at the surface before clouds correction after clouds correction 62R. RigonMonday, December 10, 12
  • 80. Considering Clouds Hydrologists (and not only them) treat the influence of clouds separately An analogous formulation holds for diffuse radiation: 63R. RigonMonday, December 10, 12
  • 81. Considering Clouds Hydrologists (and not only them) treat the influence of clouds separately An analogous formulation holds for diffuse radiation: Correction coefficient for diffuse radiation 63R. RigonMonday, December 10, 12
  • 82. Considering Clouds Estimation of the reduction coefficients (decomposition model) These reduction coefficients can be determined when we have ground measurements of total radiation, diffuse plus direct: 64R. RigonMonday, December 10, 12
  • 83. Considering Clouds Estimation of the reduction coefficients (decomposition model) These reduction coefficients can be determined when we have ground measurements of total radiation, diffuse plus direct: Measured total radiation at the ground station i 64R. RigonMonday, December 10, 12
  • 84. Considering Clouds Estimation of the reduction coefficients (decomposition model) These assumption that is often made is that, the diffuse solar radiation measured at the station is proportional to the total radiation: 65R. RigonMonday, December 10, 12
  • 85. Considering Clouds Estimation of the reduction coefficients (decomposition model) These assumption that is often made is that, the diffuse solar radiation measured at the station is proportional to the total radiation: reduction coefficient for diffuse radiation 65R. RigonMonday, December 10, 12
  • 86. Considering Clouds Estimation of the reduction coefficients (decomposition model) Therefore when substituting this diffuse radiation expression in the total radiation equation of previous slides, it results at stations: 66R. RigonMonday, December 10, 12
  • 87. Considering Clouds Estimation of the reduction coefficients (decomposition model) And, for the direct radiation, at stations: 67R. RigonMonday, December 10, 12
  • 88. Considering Clouds The key factor is the to determine the above coefficient, on which the procedure followed so far has moved all the unknown. Its estimation pass through various parameterizations: Among the most known: •Erbs et al., 1982 •Reindl et al. 1990 •Boland et al. 2001 please find the details in Formetta et al., 2012 68R. RigonMonday, December 10, 12
  • 89. Considering Clouds One more issue With the help of the parameterizations above, the correction facotrs are determined for the stations. Which are a few points in a rugged terrain. How do you solve the problem to transport it everywhere ? 69R. RigonMonday, December 10, 12
  • 90. Considering Clouds We need to use some interpolation technique Like Kriging* or the Inverse distance weighting method** which is not the matter of the present slides. * Goovaerts, 1997 **Shepard, 1968 70R. RigonMonday, December 10, 12
  • 91. Hitting the terrain Finally the residual radiation hits the terrain The terrain is not a plane but it is inclined. Therefore, besides correcting radiation for latitude, longitude and hour, it is necessary to account for slope and aspect 71 R. RigonMonday, December 10, 12
  • 92. Hitting the terrain In the presence of topographic surfaces In the northern hemisphere, slopes that face South receive a greater insolation and, therefore, the water in the soil evaporates more quickly or the snow melts faster. Slopes with differing aspects are often characterized by different species and densities of plants and trees. 72 R. RigonMonday, December 10, 12
  • 93. Hitting the terrain Projection of radiation onto an inclined surface After Corripio, 2003 First we calculate the normal to the surface 73 R. RigonMonday, December 10, 12
  • 94. Hitting the terrain Projection of radiation onto an inclined surface Unit normal vector: ⇥ After Corripio, 2003 1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) ) ⇧ ⌃ 1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j) ⌃ ⇧u = n ⇧ z(i,j+1) z(i+1,j+1) ) ⌃ ⌃ |⇧ u | ⇤ n ⌅ l2 where z are the elevations of the four points used and l2 is the are of the cell - of side l. 74 R. RigonMonday, December 10, 12
  • 95. Hitting the terrain After Corripio, 2003 Representation of the vector normal to the surface of Mount Bianco 75 R. RigonMonday, December 10, 12
  • 96. Hitting the terrain Projection of radiation onto an inclined surface After Corripio, 2003 And we compare with the solar vector, indicating the direction of the Sun 76 R. RigonMonday, December 10, 12
  • 97. Hitting the terrain Projection of radiation onto an inclined surface ⇥ sin ⇥ cos ⌥ = ⇤ sin ⇤ cos ⇥ cos s cos ⇤ cos ⌅ cos⇤ cos ⇥ cos + sin ⇤ sin Where all the quantities were already defined previously 77 R. RigonMonday, December 10, 12
  • 98. Hitting the terrain Projection of radiation onto an inclined surface s After Corripio, 2003 Then we calculate the angle between the sun vector and the normal 78 R. RigonMonday, December 10, 12
  • 99. Hitting the terrain Projection of radiation onto an inclined surface We can define then the angle s of solar incidence After Corripio, 2003 79 R. RigonMonday, December 10, 12
  • 100. Hitting the terrain Projection of radiation onto an inclined surface Angle of solar incidence cos s = ⌅ · ⌅u s n ⇥ 1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) ) ⇧ ⌃ 1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j) ⌃ ⇧u = n z(i,j+1) z(i+1,j+1) ) ⌃ |⇧ u | ⇧ n ⇤ ⌃ ⌅ l2 ⇥ sin ⇥ cos ⌥ = ⇤ sin ⇤ cos ⇥ cos s cos ⇤ cos ⌅ cos⇤ cos ⇥ cos + sin ⇤ sin 80 R. RigonMonday, December 10, 12
  • 101. Hitting the terrain Projection of radiation onto an inclined surface The above angles needs to be compared with those of the terrain: Slope s = cos 1 nu.z Aspect (from the North anti-clockwise) 81 R. RigonMonday, December 10, 12
  • 102. Hitting the terrain Projection of radiation onto an inclined surface Remarkably the form of formula for the incident radiation is the same that for a flat surface when the projection angle is accounted: 82 R. RigonMonday, December 10, 12
  • 103. Hitting the terrain Solar radiation transmitted to the ground under clear sky conditions S Therefore, for the direct shortwave radiation: Corripio, 2002 as, it was before 83 R. RigonMonday, December 10, 12
  • 104. Hitting the terrain However, it is not just matter of light but also of shadows 84 R. RigonMonday, December 10, 12
  • 105. Hitting the terrain Incident radiation Topographic effects: shading More schematically light shadow 85 R. RigonMonday, December 10, 12
  • 106. Hitting the terrain Incident radiation Topographic effects: shading More schematically 86 R. RigonMonday, December 10, 12
  • 107. Hitting the terrain Incident radiation Topographic effects: shading More schematically shadow 86 R. RigonMonday, December 10, 12
  • 108. Hitting the terrain Incident radiation Topographic effects: shading More schematically light shadow 86 R. RigonMonday, December 10, 12
  • 109. Hitting the terrain Incident radiation Therefore the direct solar radiation must be corrected to include shading Details in Corripio, 2003 87 R. RigonMonday, December 10, 12
  • 110. Hitting the terrain What about diffuse radiation ? Topographic effects: angle of view 88 R. RigonMonday, December 10, 12
  • 111. Hitting the terrain What about diffuse radiation ? Topographic effects: angle of view sky view factor 88 R. RigonMonday, December 10, 12
  • 112. Hitting the terrain What about diffuse radiation ? Topographic effects: angle of view sky view factor diffuse radiation due to Rayleigh scattering 88 R. RigonMonday, December 10, 12
  • 113. Hitting the terrain What about diffuse radiation ? Topographic effects: angle of view sky view factor diffuse radiation due to diffuse Rayleigh radiation due to scattering aerosols 88 R. RigonMonday, December 10, 12
  • 114. Hitting the terrain What about diffuse radiation ? Topographic effects: angle of view sky view factor diffuse diffuse radiation due radiation due to diffuse multiple Rayleigh radiation due to scattering scattering aerosols 88 R. RigonMonday, December 10, 12
  • 115. Hitting the terrain Incident radiation Topographic effects: angle of view Any point in a rugged landscape see just a part of the sky sphere. Its fraction says which portion of the sky contribute to diffuse shortwave radiation. 89 R. RigonMonday, December 10, 12
  • 116. Hitting the terrain Incident radiation Topographic effects: angle of view Different points view a different sky 90 R. RigonMonday, December 10, 12
  • 117. Hitting the terrain The sum 91 R. RigonMonday, December 10, 12
  • 118. Hitting the terrain Now it really hits the terrain and, in part, it is reflected away After Corripio, 2003 92 R. RigonMonday, December 10, 12
  • 119. Hitting the terrain Finally a map After Corripio, 2003 Insolation received by Mont Blanc at Spring Equinox 93 R. RigonMonday, December 10, 12
  • 120. Albedo Typical albedo values http://en.wikipedia.org/wiki/Albedo 94R. RigonMonday, December 10, 12
  • 121. Albedo Typical albedo values http://en.wikipedia.org/wiki/Albedo 95R. RigonMonday, December 10, 12
  • 122. Spectral response Spectral Signature (or Response) The percentage of radiation that is reflected (reflectance) depends on wavelength of the radiation, and on the geometry, nature, and structure of the surface under investigation. 51 96R. RigonMonday, December 10, 12
  • 123. Spectral response •In the case of solar radiation, the spectral signature is defined as the reflectance of the surface in function of the wavelength. 97R. RigonMonday, December 10, 12
  • 124. Spectral response •Every type of surface can be statistically characterised by a spectral signature. 98R. RigonMonday, December 10, 12
  • 125. Spectral response Factors •The spectral signature of a specific element of a territory will vary due to the variability of local environmental factors. •Given a certain type of ground cover, static elements, such as slope and exposition, and dynamic elements, such as surface ground humidity, the phenological state of the vegetation, the atmospheric transparence, etc., will cause variations in the spectral signature curve. 99R. RigonMonday, December 10, 12
  • 126. Spectral response Radiation that hits the terrain, heats it. Or causes changes of phase water to vapor ice to water 100R. RigonMonday, December 10, 12
  • 127. Spectral response Or is used for photosynthesis or other chemical reactions 101R. RigonMonday, December 10, 12
  • 128. Long wave radiation Earth “is” a gray body Having a temperature emits radiation A. Adams - Part of the snake river picture 102R. RigonMonday, December 10, 12
  • 129. Long wave radiation Gray Bodies • Plank’s Law for gray bodies: 2⇡c h 2 5 W =✏ ch [W cm 2 µm 1 ] e KT 1 • The Stefan-Boltzmann equation for gray bodies: W = ✏ T [W cm 4 2 ] where ε is the average emissivity calculated over the entire electromagnetic spectrum. 103R. RigonMonday, December 10, 12
  • 130. Long wave radiation Gray Bodies The behavior of a real (gray) body is related to that of a black body by means of the quantity ελ, known as the emission coefficient or emissivity, which is defined as: W (real body) ✏ = W (black body) Kirchhoff (1860) demonstrated that a good “radiator” is also a good “absorber”, that is to say: ↵=✏ ⇢+⌧ +✏=1 104R. RigonMonday, December 10, 12
  • 131. Long wave radiation Comparison of blackbody and gray body In reality emissivity depends, at least, on wavelength. Earth should be probably defined a selective radiator 105R. RigonMonday, December 10, 12
  • 132. Long wave radiation See the Earth as gray body and given that the temperature of the Earth’s surface is, on average, about 288 K, it obviously emits a spectrum of radiation in the infrared band. 106R. RigonMonday, December 10, 12
  • 133. Long wave radiation Radiation emitted by the Sun and the Earth Yochanan Kushnir 107R. RigonMonday, December 10, 12
  • 134. Long wave radiation See the Earth as gray body and given that the temperature of the Earth’s surface is, on average, about 288 K, it obviously emits a spectrum of radiation in the infrared band. Atmosphere is not anymore transparent to at these wavelengths. 108R. RigonMonday, December 10, 12
  • 135. Long wave radiation The atmosphere is warmed from below Therefore the temperature is higher at ground level than it is at higher altitudes. 109R. RigonMonday, December 10, 12
  • 136. Long wave radiation Greenhouse Effect In the absence of atmospheric absorption the average temperature of the Earth’s surface would be about -170C. 110R. RigonMonday, December 10, 12
  • 137. Long wave radiation Greenhouse Effect Instead the average temperature is about 15 0C 111R. RigonMonday, December 10, 12
  • 138. Long wave radiation Radiative heating is completed by convective heat transfer, and by water vapor fluxes (latent and sensible heat). But this you can see better on the energy budget slides. 112R. RigonMonday, December 10, 12
  • 139. Long wave radiation But now concentrate on the surroundings of a point After Helbig, 2009 Any point being at a certain temperature emits long wave radiation which must be accounted for 113R. RigonMonday, December 10, 12
  • 140. Long wave radiation The atmosphere emits infrared itself bacause of its temperature 114R. RigonMonday, December 10, 12
  • 141. Long wave radiation All the contributions Long-wave radiation is given by the balance of incident radiation from the atmosphere and the radiation emitted by the ground. Both values are calculated with the Stefan- Boltzmann law. 115R. RigonMonday, December 10, 12
  • 142. Long wave radiation Longwave (infrared) raditation Topographic effects: angle of view 116R. RigonMonday, December 10, 12
  • 143. Long wave radiation Longwave (infrared) raditation Topographic effects: angle of view Longwave radiation coming from sky 116R. RigonMonday, December 10, 12
  • 144. Long wave radiation Longwave (infrared) raditation Topographic effects: angle of view Longwave radiation Longwave radiation coming from sky coming from surrounding 116R. RigonMonday, December 10, 12
  • 145. Long wave radiation Longwave (infrared) raditation Topographic effects: angle of view Longwave radiation Longwave radiation Radiation losses coming from sky coming from by the area under surrounding exam 116R. RigonMonday, December 10, 12
  • 146. Long-wave radiation The first component should be calculated by integrating the formula over the entire atmosphere, but, given how complex this process is, typically an empirical formula is used that uses the value of air temperature as measured near ground level (2m) and a value of the atmospheric emissivity based on specific humidity, temperature, and cloudiness. The second component, on the other hand, is function of the surface temperature and its emissivity. 117R. RigonMonday, December 10, 12
  • 147. Long wave radiation Long-wave radiation The real process: The hydrological parameterisation: 118R. RigonMonday, December 10, 12
  • 148. Long wave radiation Long-wave radiation The real process: The hydrological parameterisation: Global emissivity of the atmosphere 118R. RigonMonday, December 10, 12
  • 149. Long wave radiation Long-wave radiation The real process: The hydrological parameterisation: Temperature at 2 m from ground Global emissivity of the atmosphere 118R. RigonMonday, December 10, 12
  • 150. Long wave radiation Parameterisation of Long-wave radiation The hydrological parameterisation: 6 4 εatm = εBrutsaert (1− N ) + 0.979N Brutsaert (1975) + Pirazzini et al. (2000) εatm = εBrutsaert (1+ 0.26N) Brutsaert (1975) + Jacobs (1978) 6 4 εatm = εIdso (1− N ) + 0.979N Idso (1981) + Pirazzini et al. (2000) 6 4 εatm = εIdso,corr (1− N ) + 0.979N Hodges et al. (1983) + Pirazzini et al. (2000) where N is the fraction of sky covered by clouds 119R. RigonMonday, December 10, 12
  • 151. Net Radiation The sum of longwave and shortwave ratio is called net radiation 120R. RigonMonday, December 10, 12
  • 152. 1Thank you for your attention ! G.Ulrici - 2000 ? 121 R. RigonMonday, December 10, 12
  • 153. Table of symbols 122R. RigonMonday, December 10, 12
  • 154. Table of symbols 123R. RigonMonday, December 10, 12
  • 155. Table of symbols 124R. RigonMonday, December 10, 12
  • 156. Projection of radiation onto an inclined surface 125R. RigonMonday, December 10, 12
  • 157. The geometry of radiation 126R. RigonMonday, December 10, 12