The non-LTE problem has two primary constituents: (1) the statistical equilibrium equations (SEEs), which express the equality of the total population and de-population rates for each molecular level, and (2) the radiative transfer equation (RTE), which relates the radiation field to the populations of levels, at all altitudes in the atmosphere . Hence, the system of equations for the level populations is non-local (and non-linear). The most obvious way of dealing with this situation is to iterate between the SEEs and RTE. This process, traditionally called “lambda iteration” (LI), has been investigated in the astronomical context since the 1920s . It inverts ND matrices NL×NL at each iteration step. This simple approach has been applied in Earth's and planetary atmosphere radiative transfer; see, for instance and . If the optical depths are large (as is the case for the CO2 15 µm band), the algorithm converges slowly. studied several LI schemes and showed that for the CO2 non-LTE problem, the number of iterations ILI for these algorithms may reach ∼ 200 for even moderate convergence criterion 1×10-3. This slow convergence is caused by the photons trapped in the cores of the most optically thick lines and by the strong non-linearity of SEEs related to quasi-resonant exchange of vibrational energy by the molecular collisions.

An alternative way of dealing with the non-LTE is a joint treatment of SEEs and RTE, when RTE is discretized with respect to the optical depth or altitude grid to get a matrix representation of radiative terms in SEEs. This approach, known in the atmospheric science as the Curtis matrix (CM) technique (e.g., ), leads to a matrix of dimensions (NL×ND)×(NL×ND). In stellar atmosphere studies, the generalized version of this technique is known as the Rybicki method . The time required for the solution of the non-LTE problem using the CM technique is controlled by the number of operations for matrix inversion. The advantage of the classic matrix method lies in the simultaneous determination of all populations at all altitudes instead of the sequential evaluation of populations step by step at each altitude using the radiative field from the previous iteration. Therefore, “matrix iteration” usually converges better than lambda iterations. However, the convergence of both algorithms depends strongly on how the local non-linearity is treated; see the next section. To construct an adequate model, one must account for a large number of excited levels of various molecular species plus use a detailed model of atmospheric stratification. As a result both NL and ND can become very large. The dimensions of primary matrices are reduced by introducing various assumptions (for instance, the LTE assumption for rotational sublevels as well as, for example, LTE in the groups of vibrational levels closely spaced in energy; see also the discussion of GRANADA – Generic RAdiative traNsfer AnD non-LTE population Algorithm – code in Sect. ). Nevertheless, usually the time to solve the non-LTE problem using the CM method significantly exceeds the time needed when an LI algorithm is applied to the same problem (see Sect.  for more details).

In the 1990s, stellar astrophysicists developed a family of powerful techniques that utilize lambda iteration with an approximate (or accelerated) lambda operator (see ; ; and references therein). In these so-called ALI techniques (for accelerated lambda iteration), the integral lambda operator, which links the radiation intensity at a given point with the source function at all points, is approximated by a local (or nearly local) operator. With a local operator, the largest matrices again, as in the LI case, have dimensions NL×NL. However, in this case, the convergence is rapid, since most of the transfer in cores of the lines (described by the local part of lambda operators) cancels out analytically and only the difference between exact and approximate radiative terms in the SEEs is treated iteratively. shoved that for the CO2 non-LTE problem IALI≪ILI (see Sect.  for more details).

Strong local non-linearity of the non-LTE radiative transfer problem in molecular bands caused by intensive near-resonant exchange of vibrational energy between molecules was studied by . They showed that this non-linearity causes a dramatic deceleration of the convergence. Various schemes of additional “internal” iterations (without recalculating radiative excitation rates) aimed at adjusting populations of levels coupled by strong inter- and intramolecular vibrational–vibrational (VV) exchange did not bring any help. To accelerate the convergence, suggested “decoupling”, which utilizes the approach of treating the “source function equality in the line multiples”. The SEE terms, which describe the VV coupling, depend on the products nvnv′, where nv is the population of vibrational level v of one molecular species, whereas nv′ is the population of level v′ of the same or another molecular species. In the iteration process, one needs to present these terms as nvnv′†, where † denotes the population of the level with the lower degree of excitation, which is taken from the previous iteration. showed that this stops “the propagation of errors” by iterations and guarantees the fastest convergence. This decoupling requires only slight modification of matrices to be inverted (without additional linearization of the non-LTE problem and, therefore, additional programming efforts). It provides, however, the same acceleration of convergence as the application of the Newton–Raphson method for the solution of a system of non-linear equations .

With line-by-line (LBL) calculations, a very large number of frequency points to be accounted for significantly decelerate calculations of the 15 µm CO2 radiative fluxes. In LTE the reduction in frequency points is usually achieved by utilizing the so-called CKD (for correlated k distribution) method, which is based on grouping the gaseous spectral transmittances in accordance with the absorption coefficient k. The accuracy of this approach is better than 1 %; see, for instance, . However, the k correlation is not applicable under the non-LTE conditions because the vibrational level populations involved in the k distributions are unknown and depend themselves on the solution of the radiative transfer equation.

To overcome this problem, stellar astrophysicists developed the ODF technique. In this approach, they treat the non-LTE radiative transfer in “super-lines” associated with multiplets of very large line numbers (e.g., ). They re-sample the normalized absorption and emission cross sections of super-lines, consisting of hundreds or thousands of lines, to yield a monotonic function of frequency that can be represented by relatively small numbers of frequency points. Though the idea is like k correlation, these normalized absorption and emission profiles do not depend on the total populations of upper and lower “super-levels” but only on the relative population of sublevels that are closely spaced in energy within each super-level, which are supposed to be in LTE.

described the adaptation of the ODF technique to the solution of the CO2 non-LTE problem. They treated each CO2 band branch as a super-line. Therefore, each CO2 band was presented by only three lines for perpendicular bands and two lines for parallel bands. This way of treating the radiative transfer in the molecular band is about 50–100 times faster than the classic LBL approach. Whereas in the LBL approach the radiation transfer equation is solved for each of NF frequency grid points within each rotational–vibrational line, in the ODF techniques the same number of frequency grid points as applied only to each super-line. Thus, the acceleration factor is approximately equal to the number of rotational–vibrational lines in the branch. As show, the ODF approach introduces very small errors into the 15 µm cooling. For current CO2 density (400 ppm in the lower atmosphere), these errors do not exceed 0.3 K d-1 in a broad range of temperature variations; see Fig. 18 of . They increase roughly linearly with the CO2 increase.

Since the 1960s, the Curtis matrix algorithms, with the rare exceptions for LI mentioned above in Sect. , have been dominating the solution of the non-LTE problems in Earth's and planetary atmospheres, including the studies of the 15 µm CO2 cooling (see, for instance, the work by , who developed Curtis matrix parameterization of CO2 cooling of MLT, which was very advanced for that time). Numerous non-LTE studies of the research group from the Institute of Astrophysics of Andalusia, Granada, applied the GRANADA (Generic RAdiative traNsfer AnD non-LTE population Algorithm) code described by . The core of it is a standard CM algorithm, which the authors in this and their earlier publications prefer to called MCM (for modified Curtis matrix). The cited paper discusses various ways of splitting large matrices into blocks, solving the non-LTE problem for selected sub-sets of levels and iterating to get the solution for all vibrational levels. In stellar astrophysics , this approach is known as the “generalized equivalent two-level approach for multi-level problems”. For many years it has had no use because of convergence problems. The GRANADA code also includes LI but not the ALI technique, although the transformation of LI into ALI requires a minimum of programming efforts but speeds up the convergence for optically thick problems at least 10-fold (e.g., ). Additionally, neither compared the computational performance of MCM and LI algorithms nor described the handling of a strong local non-linearity caused by VV coupling.

successfully applied the ALI technique to study the non-LTE emissions of molecular gases in planetary atmospheres (the 4.3 µm CO2 band in the Martian atmosphere and the 4.7 µm CO band in Earth's atmosphere, respectively). and described in detail the adaptation of the ALI code developed by for stellar atmospheres to the solution of the non-LTE problem for molecular bands of planetary atmospheres. They studied the performance of the new ALI-ARMS code and demonstrated computational superiority of ALI-ARMS compared to various LI and CM/MCM techniques. The ALI-ARMS code and its applications were described by . Later it was applied to study the rotational non-LTE in the CO2 4.3 µm band in Martian atmosphere, observed by the Planetary Fourier Spectrometer (PFS) in the Mars Express mission (, and references therein); to study self-consistent two-channel CO2 and temperature retrievals from the limb radiance measured by the SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) instrument on board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) Mission ; and to explain the SABER nighttime CO2 4.3 µm limb emission enhancement caused by a recently discovered new channel of energy transfer from OH(ν) to CO2 and to simultaneous retrievals of O(3P) and total OH densities in the nighttime MLT . Earlier, the ALI-ARMS code was used to pinpoint an important missing process of strong VV coupling between the isotopes in the CO2 non-LTE model of the SABER operational algorithm and to model the H2O 6.3 µm emission and the H2O density retrievals in MLT from the SABER 6.3 µm limb radiances . As we show below in Sects.  and , the ALI-ARMS code also provides an efficient way of calculating the 15 µm CO2 cooling–heating in the GCMs.

To optimize the cooling calculations, we used as a reference our working line-by-line non-LTE model in CO2, which comprises 60 vibration levels of five CO2 isotopic species and two levels of N2, O2 and O(3P). In Fig. , we show the lower levels of this model (up to 5000 cm-1). The set of collisional rate coefficients for the vibrational–translational (VT) and VV exchange that we apply is described by and is similar to rates used by . However, it relies on different scaling rules based on first-order perturbation theory. Compared to our extended line-by-line model , which includes a total of about 350 vibrational levels of seven CO2 isotopes and over 200 000 rotational–vibrational lines, the CO2 cooling of the 60-level model differs by less than 0.05 and 0.5 K d-1 for the CO2 mixing ratios of 400 and 4000 ppmv, respectively, for both daytime and nighttime conditions and for any temperature profile.

In the next steps, we gradually reduced the number of levels and bands in the model to optimize the calculations, keeping the cooling rate errors smaller than 1 K d-1 compared to the reference model.

In Table 1, we show vibrational levels included in the optimized day- and nighttime models. The isotopes in the table are marked using the lower digit of the atomic weight: 626 corresponds to 16O12C16O, 636 corresponds to 16O13C16O and so on. We account for 28 and 18 vibrational levels in the day- and nighttime models, respectively, which include the levels of the four most abundant CO2 isotopes and N2 and O2 levels (two and one for each in the daytime and nighttime, respectively). We also include O(3P), but we do not include O(1D) because its effect on the total CO2 cooling–heating is negligible for both nighttime and daytime conditions. The model uses the CO2 spectroscopic information for all transitions available from HITRAN2016 for the levels listed in Table 1. In Table 2, we provide the numbers of bands, band branches and lines used for daytime and nighttime calculations.

In the astronomical context, detailed analysis of the operation numbers and times needed for the solution of the non-LTE problem is a must because of the complexity of the problem . Even though the numbers of levels and lines involved in the non-LTE problem of planetary atmospheres are smaller, speed is a crucial parameter for the GCMs, so we perform the same type of analysis in this section below.

We compared three algorithms of the non-LTE problem solution, namely LI, ALI and the matrix method (hereafter MM); see Sect. 2 for details. Although the LI technique is much more computationally expensive compared to ALI algorithms , one cannot say a priori the same about the MM approach applied to a problem with a reduced number of levels because of the low number of iterations it usually requires, and this required testing. We generated the matrices of the MM algorithm from the matrix presentations of lambda operators, as discussed by . We used the discontinuous finite element (DFE) algorithm as the most efficient way of solving the radiative transfer equation , and we compared the LBL and ODF techniques (see Sect. 2.3). Below in this section, we discuss the performances of all algorithms only with the non-linearity caused by near-resonance VV energy exchanges resolved as outlined in Sect. . Without this, the number of iterations of all considered algorithms would be a few times higher.

For each of the three techniques, we checked the numbers of operations and times needed for each single iteration and then accounted for a number of iterations and compared total numbers of operations and times for the entire non-LTE problem solution.

We found that the time required for each iteration of the algorithms we studied is dominated by three components: (1) time for auxiliary operations TAux (such as the filling of large arrays like vectors or matrices to be inverted); (2) time for solving the radiative transfer equation (TRad) and forming the radiative rate terms in the SEEs; and (3) time for matrix inversions (TInv). The cooling itself is the by-product of the non-LTE problem solution and is estimated nearly instantaneously as in : 3h=∑bnloBlo,upJlo,up‾-Aup,lonlo×(Eup-(Elo)b, where Blo,up and Aup,lo are the band Einstein coefficients and Jlo,up‾ is the mean intensity in the band, which enters radiative rate coefficients of SEE matrices, whereas nlo/Elo and nup/Eup are the populations/energies of lower and upper vibrational levels in the band, respectively. The sum in Eq. () applies to all CO2 transitions in the model.

In Table 2, we present a summary of our study. The table gives the main parameters of the non-LTE models for day and night conditions described in Sect. , operation numbers, and the times in seconds (measured with the help of a timing routine within the code), which are required for each calculation part. We performed this study on two different machines, with x86_64 Intel and Intel Xeon Gold processors operating at 2.2 and 2.5 GHz, respectively. We compiled the ALI-ARMS code to be used in 64 bit architecture with the help of a standard GNU Compiler Collection (GCC) compiler, and we ran it on a single processor. We provide the results only for one 2.2 GHz Intel processor; the timing for the second processor is roughly 1.4 times shorter.

Compared to the reference code, we ran the routine using the convergence criterion 1.0×10-2 instead of 1.0×10-4. This allowed a reduction in the number of iterations by a factor of about 2 without sacrificing accuracy.

Similarly to the study by , we found that the time T required for any procedure like the radiative transfer equation solution or matrix inversion may be presented as 4T=CN, where N is the number of operations. N is defined by the mathematical nature of the problem and the algorithm applied for its solution. C may, however, depend on many other factors like the quality of programming, language used, operational system, interpreter, computer architecture and performance.

We found that the number of operations for the solution of the radiative transfer equation NRad in the case of the non-overlapping lines may be approximated by the expression 5NRad≃ND×NRT×NF×NA and is the same for all algorithms compared. Here NRT is the total number of lines (or band branches in the ODF case), and NF and NA are the numbers of points in the frequency and angle integrals used, respectively. The coefficient C in Eq. (), which links radiative transfer operation numbers and corresponding times, was found to be ≃1.0×10-8 s.

We found that with the LI/ALI algorithms, the number of auxiliary operations NAux is well approximated by the following expression: 6NAuxLI/ALI≃NL2×ND. This expression gives the number of terms to be filled in the block-diagonal matrix comprising ND blocks NL×NL, where NL is the number of vibrational levels. In the case of the LI/ALI techniques, these are the ND matrices generated and inverted one after another at each iteration step. The coefficient C in Eq. (), which links auxiliary operations and corresponding times, is C≃1.7×10-7 s.

In the case of the MM technique, the matrix to be generated at each iteration is much larger; namely it has the size (NL×ND)×(NL×ND) and consists of NL fully filled diagonal blocks ND×ND, which represent non-local radiative terms, whereas the same as for LI/ALI case collisional terms are now spread over non-diagonal parts of this large matrix. We found that when we present the number of operations to fill this matrix as 7NAuxMM≃ND2×NL, then approximately the same coefficient C≃1.7×10-7 s links this number with the time needed for its filling.

The number of operations needed for matrix inversion NInv is approximately N3, where N is the matrix dimension. Thus, we have the following expressions: 8NInvMM≃(NL×ND)3 for the MM algorithm and 9NInvLI/ALI≃(NL×NL)3×ND for the LI/ALI techniques. In the latter case the number of operations is ND2 times lower, since only ND matrices NL×NL are inverted one after another at each iteration. This is the great advantage of these techniques compared to MM, where the entire huge matrix needs to be inverted at once, since it has non-zero elements outside the diagonal blocks.

In the ALI-ARMS code we use ludcmp (lower–upper decomposition), lubksb (back substitution) and mprove (iterative improvement) as matrix inversion routines . We found that, for these routines applied to the non-LTE problems studied here, the coefficient between the number of operations and time for matrix inversion depends on the matrix dimension N and may be approximated by the following expression: 10C=1.0×10-7⋅(0.04+1.2/N).

One may see in the upper part of Table 2 that, for night conditions, the application of the MM technique causes the matrix inversion to be the most time-consuming calculation part at each iteration, although the number of iterations NIter=2 is low. Applying LI–LBL techniques provides a strong reduction in the matrix inversion time per iteration but gives only a moderate reduction in the total time (by a factor of ∼5) compared to MM–LBL due to a large number of iterations (60). The number of iterations for the LI/ALI techniques slightly depends on the atmospheric pressure and temperature distribution. The numbers, which are given in the table, are mean values for a few hundreds of runs for different atmospheric conditions. Applying ALI instead of LI significantly reduces the number of iterations (5 instead of 60), providing additional acceleration of calculations by a factor ≳10. We note that for LI/ALI–LBL, the most time-consuming part for each iteration is now the radiative transfer solution, which is more than 15 times slower than two other parts of calculation together. The ODF technique allows a reduction in TRad by a factor of 50–60. This provides total additional acceleration by a factor of ≳10. The last column in the table gives the acceleration factor K=Ttot/Ttot,  ALI–ODF, which shows how much faster the ALI–ODF combination works compared to other techniques: it is about 900 and 160 times faster than the MM–LBL and LI–LBL techniques, respectively.

The lower part of Table 2 shows the number of operations and times for various parts of calculations for the daytime non-LTE model described in Sect. . Compared to the nighttime, daytime calculations require about 2.5 times more time due to an increased number of vibrational levels and bands accounted for. Nevertheless, the main points discussed above in this section for the nighttime runs remain valid for the daytime: (a) the main decelerating factor for the MM technique is the matrix inversion, notwithstanding the low number of iterations; (b) the LI technique, although it reduces the matrix inversion time by a factor of ≳7000, provides only a moderate decrease in total time (by a factor of ∼ 10) because of the large number of iterations; (c) the ALI technique is more than 100 times faster than MM, with the slowest part of calculations being the LBL solution of RTE; and (d) the ODF provides acceleration of radiative transfer calculations by a factor of 50. Finally, the ALI–ODF technique appears to be over 1000 times faster than the MM–LBL approach.

We measured the time the F98 routine requires on x86_64 Intel 2.2 GHz processors and found it to be around 3×10-4 s. This means that for the nighttime, KF23 is about 300 times slower than F98. In the daytime, when the solar heating parameterization of ) is accounted for, our version of F98 requires about 30 % more time. Still, it remains about 600 times faster than the daytime KF23 routine. In the next section we discuss in detail the accuracy of both routines.

In Fig. , we show five typical temperature profiles, which demonstrate the superposition of different meso-scale waves. These profiles, as well as corresponding pressure, O(3P) and CO2 distributions (shown in Fig. ), and other constituents from the Whole Atmosphere Community Climate Model Version 6 (WACCM6) runs were kindly provided by Daniel Marsh (personal communication, 2022). For the 15 µm cooling calculations, this model uses the F98 parameterization. We show below the calculation results for these atmospheric model inputs because WACCM is widely used by the model community. Generally, any pressure/temperature profiles disturbed by strong waves give similar results, as we observed in our tests. These may be p–T distributions generated by modern GCMs, like in our case here; those retrieved from ground-based or space observations; or artificial wavy p–T distributions.

Figure  shows the CO2 15 µm cooling rates for the new KF23 routine, for the F98 parameterization, and for our reference non-LTE model for temperatures in Fig.  and the 400 ppmv CO2 profiles. One may see in this figure that the new routine errors do not exceed 0.5 K d-1. On the other hand the F98 routine errors reach up to 13 K d-1. The altitude range, where the F98 routine demonstrates significant errors, is broad starting just above the altitude of 60 km.

In Fig.  we show the same as Fig.  – cooling rates and their differences – but for the twice higher CO2 of 800 ppmv in the well-mixed range. We note here that both maximal absolute values of cooling rates and the new and the F98 routine errors are roughly twice higher compared to those of Fig. . For the new routine they do not exceed 1 K d-1, whereas for the F98 routine they reach up to 23 K d-1.

Finally, in Fig. , we show in panel (a) the cooling rates produced by the reference model and by the KF23 routine for the CO2 VMR of 4000 ppmv, which is 10 times higher than the reference one. The new routine errors are shown in panel (b). The F98 routine was not tested for these inputs, since it was not designed to work with the CO2 VMRs higher than 720 ppmv. One may see in this figure that absolute cooling rates (maximal values) are approximately 10 times higher than those for 400 ppmv of CO2. The same is roughly true for the new routine errors, which now reach values of up to 8 K d-1 in upper parts of the tested region.

In Fig.  for night, we show the contributions of major and minor CO2 isotopes included in the model (see Sect. ), for a temperature at 33.5° N for 400 and 1600 ppmv of CO2. One may see in this figure that, for the 400 CO2 ppmv, this contribution does not exceed ∼ 2 and ∼1 K d-1 for the 626 hot bands and for all minor isotope bands, respectively. This effect of hot bands and minor species is increasing with the CO2 density; see the right panel of Fig. , particularly for altitudes affected by waves.

As we mentioned in Sect. , we use all CO2 bands available in HITRAN2016 for the night set of levels in Table 1. This minimizes errors compared to reference calculations to ≤1 K d-1 for 400 ppmv of CO2. The routine allows for using fewer levels and bands to accelerate calculations but at the expense of error increase (see Appendix  for more details). For instance, excluding (see Fig. ) weak first hot bands, (10001,02201,10002)→01101, and second hot bands, (11101,03301,11102)→(10001,02201,10002), of 626 and 636 isotopes makes the total number of bands twice lower. Our tests show that in this case the routine works only about 10 % (see also Table 2) faster; however the maximal cooling rate error for 400 ppmv increases to up to 3 K d-1.

In the daytime, the near-infrared heating due to the absorption of solar radiation in the CO2 bands around 2.0–4.3 µm represents a small but non-negligible reduction in the total CO2 cooling (up to 1–2 K d-1 for the current CO2). The complex mechanisms of the absorbed solar energy assimilation into heat have been investigated in detail in a number of studies summarized by . studied this heating for smooth temperatures for various CO2 densities and solar zenith angles (SZAs) and suggested the use of a lookup table, which allows a quick estimate of this heating in GCMs. Due to its reasonable accuracy (∼ 0.5 K d-1 for current CO2), this table has been used as a daytime supplement to the F98 nighttime cooling parameterization. Unfortunately, with increasing CO2 density, the errors in this table increase rapidly above ∼ 70 km: for 720 ppmv CO2 it underestimates the heating around the mesopause by more than 50 % (see Fig. 5 of , for daily averaged heating).

In Fig.  the heating of the atmosphere due to the daytime absorption of solar radiation in the CO2 bands at 2.0–4.3 µm for SZA = 45° at 33.5° N produced by our new routine is shown. To prevent increasing errors for the daytime due to an inadequate treatment of solar radiation absorption and assimilation, KF23 utilizes, in the daytime, an extended non-LTE model which, compared to the nighttime, includes 10 more CO2 vibrational levels (see Table 1) and a comprehensive system of radiative and collisional VT and VV energy exchanges as described by and . A higher number of vibrational levels and more than twice the number of bands lead to a 2.5-fold longer time for the daytime cooling–heating calculation (see Table 2). However, the daytime errors are of the same order of magnitude (less than 1 K d-1 for 400 ppmv) as those for the nighttime (Figs. –) even for strongly perturbed temperatures and increased CO2. We do not present these comparisons. As in the night case, removing half of bands (various hot and combinational bands) in the daytime model gives only about 10 % speed gain; however, for 400 ppmv, maximal errors in cooling may reach 4 K d-1.

In Fig. , we compare the cooling rates obtained with the KF23 routine and with F98 parameterization with those produced by the reference model. For these tests, we used the temperature–pressure distributions affected by the diurnal tides at the Equator; p, T and the atmospheric constituent densities as well as local zenith angles for local times at 1.0° N and 12.0° E (15 March 2019) correspond to the WACCM-X (Whole Atmosphere Community Climate Model with thermosphere and ionosphere extension) equatorial simulations constrained by meteorological analyses of the NASA Goddard Earth Observing System version 5 (GEOS-5) below the stratopause, as discussed in . The temperature distributions at various local times are shown in panel (a). Panel (b) presents the CO2 15 µm cooling rates calculated using the reference model (thick solid lines) and the F98 routine (thin solid lines with diamonds). We do not show here the cooling rates obtained using the KF23 routine because the difference between them and the reference data does not exceed 1 K d-1 ; see Fig. c. Figure d shows the differences between the cooling produced by the F98 routine and the reference calculations. In contrast to Fig. c, these differences exceed 20 K d-1 for some temperature profiles.

One may see in Fig.  that the accuracy of the F98 routine improves above 100 km and below 80 km. Above 100 km, the F98 parameterization uses a recursive expression, the accuracy of which increases with height . Below 80 km, the F98 routine is based on the LTE matrix algorithm for cooling calculations. This algorithm accounts for the radiative interaction of the neighboring levels and the escape of radiation to above and, therefore, provides a good accuracy of cooling in this optically thick layer. In the layer between 80 and 100 km, the F98 routine merges the cooling values calculated by the two methods outlined above. This merging works reasonably well for smooth temperature distributions tested by , but it fails with wavy temperatures. One also needs to keep in mind that this layer is the transition region between the LTE and non-LTE state of the CO2(ν2) vibrations, where the physics of formation of the non-equilibrium vibrational distribution must be considered in all aspects (e.g., ). Compared to the KF23 routine, which rigorously models the non-LTE, the F98 routine fails in this situation, which warned about when they presented the first version of this parameterization.

The results above in Sect. – were obtained for the temperature-dependent CO2(ν2)+O(3P) quenching rate coefficient k=3.0×10-12 s-1 cm3×(T/300). The multiplier in this expression is the median value of the rate coefficient from the range of (1.5–6.0) ×10-12 s-1 cm3, which spans the low laboratory data up to high data obtained from the space observations of the CO2 15 µm emission; see, for example, , and references therein. This value is currently accepted for usage in the GCMs for calculation of the 15 µm cooling. The dependence of MLT cooling on k has been investigated in many previous works summarized by and also references therein. It is known that the maximum value of cooling, which is usually reached at the altitudes of 100–140 km, is roughly proportional to the k value used in calculations. The KF23 routine works well for any k value from the range given above, and it also allows varying the temperature dependence of the rate coefficient (e.g., ); see Appendix . For our calculations, we used the O(3P) densities shown in Fig. .

The accuracy tests of the KF23 routine were performed with the upper boundary of the atmosphere at 130 km. The routine may work with any upper boundary in the upper mesosphere and above. Putting the upper boundary below ∼ 110 km may cause, however, increasing cooling errors due to not accounting for the upper atmospheric layers. The lower boundary can be placed at any altitude below ∼ 50 km where all CO2 15 µm bands are in LTE. This will justify the LTE lower boundary condition for the radiative transfer equation solution. However, it is not recommended to use the routine results below ∼ 20 km because of increasing errors by not accounting for the line overlapping in a current version of the ODF approach.