2.1 Energy Balance, Infrared Cooling, and O_{2} Solar Heating
[5] In this section, the approach for deriving the global annual mean atomic oxygen concentration consistent with the longterm energy balance of the mesopause region is presented. The main terms in the radiative/chemical energy balance of the mesopause region are infrared radiative cooling by CO_{2}, solar heating due to the absorption of ultraviolet radiation by O_{2} and O_{3}, and exothermic chemical reactions. We express the energy balance due to the radiative and chemical heating rates in the mesopause region as follows:
 (1)
[6] Additional heating due to absorption of solar energy by carbon dioxide in the nearIR bands and cooling due to IR emission by water vapor are essentially of the same magnitude [Fomichev et al., 2004] and are not considered at this time. The ultraviolet heating in O_{2} occurs at the Lyα wavelength, in the SchumannRunge continuum, and in the SchumannRunge bands, spanning 121.5;204.5 nm. The nearinfrared O_{2} atmospheric bands also make a small contribution. The O_{3} solar heating is almost entirely in the Hartley band. The chemical heating is a consequence of seven exothermic chemical reactions [Mlynczak and Solomon, 1993].
[7] The main presumption in equation (1) is that, over annual and longer time scales, the global average heating within the mesopause region cannot exceed the radiative cooling; otherwise, the atmosphere would continually warm. If the total heating exceeds the total cooling over the long term, then some mechanism would be required to continually remove heat from the atmosphere to keep it from perpetually warming. In this paper, it is presumed that such a mechanism does not exist and that equation (1) is correct on annual and longer time scales. Furthermore, dissipation of gravity waves likely add heat to the mesopause region [e.g., Chandran et al., 2010]. If such heating is significant on the long term, globally averaged scale, then the radiatively constrained atomic oxygen derived from equation (1) will be smaller.
[8] Rearranging equation (1) to leave the solar O_{3} and chemical heating terms on the left hand side, we obtain:
 (2)
[9] The CO_{2} cooling rates in the ν_{2} bands at 15 µm are produced as operational data products as part of retrieving the kinetic temperature profile [Mertens et al., 2001] in the mesosphere using the CurtisMatrix formulation of radiative transfer to solve for the nonlocal thermodynamic equilibrium (nonLTE) state populations of the CO_{2} vibrationrotation bands [LopezPuertas et al., 1986]. Although radiative cooling in the mesopause region is essentially due to the fundamental band of the primary isotope, a total of nine different vibrationrotation bands are included here. As shown by Mlynczak et al. [2010], the cooling rates are essentially constrained by the infrared limb radiance measurements made by the SABER instrument and are thus dependent on the absolute calibration of SABER.
[10] Solar heating rates due to the absorption of ultraviolet radiation by O_{2} are computed using temperature and pressure profiles from SABER and daily solar irradiance values from the SORCE satellite [Rottman, 2005]. Solar heating rates are computed for each daytime temperature and pressure profile. As with the CO_{2} cooling, the O_{2} solar heating rates are binned by hour and averaged for each day, and then the daily averages are used to compute annual average heating and cooling rates. Solar heating rates for O_{2} are computed over continuous wavelength span from 121.5 to 204.5 nm. The heating rates are computed directly for each profile using the solar zenith angle and absorption crosssections for Lyα through the SchumannRunge continuum (to 173.5 nm). The technique of Koppers and Murtagh [1996] is used to compute heating rates in the SchumannRunge bands. The nearIR bands of O_{2} (the atmospheric “A” band, plus the “B” and “gamma” bands) are taken from Mlynczak and Marshall [1996]. The Herzberg continuum is not considered, as it is negligible at these altitudes. A fractional length of day (f_{day}) of 0.5 is used for all solar heating rate calculations.
[11] SABER continuously observes the latitudes between approximately ±55° due to its location on the TIMED spacecraft as discussed in Paper 1. Thus, we refer here to “global” averages as cosinelatitude weighted averages between this range, noting that it accounts for 82% of the total atmospheric area. Figure 1 shows the global mean O_{2} solar heating rate (curve 1) and the global mean CO_{2} cooling rates (curve 2) between 0.01 and 0.0001 hPa for the year 2004. The remaining curves in Figure 1 are discussed below.
Table 1. Chemical Reactions, Rate Coefficients, and Enthalpies of Reaction for the Seven Reactions That Are Important in the Energy Budget of the Mesopause RegionaReaction  Rate Coefficient  Enthalpy (cm^{1}) 


1: O + O + M ➔ O_{2} + M  k_{1}, 4.7 × 10^{33} (300/T)^{2} cm^{6} s^{1}  41700 
2 O + O_{2} + M ➔ O_{3} + M  k_{2}, 6 × 10^{34} (300/T)^{2.4} cm^{6} s^{1}  8915 
3: O + O_{3} ➔ 2 O_{2}  k_{3}, 8 × 10^{12} exp(−2060/T) cm^{3} s^{1}  32790 
4: H + O_{3} ➔ OH + O_{2}   29615 
5: O + OH ➔ H + O_{2}   5870 
6: O + HO_{2} ➔ OH + O_{2}   18645 
7: H + O_{2} + M ➔ HO_{2} + M   17190 
2.2 Solar and Exothermic Chemical Heating Expressions for the Derivation of Radiatively Constrained Atomic Oxygen
[12] The next step is to express the heating rates on the left hand side of equation (2) in terms of atomic oxygen. The purpose again is to create expressions that are functions of atomic oxygen and whose sum, when equated to the right hand side of equation (2), can be solved for the Oatom concentration. Beginning with the heating due to the absorption of UV radiation by ozone in the Hartley band, it is assumed that daytime ozone is in photochemical steady state with formation by recombination of O and O_{2} balanced by photolysis of O_{3}. This is expressed as:
 (3)
[13] In equation (3), J is the ozone photolysis rate, k_{2} the rate coefficient for the recombination of O and O_{2}, and M is the total number density. However, the instantaneous rate ∂Q/∂t of energy deposition due to ozone photolyis is given by:
 (4)
[14] In equation (4), hν is the energy of an ultraviolet photon absorbed by ozone, E_{B} is the energy required to dissociate the ozone molecule, and ε is the heating efficiency accounting for energy radiated by the electronically excited products of ozone photolysis [Mlynczak and Solomon, 1991, 1993]. From equation (3), ∂Q/∂t is then written:
 (5)
[15] The heating rate ∂T/∂t in Kelvin per day is derived from the first law of thermodynamics, expressed here as:
 (6)
[16] In equation (6), k_{b} is Boltzmann's constant, and C_{p} the specific heat at constant pressure. Below ~ 100 km, the atmosphere is composed essentially of diatomic molecules, and C_{p} is 7/2 times the specific gas constant, R. Solving the above for ∂T/∂t and averaging over the day to get the heating in Kelvin per day yields:
 (7)
[17] Equation (7) is the expression for the daily average heating in the Hartley band of ozone expressed as a function of atomic oxygen through the assumption of photochemical steady state. L_{D} is the length of day (86,400 s = 1 day), and f_{day} is the fractional length of day for which the Sun heats the atmosphere, which for the analysis of global annual mean heating is taken to be 0.5.
[18] Equation (7) depends only on the atomic oxygen concentration as all other terms are either constant (e.g., L_{D}) or are provided by SABER (e.g., the O_{2} density). Equation (7) can be simplified by the insertion of the following terms: For hν, 255 nm (the center of the Hartley band, 39215 cm^{−1}); E_{B} is 8800 cm^{−1}; ε is 0.66 in the mesopause region; thus, enabling the equation to be written as
 (8)
where ∂T/∂t is in Kelvin per day.
[19] The next step is to develop expressions similar to equation (8) for the heating due to the exothermic reactions that are important in the energy budget of the mesopause region. Listed in Table 1 are the seven reactions and their associated reaction rate coefficients and enthalpies of reaction. The objective here is to develop expressions that depend on atomic oxygen and standard reaction rate coefficients.
[20] For the recombination of atomic oxygen (the first reaction listed in Table 1), the rate of energy deposition is given by:
 (9)
[21] In equation (9), k_{1} is the rate coefficient for recombination of atomic oxygen, M is the total number density, and H_{1} is the enthalpy of reaction for this process. Applying the first law, averaging over 1 day, and assuming that this rate is diurnally invariant (due to the long lifetime of atomic oxygen in the mesopause region), the expression for the daily average heating in Kelvin per day is obtained:
 (10)
[22] Proceeding in a similar fashion, the following expression is obtained for the recombination of O and O_{2}, the second reaction listed in Table 1, again assuming that the rate of recombination is diurnally invariant:
 (11)
[23] Reactions 3 through 5 are treated similarly, although because ozone and the hydroxyl radical (OH) are reactants, there is a significant diurnal variation that must be taken into account. Separate expressions are developed for day and night, and these are averaged to yield an expression for the mean heating. For the reaction of O and ozone (reaction 3 in Table 1) in the daytime, we employ equation (3) to express ozone as a function of atomic oxygen. Doing so yields the expression (and taking f_{day} = 0.5 and J = 0.0833 s^{−1}) for daytime:
 (12)
[24] As with the recombination of atomic oxygen in equation (10), heating due to the reaction of O and O_{3} depends on the square of the O concentration. In equation (12), k_{3} is the reaction rate coefficient for the reaction of O and ozone, and H_{3} is the enthalpy of reaction. This process is a minor source of heat in the mesopause region as the reaction is relatively slow as indicated by the value of k_{3} in Table 1. The maximum daytime heating rate is around ~ 0.05 K/d. Ozone increases substantially in the mesopause region and the night value can be up to 10 times larger in the upper mesopause region. We estimate the night heating as 10 times that shown in equation (12) so that the daily average heating due to reaction 3 is given by:
 (13)
[25] Equation (13) incorporates the recommended value for the rate coefficient for k_{3} shown in Table 1.
[26] Reaction 4 in Table 1, the reaction of atomic hydrogen (H) and O_{3}, is the dominant source of heat in the mesopause region. This is a fast reaction that liberates a large amount of energy, some of which is radiated away through the Meinel band emission, reducing the amount of energy available for heat [Mlynczak and Solomon, 1991]. This reaction has a strong diurnal variation, so we again develop expressions for day and night and combine them to obtain the mean heating rate. At night, under chemical equilibrium, it is assumed that production of ozone through recombination (reaction 2) is balanced by loss of ozone reacting with atomic hydrogen (H):
 (14)
[27] Following the approaches outlined above and including the appropriate enthalpy of reaction for reaction 4 in Table 1, the heating rate for reaction 4 at night is expressed as:
 (15)
[28] In equation (15), H_{4} is the enthalpy of reaction 4, and ε_{4} is the efficiency (0.75) of the reaction accounting for the loss of energy to space by the total Meinel band emission.
[29] Daytime heating by reaction 4 is not able to be expressed directly as a function of atomic oxygen. As with reaction 3, a day/night ratio of 1:5 (or 0.2) is used to estimate the daytime heating for this reaction, which is then averaged with the night value above, resulting in one expression for the heating rate that is a function of atomic oxygen.
[30] The last reaction that may be expressed as a function of atomic oxygen is reaction 5 in Table 1, involving the destruction of OH through reaction with O. Hydroxyl is produced by reactions 4 and 6, resulting in the steady state expression:
 (16)
[31] We estimate the heating at night due to the reaction of O and OH with the first term on the right hand side of equation (16), using equation (14) to replace k_{4} [H] [O_{3}]:
 (17)
[32] As with equation (15), a day/night ratio of 0.2 is assumed in the ozone concentration to provide the daytime heating, which is added to the night heating to obtain the total heating. As with all of the above expressions, the heating is dependent only on atomic oxygen as the other terms are constants or are derived from SABER data (e.g., the O_{2} density).
[33] Heating by reactions 6 and 7 is of lesser importance, and primarily in the daytime, between ~ 80 and 85 km (in actuality, extending down to nearly 60 km, but that is out of the range of this study). These two reactions are not able to be expressed directly as functions of O. Heating due to these reactions is included here by specifying a global mean atomic hydrogen concentration of 1.5 × 10^{8} cm^{−3} from 80 to 86 km. The heating due to reaction 7 is computed (proportional to k_{7} [H][O_{2}][M]H_{7}) as described for the reactions above. For reaction 6, photochemical steady state for the hydroperoxyl radical (HO_{2}) implies k_{7} [H][O_{2}][M] is balanced by k_{6} [O][HO_{2}] so that the heating for reaction 6 is proportional to k_{7} [H][O_{2}][M] H_{6}. Heating for reactions 6 and 7 therefore depend only on the provided atomic hydrogen abundance, SABER density and temperatures, and the respective enthalpies of reaction for each process. The heating rates for these reactions is computed directly and subtracted (along with the O_{2} solar heating) from the CO_{2} cooling rates. The estimated heating due to reactions 6 and 7 is shown in Figure 1 as the curve labeled 3.
[34] The heating rates due to absorption in the Hartley band and to the first five exothermic chemical reactions listed in Table 1 have now been expressed as functions of atomic oxygen. Reactions 1 and 3 depend on the square of the atomic oxygen concentration, while the remaining processes depend linearly on the atomic oxygen. Using equations (5), (9), (11), (13), (15), (17), and (2) above, a quadratic equation in atomic oxygen is obtained:
 (18)
[35] The coefficients a and b in equation (18) depend on the rate coefficients k_{1} and k_{2}, as well as the O_{2} density and temperature. The term c is the difference between the CO_{2} cooling rates and the sum of the O_{2} solar heating rates and the chemical heating rates for equations (6) and (7). This difference is shown as the dashed curve labeled 4 in Figure 1 and also represents the upper limit for the sum of all other heat sources. Equation (18) is solved at each pressure level to derive the O concentration consistent with global annual mean radiative balance and in particular, the heating rate in the curve labeled 4 in Figure 1. The SABER global mean temperatures and pressures are used as inputs to evaluate rate coefficients k_{1}, k_{2}, k_{3}, and the atmospheric number density. Calculations are carried out on 21 separate pressure levels between 10^{−2} and 10^{−4} hPa, approximately 1 km apart in the vertical.