Influence of Atmospheric Solar Radiation Absorption on Photodestruction of Ions at D-Region Altitudes of the Ionosphere

Abstract

The influence of atmospheric solar radiation absorption on the photodetachment, dissociative photodetachment, and photodissociation rate coefficients (photodestruction rate coefficients) of O⁻, Cl⁻, O₂ ⁻, O₃ ⁻, OH⁻, NO₂ ⁻, NO₃ ⁻, O₄ ⁻, OH⁻(H₂O), CO₃ ⁻, CO₄ ⁻, ONOO⁻, HCO₃ ⁻, CO₃ ⁻(H₂O), NO₃ ⁻(H₂O), O₂ ⁺(H₂O), O₄ ⁺, N₄ ⁺, NO⁺(H₂O), NO⁺(H₂O)₂, H⁺(H₂O) _n for n = 2–4, NO⁺(N₂), and NO⁺(CO₂) at D-region altitudes of the ionosphere is studied. A numerical one-dimensional time-dependent neutral atmospheric composition model has been developed to estimate this influence. The model simulations are carried out for the geomagnetically quiet time period of 15 October 1998 at moderate solar activity over the Boulder ozonesonde. If the solar zenith angle is not more than 90° then the strongest influence of atmospheric solar radiation absorption on photodestruction of ions is found for photodissociation of CO₄ ⁻ ions when CO₃ ⁻ ions are formed. It follows from the calculations that decreases in the photodestruction rate coefficients of ions under consideration caused by this influence are less than 2 % at 70 km altitude and above this altitude if the solar zenith angle does not exceed 90°.

Appendices

Appendix 1

1.1 One-Dimensional Time-Dependent Neutral Atmospheric Composition Model

The one-dimensional time-dependent atmospheric neutral composition model calculates the O(³P), O(¹D), O₃, O₂(¹Δ_g), H, OH, HO₂, H₂O₂, N, N(²D), NO, NO₂, NO₃, N₂O₅, HNO₃, and HO₂NO₂ number densities at altitudes from the ground to 400 km altitude. The continuity equation determining the neutral number density, N _n , of each species, n, is given by

$$ \frac{\partial }{\partial t}N_{n} + \frac{\partial }{\partial z}\left[ {N_{n} \left( {V_{nz} + V_{nz}^{t} } \right)} \right] = P\left( n \right) - L\left( n \right), $$

(15)

where V _nz is the vertical component of a hydrodynamic diffusion velocity, V _n ; V ^t _nz is the vertical component of an eddy diffusion velocity, V ^t _n , due to turbulent mixing of neutral species, P(n) and L(n) are the neutral production and loss rates by dissociation and dissociative ionization of neutral components of the atmosphere by solar radiation, galactic cosmic rays and by chemical reactions of neutral species, z is an altitude, and t is a time.

Processes of photodissociation of O₂, O₃, H₂O, N₂, NO, NO₂, NO₃, N₂O₅, HNO₃, N₂O and dissociative ionization of N₂ by solar radiation under consideration are presented in Table 2. The production rate, P _sol(n,r), of a neutral species, n, due to photodissociation or dissociative ionization of a neutral species, r, by solar radiation is calculated as

$$ P_{\text{sol}} \left( {n,r} \right) = N_{r} \sum\limits_{j} {\sigma_{\text{euv}}^{\text{dis}} (r,n,\lambda_{j} )\varPhi (\lambda_{j} ),} $$

(16)

where Φ(λ _j ) is the solar flux in the wavelength range λ _j at a point of the atmosphere under consideration determined by Eqs. (3)–(11), $\sigma_{\text{euv}}^{\text{dis}} (n,r,\lambda_{j} )$ is the photodissociation or dissociative ionization cross section of a neutral species, r, with the formation of a neutral species, n.

Table 2 Model reactions of O₂, O₃, H₂O, N₂, NO, NO₂, NO₃, N₂O₅, HNO₃, HO₂NO₂, and N₂O photodissociation, dissociation of N₂ by cosmic rays, p, and dissociative ionization of N₂ by solar radiation and cosmic rays

The loss rate, L _sol(r,n), of a neutral species, r, caused by photodissociation or dissociative ionization of this neutral species with formation of a neutral species, n, is equal to P _sol(n,r).

It follows from the detailed theoretical energy degradation computations (Porter et al. 1976) that collisions of energetic protons with N₂ in air result in production of 0.538 N(⁴S) atoms per ion pair and 0.381 N(²D) atoms per ion pair, and these N(⁴S) and N(²D) branching ratios are used in the model for collisions of cosmic rays with N₂ in the atmosphere. At altitudes above 18 km, the total ion production rate due to the galactic cosmic rays is calculated using the parameterization formula given by Heaps (1978) and described more precisely by Brasseur and Solomon (2005). This total production rate is taken as zero at the Earth’s surface and is assumed to decrease linearly below 18 km altitude.

It is necessary to point out that the simplified approach by Heaps (1978) was used, for example, in model simulations of electron and ion densities at altitudes of the stratosphere (Beig et al. 1993a, b) and mesosphere (Garcia et al. 1987; Turunen et al. 1996).

The model uses the parameterizations of the mean photoabsorption cross section of O₂ at 121.6 nm wavelength (Lyman-α) and the O(¹D) and O(³P) quantum yields given by Reddmann and Uhl (2003). Laboratory measurements (Yoshino et al. 2005) of the absorption cross section of the Schumann–Runge continuum of O₂ in the wavelength region of 129.62–175 nm at the temperatures of 90 and 295 K are used, and the value of this cross section at other temperatures and in the wavelength range of 126–129.62 nm is found by linear interpolation or extrapolation of these measurements. The O(¹D) and O(³P) quantum yields given by Lee et al. (1977) and Nee and Lee (1997) are used in the Schumann–Runge continuum wavelength region. The Kockarts (1994) approach is employed to calculate the photodissociate rate of O₂ in the Schumann–Runge bands of molecular oxygen (175–205 nm). The O₂ photodissociation cross section at wavelengths from 190 to 240 nm (Herzberg continuum) (Yoshino et al. 1988, 1992) and in the wavelength range of 240–245 nm (Fally et al. 2000) is included in the model. It is taken into account in the model that the O₂ photodissociation results in two atoms of O(³P) at wavelengths between 175 and 245 nm (Sander et al. 2011).

The photodissociation cross section of O₃ measured by Serdyuchenko et al. (2013) in the wavelength range of 213–1100 nm at 293, 283, 273, 263, 253, 243, 233, 223, 213, 203, and 193 К temperatures is taken by Internet from www.iup.uni-bremen.de/gruppen/molspec/databases/referencespectra/. The value of this cross section at other temperatures is found in this wavelength range by linear interpolation or extrapolation of these measurements. The O₃ photodissociation cross section measured by Molina and Molina (1986) at 226 and 298 K temperatures are used in the 185- to 213-nm wavelength range, and the parameterization approach (Bogumil et al. 2003) applied to the cross section given by Molina and Molina (1986) is employed to calculate this cross section at other temperatures in this wavelength range. The O(¹D) and O(³P) quantum yields in the ozone photodissociation reactions shown in Table 2 are taken according to the recommendations given by Sander et al. (2011).

Ozone photodissociates in the 200–310 nm Hartley band to give a yield of O₂(¹Δ_g) close to 0.9 (Slanger and Copeland 2003). It was pointed out by Matsumi and Kawasaki (2003) that there is no information on the O₂(b¹Σ ⁺_g ) quantum yield in ozone photodissociation below 267 nm wavelength. The probabilities of the O₂(¹Δ_g) + O(¹D) and O + O₂ ozone photodissociation channels are calculated to be equal to 90–95 and 10–5 % in the Hartley band, respectively (Schinke and McBane 2010). As a result, the equality of the O₂(¹Δ_g) and O(¹D) quantum yields for the O₃ + hν channel in the Hartley band is assumed, and only this production of O₂(¹Δ_g) by photodissociation of O₃ is taken into account in the model. It should be noted that this approach was used, for example, by Mlynczak et al. (2007) and Zhu et al. (2007).

The dissociation cross section of H₂O by solar radiation and the O(¹D), O(³P), H, and OH quantum yields recommended by Sander et al. (2011) for the wavelength range of 121–198 nm are used in the model. Absorption of solar radiation in the Schumann–Runge band of molecular oxygen is taken into account in the dissociation rate of water in the frame of the Kockarts (1994) approach.

The total photodissociation cross section of H₂O₂ and the 2OH and H + HO₂ product yields recommended by Sander et al. (2011) for the wavelength range of 190–350 nm are employed in the model simulations.

The total dissociation cross section of HO₂ by solar radiation is taken according to the recommendations given by Sander et al. (2011) for the wavelength range of 190–260 nm.

Detailed calculations of high-resolution photodissociation cross sections of N₂ computed using a coupled-channel Schrödinger equation quantum–mechanical model were carried out by Lavvas et al. (2011). It was shown by Lavvas et al. (2011) that photodissociation of N₂ produces N + N(²D) and N + N(²P) in the wavelength ranges of 89.1–102.1 and 85.4–89.1 nm, respectively, and N(²D) + N(²D) at wavelengths shorter than 89.1 nm. To the best of the author’s knowledge, there are no published measurements or theoretical estimates of the quantum yields of N(⁴S) and N(²D) in the N₂ dissociative photoionization reaction, and the branching for N(⁴S) and N²(D) productions is assumed to be 50 and 50 % in this reaction. This simplified approach and the dissociation and dissociative ionization cross sections of N₂ by solar radiation (Solomon and Qian 2005) are used in the model.

Photodissociation of nitric oxide in the δ(1,0) and δ(0,0) bands is calculated in the frame of the Nicolet (1979) approach.

The photodissociation cross section of NO₂ in the wavelength range of 205–650 nm at the temperature of 298 K and the O and O(¹D) quantum yields in the NO₂ photodissociation reaction for the wavelength range of 300–417 nm at the temperatures of 248 and 298 K (Warneck and Williams 2012) are used in the model. The values of the O and O(¹D) quantum yields at other temperatures are found by linear interpolation or extrapolation of these quantum yields. It is believed that photodissociation of NO₂ produces only O and NO at wavelengths shorter than 300 nm and only O(¹D) and NO at wavelengths longer than 422 nm (Sander et al. 2011; Warneck and Williams 2012).

The total photodissociation cross section of NO₃ is given by Sander et al. (2011) in the wavelength range of 403–640 nm. The NO + O₂ and NO₂ + O quantum yields in the NO₃ photodissociation reaction are recommended for use by Sander et al. (2011) for the wavelength range of 585–640 nm at temperatures of 190, 230, and 298 K. The linear interpolation or extrapolation approach is used to find the values of the cross section and quantum yields at other temperatures. The NO + O₂ and NO₂ + O quantum yields in the NO₃ photodissociation reaction are equal to zero at wavelengths shorter than 403 nm and longer than 640 nm (Sander et al. 2011).

The total dissociation cross section of N₂O₅ by solar radiation and the NO₃ + NO₂ and NO₃ + NO + O photoproduct yields are taken according to the recommendations given by Sander et al. (2011) for the wavelength range of 200–420 nm. The photodissociation cross sections of HNO₃, N₂O, and HO₂NO₂ are calculated in the wavelength ranges of 192–530, 160–240, and 190–350 nm in accordance with the recommendations given by Sander et al. (2011). The HO₂NO₂ photolysis quantum yields of the HO₂ + NO₂ and OH + NO₃ products used are the same as those recommended by Sander et al. (2011).

The set of chemical reactions included in the model and the corresponding rate coefficients are shown in Table 3. The reaction rate coefficients recommended by Sander et al. (2011) are used in the model, except for reactions 1–3, 10, 11, 15, 26, 31, 34, 49, 50, and 55–67 of Table 3 where rate coefficients are taken from Smith and Robertson (2008), Lin and Leu (1982), Kalogerakis et al. (2009), Fischer and Tachiev (2004), Baulch et al. (2005), Altinay and Macdonald (2014), Lafferty et al. (1998), Smith and Newnham (2000), Nair and Yee (2009), Atkinson et al. (2003), Duff et al. (2003), Li et al. (2014), Herron (1999), Homayoon et al. (2014), Fell et al. (1990), Butler and Zeippen (1984), Alecu and Marshall (2014), Mellouki et al. (1981), Wine et al. (1981), and Gierczak et al. (2004). It should be noted that the low-pressure-limiting rate coefficients are presented in Table 3 for termolecular association reactions 1–3, 7, 12, 27, 41, 42, and 47–50 marked by asterisks. The high-pressure-limiting rate coefficients of reactions 12, 27, 41, 42, and 47–50 are given by Sander et al. (2011), and the generally excepted formula (see, e.g., Brasseur and Solomon 2005; Sander et al. 2011) is used in the model to calculate the resulting rate coefficients of these reactions. The termolecular rate coefficient of reaction 71 is represented by Sander et al. 2011 as K ₇₁ = C ₀(1 + Y)⁻¹ × 0.6^Z, where Y = C ₀[M]/C ₁, C ₀ = 1.5 × 10⁻¹³(T _n /300)^0.6, C ₁ = 2.1 × 10⁹(T _n /300)^6.1, Z = [1 + log²(Y)]⁻¹, M = N₂ and O₂, and the units of K ₇₁ and [M] are cm⁶s⁻¹ and cm⁻³, respectively.

Table 3 Sources and sinks of O₃, O, O(¹D), H, OH, HO₂, H₂O₂, N, N(²D), NO, NO₂, NO₃, N₂O₅, HNO₃, and HO₂NO₂ in atmospheric chemical reactions included in the model

The vertical velocity due to eddy diffusion can be written as (Banks and Kockarts 1973; Brasseur and Solomon 2005)

$$ V_{nz}^{t} = - K_{t} \left( {N_{n}^{ - 1} \frac{\partial }{\partial z}N_{n} + H^{ - 1} } \right), $$

(17)

where H = kT _n (〈m〉g)⁻¹, g = |g|, g is the acceleration due to gravity, $\left\langle m \right\rangle = N^{ - 1} \sum\nolimits_{n} {m_{n} N_{n} }$, $N = \sum\nolimits_{n} {N_{n} , \, m_{n} }$ is the mass of neutral species n, K _t is the vertical eddy diffusion coefficient.

The model uses the approximation of K _t given by Qian et al. (2009) at altitudes above 97 km. The altitude profile of K _t taken from Shimazaki (1971) was normalized to the vertical eddy diffusion coefficient approximation (Qian et al. 2009) at 97 km altitude by introducing the corresponding fitting factor for the Shimazaki (1971) vertical eddy diffusion coefficient, and this normalized vertical eddy diffusion coefficient was used in the model simulation below 97 km altitude.

The vertical hydrodynamic diffusion velocity can be presented as (Banks and Kockarts 1973)

$$ V_{nz} = - D_{n} \left[ {N_{n}^{ - 1} \frac{\partial }{\partial z}N_{n} + H_{n}^{ - 1} + (1 + \alpha_{n} )T_{n}^{ - 1} \frac{\partial }{\partial z}T_{n} } \right], $$

(18)

where $D_{n}^{ - 1} = N^{ - 1} \sum\nolimits_{q \ne n} {N_{q} D\left( {n,q} \right)^{ - 1} }$, D(n,q) is the binary diffusion coefficient of a mixture of neutral species n and q, α _n is the thermal diffusion factor of species n in the multicomponent atmosphere (Pavlov 1979, 1981a), H _n  = −kT _n (m _n g)⁻¹.

The value of D(n,q) is related to the reduced Chapman–Cowling collision integral, $\varOmega_{nq}^{(1,1)*}$, in the following manner (e.g., Ferziger and Kaper 1972):

$$ D\left( {n,q} \right) = D\left( {q,n} \right) = 3(2\pi m_{nq} kT_{n} )^{0.5} (16Nm_{nq} \sigma_{nq}^{2} \varOmega_{nq}^{(1,1)*} )^{ - 1} , $$

(19)

where k is Boltzmann’s constant, m _nq  = m _n m _q /(m _n  + m _q ), σ _nq  = 0.5(σ _n  + σ _q ) is the separation of the centres of two particles whose diameters at the instant of collision are σ _n and σ _q , respectively.

N₂, O₂, and O are the major constituents of the atmosphere at altitudes under consideration, and it is enough to use only D(n,N₂), D(n,O₂), and D(n,O) to calculate the value of D _n :

$$ D_{n}^{-1} = N^{-1} \left\{ {\left[ {{\text{N}}_{2} } \right]D\left( {n,{\text{N}}_{2} } \right)^{-1}\, +\, \left[ {{\text{O}}_{2} } \right]D\left( {n,{\text{O}}_{2} } \right)^{-1}\, +\, \left[ {\text{O}} \right]D\left( {n,{\text{O}}} \right)^{-1} } \right\}, $$

(20)

The temperature dependences of some measured binary diffusion coefficients were given by Mason and Marrero (1970), and these approximations are used in the model for pairs of O–N₂, O–O₂, H₂–N₂, H₂–O₂, H₂O–N₂, H₂O–O₂, and N–N₂. The model uses the binary diffusion coefficients of mixtures of H₂–O, NO–O₂, NO–O, H–O, and H–O₂ in the Lennard–Jones (12–6) interaction potential approximation for $\varOmega_{nq}^{(1,1)*}$, NO–N₂, and H–N₂ mixtures in the Buckingham interaction potential approximation for $\varOmega_{nq}^{(1,1)*}$, and pairs of N–O and N–O₂ in the exponential interaction potential approximation for $\varOmega_{nq}^{(1,1)*}$ (Pavlov 1981b). The binary diffusion coefficients for pairs of O(¹D)-N₂, O(¹D)-O₂, and O(¹D)-O are assumed to be equal to D(O,N₂), D(O,O₂), and D(O,O), and the N(²D)-N₂, N(²D)-O₂, and N(²D)-O binary diffusion coefficients are assumed to be equal to the N-N₂, N-O₂, and N-O binary diffusion coefficients, respectively. The approximation of the atomic oxygen coefficient of self-diffusion given by Pavlov (1997) is used in the model calculations of the O(¹D)-O binary diffusion coefficient. It is assumed that the O₂(¹Δ_g)-N₂, O₂(¹Δ_g)-O₂, and O₂(¹Δ_g)-O binary diffusion coefficients are equal to D(O₂,N₂), D(O₂,O₂), and D(O₂,O), respectively. The explicit form of the molecular oxygen coefficient of self-diffusion is taken in the Pavlov (2011) approximation. The value of $\varOmega_{nq}^{(1,1)*}$ is equal to unity in the rigid sphere interaction potential approximation, and this approach with σ _nq  = 0.3 nm is often used in atmospheric studies to calculate binary diffusion coefficients of neutral species (e.g., Banks and Kockarts 1973; Brasseur and Solomon 2005). The model includes the values of D(H₂,O), D(H₂O,O₂), D(H₂O,O), D(O₃,N₂), D(O₃,O₂), D(O₃,O) in this rigid sphere interaction potential approximation.

The thermal diffusion factors of atmospheric neutral species were calculated by Pavlov (1979). It follows from this study that the average values of α _n are equal to −0.28, −0.28, −0.10, and 0.09 for H, H₂, O, and O₂, respectively, and these average values of α _n are used in Eq. (18). Zero thermal diffusion factors of the other neutral species are used in Eq. (18).

The neutral temperature, T _n , and number densities of N₂ and O₂ are taken from the semiempirical NRLMSISE-00 neutral atmosphere model (Picone et al. 2002).

Because of long chemical lifetimes of H₂O, H₂ and CO, number densities of these neutral species should be expected to exhibit variations which are related to horizontal wind transport and vertical hydrodynamic and turbulent diffusion processes below 110 km altitude (Brasseur and Solomon 2005), and these number densities cannot be calculated in the frame of a one-dimensional time-dependent atmospheric neutral composition model. As a result, the zonally, latitudinally, and annually averaged altitude profiles of the H₂O, H₂, and CO number densities presented in Table A.6.2.a of Brasseur and Solomon (2005) below 108.4 km altitude are used to calculate the values of [H₂O], [H₂], and [CO] on the numerical grid of the model. Above 108 km altitude, the values of [H₂O] and [H₂] are found from the condition of diffusive equilibrium

$$ V_{nz} + V_{nz}^{t} = 0 $$

(21)

Carbon monoxide is included in the model only by reaction 71 of Table 3, and this reaction is not taken into account in the model calculations above 108 km altitude.

N₂O is formed from biological activity on the Earth’s surface, and its number density altitude profile depends critically on vertical and horizontal wind transport below 70 km altitude (Brasseur and Solomon 2005). As a result, the value of [N₂O] cannot be calculated in the frame of any one-dimensional time-dependent atmospheric neutral composition model, and the average altitude profile of [N₂O] given by Fabian et al. (1979) below 36 km altitude is used in the model calculations. It is assumed that the value of [N₂O] is equal to zero above 40 km altitude and varies linearly in value between 36 and 40 km altitude.

It is necessary to determine the lower and upper boundary conditions to solve Eq. (15) for [O], [O(¹D)], [O₃], [O₂(¹Δ_g)], [H], [N], [N(²D)], and [NO]. Diffusive equilibrium is considered to exist at the upper boundary for O, O(¹D), O₃, O₂(¹Δ_g), N, N(²D), and NO. The value of [H] produced by the semiempirical NRLMSISE-00 neutral atmosphere model (Picone et al. 2002) at the upper boundary is used as the upper boundary condition for [H]. As a result, the escape of H from the atmosphere (i.e. a nonzero value of V _nz at the upper boundary) is approximately taken into account by this approach. The diffusion processes are neglected in Eq. (15) at the lower boundary for O, O(¹D), O₂(¹Δ_g), H, N, N(²D), and NO.

Chemical lifetimes of OH, HO₂, H₂O₂, NO₂, NO₃, N₂O₅, and HO₂NO₂ are much less than the corresponding diffusive lifetimes below 110 km altitude (Brasseur and Solomon 2005). As a result, the diffusion processes are not included in Eq. (15) when solving this equation for OH, HO₂, H₂O₂, NO₂, NO₃, N₂O₅, and HO₂NO₂. Zero values of [OH], [HO₂], [H₂O₂], [NO₂], [NO₃], [N₂O₅], and [HO₂NO₂] are taken above 110 km altitude.

Transport of O₃ by means of horizontal circulation plays an important role in determining ozone number densities in the troposphere and lower stratosphere and in a part of the middle stratosphere (Brasseur and Solomon 2005; Seinfeld and Spyros 2006), and any one-dimensional time-dependent atmospheric neutral composition model is not capable to reproduce correctly altitude profiles of [O₃] at these altitudes of the atmosphere. As a result, the average values of [O₃] in the altitude range from 20 km to an altitude, z _O3, of a middle stratosphere can be taken from the empirical model (Keating et al. 1996) at low and middle latitudes. Average values of [O₃] in the altitude range of 0–20 km can be calculated by interpolation using the empirical model ozone number densities at 20 km altitude and surface ozone measurements ore ozonesonde data (see, e.g., Logan (1999) and Oltmans et al. (2006)). If there are ozonesonde measurements of [O₃] below z _O3 altitude then these values of [O₃] can be used as the input atmospheric neutral composition model parameter. The value of [O₃] at z _O3 altitude is used as the low boundary condition in calculations of [O₃] by the one-dimensional time-dependent atmospheric neutral composition model. It should be noted that diurnal variations of [O₃] are not taken into account below z _O3 altitude if average values of [O₃] are used in this altitude range. Such an approach cannot produce noticeable errors in [O₃] in the stratosphere because small diurnal variations are expected to occur in stratospheric ozone number densities (Brasseur and Solomon 2005).

The diffusion processes are not taken into account in Eq. (15) at the low boundary and at altitudes from 50 to 110 km for HNO₃, because the diffusive lifetime is much larger than the chemical lifetime at these altitudes for HNO₃ (Brasseur and Solomon 2005). The chemical and diffusive lifetimes of HNO₃ are comparable in value in the altitude range of about 20 to 30 km (Brasseur and Solomon 2005), and the eddy diffusion processes are included in Eq. (15) when solving this equation for HNO₃ for 0 < z < 50 km. Zero value of [HNO₃] is taken above 110 km altitude.

The numerical technique for the numerical solution of the one-dimensional, time-dependent continuity equations (15) described by Marov and Kolesnichenko (1987) is used in the model calculations. This numerical technique is similar to that described by Hastings and Roble (1977). The solving patterns of such model calculations were given, for example, by Marov and Kolesnichenko (1987), Pavlov (1994), Pavlov et al. (1999), Pavlov and Foster (2001), Pavlov (2012), and Pavlov and Pavlova (2013). The model calculates the neutral number densities in the altitude range from the Earth’s surface to 400 km altitude with a time step of 2 min. An altitude step of 1 km is used below 98, 2 and 10 km are taken from 98 to 130 and between 150 and 400 km, respectively, and this altitude step increases from 2 to 10 km if the altitude rises from 130 to 150 km.

Appendix 2

2.1 Contribution of Multiple Scattering Solar Radiation to the Average Solar Flux

If I(λ) is the intensity of solar radiation at the wavelength, λ, averaged over all azimuthal angles at a point of the atmosphere under consideration and the photodestruction reactions presented in Table 1 are considered then a photodestruction rate coefficient of ions (see, e.g., Kylling et al. 1995)

$$ J\left( N \right) = 4\pi \int\limits_{0}^{\infty } {\sigma_{\text{des}}^{N} (\lambda )I(\lambda ){\text{d}}\lambda ,} $$

(22)

where N is the number of a photodestruction reaction presented in Table 1, $\sigma_{\text{des}}^{N}$ is the photodestruction cross section.

If the average flux, Φ(λ _j ), of solar radiation that represents a number of photons passing through an unit surface per unit time in the wavelength range of λ _j averaged over all azimuthal angles at a point of the atmosphere under consideration, is used instead of I(λ _j ) in the same wavelength range then Eq. (22) is transformed to

$$ J\left( N \right) = \sum\limits_{j} {\sigma_{\text{des}}^{N} (\lambda_{j} )\varPhi (\lambda_{j} ),} $$

(23)

where the value of Φ(λ _j ) is related to I(λ _j ) in each wavelength interval, Δλ _j , of numerical integration of Eq. (22) by Φ(λ _j ) = 4πI(λ _j )Δλ _j .

In the atmosphere, solar radiation undergoes multiple scattering by air molecules and aerosol particles, and the average intensity of solar radiation can be presented as a sum of direct average intensity, I ₀, and average multiple scattering (diffuse) intensity, I _d , which consists of multiple scattered light (see, e.g., Thomas and Stamnes 1999; Brasseur and Solomon 2005; Timofeyev and Vasi′lev 2008):

$$ I(\lambda_{j} ) = I_{0} (\lambda_{j} ) + I_{d} (\lambda_{j} ). $$

(24)

Therefore, the average flux of solar radiation can be expressed in the form

$$ \varPhi (\lambda_{j} ) = \varPhi_{0} (\lambda_{j} ) + \varPhi_{d} (\lambda_{j} ). $$

(25)

where the value of Φ ₀(λ _j ) given by Eq. (4) is related to I ₀(λ _j ) by Φ ₀(λ _j ) = 4πI ₀(λ _j )Δλ _j , and the average multiple scattering (diffuse) flux, Φ _d (λ _j ), is related to I _d (λ _j ) by Φ _d (λ _j ) = 4πI _d (λ _j )Δλ _j .

In the two-stream approximation, the average intensity of solar radiation is divided into upward-propagating radiation with the average intensity, I ⁺(λ _j ), and downward-propagating radiation with the average intensity, I ⁻(λ _j ), as (Kylling et al. 1995):

$$ I(\lambda_{j} ) = 0.5[I^{ + } (\lambda_{j} ) + \, I^{ - } (\lambda_{j} )]. $$

(26)

In a similar manner, the average diffuse flux can be presented as

$$ \varPhi_{d} (\lambda_{j} ) = 0.5\left( {F_{j}^{ + } + \, F_{j}^{ - } } \right), $$

(27)

where F ⁺ _j  = 4πI ⁺ _d (λ _j )Δλ _j , F ⁻ _j  = 4πI ⁻ _d (λ _j )Δλ _j , I ⁺ _d (λ _j ) and I ⁻ _d (λ _j ) are average diffuse intensities of upward- and downward-propagating radiation, respectively, and I _d (λ _j ) = 0.5[I ⁺ _d (λ _j ) + I ⁻ _d (λ _j )].

It is necessary to solve Eqs. (7) and (8) given by Kylling et al. (1995) to determine the values of I ⁺(λ _j ) and I ⁻(λ _j ). These equations can be transformed to the equations for calculations of F ⁺ _j and F ⁻ _j :

$$ - \mu \beta_{j}^{ - 1} \frac{\text{d}}{{{\text{d}}z}}F_{j}^{ + } = F_{j}^{ + } \left( {1 - C_{j} } \right) - a_{j} b[F_{j}^{ - } + \varPhi_{0} (\lambda_{j} )] - Q_{j} , $$

(28)

$$ \mu \beta_{j}^{ - 1} \frac{\text{d}}{{{\text{d}}z}}F_{j}^{ - } = F_{j}^{ - } \left( {1 - C_{j} } \right) - a_{j} b[F_{j}^{ + } + \varPhi_{0} (\lambda_{j} )] - Q_{j} , $$

(29)

where z is an altitude, μ = 3^−0.5, C _j  = a _j (1 − b), Q _j  = 4πΔλ _j (1 − a _j )B(λ _j ,T _n ), B(λ _j ,T _n ) is the Planck function, T _n is the temperature of neutral species of the atmosphere, $\beta_{j} = \beta_{j}^{\text{a}} + \beta_{j}^{\text{sc}}$, $\beta_{j}^{\text{sc}} = \sum\nolimits_{n} {N_{n} \sigma_{n}^{\text{sc}} (\lambda_{j} )}$, $\beta_{j}^{a} = \sum\nolimits_{n} {N_{n} \sigma_{n}^{a} (\lambda_{j} )}$, $a_{j} = \beta_{\text{j}}^{\text{sc}} /\beta_{j}$, b = 0.5(1 − 3 gμcos χ), g is the asymmetry factor (g = 0 for isotropic scattering, g = 1 for complete forward scattering, and g =  − 1 for complete backward scattering), χ is the solar zenith angle, N _n , $\sigma_{n}^{a}$, and $\sigma_{n}^{\text{sc}}$ are the number density, absorption cross section, and scattering cross section of neutral species n, respectively.

If the scattering is assumed to be isotropic and the thermal radiation internal source Q _j is not taken into account then Eqs. (28) and (29) take the forms

$$ - \mu \beta_{j}^{ - 1} \frac{\text{d}}{{{\text{d}}z}}F_{j}^{ + } = F_{j}^{ + } \left( {1 - 0.5a_{j} } \right) - 0.5a_{j} [F_{j}^{ - } + \varPhi_{0} (\lambda_{j} )], $$

(30)

$$ \mu \beta_{j}^{ - 1} \frac{\text{d}}{{{\text{d}}z}}F_{j}^{ - } = F_{j}^{ - } \left( {1 - 0.5a_{j} } \right) - 0.5a_{j} [F_{j}^{ + } + \varPhi_{0} (\lambda_{j} )], $$

(31)

Downward-propagating multiple scattering radiation is assumed to be negligible at the top of the atmosphere determining the upper boundary condition used for solution of Eqs. (30) and (31):

$$ F_{j}^{ - } = 0. $$

(32)

The low boundary condition is determined at the Earth’s surface, and it states that the upward-propagating scattering radiation with average diffuse intensity is the sum of the reflected average downward-propagating diffuse intensity and reflected average direct intensity. It is assumed in this work that the intensity of solar radiation reflected from the Earth’s surface is completely uniform with angle of observation (Lambertian reflector). As a result, Eq. (30) of Kylling et al. (1995) that establishes the relationship between I ⁺(λ _j ) and I ⁻(λ _j ) at the Earth’s surface is used to derive the low boundary condition for Eqs. (30) and (31). The thermal emittance of the Earth’s surface that is proportional to the Planck function is not taken into account in this low boundary condition.

If the solar zenith angle does not exceed 90° then the low boundary condition for Eqs. (30) and (31) is given by

$$ F_{j}^{ + } = 2\mu A_{\text{s}} [\varPhi_{0} (\lambda_{j} ) + 0.5F_{j}^{ - } ], $$

(33)

where A _s is the Earth’s surface albedo.

The low boundary condition is written for χ ≥ 90° as

$$ F_{j}^{ + } = \mu A_{\text{s}} F_{j}^{ - } . $$

(34)

Equations (27), (30), and (31) with the upper and low boundary conditions (32)–(34) are solved in this paper to calculate the value of Φ _d (λ _j ) from the Earth’s surface to 400 km altitude in the isotropic Rayleigh scattering atmosphere using the Rayleigh scattering cross section, σ _sc(λ _j ), given by Bucholtz (1995) for the terrestrial atmosphere, and, as a result, $\beta_{j}^{\text{sc}} = \sigma_{\text{sc}} (\lambda_{j} )\sum\nolimits_{n} {N_{n} }$. The absorption cross sections of O₂ and O₃, and the temperature and number densities of neutral species used in the model calculations are discussed in Appendix 1 and Sect. 3.