Thermospheric resistance to greenhouse cooling : Effect of the collisional excitation rate by atomic oxygen on the thermal response to CO 2 forcing

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A7, 1292, doi: /2003ja009896, 2003 Thermospheric resistance to greenhouse cooling : Effect of the collisional excitation rate by atomic oxygen on the thermal response to CO 2 forcing R. A. Akmaev 1 Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado, Boulder, Colorado, USA Received 13 February 2003; revised 9 April 2003; accepted 5 May 2003; published 19 July [1] Infrared radiation in the 15-mm band of carbon dioxide is the major cooling mechanism of the middle and upper atmosphere. Increasing amounts of atmospheric CO 2 impose anthropogenic influence ( greenhouse cooling ) all the way through the mesosphere and thermosphere. Collisions with atomic oxygen are the primary excitation mechanism of CO 2 molecules in the thermosphere. Negative radiative forcing due to the CO 2 increase is roughly proportional to the rate of collisional excitation which in turn is proportional to the rate constant of collisional deactivation. The rate constant is still somewhat uncertain at present with various measurements and estimates varying within about a factor of 4. In light of recent laboratory measurements, two sets of numerical simulations have been performed to estimate the thermospheric response to doubling and a 15% increase of CO 2 for two values of the rate constant that differ by a factor of 2. Surprisingly, the temperature and density changes due to the CO 2 increases are practically independent of the rate constant. Simple diagnostics show that two physical mechanisms are primarily responsible: the strong temperature dependence of the radiative forcing itself in a combination with a temperature dependence of molecular heat conduction. Since the scenarios considered for the higher rate constant generally correspond to colder temperatures, the two physical mechanisms combined provide sufficiently strong negative feedbacks to entirely offset the initially stronger radiative forcing. INDEX TERMS: 1610 Global Change: Atmosphere (0315, 0325); 0358 Atmospheric Composition and Structure: Thermosphere energy deposition; 0325 Atmospheric Composition and Structure: Evolution of the atmosphere; 0350 Atmospheric Composition and Structure: Pressure, density, and temperature; KEYWORDS: greenhouse cooling, anthropogenic effects, mesosphere and thermosphere, collisional excitation, non-lte radiation, carbon dioxide Citation: Akmaev, R. A., Thermospheric resistance to greenhouse cooling : Effect of the collisional excitation rate by atomic oxygen on the thermal response to CO 2 forcing, J. Geophys. Res., 108(A7), 1292, doi: /2003ja009896, Introduction [2] Under the collisionally dominated conditions of local thermodynamic equilibrium (LTE) [Andrews et al., 1987; López-Puertas and Taylor, 2001], excited states of molecules with internal (vibrational or rotational) degrees of freedom are populated according to the Boltzmann distribution. The general principle of detailed balancing of statistical physics [Chapman and Cowling, 1970; Lifshitz and Pitaevskii, 1981] states that in equilibrium the collisional rate of excitation is exactly balanced by the reverse process of collisional deactivation. Within a standard two-level model for a particular mode of excitation, it then follows that g 1 k e ¼ k d exp E ð1þ g 0 k B T 1 Also at Space Environment Center, National Oceanic and Atmospheric Administration, Boulder, Colorado, USA. Copyright 2003 by the American Geophysical Union /03/2003JA where k e and k d are the rate constants for excitation and deactivation, the latter possibly also depending on kinetic temperature T; g 1 and g 0 are the statistical weights or quantum degeneracies (i.e., the total number of independent quantum states within a given energy level) of the excited and ground states, respectively; E is the energy of the excited state relative to the ground level; and k B is the Boltzmann constant. The rate at which energy is emitted away, say, per unit mass of air then depends only on the temperature and the mixing ratio of the radiatively active gas; the net radiative heating rate is determined by the balance between the energy emission and absorption. [3] Infrared radiation in the 15-mm CO 2 band is the major cooling mechanism of the middle and upper atmosphere and is produced by photon emission from excited levels of the bending vibrational mode n 2. Collisions with the major atmospheric species N 2 and O 2 support the LTE conditions for this vibrational mode up to about km [Andrews et al., 1987; López-Puertas and Taylor, 2001]. In the upper mesosphere and thermosphere the molecular collisions occur less frequently and other processes contribute significantly to the population of excited molecules. For example, SIA 5-1

2 SIA 5-2 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE the very process of photon emission, both spontaneous and induced by the ambient radiation field, becomes a substantial deactivation process. As a result, the population of excited states is reduced, less energy is radiated away, and the cooling rate generally decreases compared with what it would be if the LTE conditions and the associated Boltzmann distribution continued to hold. Still other processes, such as absorption of the radiation incoming from other regions of the atmosphere (primarily from below), may increase the number of excited molecules. However, the net thermal effect of the absorption is more local heating or, again, less cooling. [4] It was suggested by Crutzen [1970] that atomic oxygen, although still a minor species in the mesosphere and lower thermosphere, might be able to drive the n 2 mode toward LTE due to the much more efficient conversion between translational and vibrational energies in O-CO 2 collisions as expressed by a higher collisional excitation rate. According to equation (1), the rate of excitation k e for the O-CO 2 collisions is proportional to k d. Of course, this relation between k e and k d does not depend on the assumption of equilibrium and it holds under nonequilibrium conditions at least so far as the kinetic temperature T may be unambiguously defined. It is commonly said therefore that a higher deactivation rate of CO 2 by atomic oxygen results in stronger cooling in the upper atmosphere. This statement may be confusing for the higher deactivation rate per se may only result in more heating and less cooling. What is meant here is that the higher k d corresponds to a higher k e via equation (1), and the more efficient collisional processes then drive the n 2 system populations closer to LTE and so result in stronger cooling. [5] The rate constant for deactivation by atomic oxygen is still somewhat uncertain especially at relatively low temperatures typical for the upper mesosphere and lower thermosphere. Available laboratory measurements at room temperatures [Shved et al., 1991; Pollock et al., 1993] have suggested values of k d = ( ) cm 3 s 1 that are at least a factor of 4 smaller than the estimates from in-situ rocket measurements of the 15-mm emission in the upper atmosphere [Sharma and Wintersteiner, 1990]. In their detailed theoretical analysis of non-lte CO 2 radiation, Shved et al. [1998] recommend k d = cm 3 s 1 which is halfway between the two extremes and close to the estimates derived by López-Puertas et al. [1992] from satellite observations [see also López-Puertas and Taylor, 2001]. Recently, Khvorostovskaya et al. [2002] have repeated the laboratory measurements of Shved et al. [1991] with better accuracy and for the temperature range K. They have found essentially no significant temperature dependence in this range with an average value of k d = cm 3 s 1. [6] Stronger cooling by increasing amounts of carbon dioxide and other greenhouse gases imposes a major anthropogenic influence on the thermal structure of the mesosphere and thermosphere [Roble and Dickinson, 1989; Rishbeth and Roble, 1992; Berger and Dameris, 1993; Akmaev and Fomichev, 1998, 2000]. The greenhouse cooling, as it was called by Cicerone [1990] and Roble [1995a], should also manifest in secular changes of other atmospheric parameters, in particular, in significant neutral density reductions in the thermosphere [Roble and Dickinson, 1989; Rishbeth and Roble, 1992; Akmaev and Fomichev, 1998; Keating et al., 2000; Akmaev, 2002]. Analyses of long-term ionospheric sounding data have also indicated downward displacements of heights of various ionospheric layers [e.g., Bremer, 1992; Ulich and Turunen, 1997; Bremer and Berger, 2002], which are generally consistent with model predictions of atmospheric cooling [e.g., Rishbeth, 1990; Rishbeth and Roble, 1992; Akmaev, 2002]. [7] At high altitudes where the atmosphere is optically thin, the cooling rate is roughly proportional to the collisional excitation rate [Andrews et al., 1987] and so to the CO 2 mixing ratio. At heights where collisions with atomic oxygen are the dominant excitation mechanism, the cooling rate becomes also proportional to k e for these collisions (and to k d if temperature is fixed). It follows then that the radiative forcing due to increasing amounts of CO 2 in the thermosphere should also be proportional to the excitation (or deactivation) rate constant. This is illustrated in Figure 1 representing the radiative forcing due to doubling of CO 2 for solstice and equinox temperature distributions according to the MSISE-90 empirical model [Hedin, 1991] and for two values of k d = and cm 3 s 1 as recommended by Shved et al. [1998] and Khvorostovskaya et al. [2002], respectively. Here and in what follows, the radiative and, more generally, thermal forcing denotes the difference between the heating rates due to a particular physical process in the perturbed case (doubled CO 2 as in Figure 1, for example) and in the control case. As is clearly seen in Figure 1, the forcing remains approximately the same below about 100 km. Above this level, it becomes stronger for the higher value of k d and almost exactly doubles above approximately 120 km. [8] It would then appear natural to suggest that the atmospheric response to a perturbation in CO 2 should be stronger for higher values of k d [e.g., Akmaev and Fomichev, 1998; Keating et al., 2000]. Owing to its uncertainty, previous modeling experiments have used widely ranging values of k d with almost a factor of 15 difference between the studies of Berger and Dameris [1993] and Akmaev and Fomichev [1998, 2000], for example. In our previous simulations, k d = cm 3 s 1 has been used following Shved et al. [1998] to evaluate the effects of future doubling of CO 2 [Akmaev and Fomichev, 1998] and a 15% increase of CO 2 [Akmaev and Fomichev, 2000] which has occurred over the last 4 decades. The purpose of the present work is to investigate the possible effect of k d on the temperature and density response in the thermosphere in light of the new measurements by Khvorostovskaya et al. [2002]. This is done here by essentially repeating the numerical experiments of Akmaev and Fomichev [1998, 2000] for the two values of k d = and cm 3 s 1. [9] Contrary to the naive expectations, the temperature and density changes due to CO 2 increases turn out to be practically independent of the value of k d provided this is the only parameter changed. The primary reason for such a response is that with a higher k d, for example, the thermosphere is generally colder and its ability to respond to a CO 2 perturbation is suppressed. Simple diagnostics show that two physical mechanisms are responsible: the strong temperature dependence of the radiative forcing itself via the excitation

3 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE SIA 5-3 Figure 1. Zonal mean radiative forcing due to doubling of CO 2 for January (top) and March (bottom) MSISE-90 temperature distributions and for k d = cm 3 s 1 (left) and cm 3 s 1 (right). Positive areas shaded. rate constant k e in equation (1) and the temperature dependence of the process of molecular heat conduction. [10] Section 2 briefly describes the Spectral Mesosphere/ Lower Thermosphere Model (SMLTM) and the numerical experiments conducted. Results of the experiments are presented in section 3, and section 4 discusses the diagnostics of physical processes important in the thermal balance under perturbations of CO 2 amounts in the thermosphere. A more detailed discussion of negative feedbacks in the thermosphere is presented in Appendix A. 2. Numerical Model and Experiments [11] The three-dimensional global SMLTM extends from the pressure level of 100 mb (about 16 km) up to approximately 220 km with the vertical resolution of half a pressure scale height. The current version incorporates the parameterization of radiative transfer in the 15-mm CO 2 band of Fomichev et al. [1998] with the recent modifications by Ogibalov et al. [2000]. A horizontally uniform vertical profile of CO 2 volume mixing ratio (VMR) is assumed similar to that compiled from observations by Fomichev et al. [1998] with the VMR decreasing sharply above approximately 85 km due to the diffusive separation. The standard surface VMR of CO 2 is assumed to equal 360 ppm, and for the experiments with higher or lower concentrations of CO 2 the whole vertical profile is scaled according to its surface value. [12] The nonuniform vertical distribution in combination with a global pole-to-pole circulation during solstice produces a nonuniform latitudinal distribution of CO 2 with lower values in the winter hemisphere and higher values in the summer hemisphere [Chabrillat et al., 2002]. This in turn results in weaker cooling in winter and stronger cooling in summer. The same circulation cell also brings more atomic oxygen to the upper mesosphere and thermosphere in winter and depletes its abundance in summer [Kellogg, 1961; Johnson, 1964]. The latitudinal distribution of atomic oxygen thus enhances the radiative cooling in the winter hemisphere and diminishes it in the summer hemisphere [Fomichev et al., 1996] with the two effects compensating each other to some extent. Although the assumption of a horizontally uniform vertical profile of CO 2 is only an approximation, it appears more consistent to use it along

4 SIA 5-4 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE Figure 2. Zonal mean thermal response to the doubling of CO 2 for January (top) and March (bottom) and for k d = cm 3 s 1 (left) and cm 3 s 1 (right) as a function of height. Positive areas shaded. with the assumption of a horizontally uniform distribution of O, rather than specifying a nonuniform distribution for either of the two species. The current version of the SMLTM does not incorporate interactive composition and the global mean VMR profile of O as a function of pressure is specified according to the MSISE-90 model [Hedin, 1991]. [13] Another modification of the model introduced after the study of Akmaev and Fomichev [1998] is the incorporation of the Doppler-spread parameterization of gravity waves [Hines, 1997a, 1997b, 1999]. The parameterization allows to produce more realistic temperatures in the mesosphere and lower thermosphere, in particular the low and very cold summer mesopause [Akmaev, 2001]. This results in a positive CO 2 forcing (even stronger than for the MSISE-90 model in Figure 1) and a positive thermal response (local heating) to enhanced CO 2 levels around the summer mesopause [Akmaev and Fomichev, 2000; Akmaev, 2002]. Other particulars of the model have been documented in detail elsewhere [Akmaev and Fomichev, 2000; Akmaev, 2001, 2002]. [14] Here the doubled-co 2 experiments of Akmaev and Fomichev [1998] are repeated for solstice and equinox conditions with the new configuration of the model and for the two values of k d = and cm 3 s 1. In another set of experiments the two values of k d are used to estimate the effects of a 15% increase of the CO 2 surface mixing ratio from 313 to 360 ppm observed in the troposphere approximately over the last 4 decades [Keeling and Whorf, 1999]. 3. Simulation Results [15] Figures 2 and 3 present the zonal mean thermal response to the doubling and the 15% increase of CO 2, respectively, as a function of height. As has been done in the past [Akmaev and Fomichev, 1998, 2000; Akmaev, 2002], it should be emphasized here again that the response takes quite a different form depending on the vertical coordinate used to compare temperatures. Most upper-atmospheric models use pressure (or log-pressure) as the vertical coordinate while most observations, both satellite and groundbased, are performed with respect to height and so it seems more natural to compare temperatures at a given height instead of a given pressure. The transformation from one coordinate to the other has to account for the hydrostatic shrinking of the atmosphere as a whole, associated with an

5 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE SIA 5-5 Figure 3. Same as in Figure 2 but for the 15% increase of CO 2. overall cooling at fixed pressure. This is especially important in the thermosphere where the downward displacement of constant-pressure levels becomes relatively large [Akmaev, 2002]. As a result, a layer of an apparent heating is seen between about 100 and 130 km where, curiously, the negative radiative forcing maximizes (Figure 1). This is simply associated with a substantial downward displacement in the region of a strong positive temperature gradient in the lower thermosphere. [16] During solstice (upper panels in Figures 2 and 3) there is also a region of weak cooling or even heating near the summer mesopause as was previously noted by Akmaev and Fomichev [2000] for the 15% CO 2 increase. This is consistent with the observational estimates of Lübken [2000] and is caused by a positive radiative forcing near the cold summer mesopause (see Figure 1). This effect has also been reproduced in the simulations by Bremer and Berger [2002]. New in the present results is that the same feature is seen in the doubled-co 2 case as well, as opposed to our previous simulations [Akmaev and Fomichev, 1998]. As discussed above, the reason for that is a more realistic thermal structure effected by the incorporation of the Doppler-spread parameterization of gravity waves into the model. It is also noteworthy that the shapes of the temperature response to a strong and a relatively weak CO 2 forcings (compare Figures 2 and 3) are very similar [Akmaev and Fomichev, 2000]. [17] Perhaps the most surprising result is that contrary to what might be expected from Figure 1, the thermal response is not weaker with the smaller rate constant k d = cm 3 s 1 than with the greater constant k d = cm 3 s 1 (compare the right and left columns in Figures 2 and 3). The same is true for the relative density changes induced by the cooling (Figures 4 and 5). In some instances the temperature and density changes are even slightly stronger for the smaller value of k d, although the differences are probably negligible compared with the uncertainty of the simulations. The primary reason for a stronger than expected response with a smaller k d is of course the fact that in this case the base cooling rates are smaller and, correspondingly, the temperatures are generally warmer, including the control experiments and all the scenarios with perturbed CO 2 amounts. This is illustrated in Figure 6, presenting the thermal effect of the halving of k d in the control experiments corresponding to the present-day conditions (CO 2 surface VMR of 360 ppm). [18] It should not be unexpected that the change of k d by a factor of 2 produces temperature changes comparable in

6 SIA 5-6 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE Figure 4. Zonal mean percent density change due to the doubling of CO 2 for January (top) and March (bottom) and for k d = cm 3 s 1 (left) and cm 3 s 1 (right). magnitude with those due to doubling of CO 2. This follows from the fact that the cooling rate is proportional to both the CO 2 VMR and k d in the thermosphere. Although the absolute temperature changes are large, the relative changes remain comparatively small, of the order of a few percent, which means that the control cases for both values of k d are in reasonable agreement with empirical climatologies. These changes in temperature, however, substantially affect the ability of the thermosphere to respond to the CO 2 induced forcing. The colder thermosphere corresponding to the greater value of k d turns out to be more resistant to further cooling by the increased CO 2 amounts as is illustrated in the next section. 4. Global Mean Diagnostics [19] It is convenient to analyze the contributions of various physical processes to the thermal balance averaged globally, setting aside possible local seasonal and latitudinal variations. It is also convenient to compare these contributions at the same pressure levels. This is not only because the model uses pressure as the vertical coordinate but also because with the strong radiative cooling imposed, many important characteristics such as the composition, molecular mass, and other quantities related to composition are expected to drift with pressure levels rather than remain attached to a fixed height [Rishbeth, 1990]. [20] Figure 7 presents the thermal response T to the doubling and the 15% increase of CO 2 for both values of k d but averaged globally and as a function of both height and pressure. Pressure height used in the bottom panels of Figure 7 and in the following figures refers to globally averaged heights of pressure surfaces according to the MSISE-90 reference atmosphere and so is a convenient measure of pressure, not of the real height. The temperature changes are approximately equal for the two values of k d as a function of altitude. At constant pressure, they even appear stronger for the smaller value of k d at almost all levels. The differences are relatively small, approximately of the same order as the seasonal variability, and may not be significant everywhere with respect to that variability and model uncertainties. The vertical shapes of T are very similar for the two scenarios of CO 2 change, suggesting a universal spatial mode of the response [Akmaev and Fomichev, 2000]. Calculated as a function of the two different vertical coordinates, however, the vertical shapes differ drastically. At fixed height, the global mean responses exhibit a typical layer of apparent heating between about 100 and 130 km, while

7 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE SIA 5-7 Figure 5. Same as in Figure 4 but for the 15% increase of CO 2. Figure 6. Zonal mean thermal response to the change of k d from cm 3 s 1 to cm 3 s 1 in January (left) and March (right). Negative areas shaded.

8 SIA 5-8 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE Figure 7. Global mean thermal response to the doubling (left) and the 15% increase (right) of CO 2 as a function of height (top) and pressure (bottom). Figure 8. Ratios of global mean collisional excitation rates k e corresponding to k d = cm 3 s 1 and cm 3 s 1, respectively, for the doubling (left) and the 15% increase (right) of CO 2.

9 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE SIA 5-9 Figure 9. (top) Doubled-CO 2 global mean residual thermal forcing (see text) due to CO 2 radiative transfer (left), molecular heat conduction (middle), and the sum of the two (right). (bottom) Differences of the residual forcings corresponding to k d = cm 3 s 1 and cm 3 s 1. they are negative everywhere and substantially stronger at fixed pressure. [21] As the atmosphere responds to a negative radiative forcing, similar to that presented in Figure 1, it cools down. Since the cooling rate is a strong monotonically increasing function of T, via the collisional excitation rate k e (equation (1)) in the thermosphere, for example, the temperature decrease reduces the excess radiative forcing. If there were no other heating mechanisms depending on temperature and providing negative feedbacks, the cooling would continue until the radiative forcing were compensated entirely. In this case one would generally expect stronger cooling in response to a stronger forcing. [22] As was already mentioned, there are two main factors that prevent this from happening in the present numerical experiments. One of them is the fact that the scenarios corresponding to the smaller value of k d generally operate at warmer temperatures (Figure 6). This immediately results in substantial reduction of the difference between the CO 2 radiative forcings corresponding to the two values of k d. As illustrated in Figure 8, the ratio of the excitation rate constants k e corresponding to k d = and cm 3 s 1, respectively, substantially exceeds 0.5, especially between about 100 and 130 km, that is in the layer where the thermospheric CO 2 forcing maximizes. [23] There are also other physical mechanisms providing negative feedbacks as the temperature goes down in response to the CO 2 radiative forcing [Akmaev and Fomichev, 1998]. In the thermosphere the strongest feedback is apparently provided by molecular heat conduction, as illustrated in Figures 9 and 10 for the doubling and the 15% increase of CO 2, respectively. The upper rows in these figures present what is designated here as residual thermal forcing, defined as the difference between the heating rates in a perturbed CO 2 case and the corresponding control case after the atmosphere relaxes to equilibrium. Without the feedbacks, the residual radiative forcing (upper left panels in Figures 9 and 10) would be close to zero, as discussed above. A feedback mechanism compensates for part of the initial radiative forcing in such a way that the atmosphere no longer has to cool down as much as it would otherwise. The upper middle panels in Figures 9 and 10 present residual forcings due to turbulent and molecular heat conduction combined, with the latter being the primary mechanism above about 100 km [Roble, 1995b]. It is clearly seen that in all scenarios the molecular heat conduction compensates the residual radiative forcing to a large extent (see the upper right panels presenting the sum of the two). [24] In the context of this study it is important to note that in all cases the residual radiative forcings for k d = cm 3 s 1 are greater in magnitude than for k d =1.5

10 SIA 5-10 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE Figure 10. Same as in Figure 9 but for the 15% increase of CO cm 3 s 1 in the thermosphere above about 110 km. The reason for that is that the corresponding negative feedback (positive residual forcing) due to molecular heat conduction is greater for the larger value of k d or at colder temperatures. In other words, the molecular thermal conductivity is more efficient at lower temperatures in the terrestrial atmosphere (see Appendix A for a more detailed discussion). This can be seen in the bottom panels of Figures 9 and 10 presenting, on the same scale, the differences of the residual forcings corresponding to the larger and smaller values of k d. Without the feedback, the thermosphere would cool down further until the residual forcing is entirely balanced. The cooling would have to be stronger for the greater value of k d to compensate for the stronger CO 2 forcing. With the feedback the thermosphere does not have to, or cannot, cool down as much. In fact, even a slightly smaller cooling is sufficient in many places as is evidenced in Figure 8. [25] The two mechanisms identified here provide, in combination, sufficient negative feedbacks to offset the effect of an initially stronger radiative forcing in the case of a higher k d. Thus as the thermosphere cools down, it becomes more resistant to further cooling. This kind of response is consistent with the general Le Chatellier principle of statistical physics [Landau and Lifshitz, 1980] stating that an external forcing imposed on a system in equilibrium brings about processes in the system which tend to reduce the effect of the external forcing. [26] It should be noted that there exist still other processes not explicitly represented in the SMLTM that may further enhance the stability of the thermosphere to external thermal forcings. One such process is the radiative cooling by NO [Kockarts, 1980] with the cooling rate also strongly dependent on temperature similar to the CO 2 cooling rate and peaking at about 150 km [Roble, 1995b]. This process is not explicitly represented in the current version of the model since it would also require an interactive ionospheric photochemistry module. Instead, the NO cooling is accounted for via a prespecified efficiency coefficient of the solar extreme ultraviolet heating as recommended by Roble [1995b]. This is an approximation since the constant efficiency coefficient does not account for the strong temperature dependence of the NO cooling. [27] Of course, the stabilizing effect of the negative feedbacks also applies to the case of positive forcing. This study has concentrated on the negative forcing in the context of possible anthropogenic greenhouse cooling. 5. Conclusion [28] The thermospheric response to CO 2 increases has been estimated for two values of the rate constant for collisional deactivation by atomic oxygen: k d = and k d = cm 3 s 1 as recommended previously by Shved et al. [1998] and from the recent laboratory

11 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE SIA 5-11 measurements of Khvorostovskaya et al. [2002], respectively. Contrary to expectations, the temperature and density changes due to a doubling and a 15% increase of CO 2 turn out to be practically independent of k d provided this is the only parameter changed. [29] Simple global mean diagnostics show that two physical mechanisms are primarily responsible: the strong temperature dependence of the radiative forcing itself via the collisional excitation rate constant in equation (1) and additional negative feedbacks due to the temperature dependence of the process of molecular thermal conduction. With higher k d the thermosphere becomes colder and the feedbacks provided by the two mechanisms fully compensate for the initially stronger radiative forcing. The colder thermosphere is therefore more resistant to further cooling due to the CO 2 increases. [30] Although previous modeling studies of CO 2 greenhouse cooling have used the rate constants varying by more than one order of magnitude, there has not been that much variation in the estimated response [cf. Roble and Dickinson, 1989; Rishbeth and Roble, 1992]. The present simulations and diagnostics may explain the robustness of these estimates. Appendix A: Negative Feedbacks Due to Molecular Heat Conduction [31] Here we follow the line of reasoning employed previously [Akmaev and Fomichev, 1998] to explain the fact that the negative thermal response continues to grow in magnitude with height in the thermosphere (Figure 7) although the radiative forcing peaks at about 120 km (see Figures 1, 9, and 10). Consider a one-dimensional (e.g., globally averaged) model of the thermosphere in equilibrium, where the net radiative heating is largely balanced by the downward heat transport due to molecular heat conduction: where 1 df rc p dz ¼ h; F ¼ l dt dz ða1þ ða2þ is the downward (negative) heat flux; h is the heating rate; z is altitude; and r, c p, and l are the atmospheric mass density, specific heat at constant pressure, and coefficient of thermal conductivity, respectively. Using equation (A2) and converting the vertical coordinate to log-pressure x ¼ ln p s p where p s is some reference pressure, equation (A1) may be rewritten in the following form g d gmlðx; T c p p dx RT Þ ; dt dx ¼ hx ðþ; ða3þ where R is the universal gas constant, g is acceleration of gravity, and m is molecular mass. As in the main text, it is assumed here that the composition is tied to pressure [Rishbeth, 1990] and, consequently, so are c p, m, and l. The possible dependence of h on temperature, as in the case of atmospheric infrared radiation, is hidden for now. The temperature dependence will be reintroduced later with a discussion of its possible effects, especially in the presence of negative feedbacks. For now, h(x) may be thought of as solar heating depending, in addition to common astronomical, geometrical, and seasonal factors, only on composition and so only on x. It is also assumed that g depends only on x, that is the effect on g of possible vertical displacement of pressure levels is neglected. Equation (A3) may now be rewritten as Ax ðþ dfðþ x dx ¼ hx ðþ: ða4þ [32] Assuming a power-law temperature dependence for l [Chapman and Cowling, 1970; Banks and Kockarts, 1973] lðx; TÞ ¼ ðþt x a ; the heat flux in equation (A2) may also be rewritten as a 1 dt Fx ðþ¼ Bx ðþt dx : ða5þ ða6þ In equations (A4) and (A6), A(x) and B(x) are positive functions of x only. Equation (A4) is readily integrated assuming a zero heat flux at the top of the atmosphere [F(1) =0] Z 1 Fx ðþ¼ x hx ð 0 Þ Ax ð 0 Þ dx0 : [33] With a lower boundary condition, T(x 0 ) = T 0, equation (A6) may now be integrated to yield Z x T a ðþ¼t x 0 a þ a dx 0 x 0 Bx ð 0 Þ Z 1 hx 00 x 0 Ax ð 00 ð Þdx 00 ; a 6¼ 0: ða7þ Þ The case of a = 0 needs special treatment [Akmaev and Fomichev, 1998], and the solution has a different form but it is inconsequential for the following discussion. The double integral in equation (A7) is a linear operator, L(h), acting on h and spreading the radiative heating over the thermosphere. [34] Suppose now that there is a perturbation h(x) imposed on the radiative heating which is small enough for the following linear approximations to be valid. Assuming also that the lower boundary x 0 is deep enough in the atmosphere that T 0 does not change, the corresponding perturbation in temperature takes the following form in the linear approximation: TðÞ¼T x 1 a ðþl x ðhþ ða8þ which is also valid under the same assumptions in the case a = 0. Note that L is positive definite, in the sense that if h(x) has the same sign everywhere, T(x) has the same sign.

12 SIA 5-12 AKMAEV: GREENHOUSE COOLING IN THE THERMOSPHERE [35] Equation (A8) states essentially that the thermal response T(x) is a linear function of the forcing h(x), as is to be expected in a linear approximation, with the operator L projecting the forcing (Figure 1) onto its characteristic response modes (see the lower panels of Figure 7). The response is also nonlinear in the sense that its magnitude depends on temperature via the factor T 1 a. For a <1 this is a monotonically increasing function of temperature and so, under the balance of radiative heating and molecular heat conduction, the thermal response to the same forcing is smaller at colder temperatures. [36] According to elementary mean-free-path theory, a = 0.5, reflecting the temperature dependence of the mean molecular velocity [Chapman and Cowling, 1970; Lifshitz and Pitaevskii, 1981]. More realistic models of molecular interactions and experimental data suggest somewhat greater values for a which still remain smaller than 1 for the main components of the terrestrial thermosphere [Chapman and Cowling, 1970; Banks and Kockarts, 1973]. As was shown by Akmaev and Fomichev [1998], the same condition, a < 1, is necessary for the thermal response to increase in magnitude with height as observed in present simulations (Figure 7). [37] There exist no direct measurements of l for atomic oxygen. On the basis of estimates of its viscosity coefficient, Banks and Kockarts [1973] suggested a parameterization in the form of equation (A5) with a = 0.69, which agreed well with the theoretical calculations of Dalgarno and Smith [1962]. Kharchenko et al. [2000] have recently updated the calculations, suggesting a slightly larger value of a = The new estimates are still within a few percent of the parameterization by Banks and Kockarts [1973] for a typical range of temperatures in the thermosphere. Thermophysical properties of the main molecular species N 2 and O 2 are known relatively well [e.g., Banks and Kockarts, 1973; Younglove, 1982; Laesecke et al., 1990; Uribe et al., 1990]. Owing to the contribution of internal degrees of freedom, the rate of increase of l with temperature is usually greater for polyatomic gases compared with single atoms. It was shown by Banks and Kockarts [1973], however, that the same parameterization in equation (A5) with a = 0.69 may be used for these gases in the thermosphere. [38] Formally, thermal conductivity of a gas mixture is a complicated function of the properties of individual species since collisions between all possible pairs of molecules have to be taken into account [Chapman and Cowling, 1970]. Banks and Kockarts [1973] suggested parameterizations of variable complexity and showed that a simple VMR-weighted average provided an adequate description for typical thermospheric conditions. This simple parameterization is used in the SMLTM with the composition prescribed as a function of pressure according to the MSISE-90 model [Hedin, 1991] to calculate (x) in equation (A5). There may, in principle, be a compositional effect on l as the thermosphere evolves in response to perturbations in the CO 2 radiative forcing. However, it is not expected to be a major factor since, as already mentioned, the composition tends to drift with pressure surfaces and so remain approximately constant as a function of pressure [Rishbeth, 1990]. [39] If the perturbation heating rate depends on temperature as in the case of infrared atmospheric radiation, then in the linear approximation it should be expandable into the following form hðx; T þ T Þ ¼ hðx; T Þ½1 þ fðx; TÞTŠ ða9þ where f(x,t) is typically positive for the local temperature dependence similar to that in equation (1). If the radiative cooling rate depends on T nonlocally, the factor f (x,t) will be replaced in equation (A9) by a linear operator acting on T. Formally substituting this expansion into equation (A8) and solving for T, a generalized solution is obtained TðÞ¼ x 1 T 1 a 1T L½hðx; TÞfðx; TÞŠ 1 a L½hðx; TÞŠ: ða10þ The operator in curly braces describes the additional damping of the response due to the presence of the radiative feedback, f (x,t) > 0. The damping is stronger for a stronger initial negative forcing, h(x,t) < 0. 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