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 We present a new approach to constrain and validate atomic oxygen (O) concentrations in the mesopause region (~ 80 to ~ 100 km). In a prior companion paper [Mlynczak et al., 2013], we presented O-atom concentrations in the mesopause region inferred from measurements of day ozone and night hydroxyl emission rates made by the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument. The approach presented here uses the constraint of global, annual mean energy balance to derive atomic oxygen concentrations, consistent with rates of radiative cooling by carbon dioxide (CO2) and solar heating due to molecular oxygen (O2). The mathematical difference between these cooling and heating rates, on a global annual mean basis, effectively constrains the maximum heating rate for the sum of all other processes. The remaining terms, solar heating due to ozone plus a series of exothermic chemical reactions can be expressed as functions of O. This new approach enables a simple mathematical expression that yields the vertical profile of global annual mean “radiatively constrained” atomic oxygen in the mesopause region. The radiatively constrained atomic oxygen depends only on the CO2 cooling rates, O2 solar heating rates, and standard reaction rate coefficients and enthalpies. Radiative cooling and solar heating rates used in these analyses are derived from measurements made by the SABER instrument on the NASA Thermosphere Ionosphere Mesosphere Energetics and Dynamics satellite. There is excellent agreement between the SABER radiatively constrained atomic oxygen and that derived from the SABER ozone and OH emission measurements over most of the mesopause region. Radiatively constrained atomic oxygen represents an upper limit on the global average O-atom concentration in the mesopause region.
 A fundamental problem in the chemical aeronomy of the Earth's mesopause region is the determination of the atomic oxygen (O) concentration. Atomic oxygen plays a critical role in the photochemistry and energy balance of the mesopause region, which we define as the atmosphere between 10−2 and 10−4 hPa or approximately 80 and 100 km in altitude. The Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument on the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite provides O-atom concentrations as a routine data product. The SABER approaches involve deriving O from measurements of ozone in the day and of the hydroxyl emission at night, as described in Mlynczak et al. , hereafter Paper 1. Both the day and night SABER atomic oxygen data products are highly inferred, as they are derived from emission features that are not in local thermodynamic equilibrium and hence are dependent on the provision of various rate coefficients for physical (collisional) quenching of the quantum states of the observed emissions in addition to radiative lifetimes and other parameters. In addition, the derivations of day and night O rely on separate assumptions of photochemical equilibrium. Because of the stated difficulty in measuring O by any means, validation of the SABER measurements, or any similarly derived O, is challenging.
 This paper presents a new method by which the atomic oxygen may be constrained and validated based on considerations of the global annual energy budget in the mesopause region. SABER provides measurements of the CO2 infrared (IR, 15 µm) radiative cooling rates as routine data products, and SABER data used in conjunction with solar irradiance measurements provide solar heating rates due to absorption by molecular oxygen. The mathematical difference between the infrared cooling rates and the solar heating rates for molecular oxygen, on a global annual mean basis, effectively constrains the maximum heating rates for the sum of all other processes. In the mesopause region, the heating due to the sum of all other processes can be expressed as a relatively simple function of atomic oxygen. This enables the determination of the radiatively constrained vertical profile of global annual mean atomic oxygen in the mesopause region, consistent with the observed long-term radiative balance and global mean thermal structure. As will be shown below, the radiatively constrained O is dependent only on standard reaction rate coefficients and enthalpies and is completely independent of the non-LTE parameters required to infer SABER daytime or nighttime atomic oxygen.
 In the next section, we describe the approach to deriving the radiatively constrained O including the infrared cooling and O2 solar heating rates. We follow that with a comparison between day and night atomic oxygen inferred from SABER ozone and OH emission rate measurements. In section 4, we discuss the implications of additional energy sources, such as gravity wave dissipation, and then summarize the paper.
2.1 Energy Balance, Infrared Cooling, and O2 Solar Heating
 In this section, the approach for deriving the global annual mean atomic oxygen concentration consistent with the long-term 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 CO2, solar heating due to the absorption of ultraviolet radiation by O2 and O3, and exothermic chemical reactions. We express the energy balance due to the radiative and chemical heating rates in the mesopause region as follows:
 Additional heating due to absorption of solar energy by carbon dioxide in the near-IR 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 O2 occurs at the Ly-α wavelength, in the Schumann-Runge continuum, and in the Schumann-Runge bands, spanning 121.5-;204.5 nm. The near-infrared O2 atmospheric bands also make a small contribution. The O3 solar heating is almost entirely in the Hartley band. The chemical heating is a consequence of seven exothermic chemical reactions [Mlynczak and Solomon, 1993].
 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.
 Rearranging equation (1) to leave the solar O3 and chemical heating terms on the left hand side, we obtain:
 The CO2 cooling rates in the ν2 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 Curtis-Matrix formulation of radiative transfer to solve for the nonlocal thermodynamic equilibrium (non-LTE) state populations of the CO2 vibration-rotation bands [Lopez-Puertas 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 vibration-rotation bands are included here. As shown by Mlynczak et al. , 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.
 Solar heating rates due to the absorption of ultraviolet radiation by O2 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 CO2 cooling, the O2 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 O2 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 cross-sections for Ly-α through the Schumann-Runge continuum (to 173.5 nm). The technique of Koppers and Murtagh  is used to compute heating rates in the Schumann-Runge bands. The near-IR bands of O2 (the atmospheric “A” band, plus the “B” and “gamma” bands) are taken from Mlynczak and Marshall . The Herzberg continuum is not considered, as it is negligible at these altitudes. A fractional length of day (fday) of 0.5 is used for all solar heating rate calculations.
 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 cosine-latitude weighted averages between this range, noting that it accounts for 82% of the total atmospheric area. Figure 1 shows the global mean O2 solar heating rate (curve 1) and the global mean CO2 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 Regiona
Only the rate coefficients used in the derivation of radiatively constrained atomic oxygen are given.
1: O + O + M ➔ O2 + M
k1, 4.7 × 10-33 (300/T)2 cm6 s-1
2 O + O2 + M ➔ O3 + M
k2, 6 × 10-34 (300/T)2.4 cm6 s-1
3: O + O3 ➔ 2 O2
k3, 8 × 10-12 exp(−2060/T) cm3 s-1
4: H + O3 ➔ OH + O2
5: O + OH ➔ H + O2
6: O + HO2 ➔ OH + O2
7: H + O2 + M ➔ HO2 + M
2.2 Solar and Exothermic Chemical Heating Expressions for the Derivation of Radiatively Constrained Atomic Oxygen
 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 O-atom 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 O2 balanced by photolysis of O3. This is expressed as:
 In equation (3), J is the ozone photolysis rate, k2 the rate coefficient for the recombination of O and O2, and M is the total number density. However, the instantaneous rate ∂Q/∂t of energy deposition due to ozone photolyis is given by:
 In equation (4), hν is the energy of an ultraviolet photon absorbed by ozone, EB 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:
 The heating rate ∂T/∂t in Kelvin per day is derived from the first law of thermodynamics, expressed here as:
 In equation (6), kb is Boltzmann's constant, and Cp the specific heat at constant pressure. Below ~ 100 km, the atmosphere is composed essentially of diatomic molecules, and Cp 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:
 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. LD is the length of day (86,400 s = 1 day), and fday 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.
 Equation (7) depends only on the atomic oxygen concentration as all other terms are either constant (e.g., LD) or are provided by SABER (e.g., the O2 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); EB is 8800 cm−1; ε is 0.66 in the mesopause region; thus, enabling the equation to be written as
where ∂T/∂t is in Kelvin per day.
 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.
 For the recombination of atomic oxygen (the first reaction listed in Table 1), the rate of energy deposition is given by:
 In equation (9), k1 is the rate coefficient for recombination of atomic oxygen, M is the total number density, and H1 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:
 Proceeding in a similar fashion, the following expression is obtained for the recombination of O and O2, the second reaction listed in Table 1, again assuming that the rate of recombination is diurnally invariant:
 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 fday = 0.5 and J = 0.0833 s−1) for daytime:
 As with the recombination of atomic oxygen in equation (10), heating due to the reaction of O and O3 depends on the square of the O concentration. In equation (12), k3 is the reaction rate coefficient for the reaction of O and ozone, and H3 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 k3 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:
 Equation (13) incorporates the recommended value for the rate coefficient for k3 shown in Table 1.
 Reaction 4 in Table 1, the reaction of atomic hydrogen (H) and O3, 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):
 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:
 In equation (15), H4 is the enthalpy of reaction 4, and ε4 is the efficiency (0.75) of the reaction accounting for the loss of energy to space by the total Meinel band emission.
 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.
 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:
 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 k4 [H] [O3]:
 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 O2 density).
 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 × 108 cm−3 from 80 to 86 km. The heating due to reaction 7 is computed (proportional to k7 [H][O2][M]H7) as described for the reactions above. For reaction 6, photochemical steady state for the hydroperoxyl radical (HO2) implies k7 [H][O2][M] is balanced by k6 [O][HO2] so that the heating for reaction 6 is proportional to k7 [H][O2][M] H6. 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 O2 solar heating) from the CO2 cooling rates. The estimated heating due to reactions 6 and 7 is shown in Figure 1 as the curve labeled 3.
 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:
 The coefficients a and b in equation (18) depend on the rate coefficients k1 and k2, as well as the O2 density and temperature. The term c is the difference between the CO2 cooling rates and the sum of the O2 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 k1, k2, k3, 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.
3 Results and Comparison With SABER Atomic Oxygen
 Equation (18) is solved to yield the “radiatively constrained” atomic oxygen concentration, for which the associated chemical heating in reactions 1 through 5 in Table 1 is equal to the difference between the radiative cooling and O2 solar heating (and the minor chemical heating estimated for reactions 6 and 7). We assess the radiatively constrained atomic oxygen by comparing the profiles derived from equation (18) with the global annual mean atomic oxygen derived from SABER ozone and OH measurements described in Paper 1. Figures 2 and 3 show the comparisons for the years 2004 and 2008. Comparisons have been carried out for years 2004 to 2011, and the agreement between SABER O and the radiatively constrained O is the same in all other years.
 The results shown in Figures 2 and 3 are remarkable. Between ~ 82 and ~ 102 km, the radiatively constrained atomic oxygen is within ~ 25% of the global annual mean atomic oxygen derived from SABER ozone and OH emission measurements, i.e., within the estimated uncertainty of the SABER inferred atomic oxygen described in Paper 1. This result is shown in Figure 4 with the vertical dotted line indicating ±20% difference. The years 2004 and 2008 are shown specifically because 2004 is the first full year when SORCE data were available to provide accurate measurements of solar irradiance for the O2 solar heating calculations, and 2008 is chosen because it is the year of deep solar minimum. Figures 2-4 demonstrate agreement between the three independent SABER techniques (O from O3, O from OH emission, and O from global energy balance) within the uncertainty of each technique over most of the mesopause region. We conclude this section by emphasizing that while agreement (or uncertainty) within 20% between the various approaches may seem quite good, it still implies a difference of (1.20)2 or 1.44 (i.e., 44%) in the heating rate for the recombination of atomic oxygen.
 It is also instructive to assess the uncertainty in the radiatively constrained atomic oxygen concentrations derived here. As in Paper 1, we have carried out an analysis of the uncertainty by perturbing the key terms used in the derivation. The specific terms considered are the rate coefficients for recombination k1 and k2, the solar heating rates, and the infrared cooling rates. In principle, k3 should also be included but studies showed variation by its 75% uncertainty [Sander et al., 2011] made less than 1% difference in the radiatively constrained atomic oxygen. For the rate coefficient k1 (recombination of atomic oxygen), we use an uncertainty of 30% [Smith and Robertson, 2008]. For the rate coefficient k2 (recombination of atomic oxygen and molecular oxygen), we use an uncertainty of 20% [Sander et al., 2011]. For the solar heating rates and infrared cooling rate, we use a 5% uncertainty. Each term is changed by the prescribed amount, and the change from the unperturbed atomic oxygen is computed. It is assumed further that the uncertainties are uncorrelated so that the root-sum-square (RSS) of the differences is a reasonable estimate of the total uncertainty in the radiatively constrained atomic oxygen. As in Paper1, this approach provides the one standard deviation (“1-sigma”) uncertainty.
 The results are shown in Table 2. Over the altitude region from 80 to 100 km, we find that the overall uncertainty (the RSS) in the radiatively constrained atomic oxygen ranges from 12% to 20%. As with the day and night atomic oxygen in Table 2 of Paper 1, the rate coefficient k2 is the largest source of uncertainty over most of the mesopause region. From 95 to 100 km, uncertainty in k1 dominates the uncertainty in the radiatively constrained atomic oxygen. The uncertainties in radiatively constrained atomic oxygen are comparable to or smaller than the uncertainties in the SABER day and night atomic oxygen given in Paper 1.
Table 2. Percentage Uncertainty in Radiatively Constrained Atomic Oxygen Due to the Specified Uncertainties in Recombination Rate for Atomic Oxygen (k1, 30%), the Recombination Rate for Atomic Oxygen and Molecular Oxygen (k2, 20%) to the Total Solar Heating Rate (QSOL, 5%) and to the Rate of Radiative Cooling Due to CO2 (QCO2, 5%)a
The root-sum-square of the individual uncertainties is in the right-most column.
4 Discussion of Additional Heat Sources and Summary
 The results presented in Figures 2-4 demonstrate significant potential for using the principle of long-term energy balance as a way to constrain and validate the concentration of atomic oxygen in the mesopause region. The radiatively constrained atomic oxygen presented here is an upper limit to the O-atom concentration. To fully exploit the concept of this “energy balance chemistry model,” the potential influence of other heat sources in the mesopause region must be considered. In particular, heating due to dissipation of gravity waves must be assessed. If it is significant, then the radiatively constrained atomic oxygen derived here would be smaller than shown, all else being equal. However, reliable and accurate parameterizations of global annual mean gravity wave heating are required. Finally, as noted in Paper 1, a reduction in the SABER atomic oxygen is also possible through a reduction in the collisional quenching rates for vibrationally excited OH and vibrationally excited ozone. The radiative constrained approach presented herein enables the ability to distinguish between dynamical and chemical kinetic factors in the derivation of atomic oxygen.
 Some comments on the SABER heating and cooling rates are also in order. In particular, the SABER cooling rates in the Version 1.07 data set used in this study are frequently averages over several kilometers in altitude. This is necessary to stabilize the temperature retrieval. As a consequence, the cooling rates may be smaller at some altitudes than the actual cooling rate. Version 2 of the SABER data will eliminate this possibility, because a final, stand-alone calculation of the radiative cooling will occur using the retrieved temperature profiles after the retrieval of temperature converges. The Version 2 data set will be released starting sometime late in 2012/early 2013, and we will update these calculations at that time.
 The radiatively constrained atomic oxygen offers an additional benefit in that it allows for the assessment of various kinetic and spectroscopic rate coefficients used in the inference of atomic oxygen from ozone or airglow measurements. As discussed in Paper 1, there is still uncertainty in some key rate coefficients, particularly the rate of quenching of vibrationally excited OH with atomic oxygen. An independent constraint on the atomic oxygen concentration will be a substantial aid in resolving questions regarding rates and mechanisms for quenching of vibrationally excited OH when assessed in concert with the excellent SABER OH radiance and OH volume emission rate profiles. Radiatively constrained atomic oxygen also offers the possibility for overall improvement of photochemical modeling of the mesopause region. It may be possible to develop “energy balance chemistry models” to further constrain concentrations of other chemical species in the mesopause region.