As discussed in the Introduction, both the background stratospheric aerosol layer and its perturbation brought by the Hunga Tonga–Hunga Ha'apai eruption can be solely characterised, in terms of composition, as secondary SA particles. Thus, we model both the background and Hunga Tonga–Hunga Ha'apai-perturbed stratospheric aerosol layers as composed of SA. These particles are usually represented as spherical liquid droplets of binary aqueous solution of sulfuric acid (H2SO4) as done by e.g. Sellitto and Legras (2016). The link between the chemical composition and the optical properties of a specific aerosol particle is provided by the complex refractive index (CRI). Among the available laboratory measurements of SA CRI, we have selected, for this work, the one from Hummel et al. (1988). This is one of the few datasets that extend the CRI spectra in both the shortwave (SW) and the longwave (LW) spectral ranges, from the ultraviolet (UV) to part of the far infrared (FIR), for representative stratospheric conditions in terms of the H2SO4 mass mixing ratio and temperature. Both the background stratospheric SA (e.g. Kremser et al., 2016) and the Hunga Tonga–Hunga Ha'apai perturbations (Duchamp et al., 2023) are characterised by very acidic particles with a H2SO4 mass mixing ratio of typically 70 %–80 %. Thus, a 75 % H2SO4 mixing ratio is selected in this work. Among the available particle temperatures in the Hummel et al. (1988) database, a temperature of 215 K is selected as the most suitable to represent lower- and mid-stratospheric conditions. Real and imaginary parts of CRI used in this work are shown in Fig. 1.

The number density SD n(r) is defined so that n(r)dr is the number of particles per unit volume, in the aerosol layer, with a radius between r and r+dr. All SDs in this work are modelled as mono-modal log-normal distributions (Eq. 1). In Eq. (1), N0 is the total number concentration (in particles cm-3); rm is the median radius; and S=ln⁡σ is the SD spread, i.e. the unitless standard deviation of ln⁡(r/rm). 1n(r)=N0rln⁡σ2πe-12ln⁡rrmln⁡σ2 Both background and Hunga Tonga–Hunga Ha'apai-perturbed typical SDs are derived from the results of Duchamp et al. (2023). In that work, SD parameters of a mono-modal log-normal SD, i.e. N0, rm, and σ in Eq. (1), are obtained using Stratospheric Aerosol and Gas Experiment III on the International Space Station (SAGE III/ISS) satellite observations with the method initially developed by Wrana et al. (2021). Duchamp et al. (2023) applied that method to the Hunga Tonga–Hunga Ha'apai plume and extended these retrievals back to immediate previously unperturbed periods, so as to also derive SD parameters for a representative background stratospheric aerosol layer. It is usually convenient, in remote sensing applications, to define an effective radius re (the cube particle radius divided by the square particle radius, averaged over the SD); re is directly linked to the extinction of the aerosol layer. For a mono-modal log-normal SD, an effective radius can be defined as re=rme2.5ln⁡2σ. For the background stratospheric aerosol layer, a typical combination of rm=0.20 µm and σ=1.50 is considered (Fig. 2a and Table 1). The Hunga Tonga–Hunga Ha'apai eruption produced an increase in the mean particle size and a decrease in the spread of the particle SD with respect to background conditions (Duchamp et al., 2023). Typical values of Hunga Tonga–Hunga Ha'apai-perturbed stratospheric aerosol layers are used here, with re varying between 0.35 and 0.45 µm and σ varying between 1.20 and 1.30 (Fig. 2a and Table 1). In Fig. 2, N0 is fixed to 1 particle cm-3 for the background and all Hunga Tonga–Hunga Ha'apai cases. As a further comparison, the Mount Pinatubo (Philippines) and El Chichón (Mexico) eruptions in 1991 and 1982 are also considered (Fig. 2b and Table 1). For the Pinatubo eruption, we use aerosol SD for both a relatively young plume (Asano, 1993) and an extreme case of an aged plume, i.e. several months after the eruption (Russell et al., 1996). Immediately after the Pinatubo eruption, the stratospheric aerosol perturbation was associated with relatively large particles (up to rm=0.6 µm) on average, with an extremely small SD width (down to σ=1.05) (Asano, 1993). The effective radius of the Pinatubo-perturbed aerosol layer increased during the first year after the eruption (extreme values as large as re=0.9 µm) and then decreased slowly to background levels (Russell et al., 1996). Thus, the SDs used in this work for Pinatubo perturbation are to be regarded as two extreme values for this event. For the El Chichón eruption, only a small amount of information is available for the SD and its temporal evolution. In this paper, we consider the results of Hofmann and Rosen (1983), which are representative of the aerosol SD after about 1.5 months after the eruption of El Chichón.

For an SA layer of given mono-modal log-normal SD, the effective SA mass Me (i.e. the total aerosol mass per unit volume in the layer) can be calculated as done in Eq. (2), where ρ is the mass density of SA (here taken as 1.75 gcm-3) and Ne=N0e-3ln⁡2σ is the mono-modal log-normal effective number density. The Me values for the SDs used in the present work are also indicated in Table 1. 2Me=43πre3ρNe

The optical properties of background, Hunga Tonga–Hunga Ha'apai-perturbed, El Chichón-perturbed, and Pinatubo-perturbed stratospheric aerosol layers are calculated with the scheme shown in Fig. 3. Starting from CRI and SD input data described in Sect. 2.1 and 2.2, a Mie code estimates the extinction coefficient (βext), single-scattering albedo (SSA), and asymmetry parameter (g) spectra for both cases and for Mount Pinatubo (young and aged plume) and El Chichón as a further comparison. As Mie code, the Interactive Data Language (IDL; https://www.nv5geospatialsoftware.com/Products/IDL, last access: 18 June 2025) Mie-scattering routines of the Earth Observation Data Group of the Department of Physics of Oxford University are used (https://eodg.atm.ox.ac.uk/MIE/, last access: 18 June 2025). The outputs of the Mie calculations are intended to represent the overall extinction of typical stratospheric background and volcanically perturbed layers, along with their absorption/scattering properties (through SSA) and angular distribution of the scattered radiation (through g). The SSA is the ratio of the aerosol scattering to the aerosol extinction (i.e. scattering plus absorption) efficiency factors Qsca and Qext obtained with Mie calculations (van de Hulst, 1957). Values of the SSA approaching 1.0 point towards pure scattering particles, while values of the SSA approaching 0.0 point towards pure absorbing particles. The g parameter is the mean value of the cosine of the scattering angle weighted though the scattering phase function obtained with Mie calculations (van de Hulst, 1957). Values of the g parameter approaching 1.0 point towards pure forward scattering, while values of the g parameter approaching 0.0 may point towards isotropic scattering. The overall extinction of the layers, measured by βext, is calculated with two different assumptions: a fixed N0 and a fixed Me. With the latter, extinction spectra per unit mass concentration (βext/Me) are also derived. The SSA, g, and βext/Me are intensive optical properties of the aerosol layer; i.e. they do not depend on the injected aerosol mass. It was discussed previously that these optical parameters, together with surface reflectivity information fed to a radiative transfer model (RTM), are sufficient to describe the radiative properties and impacts of an aerosol layer, like the instantaneous radiative forcing and the vertical profiles of radiative/cooling heating rates (e.g. Sellitto et al., 2022a, 2023). In this work, optical properties in both the solar shortwave (SW) and the terrestrial longwave (LW) spectral ranges are estimated in order to produce results usable in RTMs when there is interest in the whole radiant energy spectrum in Earth's atmosphere. The output spectral range of the optical properties is somewhat limited by the available spectral range of the input CRI. In our case, we are limited at 25 µm on the upper end, due to available SA CRIs in the literature, so a part of the far-infrared range is not represented in this work, likely leading to an underestimation of its impact.

Figure 4 shows βext/N0 and βext/Me calculation for the background and Hunga Tonga–Hunga Ha'apai-perturbed scenarios, as described in Sect. 2. The extinction is an extensive optical property that depends on the aerosol layer mass (the SA mass, in the present case). Thus, large differences in βext/N0 among background and Hunga Tonga–Hunga Ha'apai-perturbed scenarios can be observed (Fig. 4a). This is principally due to the larger SA mass, for a fixed number density N0=1.0 particle cm-3, of the larger Hunga Tonga–Hunga Ha'apai particles with respect to the smaller background particles. The effective mass varies from 0.12 µgcm-3 for the background layer to up to 0.60 µgcm-3 for the Hunga Tonga–Hunga Ha'apai-perturbed layers (see Table 1). On the contrary, βext/Me is much less variable depending on the scenario, except at very short wavelengths in the SW (Fig. 4b). In general, a clear exponential decrease in βext can be observed in the UV–Vis and part of the near infrared (NIR), thus in most of the SW spectral range, following the empirical Ångström law in Eq. (3), where λ is the wavelength (λref is a reference wavelength, in many cases taken as 1 µm) and AE is the Ångström exponent. 3βext(λ)=βext(λref)λ-AE

At longer wavelengths, absorption features of SA particles appear, including the peculiar mid-infrared (MIR) signatures at 8.0 to 11.0 µm. These absorption features can be associated with the rotational–vibrational absorption bands of the undissociated H2SO4 in the concentrated solution droplets discussed by e.g. Sellitto and Legras (2016). More absorption features are visible in the NIR, from about 3.0 to 6.0 µm, and in the FIR, from 15.0 to 18.0 µm. The two regimes, dominated by scattering in the SW for wavelengths shorter than about 3.0 µm and by absorption in the LW for longer wavelengths, are discussed further in the following (associated with SSA) and are linked to a large variability in βext/Me. The latter ranges between less than 1.0×10-3 km-1 per unit effective mass above 3.0 µm and 4.5×10-3 km-1 per unit effective mass below 3.0 µm.

Figure 5 shows the SSA (Fig. 5a) and g (Fig. 5b) calculations for the background and Hunga Tonga–Hunga Ha'apai-perturbed scenarios. The first takeaway message from these results is that both SSA and g are not significantly perturbed by the Hunga Tonga–Hunga Ha'apai eruption and that their absolute values and variability are similar for background and Hunga Tonga–Hunga Ha'apai-perturbed scenarios. The two scattering- and absorption-dominated regimes, discussed for the aerosol extinction, can be seen here with an SSA of approximately 1.0 (pure scattering aerosol layers) for wavelengths shorter than about 3.0 µm, sharply decreasing to values lower than about 0.2 for wavelengths longer than about 3.0 µm. The SSA approaches values of 0.0 (pure absorbing particles) for wavelengths longer than about 6.0 µm. The g parameter, starting from values of about 0.6 to 0.8, steeply decreases to values lower than 0.1 for wavelengths longer than 6.0 µm. This can be associated with a markedly dominating forward scattering in the SW and a quasi-isotropic scattering in the LW. Figures 4 and 5 are also shown on a vertical log-scale in Fig. S1 in the Supplement.

These results are summarised in Fig. 6, where spectral band average values of the intensive optical properties βext/Me, SSA, and g, for background and Hunga Tonga–Hunga Ha'apai-perturbed layers, are shown for the UV–Vis, NIR, MIR, and FIR broad-band ranges. Large values for the three intensive parameters are found in the UV–Vis, thus pointing towards very extinction-effective layers dominated by a largely forward scattering. While this characteristic behaviour of the aerosol layers stands for the whole SW, the extinction efficiency markedly decreases in the NIR. In the LW, the optical characterisation of the layers is more spectrally homogeneous, with relatively small extinction dominated by absorption and a small quasi-isotropic scattering, with small differences between the MIR and the FIR.

More generally, the most apparent effect, visible in Figs. 4–6, is that all intensive optical properties of the Hunga Tonga–Hunga Ha'apai-perturbed layer are very similar to those in background conditions. It can be concluded that, despite the significant perturbation in stratospheric aerosol SD and the relatively large mass of the injected SA, the Hunga Tonga–Hunga Ha'apai eruption had minimal effects on the intensive optical properties of the stratospheric aerosol layer. Thus, the only impact of the Hunga Tonga–Hunga Ha'apai eruption on the optical properties of the stratospheric aerosol layer is the marked increase in the overall extinction and the aerosol optical depth (AOD). For this latter property, Hunga Tonga–Hunga Ha'apai produced the largest global perturbation since the Mount Pinatubo eruption in 1991 (Sellitto et al., 2022b). The specific absorption and scattering properties of the stratospheric aerosol layer were not significantly perturbed in the SW and LW by the Hunga Tonga–Hunga Ha'apai eruption, as shown by the SSA and g calculations of Figs. 5 and 6b, c. It must be noted that Hunga Tonga–Hunga Ha'apai perturbations on the stratospheric aerosol SD, though significant, are much smaller than for other, stronger recent eruptions, such as El Chichón in 1982 (Hofmann and Rosen, 1983) or the aged Pinatubo in 1991 (e.g. Russell et al., 1996); see respective SDs in Fig. 2b. In these cases, a fraction of relatively large particles (larger than 1.0 µm) was present, which is not the case for the Hunga Tonga–Hunga Ha'apai eruption (see Fig. 2b). For the fresh Pinatubo plume, i.e. during the first few months after the eruption in 1991, larger particles were present, on average, than for the Hunga Tonga–Hunga Ha'apai eruption, but with a minimal contribution of particles larger than 1.0 µm, due to the exceptionally small width of its SD (Asano, 1993). We performed additional Mie calculations for the Pinatubo and El Chichón SDs, using the SD parameters of Asano (1993), Russell et al. (1996), and Hofmann and Rosen (1983), summarised in Table 1. For the aged Pinatubo and El Chichón cases, the UV–Vis βext/Me is significantly smaller, and the overall LW SSA and g are significantly larger, than for Hunga Tonga–Hunga Ha'apai and the stratospheric background (Fig. 6). This effect, even if still present, is less strong for the fresh Pinatubo plume. Such a change in optical regime implies a stronger LW climate warning effect that can counterbalance the SW cooling. More generally, Lacis et al. (1992) showed that a marked change in optical and radiative regime occurs for volcanic aerosol layers with mean size exceeding 1.0 µm, which is the case for the aged Pinatubo plume and El Chichón but not for the Hunga Tonga–Hunga Ha'apai perturbations. This change in the SW / LW optical properties can result in less effective cooling at top of the atmosphere or even in aerosol-related warming, which could initially have been the case for the Pinatubo and El Chichón eruptions (before a marked and relatively long-term cooling effect due to the removal of larger aerosol particles) but was likely not the case for Hunga Tonga–Hunga Ha'apai, even in the first phases after the eruption (e.g. Sellitto et al., 2022b). The mean particle size for Hunga Tonga–Hunga Ha'apai was larger than any post-Pinatubo stratospheric eruption (Wrana et al., 2023). Most post-Pinatubo eruptions perturbed the stratospheric aerosol layer with a decrease in mean particle size, rather than an increase. The radiative effects of the Hunga Tonga–Hunga Ha'apai-related aerosol SD perturbations are discussed more in Sect. 3.2.

Another intensive aerosol optical parameter is the Ångström exponent (AE); see also Eq. (3). An average AE can be obtained by combining aerosol extinction or AOD information at two different wavelengths. In Sellitto et al. (2024), the average UV–Vis to MIR AE was calculated using combination of Ozone Mapping and Profiler Suite Limb Profiler (OMPS LP) and Infrared Atmospheric Sounding Interferometer (IASI) observations at 0.7 and 8.5 µm, respectively. As for other intensive optical properties shown above, the stratospheric aerosol layer UV–Vis to MIR AE obtained using our calculations is not significantly perturbed by the Hunga Tonga–Hunga Ha'apai eruption, and background and Hunga Tonga–Hunga Ha'apai-perturbed values stay around a value of 1.0 (0.94 for the background stratospheric conditions and 0.97 ± 0.02 for Hunga Tonga–Hunga Ha'apai-perturbed conditions, with standard deviation representing the variability associated with the different Hunga Tonga–Hunga Ha'apai SDs). These results are not dissimilar to the observed value (1.13 ± 0.23), even if ∼15 % smaller. Thus, this shows the consistency of our calculated optical properties and the observed optical properties for the Hunga Tonga–Hunga Ha'apai plume. It has to be noted that Taha et al. (2022) showed that the observed Vis AE, i.e. estimated with two different Vis bands of the OMPS LP instrument (0.52 and 1.02 µm), is significantly perturbed by the Hunga Tonga–Hunga Ha'apai eruption. The AE estimation from observations is very sensitive to the selection of the two wavelengths used to derive it. Using the same wavelengths as in Taha et al. (2022), we obtain a Vis AE of 1.35 and 0.80 for the background and Hunga Tonga–Hunga Ha'apai-perturbed scenarios. This is very consistent with past theoretical studies, i.e. the one of Schuster et al. (2006) (see their Fig. 4a, with effective radii 0.30 µm and 0.40–0.45 µm, respectively). The variability in the UV–Vis AE, in terms of the selected background or Hunga Tonga–Hunga Ha'apai-perturbed scenarios, can also be seen in Fig. 4, reflected by the different slopes of the spectral variability in the extinction coefficient.

A simple parameterisation of the top-of-atmosphere (TOA) radiative forcing is used to estimate the radiative impact, through the perturbation of the optical properties, of the modification of the SD of stratospheric aerosol by the Hunga Tonga–Hunga Ha'apai eruption. For partly absorbing aerosols in a layer over a given surface of spectral reflectivity Rs and placed in a stratified atmosphere of spectral transmissivity Tatm, a broad-band TOA radiative forcing ΔF per unit SA mass can be defined as in Eq. (4). This quantity is estimated at the four broad-band spectral ranges, as defined in e.g. Fig. 6 (UV–Vis, NIR, MIR, and FIR), then these broad-band estimations are added up to obtain a total ΔF. In Eq. (4), the optical properties βext/Me and SSA are the same as in Fig. 6. The optical parameter b is the hemispheric backscatter ratio, which can be derived using our asymmetry parameter g estimations, using the method described by Marshall et al. (1995). The aerosols are put in a 1 km deep layer. In the parameterisation, S is the radiation source, taken as pure Planck functions at temperatures of 5770 K in the SW range (UV–Vis and NIR broad bands) and 300 K in the LW range (MIR and FIR broad bands), to simulate the solar and terrestrial radiation sources. The solar S component is then scaled at the mean Sun–Earth distance. The diurnal cycle of S was not considered in this example. The underlying surface is regarded as marine (Rs 0.05 in the UV–Vis and FIR; 0.00 in the NIR and MIR), and average values of Tatm are also considered (0.6, 0.5, 0.5, and 0.2 in the UV–Vis, NIR, MIR and FIR). 4ΔF=-S(λ)Tatm(λ)βext(λ)MeSSA(λ)b(λ)×(1-Rs(λ))2+2Rs(λ)1-SSA(λ)b(λ)SSA(λ)Δλ

The SW (UV–Vis + NIR), LW (MIR + FIR), and total (SW + LW) radiative forcings obtained with Eq. (4) and our estimated optical parameters are shown in Fig. 7 for a background stratospheric aerosol layer and for layers perturbed by the Hunga Tonga–Hunga Ha'apai, Pinatubo, and El Chichón eruptions. Two different radiative regimes can be observed in Fig. 7. The first regime, for smaller effective radii, is clearly dominated by the SW scattering, thus resulting in a large negative radiative forcing per unit mass. The second regime, for larger effective radii, is less and less dominated by the decreasing SW scattering with increasing mean size and the progressively increasing importance of the LW warming effect due to the LW absorbing effect. In the latter case, the total radiative forcing per unit SA mass decreases significantly with respect to background conditions. In addition, the SW / LW absolute ratio of the ΔF is approximately 99 %, 98 %, 80 %, and 60 % for the background, Hunga Tonga–Hunga Ha'apai, aged Pinatubo, and El Chichón scenarios. These results point towards very different overall radiative regimes between the background and Hunga Tonga–Hunga Ha'apai with respect to aged Pinatubo and El Chichón. For the young Pinatubo plume, the SW / LW absolute ratio of the ΔF is around 90 %, larger than for the aged Pinatubo plume while still smaller than for the Hunga Tonga–Hunga Ha'apai-perturbed scenario. The Hunga Tonga–Hunga Ha'apai eruption did not significantly modify the radiative regime of the stratospheric aerosol layer with respect to background conditions, with values around -0.17 Wm-2 (µg m⁻³ SA)⁻¹, dominated by SW scattering (pink triangle in Fig. 7). This is 1.5 to 4 times more effective, per unit SA mass, than the Pinatubo eruption (about -0.12 to -0.040 Wm-2 (µg m⁻³ SA)⁻¹, depending on the ageing of the plume; pink squares in Fig. 7) and nearly 10 times more effective than the El Chichón eruption (about -0.015 Wm-2 (µg m⁻³ SA)⁻¹; pink hexagon in Fig. 7). For the latter, the SW cooling and the SW / LW ratio markedly decreased with respect to the background and the Hunga Tonga–Hunga Ha'apai eruption. Our results are consistent with previous studies, particularly the one of Lacis et al. (1992), who showed that the cooling potential at the TOA of stratospheric aerosols significantly decreases for effective radii larger than about 1.0 µm and can even switch from negative (cooling) to positive (warming) for effective radii larger than 2.0 µm (see their Fig. 2). The effect of a significantly larger RF per unit SA mass, with respect to major recent eruptions, adds up with the larger stratospheric AOD production per unit SO2 injected mass shown by Li et al. (2024). Thus, for different factors, including the high-altitude injection and the aerosol SD, the Hunga Tonga–Hunga Ha'apai eruption produced stratospheric aerosol perturbations with a particularly large potential to produce a negative RF and a cooling of the climate system. These factors can be associated with the phreatic nature of the eruption and therefore with the interaction with seawater before the eruption and the subsequent availability of water vapour in the plume. It is important to recall that the Hunga Tonga–Hunga Ha'apai eruption is characterised by much less injected SO2 and SA mass than the Pinatubo and El Chichón eruptions, so, despite the larger radiative effectiveness of its stratospheric aerosol perturbations, in terms of the SD, its overall radiative effect is expected to be smaller than Pinatubo and El Chichón. The direct radiative effect of water vapour must also be considered. Water vapour can produce a warming at the TOA that can bias or revert the cooling effect of the sulfate aerosol layer (Sellitto et al., 2022b).

Optical properties of aerosol layers are needed, in radiative calculations with both climate models and offline radiative models, for a given source of radiative forcing to estimate its impacts (e.g. Sellitto et al., 2022a, 2023). Our results provide a ready-to-use benchmark for the radiative impact estimations of the Hunga Tonga–Hunga Ha'apai eruption versus stratospheric aerosol background. Thus, in the Supplement, we provide datasets on optical properties (1) computed at the wavelengths at which the refractive indices of Hummel et al. (1988) are available and (2) averaged over the 30 broad bands of the Rapid Radiative Transfer Model (RRTM; Iacono et al., 2008) in the ECMWF ECRAD implementation (Hogan and Bozzo, 2018) (the latter are provided as a netCDF file) for future studies on the Hunga Tonga–Hunga Ha'apai radiative impacts. Averaged values for the UV–Vis, NIR, MIR, and FIR bands are also reported in Table 2 (same as Fig. 6). Caution must be taken for the radiative impact estimation of the Hunga Tonga–Hunga Ha'apai eruption because of the additional important radiative impacts of water vapour injections, which can be even larger than aerosol impacts during the first months (Sellitto et al., 2022b). This additional radiative effect for the Hunga Tonga–Hunga Ha'apai eruption was not present in recent stratospheric volcanic eruptions from subaerial volcanoes.