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CEGE 4501 Hydrologic Design Chapter 3: Solar Radiation and Surface Energy Balance Ardeshir Ebtehaj Department of Civil, Environmental, and Geo-Engineering September 27, 2021

Outline Introduction Some Radiation Laws Radiation Partitioning Solar radiation at top of the atmosphere Net Radiation at Earth’s Surface Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj

Radiative Energy II Electromagnetic waves are formed by elementary particle, called photons , travel at the speed of light at different wavelengths λ [m]: λ = cT , where c = 3 × 10 8 [m s -1 ] is the speed of light, T is the wave period [s -1 ]. Often, we express λν = c , where ν = 1 / T is the frequency in Hertz [Hz]. Figure 1: Wavelength is the distance between two crests of a wave. The electromagnetic energy (EM) spreads across different wavelengths forming a spectrum shown in the following figure. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 2

Radiative Energy III Figure 2: Electromagnetic (EM) spectrum The wavelength of interests for hydrologists may include ultra-violet, visible, infrared, and microwave bands . It is worth nothing that almost 50% of the earth heating is coming from the Sun’s radiative energy in infrared bands , which has important implications on global warming. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 3

Some Radiation Laws I A blackbody is a hypothetical physical object that absorbs all incident EM radiation. Planck’s law: The emitted radiation by a blackbody at temperature T [K], in terms of spectral radiances, as a function of wavelength ( λ ) or frequency ( ν ) is expressed as follows: B λ = 2 hc 2 λ 5 e hc λκ B T - 1 [ Wm -2 sr -1 m -1 ] or B ν = 2 h ν 3 c 2 e h ν κ B T - 1 [ Wm -2 sr -1 Hz -1 ] , where κ B = 1 . 38064 × 10 - 23 [J K -1 ] is the Boltzmann constant and h = 6 . 626 × 10 - 34 [J s -1 ] represents the Planck’s constant . Figure 4: Planck’s law as a function of temperature and wavelength. Note that the y-axis is in logarithm scale. The yellow and red curves denote the law for average temperature of the Sun and Earth. The blue line shows the Wien’s displacement law explained below. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 5

Some Radiation Laws II Wien’s displacement law: explains the wavelengths over which the Plank’s law is maximum as a function of temperature T: ∂ B λ ∂λ = 0 ⇒ λ max = 2897 T [ μ m] , The Stefan-Boltzmann law can be obtained by integrating the Planck’s law in irradiance units (4 π B λ ) over all wavelengths or frequencies as follows: B = λ 4 π B λ d λ = σ T 4 . [Wm -2 ] As a result, one can see that the Stefan-Boltzmann law describes radiative energy as irradiance across all wavelengths , where σ = 5 . 67037 × 10 - 8 [Wm -2 K -4 ] is the Stefan-Boltzmann constant. Graybody: Unlike a blackbody, natural objects are gray bodies, as they only absorb and thus emit a fraction of incident radiation . Radiation of a graybody is different than a blackbody up to a proportionality constant called the emissivity : R λ = λ B λ , where the dimensionless coefficient 0 < λ < 1 is called spectral emissivity . As is clear, emissivity is an spectral quantity, which varies at different wavelengths so one can integrate spectral emissivity over all wavelengths = λ λ d λ to obtain the so-called broadband emissivity . As a result, the Stefan-Boltzmann law for a graybody can be expressed as follows: R = σ T 4 . [Wm -2 ] Kirchhoff’s Law: For an object with constant temperature, the absorbed radiative energy is equal to the emitted radiative energy. Therefore, absorptivity and emissivity are equal a λ = λ , when an object is in thermodynamic equilibrium. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 6

Radiation Partitioning II It can be shown that the solar radiation above 4 μ m is almost negligible . Therefore, by convention, we have: Shortwave radiation = 4 μ m 0 . 1 R λ d λ Solar , Longwave radiation = 200 μ m 4 R λ d λ Terrestrial [Wm -2 ] . Shortwave radiation contains : ultra-violet + visible + near-infrared + mid-infrared Longwave radiation contains: mid-infrared + far-infrared Radiation Budget: Any incident radiation upon a medium will be absorbed, scattered, or transmitted at the boundary. Absorption is a process that removes the radiant energy from an EM field and transfers it into the internal energy of the absorber. Scattering or Reflection is a process that does not remove the energy from the EM field, but redirects it. Scattering often occurs through a volume (atmospheric depth). However, reflection occurs at the interface of two materials that could be diffuse and/or specular (see Figure 7) Extinction or Attenuation is total effect of absorption + scattering. Transmission is the radiant energy that passes through a medium without any attenuation. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 8

Radiation Partitioning III The radiative energy balance at an interface can be written as: R λ = A λ + S λ + T λ ⇒ A λ R λ + S λ R λ + T λ R λ = 1 ⇒ a λ + s λ + τ λ = 1 , where a λ is absorptivity , s λ is reflectivity and τ λ denotes transmissivity . Clearly, the radiative balance holds for both short and longwave radiation as follows: a s + s s + τ s = 1 (shortwave) a l + s l + τ l = 1 (longwave) Figure 6: Absorption, Scattering/Reflection, and Transmission. Often some of the terms in the above radiation balance equation are negligible: Within the atmosphere longwave scattering is often negligible during the clear sky condition, hence, a l + τ l = 1 (see, Figure 8). At the Earth’s surface both long and shortwave transmissivity are negligible. Therefore, for shortwave radiation, we have a s + s s = 1. The shortwave reflectivity coefficient at the Earth surface is called albedo and denoted by α in hydrologic community, which leads to α = 1 - a s . Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 9

Solar radiation at top of the atmosphere I To understand hydrologic cycle, it is key to know that how much of the Sun’s energy will reach to the earth’s surface and what will be its distribution. Using the Stefan-Boltzmann law, we have: R = σ T 4 = 5 . 67 × 10 - 8 × 5780 4 = 6 . 33 × 10 7 [W m -2 ] . In far-field radiation, basically it is reasonable to assume that the radiation beams are parallel. Therefore, given that: d =1.496 × 10 6 [km] (Sun-Earth distance) r s =696,300 [km] (Sun radius) r e =6371 [km] (Earth radius), Assuming that r e r s d , one can easily compute the intensity of solar radiative energy at top of the atmosphere as follows: Solar Constant = I sc ≈ 6 . 33 × 10 7 × r 2 s d 2 ≈ 1367 [W m -2 ] Figure 9: Far field solar radiation and solar constant at TOA. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 11

Solar radiation at top of the atmosphere II Note: recent detailed calculation by NASA using satellite observations shows that the solar constant varies between 1361-1362 [Wm -2 ] at the top of the atmosphere. However, it is still customary to use 1367 [W m -2 ] in hydrologic analysis as the solar constant . Question: Why does the earth surface receive a non-uniform amount of radiative energy? Answers: - Earth is an ellipse (not a flat plane) and rotates around its central axis passing through its poles. - Earth’s axis of rotation is tilted by 23 . 5 ◦ with respect to its orbital plane, which is the main reason for experienced seasonality in the earth’s climate. - Earth’s orbit is slightly elliptical and thus the earth’s distance to the sun is variable and thus the lengths of days and nights keep changing throughout the year. Figure 10: Schematic of global insolation and effects of earth geometry on distribution of the solar radiation on earth surface (left). The earth axis of rotation is tilted with respect to its orbital plane, which is in fact the main reason for seasonality of earth’s climate. Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 12

Solar radiation at top of the atmosphere IV Figure 12: Summer and winter solstices and equinoxes. Summer Solstice (June 20 or 21) refers to the longest day in the Northern Hemisphere. As is evident, summer solstice occurs while the Earth Northern Hemisphere is at its closest distance to the Sun and receives its maximum annual daily radiation budget. Analogously, while the Northern Hemisphere is at its longest distance with the Sun throughout the year, it experiences the longest night of the year. The day of Dec 21 or 22 is the Winter Solstice in the Northern Hemisphere while it is summer solstice in the Southern Hemisphere. Note that while it is summer solstice in the Northern Hemisphere, it is winter solstice in the Southern Hemisphere with the longest night of the year. When the Earth’s celestial/equatorial plane passes through the center of the Sun, the day and night are equal. These days (Sep 22 or 23 or Mar 20 or 21) are called Equinoxes . Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 14

Net Radiation at Earth’s Surface I In order to understand hydrologic fluxes at the surface, we need to know that how solar radiation partitions between the Earth’s surface and its atmosphere. This partitioning is dynamic depending on the earth rotation and atmospheric composition such as cloud and snow cover . Recent analysis of the attenuation of the incoming solar radiative energy within the Earth’s atmosphere shows that, on average , earth has a planetary albedo of around 30% , while nearly 23% of solar insolation is absorbed by the atmosphere and clouds , leaving around 47% of the insolation reaching the Earth’s surface (Trenberth, 2009). Figure 13: Estimates of planetary radiation attenuation (Trenberth,2009) Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 15

Net Radiation at Earth’s Surface III The net radiation at the earth surface is partitioned into three components to create the Surface Energy Balance (SEB) : R n = LE + H + G [W m - 2 ] where LE is the latent heat flux , H is the sensible heat flux , and G is the ground heat flux . We will discuss these very important components in the lessons on evaporation. However, as way of review, below is a figure of the estimated global energy budget containing the components discussed in this section. Figure 15: The global annual mean Earth’s energy budget for 2000 to 2005 [W m -2 ] (from Trenberth et al. 2009). Physical Hydrology, Chapter 3: Solar Radiation and Earth Energy Balance Ardeshir Ebtehaj 17