The radiative impact of a perturbation, e.g., clouds, is quantified by the concept of the radiative effect (RE). The RE is defined as the net difference in the downward and upward irradiance (F↓-F↑) between the perturbed and unperturbed condition. In the case of clouds, the cloud radiative effect (CRE; denoted here as ΔF) is the difference in the fluxes between the cloud (Fc) and cloud-free (Fcf) atmosphere at a given altitude z : 1ΔF(z)=Fc(z)-Fcf(z)=F↓(z)-F↑(z)c-F↓(z)-F↑(z)cf, where the upward and downward and cloudy and cloud-free irradiances are all counted to be positive. The net RE is given by 2ΔFnet(z)=ΔFsol(z)+ΔFTIR(z), which can be split into a solar and a thermal-infrared component. Within this study, the CRE is calculated for the top of the atmosphere (TOA), which is set in the radiative transfer calculations to an altitude of 120 km, unless stated otherwise.

In addition to the RE, the albedo α describes the interaction of a cloudy scene or a surface with the solar incident radiation. The scene albedo αsol(z=TOA) at the TOA is defined as the ratio of the reflected upward irradiance Fsol↑ at the TOA in relation to the incident downward irradiance Fsol↓ at the TOA and is given by 3αsol,TOA=Fsol↑(z=TOA)Fsol↓(z=TOA). Similarly, the surface albedo αsol,srf is calculated with Fsol,srf↑ and Fsol,srf↓ to find the respective irradiances at the surface (z=0 km).

Upward and downward irradiances F↑/F↓ were simulated with the library for radiative transfer libRadtran;. The solar irradiances Fsol cover a wavelength range from 0.3 to 3.5 µm, which represents 97.7 % of the total incoming solar radiation (0–10 µm) calculated from the spectrum provided by . The thermal-infrared (TIR) irradiances include wavelengths from 3.5 to 75 µm, representing 99.3 % of the integrated blackbody radiation (3.5 to 100 µm) at 285 K (12 ∘C).

The RT simulations are performed with the one-dimensional (1D) solver DISORT , which is part of libRadtran. Clouds are assumed to be horizontally uniform, and lateral photon transport between columns is neglected, which is called the independent pixel approximation IPA;. As the main objective of this study is to map the basic dependencies of ΔF on the driving parameters, we neglect any variability in the spatial ice water content (IWC) distribution that exists in cirrus . We also restrict the simulations to fully cloud-covered scenes. The required number of streams was iteratively determined and set to 16 streams, which provides sufficient accuracy, while limiting the computational time. The trade-off between accuracy and computational time is detailed in Appendix . The spectral TOA solar irradiance is provided by . The RT simulations consider molecular absorption using the coarse-resolution REPTRAN parameterization from . Appendix  provides an uncertainty estimation related to the REPTRAN resolution. Absorption by water vapor, carbon dioxide, ozone, nitrous oxide, carbon monoxide, methane, oxygen, nitrogen, and nitrogen dioxide is included in the simulations .

The sensitivity of solar, TIR, and net cloud RE ΔF is estimated by varying eight parameters. The parameter ranges were chosen to represent commonly observed cirrus and contrail cirrus properties, as well as environmental parameters.

The daily course of the Sun position is represented by solar zenith angles θ ranging from 0 and 85∘. Larger θ values are omitted to avoid numerical instability that would require more streams in the calculation. Furthermore, RT simulations with the DISORT solver for θ>85∘ have to be interpreted with caution, as DISORT does not consider the sphericity of the Earth and treats atmospheric layers as plane parallel . In addition, differences between 1D and three-dimensional (3D) RT simulations increase significantly with values of up to 40 % .

An overview of the model configuration is given in Table  and the input parameter space is listed in Table . An example libRadtran input file is provided as Supplement.

For each of the three simulated ice crystal shapes, a NetCDF file is provided . The files include ice cloud optical thickness τice, the simulated upward and downward irradiances F at TOA with 120 km (with and without the presence of the ice cloud), and the calculated ice cloud radiative effect ΔF (solar, TIR, and net). The available cloudy and cloud-free irradiances further allow us to calculate the cirrus RE by scaling the “cloudy” RE with the required cloud cover. An overview of all variables provided in the NetCDF files is given in Table . The data set allows the user to extract ΔF values for their parameter combinations, instead of running costly RT simulations. The lookup table could in fact be coupled with models of any complexity, as long as they simulate the dimensions of the data set, namely solar zenith angle, ice cloud temperature, surface albedo, ice water content, surface temperature, ice crystal effective radius, and liquid water cloud optical thickness.

The simulations base on three relative humidity profiles, which were selected to represent sub-Arctic, mid-latitude, and tropical conditions. An estimation of the RE variability due to variations in the RH profile showed an effect of less than 1 % for ΔFsol but can range up to 4 % for ΔFtir and 8 % for ΔFnet, especially for the warm and moist tropical profile. These variations have to be considered when using the data set. We further emphasizes that the simulations are performed with a 1D RT solver, i.e., plane-parallel clouds that neglect 3D scattering and horizontal photon transport .

The liquid water content (LWC) of a liquid water cloud can be obtained by 11LWC=43⋅π⋅ρliq⋅∫0∞n(r)⋅r3⋅dr, with ρliq=1000 kg m-3 as the density of liquid water, r as the radius, and n(r) as the number of droplets with size r. Equation () assumes spherical ice crystals, so it might be valid for droxtals, which are almost spherical ice crystals, but it is invalid for other ice crystal shapes. To obtain the particle number concentration Nice for non-spherical crystals, appropriate power law–mass–dimension relations are needed. Here we employ Eq. (29) from but modify the notation to be consistent with the previous equations from the present study. Equation (29) from is then given as 12IWC=α⋅Γ(β+μ+1)⋅NiceΓ(μ+1)⋅Λβ, with Γ being the result of the numerically solved gamma function. The constants α and β are the prefactor and the power in the mass–dimensional relationship, respectively. They are related by 13m=α⋅Dβ, with m as the mass of the ice crystal, and D as the maximum dimension of the ice crystal. Both constants depend on the ice crystal shape and are, for example, listed in . Using Eq. () and assuming an exponential PSD with the special case μ=0 and Λ=3re finally leads to 14Nice=3β⋅IWCα⋅Γ(β+1)⋅reβ. Therefore, Nice is proportional to IWCreβ, with β at around 2 for aggregates, 2.4 for hexagonal-plates, and 3 for almost spherical droxtals .

Radiation in the TIR is primarily of terrestrial origin . Therefore, the TIR irradiance at the TOA has only an upward-directed component F↑,TIR, while the downward component F↓,TIR is essentially zero. The magnitude of F↑,TIR is primarily driven by the cloud absorption optical depth, the surface temperature Tsrf, and the (ice) cloud temperature Tcld,ice or ice cloud altitude zice . Assuming the Earth's surface is a blackbody, the outgoing FTIR↑ at TOA could be calculated, in a first-order approximation, by the Stefan–Boltzmann law 15Fcf↑=σ⋅ϵ⋅T4, which is obtained by integrating the Planck function over all wavelengths and 2π of a hemispheric solid angle. In Eq. (), the Stefan–Boltzmann-constant is represented by σ=5.67×10-8 W m-2 K-4 and the emissivity ϵ=1 of a blackbody. In reality, however, the Earth acts as a graybody (ϵ≠1), and the surrounding atmosphere must be taken into account.

Absorption of radiation in the atmosphere in the TIR wavelength range depends on the wavelength and atmospheric composition. The primary components that control absorption are water vapor and carbon dioxide (CO2) . While CO2 is well-mixed and thus approximately constant in space and time, WV is highly variable. Furthermore, the amount of WV is linked to the temperature in the AP via the Clausius–Clapeyron equation . The lowermost values of the AP are also influenced by Tsrf. Due to these interactions, developed a model to estimate TIR irradiances and the resulting CRE. The model was derived by fitting RT simulations, which cover a wide range of environmental conditions, to Eq. (), which leads to 16Fcf↑∗≈σ∗⋅ϵ⋅Tsrfk∗, with σ∗=1.607×10-4 W m-2 K-2.528 and k∗=2.528. Fcf↑∗ represents the surface emission with Tsrf and atmospheric absorption.

Clouds in the atmosphere can be approximated by semi-transparent blackbodies that partly absorb and re-emit radiation, according the Stefan–Boltzmann law. The emissivity ϵ of a cloud depends on τ, which in turn depends on the wavelength . FTIR↑ in the cloudy case can be estimated with 17FTIR↑=1-ϵ∗⋅σ∗⋅Tsrfk∗+ϵ∗⋅σ∗⋅Tcldk∗, with σ∗ and k∗ for cloud-free conditions. ϵ can be approximated by 18ϵ≈1-exp⁡-δ⋅τ, where δ=D⋅(1=ω̃) and D≈1.66, relying on the zero-scattering assumption , and an effective emissivity ϵ∗ that is also derived from their RT simulations. Finally, the TIR RE of a cloud above a surface can be approximated with 19ΔFtir=FTIR,c-FTIR,cf≈σ∗Tsrfk∗-Tcldk∗⋅1-exp⁡-δ∗⋅τ, with δ∗=0.75. It follows from Eq. () that the forcing of a cloud, with constant τ, is proportional to the temperature difference between cloud and surface.

One difficulty of RT simulations in ice clouds is the uncertainty about the dominating ice crystal shape, which is commonly unknown, and therefore, a general ice crystal shape has to be assumed . Scattering and absorption by an ice crystal is characterized by its orientation, complex refractive index of ice, the wavelength of the incident light, shape, size, and the resulting asymmetry parameter. The asymmetry parameter is a measure of the asymmetry of the phase function P between forward and backward scattering . P provides the angular distribution of the scattered direction in relation to the incident light. For example, in the case of idealized hexagonal ice crystals and a wavelength below 1.4 µm, the asymmetry parameter is primarily determined by the ice crystal shape / aspect ratio, but for wavelengths larger than 1.4 µm, the asymmetry parameter also depends on the ice crystal size . Consequently, the assumption of an ice crystal habit and ice crystal size, with the related aspect ratio, is vital information for the estimation of the ice cloud RE. Furthermore, the ice optical properties by , which are used for the RT simulations in the present study, based on a coupling of the maximum diameter of the ice crystal and the aspect ratio, with the latter one being different for each crystal shape. This impacts the RT of different ice clouds with varying IWC and reff.

Subsequently, the shape effect is quantified using Eq. (), and relative differences in ΔF are given with respect to crystals with the same reff in relation to the ΔF simulated for aggregates.

Figure a–c show ΔFsol as a function of IWC, separated for crystal shape, reff, and three selected θ. For simplicity, αsrf and τliq are set to zero in this discussion.

The strongest ΔFsol is found for aggregates (green) with reff=5 µm, with the Sun at zenith (θ=0∘; Fig. a). A lower cooling effect in the solar spectrum is found for droxtals (orange) and plates (blue) with the same reff. The order of ΔFsol remains constant for increasing reff.

The spread in ΔFsol across crystal shapes with the same reff and IWC can be interpreted as a potential uncertainty in ΔFsol, due to the ice crystal shape. One has to keep in mind that the differences partially result from deviating crystal size distributions, as these depend on the selected crystal shape. showed that, in the solar wavelength range, the crystal shape is the main driver, and the actual ice PSD has only a minor effect on ΔFsol. Nevertheless, and found that the PSD also has a considerable impact on ΔFtir, leading to differences of up to 48 % in the single-scattering albedo when switching between PSDs.

To quantify the deviations resulting from the ice crystal shape, Fig. d–f show the absolute and Fig. g–i present the relative differences in ΔFsol of droxtals and plates with respect to aggregates. For θ=0∘, the largest absolute deviation is found for plates with reff of 25 µm and the highest IWC with an absolute range of up to 250 W m-2 (reff=25 µm, θ=0∘, τice=6.6), corresponding to a relative difference of 58 %. Relative deviations reach even larger values, e.g., when the cloud is optically thinner and ΔFsol becomes smaller. In the case of plates, the relative deviations range from -20 % (reff=5 µm) to -82 % (reff=85 µm). The large absolute and relative deviations between plates and aggregates in ΔFsol and later ΔFnet appear because plates are characterized by the smallest reflectance and absorption efficiency . The absolute differences among droxtals and aggregates are smaller. With increasing IWC, the absolute ranges quickly reach a maximum of 27 W m-2 at an IWC of 0.024 g m-3 and decrease towards the largest IWC. The associated relative deviations are also smaller compared to plates, ranging between -3 % (reff=5 µm) and -18 % (reff=85 µm).

Another characteristic of the absolute range of ΔFsol is the steep slope for θ=0∘ over the entire range of IWC. For illumination geometries with the Sun closer to the horizon, particularly θ=70∘, the behavior of absolute range in ΔFsol is characterized by a rapid increase and convergence towards a maximum. At a certain IWC and related τice, the slant optical path and cloud–radiation interactions are dominated by multiple scattering that suppresses single-scattering effects of individual ice crystal shape and, hence, reducing the absolute and relative difference resulting from the choice of the ice crystal shape. This is supported by earlier observations and simulations, for example, by , who showed that for large θ and multiple-scattering the shape effect becomes less prominent.

Next, we consider the solar, TIR, and net ΔF at θ=30∘ (Fig. ). The leftmost column for ΔFsol is identical to the middle column in Fig. . In the TIR, the largest ΔFtir is generally found for smallest crystals (5 µm) and highest IWC in decreasing order from droxtals, plates, and aggregates. With increasing crystal size, the order changes to droxtal, aggregates, and plates, and the absolute values of ΔFtir decrease. The largest ΔFtir range of 130 W m-2 is found for clouds with IWC between 0.024 and 0.1 g m-3 caused by droxtals. For thin clouds with IWC < 0.04 g m-3, the largest absolute range RΔF,tir of around 6.5 W m-2 appears for reff of 5 and 25 µm, which is shifting towards larger IWC with increasing reff and vanishes for the largest crystals with reff of 85 µm. The relative differences are the largest for the optically thinnest clouds and decrease with increasing IWC. While droxtals are characterized by relative differences close to 0 % (reff=5 µm; IWC = 0.1 g m-3) and 18 % (reff=25 µm; IWC = 0.007 g m-3), plates lead to relative differences between 9 % (reff=5 µm; IWC = 0.007 g m-3) and -5 % (reff=85 µm; IWC 0.007 g m-3). The TIR RE of the optically thickest cloud is independent on ice crystal shape, which is addressed to multiple scattering.

For all IWC and reff, ΔFsol is generally larger than ΔFtir and, therefore, dominates the resulting ΔFnet (Fig. c, f). Consequently, ΔFnet and the absolute ranges among the ice crystal shapes follow the distributions from ΔFsol. The largest relative deviations are found for the optically thinnest clouds, where ΔFnet is generally small. In these cases of optically thin clouds consisting of the smallest crystals (reff=5 µm), the relative deviations exceed the relative difference for optically thick clouds with the same crystal size by a factor of 10.

The analysis of all simulations shows that the crystal shape assumption on the cirrus RE is small compared to other parameters, particularly IWC or reff (see Fig. ). However, we found a larger variability in ΔFsol and the resulting ΔFnet (i.e., whether a contrail has a net warming or cooling effect compared to ΔFtir). For the defined reference consisting of aggregates, a ΔFsol value of -50.2 W m-2 was simulated, while for plates and droxtals, values for ΔFsol of -8.6 and -44.3 W m-2 were obtained, respectively. The impact of the crystal shape is less pronounced in the TIR wavelength range, with ΔFtir of 46, 44.5, and 48.9 W m-2 for aggregates, plates, and droxtals, respectively. The variation in ΔFsol propagates into ΔFnet, with -4.2, 35.9, and 4.6 W m-2 for aggregate, plates, and droxtals, respectively. Based on the presented simulations, we found larger maximum variations in ΔFsol, ΔFtir, and ΔFnet of 41.6, 4.4, and 40 W m-2, respectively, compared to . They found variations in ΔFsol, ΔFtir, and ΔFnet of 2, 6, and 7 W m-2, respectively. The difference is explained by the selected reference . However, even when selecting cloud parameters similar to the reference cloud of , we still found larger maximum variations in ΔFsol, ΔFtir, and ΔFnet of 17.3, 4.2, and 17.9 W m-2, respectively. This is attributed to the remaining differences among the selected reference values.

In this section, the impact of each parameter is estimated by fixing one parameter at a time (represented by the x axis), while the others can vary. For example, in case of θ, all simulations, for steps of θ given in Table , are extracted from the eight-dimensional (8D) hypercube. The extracted sub-sample, in the example for a specific θ, is used to calculate and visualize the distributions of solar, TIR, and net ΔF. This strategy can be interpreted as a type of sub-sampling, by averaging all unfixed parameters to project ΔF onto the 1D space. The impact of each parameter is further quantified by the minimum and maximum RE. We define the full range of ΔF by 20RΔF=max⁡{ΔF}-min⁡{ΔF}, with max⁡{ΔF} and min⁡{ΔF} as the maximum and minimum of ΔF across the sub-sampled distributions, respectively. As RΔF is susceptible to outliers, we further characterize the width of a distribution by the interquartile range, which is defined as the difference between the 75th (Q75%) and 25th (Q25%) percentiles of ΔF, as follows: 21QΔF=Q75%(ΔF)-Q25%(ΔF).

Variations in θ are caused by the diurnal and seasonal cycle of the Earth or variations along the longitude at a given time.

Figure a shows distributions of solar ΔFsol for θ=0∘, ranging from -944.5 W m-2 (high IWC) to 78.0 W m-2 (high αsrf). For simulated θ<85∘, the median values range from -11.9 to -12.9 W m-2, with an intensification of ΔFsol towards larger θ. At the same time, the upper maxima of ΔFsol are shifted towards zero, which is a combination of the following three effects: (i) a decreasing downward irradiance at TOA with increasing θ, (ii) an increasing optical path length s through the cloud with increasing θ and the corresponding increase in scattering, and (iii) an increase in upward scattered radiation with increasing θ as the light rays become slanted and a larger fraction of radiation from the forward scattering range is directed upwards. Effects (i) and (ii) compete and are dominated by effect (iii). The combination of effects (i) to (iii) also reduces the interquartile range for larger θ and indicates a reduced influence of the other free parameters on ΔFsol. However, the smallest ΔFsol is calculated for θ of 85∘ and is caused by the reduced sideward scattering of ice crystals.

The value of θ where ΔFsol is most intense depends on αsrf and is typically located between 50∘ and 70∘ . The maximum in ΔFsol and the corresponding θ values are explained by the strong forward-scattering peak of ice crystals and the resulting weak backscattering .

To further elaborate on the response of ΔFsol on large θ, Fig. a shows ΔFsol as a function of θ for selected τice and αsrf. For an optically thick cirrus with τice=6.62 located over a surface with αsrf=0 (solid blue curve), the maximum ΔFsol appears around θ=50∘. For the same cloud above, a more reflective surface with αsrf=0.3 (dashed blue curve) the maximum is shifted towards θ=70∘. Further increasing αsrf to 1 (dotted blue curve), solar cooling turns into a heating, and the strongest solar cooling is found for the largest θ. Figure a also shows that the shift in the absolute maximum ΔFsol is most pronounced for optically thicker clouds. However, the largest relative change in ΔFsol by varying θ appears for optically thin clouds .

Figure b shows ΔFsol normalized with the respective ΔFsol at θ = 0∘. The sensitivity of normalized ΔFsol on θ is most pronounced for optically thin clouds, with τice=0.2 over a moderately reflective surface (αsrf=0.3; dashed black). For this combination, ΔFsol at θ=70∘ is a factor of 3 larger compared to a Sun overhead (θ=0∘). The same cloud over a non-reflective surface (αsrf=0) reduces the sensitivity leading to a factor of 1.2 in relation to ΔFsol at θ=0∘ (solid black). A similar pattern but with a generally reduced sensitivity is found for the optically thicker cloud case, with τice=6.62. In this case, ΔFsol is larger by a factor of 1.05 at θ=50∘ (solid blue) and larger by a factor of 1.7 at θ=70∘ (dashed blue) with respect to the Sun at θ=0∘. The large sensitivity for optically thin clouds is explained by the dominance of single scattering, where scattering is strongly dependent on the value of the P at a given scattering angle. When the cloud becomes optically thicker, multiple-scattering processes start to dominate the RT, and P is averaged over a range of scattering angles, thus reducing the sensitivity on θ. However, while the sensitivity might be largest for optically thin clouds, the absolute ΔFsol of optically thin clouds is small compared to clouds with higher τice.

Figure b shows that ΔFtir is unaffected by θ, leading to a constant median ΔFtir of 21.5 W m-2. The highest positive values of ΔFtir (strongest warming effect) are found for clouds with maximal IWC. The resulting ΔFnet, shown in Fig. c, is dominated by a warming in the TIR that leads to median ΔFnet between 1.4 and 11.3 W m-2, with a minimum of ΔFnet of -872.8 W m-2 and maximum of 160.1 W m-2. With increasing θ, ΔFnet increases. This is caused by the shift in the lower minima of ΔFsol towards zero, which indicates that a larger fraction of the simulations has a reduced solar cooling effect, and thus, the fraction of simulations with a positive ΔFnet (net warming) increases. The reduced variability in ΔFsol with increasing θ propagates into the distribution and variability in ΔFnet.

The influence of the underlying surface is shown in Fig. . For αsrf=0, the surface absorbs the entire incident solar radiation creating the largest contrast between αsrf and the cloud albedo αcld. When the surface is fully absorbing (αsrf=0), almost all simulated cloud combinations are characterized by a cooling in the solar range, with ΔFsol ranging from -944.5 to 80 W m-2. The cooling is reduced when the surface becomes more reflective and the contrast between surface and cloud is reduced, which shifts the distributions and their medians towards positive ΔFsol. With αsrf approaching 0.66, around 25 % of the parameter combinations lead to a solar heating. This becomes even more pronounced towards αsrf=1, where around 50 % of the simulations yield a warming effect in the solar range. ΔFtir is unaffected by changes in αsrf, as expected, and remains constant for all αsrf with a median at 21.5 W m-2. The resulting ΔFnet is dominated by a net warming effect, indicated by mostly positive median values ranging from 1.4 W m-2 (αsrf=0.25) to 18.8 W m-2 (αsrf=1). An exception is αsrf=0, where more than 50 % of the simulations lead to a net cooling, with a median ΔFnet at -2.1 W m-2.

As presented in Fig. , the IWC is the second most influencing factor that controls ΔF. For a constant crystal number concentration, the increase in IWC leads to an increase in reff and the total particle scattering and absorption cross sections. This enhances the scattering and absorption along the optical path s though the cloud.

Figure a reveals that with increasing IWC, the median of ΔFsol becomes more negative (intensification of the cooling effect in the solar part of the spectrum). The steepest increase is found for IWC < 0.012 g m-3, while for IWC ≥ 0.012 g m-3, the solar cloud RE saturates. At the same time, QΔF,sol, given by Eq. (), increases, indicating an enhanced sensitivity of ΔFsol on the free parameters. The minimum and maximum values of ΔFsol result from clouds over a highly reflective surface (αsrf = 1) and clouds containing crystals with the smallest reff = 5 µm.

For ΔFtir, the increase in IWC leads to an intensified warming effect (Fig. b). Again, this is caused by the increase in the total particle scattering and absorption cross sections. Similar to ΔFsol, the steepest increase in ΔFtir appears for IWC < 0.012 g m-3, while for larger IWC, the medians approach an almost constant level, and a further increase in IWC has only a limited effect on ΔFtir. The resulting ΔFnet (Fig. c) ranges from -543.2 to 125.5 W m-2 and is skewed to positive ΔFnet with median values between 1.1 and 12.2 W m-2.

The size of ice crystals also influences the cloud RE, with a larger sensitivity of ΔFsol on reff than ΔFtir . Figure a illustrates that cirrus with the smallest reff are associated with the most intense cooling effect in the solar range, leading to ΔFsol between -944.5 and 80.0 W m-2. Small crystals and high number concentrations lead to higher αcld,ice in the solar range compared to fewer and larger crystals . For the smallest crystals in the simulations, a median ΔFsol of -38.6 W m-2 is determined. For increasing reff, the cooling effect in the solar range decreases and tends towards ΔFsol of -1.8 W m-2. The intensified solar cooling (more negative ΔFsol) with decreasing reff is associated with an increase in the ice crystal number concentration while keeping IWC constant, which is also known as the cloud albedo effect. In addition, ice crystals with larger reff are characterized by enhanced forward scattering. Hence, less radiation is scattered to the sides or backwards into space. Figure a shows that clouds with larger reff are less sensitive to the effect of the free parameters as the interquartile range decreases strongly from QΔF,sol (reff=5 µm) = 108.7 W m-2 to QΔF,sol (reff=85 µm) = 8.0 W m-2. Similarly, Fig. b shows the strongest TIR heating for the smallest crystals/highest Nice. Such clouds have the largest total absorption cross section and act almost as blackbodies in the TIR . However, an increase in reff, while fixing IWC, leads to a reduction in ΔFtir, which is caused by the lower total particle scattering and absorption cross sections. QΔF,tir decreases from 58.8 W m-2 for reff=5 µm to 18.5 W m-2 for reff=85 µm. Median values of ΔFnet, shown in Fig. c, indicate only a net cooling for reff=5 µm, with -1 W m-2, whereby elsewhere a net warming is dominant, with ΔFnet between 2.6 and 4.5 W m-2. Simultaneously, QΔF,net slightly decreases, which indicates the reduced impact of the remaining free parameters for large crystals. The presented dependencies, especially for small reff, of solar, TIR, and net ΔF on reff and IWC agree with previous studies, e.g., from but particularly and .

The previous analysis aimed to sample the 8D hypercube in a series of 1D cross sections to focus on the general distribution of ΔF that results from a single parameter. This likely masks the dependencies of ΔF on specific parameter combinations that are closely interconnected. Subsequently, we focus on a detailed analysis, particularly in the solar wavelength range, to highlight the dependencies among Sun geometry, surface albedo, and cloud properties – especially reff and IWC.

Within this study, the atmospheric profiles, the surface temperatures Tsrf, and the vertical location of the ice cloud are coupled. For example, the selection of the US standard atmosphere is directly linked to a surface temperature of Tsrf = 288.2 K. Tsrf is equal to the lowermost temperature value in the respective AP. The vertical position of the ice cloud depends on the temperature of the AP and the selected cloud-top temperature Tcld,ice (see Appendix  and Fig. a–b therein).

Figure a shows that variations in Tsrf have an effect on ΔFsol, with the differences in the median ΔFsol of up to ±7 W m-2. Generally, larger effects are found for the TIR component, where an increase in Tsrf enhances the temperature difference between surface and cirrus, which leads to an intensification of the TIR heating (see Eq.  and ), thus shifting the median ΔFtir from 3.6 to 13.6 W m-2 (Fig. b). Simultaneously, the distributions broaden with increasing Tsrf, with QΔF,tir ranging from 1.4 to 7.4 W m-2 (257.2 K) and 5.6 to 26.0 W m-2 (299.7 K), which results from the warmer and moister tropical profile compared to the drier sub-Arctic profile. As a result of the almost constant ΔFsol and the increase in ΔFtir, the net heating effect is enhanced with medians ranging between 0.7 and 6.6 W m-2.

The effect of variations in Tcld,ice is shown in Fig. a–c. Increasing Tcld,ice reduces the temperature difference between surface and ice cloud and therefore the TIR heating effect (Fig. b). Median ΔFtir values are reduced from 29.2 to 15.3 W m-2 when Tcld,ice is increased from 219 to 243 K. Compared to the impact of Tsrf, which was varied over a range of 42.5 K, shifting the cloud in the vertical has only a minor effect on ΔFtir and ΔFnet, as the variation in Tcld,ice spanned only 24 K. The resulting net effect from variations in Tcld,ice leads to medians between 1.2 and 6.1 W m-2 for Tcld,ice of 219 and 243 K, respectively.

The previously mentioned impact of Tsrf on ΔFsol is traced back to (a) the different optical path length through the atmosphere because of variations in cloud-top altitude and (b) the different water vapor concentration due to the three applied APs. The effect of varying RH profiles was investigated by manipulating the original RH profiles by ±20 % to represent the variability in RH reported by . The RT simulations were performed for a sub-set of the parameter space with fixed Tcld,ice = 231 K, αsrf = 0, and τliq = 0. The effects on solar, TIR, and net ΔF are quantified by their absolute and relative differences. Variations in RH have only a small effect on ΔFsol, with maximal ±0.15 W m-2 (±0.4 %) among all profiles. A slightly larger impact is found for ΔFtir, with up to ±1.45 W m-2 (±4.1 %) in case of the warm and moist tropical profile (afglt). Less affected are the standard atmosphere (afglus), where ΔFtir varies by ±0.9 W m-2 (±3.2 %), and the dry sub-Arctic profile (afglsw), with variations in ΔFtir of ±0.3 W m-2 (±2.4 %). Consequently, afglt has the largest variation in ΔFnet of ±0.8 W m-2 (±8 %) and is followed by ±0.6 W m-2 (±3.8 %) for afglus and ±0.2 W m-2 (±0.6 %) for afglsw. Scaling the original RH profiles showed that variations in the RH profile explicitly influence the TIR wavelength range but particularly the net RE. This analysis suggests that the variations in RH have to be considered to be a potential source of variability when using this publicly available data set.

All simulations within this study were performed for a fixed cloud geometric thickness dz of 1000 m. In reality, however, dz is likely to vary over the cirrus lifetime, for example, due to the sedimentation of ice crystals or vertical winds. The effect of changing dz is quantified by a dedicated sensitivity analysis of ΔF for a sub-sample of the full parameter range (Table ). A similar sub-parameter space is used as was done for the RH sensitivity but additionally fixing Tsrf = 288 K, i.e., using the afglus profile. With τice being proportional to the IWP of the cloud (Eq. ), the IWP of the 1000 m reference and solar τice are kept constant, and the IWC for the clouds with dz of 500 and 1500 m clouds is scaled accordingly.

As expected from Eq. , the resulting effect on median ΔFsol, given in Fig. , is almost negligible, with ±0.1 W m-2 (±0.3 %). Differences in the median ΔFtir are up to ±0.6 W m-2 (±3.5 %), which leads to differences in the median ΔFnet of ±0.6 W m-2 (±6.2 %). The relevant relative differences in ΔFtir and ΔFnet are explained by the varying cloud base altitude, which modifies the vertical distribution of IWC and the temperature of the cloud base, which determines the amount of emitted radiation. In addition, geometrically thin clouds with low τice act as graybodies, while with an increase in dz, cirrus clouds become opaque and act as more efficient blackbodies . further reported that cirrus with small reff reflect solar radiation at the cloud top (solar cooling) but absorb TIR radiation at the cloud base (TIR warming), which creates a temperature gradient within the cloud that depends on dz. From the dz sensitivity analysis, it is found that dz can be neglected in the solar wavelength range but is of relevance for ΔFtir and especially ΔFnet, where the absolute values are small. This partly agrees with the findings from , who showed that solar, TIR, and net ΔF are only slightly sensitive to changes in dz with solar, TIR, and net ΔF below 2 W m-2 under the premise of a constant ice water path (IWP). The presented simulations indicate ΔΔFsol of 2 W m-2, which is comparable to , but we found slightly higher ΔΔFtir and ΔΔFnet of 4.5 and 3.1 W m-2, respectively.