The nitrogen cycle governs atmospheric oxidants budget through the photochemistry of nitrogen oxides (NOx= NO + NO2), which are strongly coupled with ozone (O3) and hydroxyl (OH) chemistry in the troposphere . Atmospheric nitrate is the end product of NOx oxidation, and the snowpack (and subsequently the firn and ice) acts as a sink. Temporal variations in the nitrate concentration recorded in ice cores could thus provide information about the oxidative capacity of the atmosphere in past times , or even about past solar activity . However, as illustrated by their Fig. 2, several post-deposition processes occur in the snow and hamper our current ability to interpret ice core records of nitrate. As first evidence of these post-deposition processes, NOx has been shown to be produced in sunlit snowpack . A production pathway involving nitrate photolysis in snow was rapidly elucidated afterwards . These pioneering works drove numerous field campaigns (e.g. SNOW99: ; ISCAT2000: ; ANTCI: ; CHABLIS: ; OPALE: ), as well as laboratory studies and modelling studies , in order to improve the understanding of the underlying processes responsible for the nitrogen recycling inside the snowpack. These studies focused on the nitrate photolysis in the photic zone of the snowpack and the subsequent release of NOx to the overlying atmosphere.

None of these studies investigated the physicochemical uptake processes of atmospheric nitrate into snow. However, it has been established that several physical processes also affect snow chemical composition . Numerous experimental studies of adsorption on ice surfaces have demonstrated that several chemical compounds, and especially acidic gases such as HCl and HNO3, have a great affinity for ice surface (see reviews by and ). Several small molecules, such as HCl , HNO3 , HCHO , and H2O2 (, and references therein; ), form solid solutions in ice. Thus, solid-state diffusion is able to either bury these molecules in the inner part of snow crystals or, conversely, make these molecules available for (photo)chemical reactions at the surface after migration from the bulk crystal.

Another physical process, known as co-condensation, is the simultaneous condensation of water vapour and trace gases at the air–ice interface. Water vapour fluxes in the snowpack are mainly driven by temperature gradients, leading to massive mass transfer from the warmest snow layers, which sublimate, towards the coldest parts, where vapour condenses . More generally, the subsequent change in snow morphology, called temperature gradient metamorphism, affects the whole snowpack following seasonal temperature variations , and particularly the upper part of the snowpack subjected to the diurnal temperature cycle (, and references therein). Indeed, high crystal growth rates are observed at the surface of the snowpack, and up to 10 cm under the snow surface their Fig. 8 though the exact depth is subject to debate . Along with the vapour flux, trace impurities present in the interstitial air, or temporarily adsorbed on the ice surface, might be incorporated in the crystals . This kinetic process of incorporation is much more efficient than air–ice thermodynamic equilibrium, which probably explains why measured concentrations have sometimes been shown to be of out of equilibrium .

As regards the snow composition, nitrate sinks are either the photolysis or physical release processes (desorption, sublimation), sometimes referred to as volatilisation or evaporation. An early study by concluded that the nitrate photolysis is the major loss process. A recent work from confirmed that the denitrification of the snowpack by means of physical release is negligible compared to the photochemical loss process. Thus, as regards the air composition above the snow, the nitrate photolysis occurring in the snow is the main source of NOx. The models of snow chemistry developed so far mainly intend to reproduce field measurements of NOx fluxes emitted by the snowpack. Thus, they focus on snow-to-air exchange processes driven by (photo)chemistry. However, air-to-snow physical exchange processes have been ignored in several studies . In other models, these physical processes were bypassed through ad hoc parameterisation and/or implemented using air–liquid equilibrium following Henry's law, based on the assumption that snow crystals are covered by a liquid layer .

These modelling approaches and their pitfalls were discussed in detail by . One of the problems of these models is that ignoring or using inappropriate parameterisations for air-to-snow uptake processes implies that the snow behaves mostly as an initial reservoir of chemical species but does not replenish properly. This implicit assumption can be correct when focusing on the fluxes emitted by the snowpack over a short period of time but is unable to accurately describe the evolution of the snow composition . The most striking example to illustrate the importance of air-to-snow uptake processes is revealed by the yearly pattern of nitrate concentration in surface snow (see Fig. a). It is now well documented that the nitrate concentration in the surface snow exhibits a seasonal peak during summer on the Antarctic Plateau, when the solar flux is close to its annual maximum and photolysis is strongest (, and references therein). This implies that uptake processes counteract photochemical loss and thus need to be studied in order to understand the nitrate budget of the snow. Additional evidence that snow composition is strongly linked to physical processes is shown by a recent study by . Measurements of gaseous HNO3 were carried out with a high temporal resolution of 10 min, during 4 winter months at Halley station, located on the Antarctic coast. This work reveals that HNO3 concentration is strongly correlated (R2=0.70) with the temperature, emphasising that physical air–snow exchange processes play a key role during this period of the year.

As far as we are aware, the only physically based modelling studies of air–snow exchange processes were carried out in the late 1990s to interpret multi-year firn concentration profiles of H2O2 and HCHO . Both of these series of modelling studies dealt with air–snow uptake/release through an exchange coefficient accounting for a Henry's law type partitioning between the two compartments their Fig. 1. More recently, proposed an air–snow exchange model to reproduce surface snow HCHO concentration. In that study, the surface snow is depicted as a unique spherical, layered grain whose surface concentration of HCHO is constrained by the air–ice thermodynamic equilibrium. Their model uses the measured gas-phase HCHO concentration as input and solves the spherical diffusion equation with radial symmetry to calculate the mean concentration in the whole snow grain. Their results reproduce the concentration measured in surface snow during a 36 h intensive sampling period in the course of the OASIS 2009 campaign with fairly good agreement their Fig. 4.

For the first time, we propose a process-resolving model for air–snow exchange of nitric acid (HNO3), which allows for an investigation of the above-mentioned physicochemical exchange processes. An in-depth investigation of the co-condensation process leads to the development of a physically based parameterisation of this process. Following a similar approach to that of , we consider a single spherical layered snow grain located in the uppermost ∼ 4 mm of the snowpack (“skin layer” hereinafter). This snow grain is assumed to be in direct contact with the air just above the snowpack, because the air in the skin layer pore-space rapidly equilibrates with the atmosphere. Using the atmospheric nitrate concentration measured at Dome C (DC) for about 1 year as input, the model calculates the snow nitrate concentration resulting from (i) adsorption on the snow grain surface, (ii) solubilisation into the outermost layer according to thermodynamic equilibrium and solid-state diffusion inside the snow grain, and (iii) co-condensation following vapour fluxes inside the upper snowpack. Model results are compared to year-round measurements of the skin layer nitrate concentration.

Based on the evidence that the photolysis sink is weaker than uptake processes (see Fig. a), we did not implement the photolysis process in our model. An estimation of the uptake flux of nitrate inferred from the developed parameterisation allows a comparison with photolysis loss flux. This analysis confirms that the photolysis is negligible in the skin layer due to the very strong temperature gradient driving an intense condensation flux.

The input datasets are presented in the next section, and the model is described in Sect. . The results obtained in configuration 1 (adsorption only) are presented and discussed in Sect. , and those relative to the model configuration 2 (solid-state diffusion) are presented in Sect. .

A brief summary of the current knowledge about solvation steps which lead gaseous HNO3 to form solid solution in bulk ice is presented in this section.

The uptake of trace gases on ice, and more specifically of acidic gases among which HNO3, has been the subject of numerous investigations see reviews by. Conceptually, this uptake proceeds firstly by molecular adsorption of HNO3, followed by the ionisation (or dissociation) and then progressive solvation at the surface leading to a partial solvation shell . In a second stage, thought to be much slower, the adsorbed nitrate anions sink into the innermost crystal layers, leading to a complete solvation shell, and diffuse towards the bulk crystal. Recent studies have addressed the ionisation state of HNO3 adsorbed on ice surface, either by using surface sensitive spectroscopy techniques or through molecular dynamics models . Molecular adsorbed state is found to be metastable, which happens only at very low temperatures (45 K), whilst ionic dissociation irreversibly occurs upon heating at 120 K . Molecular dynamics simulations suggest a pico- and subpicosecond ionisation of HNO3 in the defect sites , further supporting the idea that molecular adsorption of HNO3 on ice is a fleeting state prior to ionisation, at least for environmentally relevant temperatures.

Despite these recent improvements in the understanding of HNO3 ionisation following adsorption on an ice surface, the transition between surface (adsorption) and bulk (diffusion) processes still needs to be fully characterised. To the best of our knowledge, no process-scale parameterisation of the dissociation/solvation exists at the moment. Such parameterisation would be necessary to link surface and bulk concentrations, and further studies are thus needed to fully characterise the transition between these states. For this reason, both processes were treated separately in our model. Model configuration 1 (adsorption) is described in the next section, while configuration 2 (solid-state diffusion) is described in Sect. .

The HNO3 surface coverage is a function of temperature and pressure only. presented a compilation of data evaluated by a IUPAC subcommittee that characterises heterogeneous processes on the surface of solid particles, including ice. They recommend the use of a single-site Langmuir isotherm which gives the fractional surface coverage θ: θ=NNmax=KLangPPHNO31+KLangPPHNO3, where Nmax=2.7×1018 molecules m-2 is the HNO3 surface coverage at saturation, KLangP=KLinCNANmaxRT(inPa-1),KLinC=7.5×10-7exp⁡4585T (in m). KLangP and KLinC are partition coefficients expressed in different units, N is the HNO3 surface coverage (in molecules m-2), PHNO3 is the HNO3 partial pressure (in Pa), NA is the Avogadro constant, T is snow temperature (in K), and R is the molar gas constant (R=8.314 J K-1 mol-1).

This parameterisation is established for temperatures ranging from 214 to 240 K, which is almost adequate for DC temperatures, typically in the 200–250 K range (see Fig. b). The conversion of surface coverage to bulk concentration is done using SSA: HNO3=N×SSANA, where HNO3 is the nitrate concentration (in molm-3).

The results and discussion following adsorption calculation are presented in Sect. .

In configuration 2, the model computes solid-state diffusion in a layered snow grain. The outermost layer concentration or boundary condition (BC) is successively set according to three distinct parameterisations. Firstly, the NO3- concentration at the air–ice interface is set according to thermodynamic equilibrium (BC1). In a second stage, the kinetic, co-condensation process is taken into account through an empirical, diagnostic parameterisation (BC2). Then, using the results from the previous BCs, a physically based prognostic parameterisation is developed (BC3). The general diffusion scheme and specific BCs are presented in the next sections.

The evolution of the concentration of nitrate in the snow skin layer is plotted in Fig. a. Undeniably, the adsorbed concentration modelled using non-dissociative Langmuir isotherms parameterisation does not fit with the measured concentration in three ways: firstly, the modelled concentration is higher than the measured ones during most of the year. From February to August, the average modelled concentration is 2.5-fold higher than the measured one, and this ratio increases to 8.3 from September to mid-November (see vertical separations in Fig. a). However, the modelled concentration gradually decreases towards the end of January, while the measured one reaches a seasonal maximum, leading to a ratio of 0.62 between modelled and measured concentrations during this last period. Secondly, the modelled concentration shows a strong diurnal variability following temperature, with a ratio between daily maximum and minimum concentration regularly higher than 5, and with a yearly average equal to 2.6. By contrast, field measurements show weak diurnal variations in nitrate concentration in surface snow and no anticorrelation with temperature (Fig. S2 in the Supplement). The third major discrepancy is a premature seasonal maximum in the computation, starting late August and reaching maximum early November, while concentration measured in snow lags by 65 days.

The features of the modelled concentration attributed to adsorbed nitrate can be explained by the temperature and partial pressure dependencies of the adsorption isotherm. The surface coverage parameterisation strongly decreases with temperature (exponential function of the reciprocal temperature in Eq. ), whilst it increases roughly linearly with the HNO3 partial pressure when the surface coverage is well below saturation. This explains the strong diurnal variations following the temperature cycle. It also explains the yearly pattern of the modelled concentration: firstly, during the winter, very low temperature prevails over the low HNO3 partial pressures, leading to modelled concentration much higher than that measured. The influence of temperature is easily seen in April, May, and August, when temperature is the lowest (see Fig. b), leading to higher modelled concentration than in June and July, when temperature is higher and HNO3 partial pressure is alike. Then, from early September to early November, HNO3 partial pressure increases while temperature shows only a moderate increase, leading to the modelled peak of absorbed nitrate. Finally, nitrate partial pressure stays high until January, but this is counterbalanced by the temperature, which increases to its yearly maximum, forcing modelled surface coverage to fall well under the measured values.

Despite the use of the current IUPAC recommendation for the parameterisation of HNO3 adsorption on ice, the modelled quantities adsorbed on snow are clearly incompatible with the measured concentration. In order to explain this discrepancy, we compared the experimental setups used in the various studies of adsorption . A review of these studies, and of the experimental techniques used, can be found in . In brief, two main experimental techniques prevail: flow tubes, which were used in most studies , and Knudsen cells, which were used in two studies . Whatever the technique used, ice was deposited on the reactor walls either by water vapour condensation or by fast freezing of an ice film .

A first pitfall which may arise from these studies comes from the lack of quantification of the exposed surface area of ice, which was measured only once by . They carried out several experiments at 209, 213, and 220 K and found that the exposed surface was twice the geometrical surface. found that this ratio can be as high as ∼ 9 in the case of ice formed by water vapour deposition at 196 K. These authors also reported that this ratio increases with the amount of water deposited, as well as with decreasing temperature. On the other hand, in another study using ice formed by fast freezing of a film of water, concluded that the ice surface was smooth at a molecular level, implying a ratio near 1. However, except in the study by , an underestimation of the exposed surface, which leads to an overestimation of the surface coverage of ice, cannot be ruled out.

All adsorption studies assumed that diffusion in bulk ice is negligible at very low temperature. However, even if the fraction of HNO3 entering the bulk ice is small, neglecting it leads to a systematic overestimation of the surface coverage. analysed the data in to include the diffusion process. Their study brought new insight into surface vs. bulk processes, and their model performed well in reproducing adsorption curves when diffusion into the bulk was also taken into account. However, instead of using the existing parameterisation for nitrate solubility and diffusion coefficient in the ice (see Sect.  and ), they made use of a simplified scheme to consider the diffusion process, which includes an adjustable rate coefficient for diffusion and hinders a close comparison with our parameterisation. Furthermore, the desorption curves could not be well fitted by their model, especially for low surface coverage, indicating that the processes involved are still not fully understood and constrained.

The diffusion of nitrate into bulk ice could also have been further enhanced for three distinct reasons. Firstly, it is worth noting that if the exposed surface area of ice is larger than the geometric surface, this leads to a larger exchange interface, thus increasing the amount of HNO3 diffusing to bulk ice in the total uptake. On the other hand, even if the ice covering the reactor's walls was smooth in the case of a frozen liquid film, the fast-freezing process would very likely lead to a highly polycrystalline structure, where grain boundaries may act as shortcuts for the diffusion, thus enhancing bulk uptake. Lastly, several authors have pointed out that despite the careful attention to ensure that ice surface was in equilibrium with its vapour, part of the observed uptake could be ascribed to bulk incorporation of HNO3 with condensing water if the exposed ice was slightly growing because of slight supersaturation or due to the highly dynamic air–ice interface .

More generally, the question of the adsorbed state, closely linked to the ionisation process and to the reversibility of the adsorption, can also explain the mismatch between the current parameterisation and measurements. In all the uptake experiments, it was observed that the total uptake splits between reversible and irreversible components, the former being only a minor part of the total. For instance, reported that on average 20 % of the initial uptake was desorbing. Should a part of this irreversible uptake already account for a strongly bound, bulk uptake, this could explain a major part of the overestimation of the modelled absorbed concentration. New investigations are needed to gain a clearer view of the partitioning between surface and bulk.

Finally, several other uncertainties can be invoked to explain the discrepancies. The saturated surface coverages reported in the various studies range over almost 1 order of magnitude, from 1.2×1014 moleccm2 to 1.0×1015 moleccm2 . This uncertainty directly impacts the modelled surface coverage (Eq. ). Secondly, most adsorption studies have used HNO3 partial pressure between 2 and 3 orders of magnitude higher than the one relevant at DC. improved this by using partial pressures down to ∼9×10-7 Pa; however, this remains ∼25 times higher than the lowest partial pressures measured in winter at DC (∼3.5×10-8 Pa). Using their parameterisation in DC conditions thus implies a great extrapolation. The lack of data for very low partial pressures is another potential uncertainty over the relevant type of adsorption isotherms, as the behaviour in the unsaturated region (i.e. at low partial pressure) provides more constraint over the best type of adsorption isotherms than that in (or near) the saturated region. This explains why several kinds of isotherms (dissociative or non-dissociative Langmuir isotherm , Frenkel–Halsey–Hill isotherm ) have been proposed but no clear consensus has been achieved.

In order to test these different explanations, experimental setups should systematically include measurements of the exposed area of ice and use partial pressures as low as possible. Processing the raw experimental data with the approach developed by seems a promising way to discriminate between surface and bulk uptake processes. Improvements in this approach could probably be achieved by using state-of-the-art parameterisation of the diffusion process.

Regarding the present study uncertainties, snow temperature, snow SSA, and HNO3 partial pressure are the three variables controlling the adsorbed surface coverage. HNO3 partial pressure, assumed to be equal to the total atmospheric nitrate (see Sect. ), is thus the upper limit. However, as presented in the data description (see Sect. ), this assumption likely leads to an overestimation not larger than 20 % on average, which cannot explain the overestimation of the modelled concentration by a factor of 2.5–8.3. Conversely, the warm bias of modelled temperatures (see Sect.  and Supplement Sect. S1) leads to smaller modelled adsorption concentration, and the slightly reduced diurnal amplitude tends to reduce this other discrepancy between modelled and measured concentration. Lastly, the SSA was kept constant during the whole simulation, but a recent study by indicated that the SSA value adopted in our model is comparable to summer observations but 2–3 times lower than the winter SSA observations (see Sect. ). At that time of the year, the modelled adsorbed concentration is already highly overestimated, thus accounting for a higher SSA would increase the discrepancy.

To conclude this section, several reasons were invoked to explain the overestimation of the modelled adsorbed concentration. In order to estimate the actual fraction of adsorbed nitrate over total snow nitrate, we make the rough hypothesis that the current adsorption parameterisation is flawed by a constant overestimation factor. Decreasing the modelled adsorbed concentration by a constant factor of ∼ 20 so that its envelope never exceeds measured concentration leads to small adsorbed concentration during most of the year except in early spring, i.e. in the September–early November peak period (see Fig. b). In this situation, we estimate that adsorbed nitrate accounts for less than 13 % of snow nitrate on yearly average (less than 9 % when excluding the early September to early November period, and almost 30 % during these 2 months). We thus decided thereafter to put aside the adsorption process, which should only lead to a minor error, except during spring. One way to test this hypothesis is to carry out hourly measurements of nitrate concentration in surface snow during spring. Owing to the strong temperature dependency of the adsorption isotherm, if adsorbed nitrate accounts for an important fraction of snow nitrate, then significant daily variations in snow nitrate concentration should be observed.

The first attempt to model nitrate concentration in the skin layer was done using solely the thermodynamic equilibrium concentration (see Sect.  and Eq. ) to constrain the concentration of the external layer of the snow grain (BC1). The resulting concentration is plotted in Fig.  along with the measured concentration. The initial value of ∼ 500 ng g-1 and the sharp decrease at the beginning of the series (30 January 2009–7 February 2009) are due to the initialisation of the whole grain concentration to the closest measurement (point not shown, a few hours before the start of the simulation) and should not be interpreted. This spin-up duration shows that the time needed to re-equilibrate the snow grain concentration, roughly 2 weeks, compares well with the characteristic diffusion time (see Sect. ).

From mid-April to late October, the modelled concentration is in reasonable agreement with the measured concentration, with some features appearing to be reproduced by the model (a slight, steady increase lasting from July to August, followed by a trough and then a second slight increase from September to mid-October). During this winter period, the modelled concentration appears to be often slightly lower than the measurements; this point will be further discussed in the sensitivity study presented in Sect. . The modelled concentration also features smoother variations than the measurements, which can be mainly explained by the coarse time resolution of HNO3 partial pressure used as input, of roughly one week (see Sect.  and Fig. a). The good consistency between modelled and measured concentrations during winter months is an important result, as this indicates that winter concentration of nitrate in surface snow is mainly driven by the thermodynamic equilibrium solubility, coupled to solid-state diffusion.

On the other hand, this first modelling attempt clearly fails to reproduce the summer peak of nitrate concentration in snow, with values in the 50–200 ng g-1 range from November to early April, while measured concentration peaks above 1400 ng g-1. These results also show that summer concentration of nitrate in surface snow is highly enriched compared to what is expected from the thermodynamic equilibrium. These results demonstrate that another uptake process, driven by kinetics rather than thermodynamics, is needed to explain such high summer concentration.

The BC2 includes the kinetic co-condensation process, through the empirical diagnostic parameterisation presented in Sect. .

We adjusted the three coefficients in Eq. () in order to minimise the RMSE between modelled and measured snow nitrate concentration. The optimal result, plotted in Fig. , was obtained with XHNO3=XHNO30+α×PHNO30.43×PH2O1.27. The α parameter value was adjusted so that the amplitude of the modelled summer peak fit the data, but it has no physical signification. However, the most relevant point to note is that the modelled peak is well in phase with the measurements (as a main difference with the adsorption), and both time series display similar features. Furthermore, it is worth noting that including the co-condensation has not degraded the winter prediction. Indeed, because of the very low winter temperature at DC, and given the exponential dependency of water vapour pressure over temperature, the co-condensation term becomes almost negligible .

The optimum exponent for HNO3 partial pressure is 0.43, which exactly corresponds to the exponent for HNO3 partial pressure of thermodynamic equilibrium concentration (in Eq. , 1/2.3≃0.43). Even if that needs to be confirmed by further investigations, this result tends to confirm the hypothesis formulated by that thermodynamic partitioning plays a role in the co-condensation process (see Sect. ).

Because of the correct timing and shape of the modelled peak of nitrate, these results suggest that the co-condensation process is responsible for the out-of-equilibrium, high concentration of nitrate in the skin layer in summer. Among the two available laws giving Xkin (i.e. the concentration of the growing phase; see Sect. , Eq.  or ), the empirical one, whose dependency over the HNO3 partial pressure is the closest to 0.43, seems the more suited to reproduce the observations.

The last part of this work aimed at refining the parameterisation for the co-condensation process, using physically based variables. The prognostic parameterisation developed hereafter is referred to as BC3. For the sake of simplicity, and because the growth of snow grain is very slow compared to the recycling of vapour as suggested by , a constant radius (R) is assumed. However, the growth law defined in Eq. () is used in order to evaluate the equivalent radius increase ΔR resulting from the co-condensation process during the model time step Δt (Eq. ). Finally, the concentration resulting from concomitant thermodynamic process (diffusion equilibration) and kinetic process (co-condensation process) is calculated using the theoretical Eq. () at depth ΔR, which is at the surface of the modelled snow grain whose radius is supposed to be constant.

The radius growth rate ΔR/Δt as derived from Eq. () is presented in Fig. . It spans roughly 3 orders of magnitude over the year, from about 10-12 ms-1 in winter to ∼8×10-10 ms-1 in summer. The explanation of this behaviour is twofold. First, the diurnal temperature cycle has a larger amplitude in summer, which enhances the temperature gradient close to the surface. Second, the vapour pressure over ice increases exponentially with temperature. As a consequence, with a given value of the temperature gradient, the gradient of water vapour concentration used in Eq. () is larger if temperatures are higher. This also explains the diurnal variation in the grain radius growth. The most striking feature of the radius growth rate is that it peaks during the same period of the year that the peak of nitrate concentration in the skin layer. The yearly pattern of the radius growth rate predicted by our model is also consistent with independent studies focused on snow physical properties . This comes as additional evidence that snow metamorphism and co-condensation have a major influence on the snow chemical concentration.

The resulting modelled nitrate concentration in surface snow is presented in Fig. . In Table , a summary of the model runs, along with their RMSE, is presented. Simulation results are similar to those obtained with the BC2 parameterisation, but with a slightly improved RMSE. A diurnal variation in the modelled concentration is observed, as a consequence of the diurnal variation in the radius growth rate. However, the diurnal variation in the concentration is much smoother because solid-state diffusion in the whole snow grain softens the large diurnal variations in the outermost layer of the snow grain. The relative diurnal variation in the concentration is always smaller than 20 % and thus cannot be distinguished from the measurements uncertainties due to spatial heterogeneity. In this physically based parameterisation, a slight dependency of the results to the model time step arises. This is explained by the radius increase ΔR which depends on the cubic root of the time (Eq. ), and which is divided by the square root of the time in Eq. (). To compensate for this unphysical dependency, either the time step of the model needs to be adjusted for optimum results or a correction factor can be used in order to keep the time step unchanged, with a value well suited regarding the diffusion process. The exact reason of this dependency over the time step is complex to establish, but it can very likely be ascribed to the hypothesised geometry of the snow grain (a sphere) and of the condensed phase (a layer). Improving this point necessitates determination of the relationship between mean thickness of the co-condensed layer as a function of time, which is left to further work.

In Fig. , the modelled concentration shows a poorer fit with the measured concentration during spring, just before the observed peak of snow nitrate. This is confirmed by a lower correlation from September to November (Table S1), which corresponds to the period where the modelled adsorption peaks (see Fig. ). This is another indication that adsorbed nitrate may account for a noticeable part of surface snow nitrate in early spring.

As stated in the introduction, the photolysis has not been included in this study since the dramatic increase in summer nitrate concentration in the skin layer demonstrate that uptake processes overtake loss processes in this specific layer. In order to refine this comparison regarding the budget of nitrate in the skin layer, an estimation of the uptake and loss fluxes is presented here. Both calculations are based on the following assumptions: a skin layer thickness of 3 mm, with a snow density of 0.3 kg m-3. The fluxes are calculated for an area of 1 cm2.

The photolysis flux is calculated for a single nitrate concentration of 1200 ng g-1, which results in 9.7×1014 molecules in the 1 cm2×3 mm volume. reported a photolysis rate for nitrate of about 1×10-7 s-1 in Dome C surface snow, for a solar zenith angle (SZA) of 52∘, which is the maximum solar elevation at Dome C. The resulting photolytic loss flux is 9.7×107 molecules cm-2 s-1.

The uptake flux resulting from the co-condensation process is calculated by assuming that the 1 cm2×3 mm volume is filled with ice spheres of radius R=85 µm (see Sect. ) up to the prescribed density. This results in ∼ 37 200 spheres. In the theoretical study by , the average concentration in the condensed layer immediately before another layer condenses and isolates the previous one is given by the integral of Eq. () over the condensed thickness ΔR: Xaverage=Xkin+Xeq-XkinΔR∫0ΔRerfcx2Dtdx. Using the same input data as in the model, and assuming that this average concentration multiplied by the condensed volume corresponds to the quantity of nitrate actually taken up by the snow, we calculate an average uptake flux of 5.4×109 molecules cm-2 s-1 over the December 2009 to January 2010 period. The minimum and maximum values are 1.6×108 and 2.7×1010 molecules cm-2 s-1, respectively. Strong negative gradients have been reported above snow surface (see, for instance, the measurements by (their Fig. 3) at South Pole), but only one HNO3 flux measurement was found in the literature . This work was carried out in the Arctic, and due to the numerous differences between both locations (type of snowpack, temperature, and temperature gradient), a close comparison is not possible. reported an average value of 1.2×109 molecules cm-2 s-1 (interquartile range: 6.3×108–2.4×109 molecules cm-2 s-1). The uptake flux ascribed to the co-condensation has the same order of magnitude as this measured flux, which seems promising. However, HNO3 flux measurements should be carried out in Dome C in order to allow a realistic comparison.

As a conclusion, the uptake flux due to the co-condensation appears to be ∼ 56 times larger, on average, than the photolysis loss flux calculated for the highest solar elevation conditions. This confirms that photolysis loss can be neglected when studying the nitrate concentration in the skin layer. Given the numerous assumptions made in the model, the overall reproduction of the measurements by the parameterisation including co-condensation appears satisfactory.

In order to further investigate the modelling uncertainties, the sensitivity of the model to the thermodynamic equilibrium concentration, diffusion coefficient and SSA value is evaluated. A synthesis of RMSE values of the sensitivity runs is presented in Table .

As shown in Sect. , winter modelled concentration underestimates the measurements, which could be explained by an underestimated thermodynamic equilibrium solubility (Eq. ). The best fit with the data is obtained for an increase of 39 % (see Table ). This optimum increase is almost twice as much as the uncertainty reported by 20 %; however, we applied the solubility parameterisation at much lower temperature than in their study, which could explain the results.

A few measurements of the ratio of HNO3 over atmospheric nitrate presented in Sect.  suggest that HNO3 might account for roughly 70–90 % of atmospheric nitrate. Taking this ratio into account would reduce the HNO3 partial pressure used as input in the model but might be counterbalanced by a further increase in the thermodynamic solubility. New studies are needed to confirm the speciation of atmospheric nitrate and its seasonal variation. On the other hand, the current underestimation of the modelled concentration during winter can also be partly ascribed to a small adsorbed fraction amongst the total snow nitrate.

Secondly, using a diffusion coefficient lower than that suggested by their Eq.  generally improves the simulation performance. Using BC3 simulation as a reference, decreasing the diffusion coefficient by 72 % leads to the best reproduction of the results (see Table ). When the solubility value increased by 39 % is used, the diffusion coefficient is decreased by 64 %. reported a 60 % uncertainty for the diffusion coefficient and indicated that their parameterisation likely represents the upper bounds, which compares well with the present sensitivity analysis.

However, another explanation is possible: a decrease in SSA linked to an increased radius (Eq. ) has a similar effect to a decrease in the diffusion coefficient. Decreasing SSA to 23 m2kg-1 leads to almost the same result as reducing the diffusion coefficient by 64 % (see Table ). In the current version of the model, the radius of the snow grain is kept constant over time as a simple hypothesis, but it has been shown by and that snow grain size features a sharp increase at DC during December and January, when the modelled water vapour fluxes driving the co-condensation process are highest. It is remarkable that the optimum value of 23 m2kg-1 is in very good agreement with that observed in summer their Fig. 1. Future development of the current work should consider grain size change to distinguish between these two alternative hypotheses.