The clear sky instantaneous radiative forcings of the 14 halogenated gases previously shown to have the largest contribution to global warming, were found. The calculation used the absorption cross sections for the halogenated gases which are assumed to be independent of temperature as well as over 1/3 million line strengths for the five main naturally occurring greenhouse gases: H 2 O, CO 2 , O 3 , CH 4 and N 2 O, from the Hitran database. The total radiative forcing of the halogenated gases at their 2020 concentrations is 0.52 (0.67) W/m 2 at the tropopause (mesopause). Over half of this forcing is due to CFC11 and CFC12 whose concentrations are steadily declining as a result of the Montreal Protocol. The rate of total forcing change for all 14 halogenated gases is 1.5 (2.2) mW/m 2 /year at the tropopause (mesopause). The calculations assumed a constant altitude concentration for all halogenated gases except CFC11, CFC12 and SF 6 . Using the observed altitude dependence for those 3 molecules reduced their radiative forcings by about 10%. The global warming potential (GWP) values were found to be comparable to those given by the Intergovernmental Panel on Climate Change. The contribution of a gas to global warming can be estimated using the forcing power per molecule defined as the derivative of its radiative forcing with respect to its column density. For the present atmosphere, the per-molecule forcing powers of halogenated gases are orders of magnitude larger than those for the 5 naturally occuring greenhouse gases because the latter have much higher concentrations and are strongly saturated. However, the rates of concentration increase of the 5 main greenhouse gases are orders of magnitude greater than that of any halogenated gas. Assuming the temperature increase caused by each gas is proportional to its radiative forcing increase, the 14 halogenated gases are responsible for only 2% of the total global warming. Halogenated molecules frequently contain one or more carbon atoms as well as various halogen atoms and hydrogen. Chlorofluorocarbons (CFCs) are nontoxic and nonflammable. CFC11 and CFC12, commercially known as freon 11 and freon 12 respectively, were discovered about a century ago and widely used in refrigeration. When released into the atmosphere, these molecules can travel to the stratosphere where they break down releasing chlorine atoms. In the 1970s, it was found that chlorine catalyzed the destruction of the stratospheric ozone layer [1,2]. An international agreement known as the Montreal Protocol was created in 1987 to eliminate the use of CFCs [3]. Subsequent amendments seek to phase out hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs). These molecules were developed to replace CFCs. They have much less ozone destroying ability but along with the CFCs are greenhouse gases as pointed out by the Integovernmental Panel on Climate Change (IPCC) [4]. There have been a number of calculations of the radiative forcings and global warming potentials for various halogenated molecules [5,6,7]. These calculations used various radiative transfer models along with results from the Hitran database for the absorption cross sections for the halogenated gases as well as line strengths for greenhouse gases such as carbon dioxide [8]. The calculations use temperature profiles and altitudinal profiles of greenhouse gases such as water vapor at various locations on the Earth. Unfortunately, these profiles are not always readily available which would facilitate comparison of different calculations. The instantaneous radiative forcing was calculated for 14 molecules found to be the strongest greenhouse gases of the halogenated molecules. The calculations were done for a clear sky and use the standard US temperature profile for the midlatitude [9]. This expands our previous work that found the radiative forcings of the 5 main naturally occurring greenhouse gases H 2 O, CO 2 , O 3 , CH 4 and N 2 O [10]. Our calculations use the observed altitudinal profiles of these gases [11]. The radiative forcings of the halogenated gases using their 2020 concentrations, were found at the tropopause and top of the atmosphere or mesopause. The rate of change of the total halogenated gas forcing was found using the observed concentration change rates of the various gases. The warming effectiveness of each molecule was found by taking the derivative of its radiative forcing with respect to its column density to obtain the per-molecule forcing power. Finally, the global warming potentials were obtained and compared to values in the literature. Careful measurements of surface greenhouse gas concentrations have been ongoing for more than half a century. The pioneering work of C. Keeling studying carbon dioxide concentrations at Mauna Loa has been extended to include a number of halogenated species as shown in Fig. 1 [12]. The graph clearly shows the effectiveness of the Montreal Protocol as the concentrations of the two most abundant halogenated gases, CFC11 and CFC12 have decreased by about 20% and 10% respectively, since their peak concentrations in the 1990s. All of the halogenated molecules have an entirely anthropogenic origin with the exception of CF 4 whose preindustrial level is estimated to be 34 ppt [12]. Radiation transfer in the cloud-free atmosphere of the Earth is controlled by the temperature T = T (z) at the altitude z and the number densities, N {i} = N {i} (z) of the ith type of molecule. Representative midlatitude altitude profiles of temperature [9], and greenhouse gas concentrations [11], are shown in The standard US temperature profile for the midlatitude, plotted as the solid blue line in the left panel of Fig. 2, consists of 5 linear segments. The average surface temperature of the Earth is 288.7 K. This decreases to 217.2 K at the tropopause altitude, z tp , of 11 km and remains constant until 20 km. The temperature in the stratosphere increases to 229.2 K at 32 km with an additional increase to 271.2 K at 47 km as a result of absorption of incoming solar UV radiation by ozone molecules. Finally, the temperature decreases in the mesosphere to 187.5 K at the mesopause altitude, z mp , of 86 km.
The concentrations for the ith greenhouse gas, C {i} sd , based on observations [11], are shown as functions of altitude on the right of Fig. 2 for the 5 main greenhouse gases. The 2020 surface concentrations are 7, 750 ppm of H 2 O, 1.9 ppm of CH 4 , 0.33 ppm of N 2 O and 10 ppt for SF 6 [12]. The O 3 concentration peaks at 7.8 ppm at an altitude of 35 km, while the concentrations of CO 2 and CF 4 were 413 ppm and 86 ppt, respectively, at all altitudes. The dependence of the concentrations relative to the surface of CFC11 and CFC12 with altitude is shown in Integrating the number density, N {i} , over an atmospheric column having a cross sectional area of 1 cm 2 yields the column number density of the ith type of molecule N {i} sd listed in Our previous publications describe in detail how to calculate radiation transport from the Earth's surface through the atmosphere [10,16] and is here only briefly discussed. Radiation transport is governed by the Schwarzschild equation [17] in cloud-free air where scattering Here I = I(, z, ) is the spectral intensity of a pencil of radiation of frequency between  and  + d at altitude z. The pencil makes an angle  to the vertical. The Planck intensity is given by The radiation frequency,  has units of cm 1 , h is Planck's constant, k B is Boltzmann's constant and c is the speed of light. The optical depth is defined by where the net attenuation coefficient due to molecules absorbing and reemitting light of frequency  at altitude z is given by Here N {i} (z) is the density of a greenhouse gas molecule of type i and  {i} =  {i} (z, ) is its absorption cross section for radiation of frequency  at the altitude z. The cross section can depend strongly on altitude because temperature and pressure are functions of altitude. Temperature controls the distribution of the molecules between translational, rotational and vibrational states. Pressure, together with temperature, determines the width of the molecular resonance lines. The halogenated gas absorption cross sections were obtained from the Hitran website [8] and are listed in The uncertainty of the Hitran cross sections is given to be about 8% [7]. Integrating over frequency gave the total cross sections listed in the final column of The cross section can be written as the sum of partial cross sections  ul , corresponding to each Bohr transition frequency  ul , The partial cross section,  ul , is the product of a lineshape function, G ul = G ul (,  ), and a line strength, S ul = S ul (T ), The absorption of the 5 main greenhouse gases, H 2 O, CO 2 , O 3 , CH 4 and N 2 O were obtained using the Hitran line strengths [10]. The optical depth from the surface to the top of the radiative atmosphere, the altitude z mp of the mesopause, is As indicated by the notation (7), we have assumed that the optical depth  mp at the mesopause altitude z mp differs negligibly from the optical depth   at infinite altitude since there is so little opacity of the atmosphere above the mesopause. The Schwarzschild equation ( 1) can be solved to find the intensity [10] For For where  = sec . Temperature Pressure  min  max Cross Section (K) (mbar) (cm 1 ) (cm 1 ) (10 can be rewritten after substituting ( 8) and ( 9) into (10) to give Here, Bs = B(T s ), is the Planck intensity evaluated at the surface temperature T s . The exponential-integral functions, E n ( ), that account for slant paths of radiation between different altitudes are defined for integers n = 1, 2, 3, . . . by The spectral forcing, F , is defined as the difference between the spectral flux  Bs through a transparent atmosphere from a surface with temperature T s , and the spectral flux Z for an atmosphere having greenhouse gases, The spectral flux Z at the mesopause altitude is shown in Fig. 4 for the case of the 5 main greenhouse gases as well as CFC12. The frequency integrals of the flux (10) and the forcing (13) are where  SB is the Stefan Boltzmann constant. Fig. 5 shows the frequency integrated flux versus altitude for three different concentrations of CO 2 in addition to the four other main greenhouse gases. A similar plot to Fig. 5 where one would show the result of increasing the concentration of any of the halogenated gases from its preindustrial level to its 2020 concentration results in an imperceptible change from the green curve. The radiative forcings of the various gases resulting from the increase in the respective concentration relative to their preindustrial levels are given in Table 2 at the tropopause and mesopause altitudes. Table 2 as well as subsequent tables do not list water vapor, the most significant greenhouse gas. As shown in Table 2 of reference [10], the total radiative forcing of water vapor is about twice that of CO 2 . Clouds further increase the importance of H 2 O for radiative transfer in Earth's atmosphere. All halogenated gases except for CF the mesopause. The total anthropogenic forcing was found relative to the preindustrial levels which was zero unless listed in brackets. The observed rates of the concentration changes of the halogenated gases is given in Fig. 1 while the observed concentration changes for CH and N O are discussed further in reference [20]. The values for CFC11, CFC12 and SF lacking altitude dependence were not considered when finding the halogenated gas total radiative forcings. were not present in the atmosphere prior to the industrial revolution. The effect of the altitude dependence on forcing was examined for CFC11, CFC12 and SF 6 .
Taking a constant concentration at all altitudes increased their forcings by as much as 8% at the tropopause and 14% at the mesopause. It is reasonable to assume that the radiative forcings of the other halogenated species are similarly affected. The effect due to all 14 halogenated species present in the atmosphere gives a forcing of 521 (667) mW/m 2 at the tropopause (mesopause) which is  0.2% of the total flux as shown in The frequency integrated forcing, F , of (15) depends on the altitude z and on the column densities of the five main greenhouse gases as well as the 14 halogenated species given in We assume the temperature T and densities N {i} have the same altitude profiles as shown in A second special case of ( 17) is the hypothetical, per molecule standard forcing, F {i} sd , when the atmosphere contains only molecules of type i at their standard column density, N {i} = N {i} sd , and the concentrations of the other greenhouse vanish, N {j} = 0 if j  = i, F {i} sd (z) = F (z, 0, . . . , 0, N {i} sd , 0, . . . , 0). We define the forcing power per added molecule as The densities of greenhouse gases j with j  = i are held constant in the partial derivative of (20). If the units of F are taken to be W m 2 and the units of N {i} are taken to be molecules m 2 , then the units of P {i} will be W molecule 1 . We define a finite forcing increment for the ith type of greenhouse molecule as Differentiating (21) with respect to f we find where P {i} sd (z, f ) is the forcing power per additional molecule of type i when these molecules have the column density N {i} = f N {i} sd and all other types of greenhouse molecules have their standard column densities. We now consider the optically thin limit, where the concentrations of greenhouse gases are sufficiently low that the optical depths  of (3) will be small,   1, for all frequencies  and at all altitudes z. The frequency integral of the spectral forcing (15) at altitude z can then be written as where the forcing power per molecule of type i is Here N {i}  = N {i} (z  ), and T  = T (z  ). The mean power absorbed by a greenhouse gas molecule of temperature T from thermal equilibrium radiation of temperature For z  > z we see from (24) that the dz  N  molecules in the altitude interval z  to z  + dz  each emit the power (T  , T  ), of which half goes to outer space and half goes down through the reference plane at altitude z, diminishing the net flux through the reference plane by (T  , T  )/2. Molecules above the reference plane can only cause positive forcing unlike molecules below the reference plane which can cause either positive or negative forcing. An important check of our work was to use (24) to calculate the forcing power per molecule for the optically thin limit. The results displayed in column 2 of Table 3 20) per additional molecule in units of 10 21 W at the altitude z tp = 11 km of the tropopause and z mp = 86 km of the mesopause. Altitude dependences were taken into account as given in Fig. 2 and Fig. 3 while all other species had altitude independent concentrations. The surface temperature was T s = 288.7 K, and the altitude profile of temperature is given in with that found by dividing the radiative forcing in the limit of zero gas concentration, by the column density. The forcing powers per molecule given in Table 3 are all on the order of 10 21 W for the halogenated gases. The values are largest for the optically thin case and slightly lower for P {i} sd (z, 0) and P {i} sd (z, 1) because of saturation. The effect of satuation is much larger for the 5 main greenhouse gas values as explained in reference [16] because their number densities are up to 10 7 times greater than those of the halogenated gases. The forcing power can be used to estimate the contribution of each gas to global warming as follows. First, one assumes that the temperature increase T {i} due to a change in the concentration of gas i is proportional to its radiative forcing change, where c is a constant of proportionality whose estimation is difficult because it includes feedback effects. The radiative forcing change at the tropopause due to a concentration change occurring over time t is given by Here, we have evaluated the forcing power per molecule at the tropopause which is given in The total temperature change due to changes in all i greenhouse gas concentrations over time t is given by Equations ( 26), ( 28) and (29) imply Surface temperatures have increased by about 1.1 K since 1850 according to the IPCC [4]. More recent satellite measurements since 1979, show a slightly higher warming rate of about 0.1 C per decade [19]. Since the observed temperature increase is comparable to previous temperature increases when there were negligible changes of greenhouse gas concentrations, there is no way to know how much is due to currently increasing greenhouse gases. But if one makes the extreme assumption that changes in greenhouse gas concentrations have driven all of this observed warming, one finds the changes in the halogenated gas concentrations are responsible for a warming of 0.002 C per decade. F Tot represents the total radiative forcing change caused by changes in the concentrations of all greenhouse gas molecules. The relative forcing was found using the observed 2020 concentration change rate of each gas i as discussed in the text.
A negative relative forcing change occurs if the halogenated gas atmospheric concentration is decreasing as a result of the Montreal Protocol. A comparison of the greenhouse warming contribution of a gas i relative to carbon dioxide is provided by the global warming potential which is defined as [20] GWP The time-integrated radiative forcing at the tropopause per unit mass over an observation time T is The units of RF {i}  are joules per square meter per unit mass. It is the amount of "forcing heat" acquired by a square-meter column of the atmosphere during time T after the pulse emission of one unit mass of greenhouse gas of type i. The forcing time  {i} is f {i} (t  ) is the fraction of the excess greenhouse gas molecules remaining in the atmosphere at a time t  after the "pulse emission." Excess greenhouse molecules can remain in the atmosphere long after the greenhouse molecules of the original pulse have exchanged with the land and oceans because they are replaced with equivalent molecules that continue to provide radiative forcing. An example is the relatively short time needed for the exchange of 14 CO 2 molecules from atmospheric nuclear weapon tests with CO 2 molecules of the ocean. It takes a much longer time for excess CO 2 concentrations to decay away. The excess atmospheric concentration of molecules, caused by the emitted pulse of the species i, can be removed from the atmosphere by various mechanisms. For example, CO 2 is absorbed by the biosphere and oceans. CH 4 is oxidized by OH radicals and other atmospheric gases. N 2 O and the various halogenated gases are photodissociated by solar ultraviolet radiation in the stratosphere, etc. It is convenient to describe the excess fraction f {i} (t) = f (t) with a sum of n + 1 exponentially decaying components, of amplitudes a j and time constants  j [21,22] a j e T / j , where n j=0 a j = 1. (34) Substituting (34) into (33) we find that the forcing time,  {i} = , is For the limiting case of infinitely long time constants,  j  , one should make the replacement  j 1  e T / j  T in (35). From inspection of (35) one can conclude that the forcing time  {i} is always less than or equal to the observation time,  {i} (T )  T . Using (35) in (32) we find that the global warming potential is Unlike the forcing powers P {i} per greenhouse molecule of type i, which can be accurately calculated, the time dependence of the fractions f {i} (T ) is not well known. So, the global warming potentials GWP {i} are of limited quantitative value and do not contribute in a clear way to estimating the warming caused by a change in a greenhouse gas concentration. To add further uncertainty, "indirect effects" are sometimes included in the GWP {i} , for example, the effects of the CO 2 and H 2 O molecules which result from the oxidation of CH 4 . For this paper only the direct forcing of the greenhouse gases was considered. For CO 2 , Harvey [21] uses a five-exponential form of (34) with the parameters Alternate parameterizations of f {CO 2 } , for example, those of Joos et al. [22], give GWPs that do not differ more than the uncertainties of the estimates. Table 5 shows the global warming potentials calculated with (36) using the lifetimes of the various greenhouse gases given in the most recent IPCC report [23] and equation (37). Our GWP values are consistent with those of the 2021 IPCC report. The somewhat larger values of GWP {CH 4 } may be due to indirect effects included in the IPCC calculations. This work found the instantaneous clear sky radiative forcings of 14 halogenated molecules that have previously been found to exert the largest greenhouse warming. CFC11 and CFC12 comprise about 60% of the total forcing generated by the 14 halogenated molecules and their concentrations have been decreasing in recent years at 2% and 4% per year, respectively. The concentrations of 6 of the 14 halogenated gases studied are decreasing as a result of the Montreal Protocol. A number of factors affect the accuracy of the results. First, the calculations used the absorption cross sections given by the Hitran database which have an uncertainty of several percent. Second, the Hitran data does not show a temperature dependence of the cross sections. This contrasts with our previous work [16] that examined the effect of temperature on the frequency integrated cross section which equals the sum of the line strengths. Negligible temperature effect was found for the 5 main greenhouse gases as well as CF 4 but the absorption cross section for SF 6 increased by a factor of 2 as temperature decreased from 300 to 200 K. Molecules made of many atoms, like SF 6 , have many low-frequency, infrared-inactive modes which can be significantly excited at higher temperatures [16]. Then the molecule Molecule Mass Lifetime [23] This Work IPCC 2021 [23] i m {i} (amu)  i (yr) GWP(0) GWP( 20) GWP(100) GWP( 20 3 with IPCC 2021 values. There are multiple pathways for CO spends more time in nonradiating modes and emits less average power at the frequencies of infrared active modes than a simpler molecule would. The temperature increase causes a corresponding decrease in the absorption cross section of the infrared active mode. The absorption cross sections of other complicated halogenated moecules can be expected to have analogous temperature dependences. Finally, the altitude dependence of the halogenated gas concentrations is poorly known. For the cases of CFC11, CFC12 and SF 6 , the radiative forcing increases by up to 14% if the altitude concentration dependence is ignored. Our calculations were done for the case of a clear sky. In reality, about 70% of the Earth's surface is covered by clouds [24].
A cloud layer will particularly perturb the so called spectral window shown between the CO 2 and O 3 absorptions in Fig. 4, where the halogenated gas absorptions most affect the radiative forcing. It is therefore reasonable that incorporating clouds, will reduce the halogenated gas radiative forcing. Some studies have found clouds reduce the halogenated gas forcing by 25% [7]. This study computed the global warming potentials which are comparable to those found by the IPCC2021 as shown in and Engineering Research Council provided financial support of one of us. and Engineering Research Council provided financial support of one of us.