We describe and validate a Monte Carlo model to track
photons over the full range of solar wavelengths as they travel into
optically thick Antarctic blue ice. The model considers both reflection and
transmission of radiation at the surface of blue ice, scattering by air
bubbles within it, and spectral absorption due to the ice. The ice surface is treated as planar whilst bubbles are considered to be spherical scattering
centres using the Henyey–Greenstein approximation. Using bubble radii and
number concentrations that are representative of Antarctic blue ice, we
calculate spectral albedos and spectrally integrated downwelling and
upwelling radiative fluxes as functions of depth and find that, relative to
the incident irradiance, there is a marked subsurface enhancement in the
downwelling flux and accordingly also in the mean irradiance. This is due to the interaction between the refractive air–ice interface and the scattering interior and is particularly notable at blue and UV wavelengths which correspond to the minimum of the absorption spectrum of ice. In contrast the absorption path length at IR wavelengths is short and consequently the attenuation is more complex than can be described by a simple Lambert–Beer style exponential decay law – instead we present a triple-exponential fit to the net irradiance against depth. We find that there is a moderate dependence on the solar zenith angle and surface conditions such as altitude and cloud optical depth. Representative broadband albedos for blue ice are calculated in the range from 0.585 to 0.621. For macroscopic absorbing inclusions we observe both geometry- and size-dependent self-shadowing that reduces the
fractional irradiance incident on an inclusion's surface. Despite this, the
inclusions act as local photon sinks and are subject to fluxes that are
several times the magnitude of the single-scattering contribution. Such
enhancement may have consequences for the energy budget in regions of the
cryosphere where particulates are present near the surface. These results
also have particular relevance to measurements of the internal radiation
field: account must be taken of both self-shadowing and the optical effect
of introducing the detector. Turning to the particular example of englacial
meteorites, our modelling predicts iron meteorites to reside at much reduced
depths than previously suggested in the literature (

Incident solar radiation varies over a range of timescales due to the predictable seasonal and daily motion of the Sun–Earth system, supra-daily stochastic influences of the changing atmosphere and longer-term climate effects. It is a key driver of the cryosphere's energy budget (van den Broeke et al., 2011; Van Tricht et al., 2015; Hofer et al., 2017), and its variability strongly affects the internal temperature profile of ice, particularly close to the surface (Liston et al., 1999). This has implications for processes such as ice sheet near-surface melting and ice shelf crack formation (Bennartz et al., 2013; van den Broeke et al., 2016; Webster et al., 2017). In addition to driving physical processes, the quantity and spectral composition of solar radiation transmitted through ice, or available within it, can have significant effects on aquatic and other polar ecosystems.

The present study was specifically initiated in response to the need for a better understanding of this glacial subsurface radiative field. Recently Evatt et al. (2016) presented a mathematical model for the vertical movement of meteorites through blue ice in Antarctica. In this work the attenuation of solar radiation through ice and the absorption of solar radiation by a meteorite were modelled using the Lambert–Beer law. Although this approach works well at certain depths, it is limited in its accuracy, particularly near the surface. This issue specifically motivated us to find a more accurate, yet still simple and easily applicable, methodology for modelling the attenuation and absorption of solar radiation through blue ice.

Despite this specific motivation, there are obvious parallels that can be drawn with several different climatologically important research foci. Examples include the impact of anthropogenic soot, pollutants, cryoconites, and other englacial absorbers near the surface of the Greenland Ice Sheet (e.g. Box et al., 2012; Dumont et al., 2014; Stibal et al., 2017). In all of these cases, including that of englacial meteorites, the inclusions have a low albedo and are subject to an atmospherically modulated near-surface radiation field, cause local heating, and thus can contribute to increased melt rates.

Notwithstanding the importance of shortwave radiative transfer in the aforementioned studies, there is a tendency, as in Evatt et al. (2016), to treat the shortwave radiative flux as a single broadband parameter (via the Lambert–Beer law) and neglect to incorporate the range of behaviours exhibited by different wavelengths of solar radiation. This is a fundamental simplification as the incident solar spectrum exhibits a great deal of structure due to terrestrial and solar processes, whilst the absorption spectrum of ice spans 8 orders of magnitude across the solar wavelength range (Brandt and Warren, 1993; Warren and Brandt, 2008). For some applications such a simplification is reasonable; however for others – such as those concerned with the near-surface heat budget – a more encompassing model will be beneficial.

It is also instructive to briefly compare, in order of increasing density, the optical characteristics of Antarctic blue ice to other cryospheric forms of water. Freshly fallen snow is loosely packed, and because it is formed of many irregular voids it is often characterised in terms of its grain size. As suggested by its high albedo, it is highly scattering. Thus even physically thin snowfalls are usually optically thick. As it graduates to firn, that is, partially recrystallised older snow that has generally survived a melt season, it becomes denser and contains fewer voids. Like snow, firn is usually optically thick, but has a longer scattering path. Sea and lake ice exhibit scattering centres that are comprised of bubbles; both often exhibit a layered structure and are additionally underlain by low scattering parent bodies of water, though sea ice contains brine pockets affecting its optical properties. In contrast blue ice is characterised by its colour and is formed through long-term compression of snow. It has a high density, is physically and optically thick, and exhibits a longer scattering path than the other forms due to relatively few scattering centres (bubbles). Accordingly its bulk optical properties are the result of both the intrinsic absorption by ice and the residual scattering by remnant bubbles.

There are three key studies that have investigated radiative transfer within various types of ice in detail and gave due attention to the aforementioned spectral issues. Mullen and Warren (1988) developed a radiative transfer model of lake ice to illustrate the processes responsible for the resulting albedo and transmission through a layer of ice. They treated bubbles as spheres, deriving the scattering coefficient and asymmetry parameter from Mie calculations, and they relied on the delta-Eddington method in their treatment of multiple scattering. They showed results across the solar waveband for the direct beam and diffuse incidence as well as the transmission for different bubble concentrations. Later, Light et al. (2003) developed a Monte Carlo model for radiative transfer in sea ice. Their focus was on cylindrical samples of ice, with the model being used to interpret backscattering from cylindrical core samples. In both these studies the emphasis is on relatively optically thin samples where the ice overlays a body of water, or where the parameter of interest is the transmission. In cases where ice overlays water there is very little change in the refractive index at the lower ice–water interface: this effectively removes any lower refractive boundary and permits downwelling photons to continue their trajectory into the water with reduced opportunities for further scattering. Therefore an assessment of the transmission and reflectance of the incident sunlight is considered an adequate summary of the interaction in these cases.

The third key study was presented by Liston et al. (1999) and relates most closely to that described here as it detailed the ice melt in blue ice vs. deep snow areas. The authors' focus, however, was on understanding the resulting subsurface temperature and melting profiles, and they relied on a two-stream approach constrained by specific atmospheric forcings and measured surface albedos. Account was taken of the spectral nature of the problem; however, to maintain consistency between the treatment of snow and blue ice, the optical properties were linked to effective grain size and a spectral extinction coefficient was calculated on this basis. We also note related studies that investigated the spectral albedo of white sea ice or snow, but not the internal radiative field (Gardner and Sharp, 2010; Ehn et al., 2011; Haussener et al., 2012; Malinka et al., 2016; Taskjelle et al., 2017), or considered internal scattering from a purely theoretical perspective (Malinka, 2014).

However to our knowledge there has been no study for optically thick blue ice where the spectral radiative transfer and albedos are derived from Monte Carlo modelling of solar radiation interacting with embedded bubbles and the underlying material properties. In the present study we therefore take this approach to address two core aims. The first is to present an in-depth investigation of the radiation field within optically thick bubbled ice at different solar wavelengths, including a range of sensitivity tests and its impact on inclusions. This then leads us to the second aim: a distillation of these results into a simple, and widely applicable, mathematical model for the net flux.

In Sect. 2 we describe the details of our Monte Carlo radiative transfer model, including its initialisation, the conditions at the boundaries, the scattering of photons by bubbles, and the eventual absorption of photons by ice. The validation of the model is also discussed. In Appendices A and B we describe our methodology for the related calculations of the incident solar spectrum and the derivation of the relevant bubble parameters. In Sect. 3 the model is used to investigate how the incident solar spectrum propagates through ice, both when considered spectrally and when integrated across solar wavelengths. We investigate the influence of varying the bubble number concentration and effective radius, the effect of varying the solar zenith angle, and the influence of different surface environments which might alter the spectral balance of the incident solar spectrum. In Sect. 4 a macroscopic inclusion (absorbing target) is added to the model in order to study how the target's geometry and size impact on the effective radiation field incident on its surface. In Sect. 5, curves are fit to the integral shortwave results for the net radiative flux in a typical blue ice area. Lastly, in Sect. 6 we discuss how these results relate to the specific application of modelling the dynamics of meteorites in Antarctic blue ice.

Whilst different approaches to utilising the general radiative transfer equation exist (for background see Thomas and Stamnes, 1999), here we choose to use an unpolarised Monte Carlo simulation approach to investigate radiative transfer within bubbled ice. We do so for its ability to represent the different physical aspects, its more intuitive nature, and its capability to study inclusions in a non-plane-parallel fashion.

Schematic illustrating model geometry (not to scale).

The core Monte Carlo model aims to calculate the downwelling and upwelling
solar irradiance fluxes,

There are three principal sets of inputs to the Monte Carlo model: the incident solar spectrum, the intrinsic optical properties of bubbled ice, and the geometric characteristics of the bubbles. These are detailed in turn below.

The incident spectral irradiance at the ice surface is calculated using the
libRadtran radiative transfer model (Mayer and Kylling, 2005) with relevant
atmospheric inputs for clouds, aerosols, and solar zenith angle. These, and
the surface altitude and broadband albedo, are initially chosen to be
appropriate for a blue ice area near the Frontier Mountain range, Antarctica
(72.95

The intrinsic optical properties of the ice are fully defined by the
wavelength of the photon being tracked (

Turning to the geometric characteristics of blue ice bubbles, there is a paucity of relevant in situ data, and thus we principally rely on a homogenisation of bubble density and radii concentration measurements carried out by Dadic et al. (2011, 2013) near the Transantarctic Mountains, Antarctica. In summary, the samples collected by Dadic et al. (2011, 2013) cover a range of ice conditions along two blue ice area (BIA) transects; cores from depths down to 0.94 m were analysed by a combination of micro-CT analysis, estimates derived with specific surface area (SSA), and caliper-and-scale measurements. We have carried out a homogenisation exercise of these and observations from the wider literature to determine sets of internally consistent parameters linked to bulk properties, including the effective scattering coefficient to retain generality. Those referred to further are shown in Table 1; more details are provided in Appendix B.

Summary of bulk ice bubble data referred to in the text and
calculated albedos, ranked by increasing bulk density. The Bintanja (1999)
(850 kg m

Individual photons are tracked from their incidence on the air–ice interface
(

Specular reflections at the surface are dealt with by calculating the wavelength and angle-dependent reflection coefficient for an unpolarised beam according to Fresnel's equations of reflectance (Hecht, 1987). If a uniform random number in the interval [0, 1] is less than the calculated reflection coefficient, then the photon is marked as being returned to the atmosphere. If it is greater, then the photon is considered to have passed into the ice and the direction vector is updated in line with the angle of refraction.

For photons that have passed into the ice the next stage is to calculate the
distance before a scattering or absorption event occurs. We construct a
cumulative exponential distribution where the mean free path represents the
total extinction of the photon from its straight line path

The updated position is checked at each iteration against the local
boundaries. First the

If the event corresponds to absorption this is flagged and the photon is no
longer tracked. If the interaction is a scattering event, then a new
direction of travel is calculated, defined by a deflection angle and an
azimuthal angle with respect to the original direction of travel. The deflection
angle is calculated using a Henyey–Greenstein phase function (Henyey and
Greenstein, 1941) with the asymmetry parameter

Spectral variation in asymmetry parameter for spherical air
bubbles in ice, calculated from Mie theory (Bohren and Huffman, 1983) for a
bubble radius of 198

As well as recording the final positions of the photons, we track them
throughout their multiply scattered passage through the ice. In order to do
so we record the depth of each photon and whether the direction of its
flight is positive (downwards) or negative (upwards) at every step. This
allows us to determine the downward and upward irradiance fluxes
as fractions of the total number of photons that were initially
released. No cosine weighting of the angle to the surface normal is required
as it is implicitly included in the photon energy (Mayer et al., 2010;
Jacques, 2011). In addition the

To validate the described model and test its predictive skill, we follow the
example of Light et al. (2003), who relied on the four-stream results of
Grenfell (1991). For the outputs of interest to us there are three key
scenarios in common. The first is a conservative non-refractive domain where
we calculate the albedo and transmissivity of a horizontally infinite slab
at a range of optical depths,

Doing so we find the discrepancy between the albedo and transmissivities
calculated with the method described herein and the four-stream solution
presented in Light et al. (2003) are typically

Comparison of albedo for

Internally we also check the reproducibility of repeated runs to ensure a
stable solution has been reached. For an example wavelength of 600 nm the
choice of tracking 10

Whilst the Monte Carlo model produces reproducible results and shows good agreement across the range of optical parameters used by Light et al. (2003, which cover the range of absorption and asymmetry parameters exhibited by blue ice areas), there are some limitations in regards to real-world applications. The first of these is that all bubbles are assumed to be spherical and thus their single-scattering behaviour is governed by Mie theory. This is a useful simplification but in reality individual samples and individual bubbles within them will deviate from perfect spheres. This will affect the asymmetry parameter and consequently the observed attenuation, but to answer the broader question we choose not to further specify the bubble geometry beyond the assumption that it is spherical. Neither do we consider any vertical or spatial inhomogeneity of bubble density that clearly exists – on a small spatial scale blue ice bubbles form at higher densities along cracks, although these cracks do not typically show any preferential orientation. To maintain the general applicability of the results, we therefore consider a continuous medium with spherical bubbles, using typical values that are relatable to bulk ice parameters. We note that vertical variations in the asymmetry parameter, the effective bubble radius, and their density could be dealt with by relatively small adaptations to the code.

The model also assumes that the ice surface is planar. In reality the surfaces of blue ice fields are often scalloped or roughened, which would complicate the modelled environment. We anticipate that such surface roughening would generally lead to a reduction in the incident downwelling radiation on most facets, but the effect on the upwelling irradiance is more difficult to assess – where the length scale of the ice surface geometry is large in comparison to the mean scattering length of photons, locally the surface would appear flat, in line with our planar assumption. A more detailed analysis of this point is therefore left for a future study. We also do not account for any partial covering by a windblown snow layer. On the whole, blue ice layers are largely free of snow, but we estimate that an intermittent snow layer of a depth of 5 cm covering approximately 10 % of the surface would reduce the visible irradiance incident on the upper ice surface by 5 % to 10 % on average (Perovich, 2007). We also anticipate some enhancement of the incident irradiance under cloudy skies due to multiple reflections between clouds and the ground. This mechanism may have not been fully captured by the use of a broadband albedo input to the atmospheric radiative transfer calculation; it is expected to be of a similar magnitude to that caused by the absence of a snow layer, but in an opposing direction.

Our modelling assumes a prescribed description of the incoming solar
irradiance spectrum and estimates of the bubble number concentration and
effective radius. In this study we use a solar irradiance based upon the
Frontier Mountain blue ice area, Antarctica (72.95

Assuming fully diffuse sky conditions (that is, the incident irradiance has no separate direct beam component), we now apply our Monte Carlo model to four air bubble parameter sets as detailed in Table 1.

In all cases the spectral attenuation by ice is controlled by the
interaction of the spectrally independent scattering coefficient,

Spectral downwelling irradiance at the ice surface and at selected
depths. The incident solar spectrum is calculated as described in Sect. 2;
within the ice, the effective bubble radius is 198

Notably for UV and the shortest visible wavelengths, there is an enhancement
of the subsurface downwelling irradiance,

To provide an upper limit for the enhancement we assume a semi-infinite
refractive ice slab that scatters but does not absorb photons. For photons
incident on the upper surface of the ice–air interface, a fraction,

Our Monte Carlo results fall within these theoretical limits (e.g. Fig. 4),
with a maximum enhancement factor of 1.735 at

Spectral albedo calculated for selected bubble parameter sets from
Table 1, corresponding to porosities (bubble volume fractions) of 1.35 %
for the original Dadic BIA unadjusted no-crack parameter set
(

The same interaction between the scattering coefficient and the absorption
coefficient that controls the wavelength dependence of irradiance also
results in a strong wavelength dependence in the total albedo. There are two
contributions to the overall albedo of blue ice: the direct, specular
reflection according to Fresnel and the contribution from internally
back-scattered photons that return to and escape from the surface. The first
only exhibits a weak dependence on wavelength, but the latter relies on
photons entering the ice not being absorbed before they travel sufficiently
far to be scattered back to the surface. Consequently for

From a visual perspective it is notable that the spectral variation in the
albedo implies that the characteristic colour of bubbled ice is predicted
(see highlighted region in Fig. 5): high albedos are found at wavelengths
associated with the human eye's short-wavelength (blue) cone response,
decreasing at visible wavelengths associated with the green cone response and
then still further at wavelengths where the long-wavelength cone is
preferentially sensitive (red). In addition, this reinforces the point that
blue ice environments with smaller scattering coefficients,

The wavelength-integrated results which are normalised by the downwelling
irradiance incident on the surface,

The results presented so far have relied on the assumption that the incoming
solar flux arrives from a diffuse hemispherical sky (with no direct
component). Atmospheric and other inputs were selected according to a
specific location, the Frontier Mountain range, Antarctica. To test the
sensitivity of the presented results to differing assumptions, we have rerun
the Monte Carlo model to include partitioning between direct and diffuse
solar components and at a range of solar zenith angles (SZAs). The model has
also been implemented with surface elevations, cloud optical depths, and
solar zenith angles appropriate for a range of geographic locations across
both polar regions. In short we find that once partitioning between diffuse
and direct components is properly accounted for, the attenuation of the
downwelling and upwelling fluxes shows a weaker dependence on SZA. The
relationship is most clearly expressed in the spectral albedo at IR
wavelengths, while broadband albedos range between 0.597 and 0.618 for the
most representative solar zenith angles of 49 to
69

Further details can be found in the Supplement.

So far the Monte Carlo model has been used to investigate the impact of
varying bubble radii, number concentration, and solar zenith angle on the
propagation of solar irradiance into blue ice; we now apply it to
investigate the energy that impacts upon and is absorbed by inclusions
within the ice. Accordingly, we adjust the model to count photons whose path
intersects with a defined volume element that represents the inclusion. For
computational reasons we restrict this test to photons whose positions fall
within a set distance of the centre of the inclusion. The path between one
scattering event and the next is then subdivided at a granularity of

To model a range of possible inclusions we construct spherical, planar, and ellipsoidal geometries. Following Evatt et al. (2016), we choose dimensions appropriate for englacial meteorites augmented by one smaller and one larger geometry (see Appendix C for further details). A single inclusion is defined during each model run to ensure independence: we calculate the fractional downwelling and upwelling irradiance incident on the inclusion at 10 geometrically spaced depths.

Figure 7a shows the fractional irradiance incident on the set of spherical
inclusions whilst Fig. 7b shows the same for planar inclusions. The most
prominent feature is that, for both cases, the fractional irradiance per
unit surface area is substantially lower than without an inclusion. We
attribute this to self-shadowing of the diffuse radiation field: downwelling
photons once absorbed by the inclusion cannot be scattered up and then down
to contribute multiply to the irradiance. Despite this, the fractional
irradiance absorbed by the inclusions is still markedly greater than the
singly scattered contribution (also shown in grey in Fig. 7a and b) as the
inclusion acts as a sink for photons in its vicinity. This self-shadowing
can also be seen to be dependent on both the geometry and dimension of the
inclusion: when a target has a larger extent in the

For absorbing inclusions, the energy balance may lead to the surrounding ice
reaching melting point. Once it does any inclusion denser than water will
move downwards under gravity and be capped by a water layer above. When the
porosity of ice is

The processes of total internal reflection by an ice–air boundary within the ice and self-shadowing also relate to the measurability of the predicted irradiance fluxes. Specifically, attempts to measure the irradiance by insertion of an optical detector into the ice would have to account for both these points. The degree of self-shadowing would be a function of both the size and geometrical shape of the detector, whilst the transmission across the interfaces between the ice, a (partial) air layer, and the outer envelope of the detector would need to be assessed carefully. If they were not included, the irradiance fluxes within the body of the ice would be underestimated.

Whilst the described Monte Carlo model aims to replicate the radiative
transfer processes occurring within the ice accurately, it is
computationally intensive. For each combination of bubble parameter sets and
SZA, we follow and track the paths of approximately

Following Marchesini et al. (1989) we can define an effective attenuation
coefficient,

Spectral variation in the effective attenuation coefficient
calculated from

In light of this, it is tempting to consider the form of the spectrally
integrated attenuation curves in Fig. 6c as quasi-exponential, thus
imitating the form of the well-known Lambert–Beer exponential relation for
attenuation through an absorbing medium. The Lambert–Beer exponential
relation holds at a single wavelength: longer wavelengths exhibiting high
attenuation and shorter wavelengths having lower attenuation. Using an
integrated form of the Beer–Lambert decay function is common practice
(Cuffey and Paterson, 2010; Evatt et al., 2016), with attenuation values
typically around 2.5 m

This figure shows the scaled net irradiance for the four different
bubble datasets. The dashed black line, which is almost identical to the
grey line, shows the triple-exponential curve fit,

As seen in Sect. 3, the attenuation of the irradiance depends upon bubble
size and distribution. If the attenuation for distinct ice samples was
highly different, then before an analytical approximation could be found,
one would first have to solve the full presented Monte Carlo model.
Fortunately, plots of the irradiance against depth for the results of Fig. 6c, when scaled against their own surface albedo, show very similar
attenuation profiles (Fig. 9) – clearly the result holds less well for the
no-crack dataset. As such, one need not necessarily run the Monte Carlo
code for each new ice sample. Instead, assuming the ice in question is
reasonably generic, one can use a triple-exponential function that has been
best-fitted to the scaled datasets presented here. For example, if the
Dadic BIA no-crack data are omitted, then the mean scaled net downwelling
irradiance,

As noted in the introduction, this study was conceived in order to provide a better understanding of the glacial subsurface radiative field in regards to the vertical movement of meteorites through Antarctic blue ice. This is reflected in our choice of inclusion dimensions. In Evatt et al. (2016) the attenuation of solar radiation through ice and the absorption of solar radiation by a meteorite were modelled using the Lambert–Beer law. However the 1-D treatment of the problem and simplifications in the assumed radiation field used in that study warrant a more involved investigation. The Monte Carlo model and results described in Sects. 2–5 are a core part of that investigation. However computational limitations mean that it is not practical to include the dynamical behaviour of meteorites directly in the Monte Carlo model. Instead we take the two-step approach described below, considering the 3-D temperature distribution around static meteorites, and then applying this to a dynamic 1-D model.

When considering the hypothesised sinking of englacially transported
meteorites, the key parameter is the temperature of the lower (basal)
interface between the meteorite and ice. If this rises to 0

In Fig. 10a both the resulting maximum temperatures experienced at the upper and lower surfaces of the meteorite are shown for iron meteorites and H and L chondritic meteorite classes. This shows that at a fixed depth, meteorites with the lowest iron abundance (L chondrites) produce the lowest basal temperatures; at the upper surface those with the lowest iron abundance produce the highest temperatures. Figure 10b gives an alternative view of this result displayed as a cross section of the difference in temperature fields surrounding an iron meteorite vs. an L chondrite. Simply put, when fixed in position, iron meteorites experience the highest basal temperatures. However, this still leaves the question of dynamics unresolved.

The second modelling step, to address the dynamical part of the problem, is to develop a numerical one-dimensional implementation of the full heat equation (not a quasi-steady approximation). Its one-dimensional nature gives a clear computational advantage, thus allowing the upward motion of blue ice, the transport and sinking of a meteorite, and a temporary melt layer to also be included whilst the model is run over several seasons. Initially, however, the meteorite's depth is fixed whilst the radiation incident upon the upper and lower surfaces of the meteorite is scaled so that the resultant basal temperature of the meteorite matches that predicted from the 3-D finite difference model. Finally, the 1-D model is run in a dynamical mode with this depth-dependent scaling of the radiation field, permitting the meteorite to produce a meltwater layer and sink accordingly within a rising column of blue ice. A more in depth description of these models and inputs can be found in Mallinson (2019).

Following this approach, and as noted in Evatt et al. (2016), we find that
seasonal melting is controlled by a meteorite's thermal conductivity, with
iron meteorites transferring heat to their lower surface more rapidly than
their stony (chondritic) counterparts. As before, the characteristic
sawtooth behaviour is still seen. Whilst ablation is often reduced during
the winter, it is still present. Accordingly, the meteorite rises with
respect to the ice surface during the winter, and during the summer when
solar heating is active a meteorite can melt the ice below it and sink,
falling faster than the ice surface is ablated. However, our modelling now
suggests the process is more nuanced. We find that melt and sinking is
initiated slightly later in the year for iron meteorites, but, as their
higher thermal conductivity permits basal melting at increased depths, melt
also continues later each season. In short, iron meteorites still experience
preferential melt and sinking, and hence generally they are predicted to lie
below the surface of the ice whilst chondrites remain exposed (see Fig. 11a). However, using the more sophisticated description of radiative
transfer described here, coupled to the 3-D finite difference model, we
predict a minimum depth

This preferential melt result is in better accordance with the relative paucity – but not total absence – of iron meteorites discovered in Antarctica. Specifically, our modelling does not rule out finding some iron meteorites during searches undertaken during the earlier part of the austral summer, or given specific meteorite dimensions and surface conditions are met. Likewise, it can explain the observation that sometimes a stony meteorite may be found partially encased in ice (Fig. 11b) and, importantly, aligns with the observation that the size distribution of iron meteorites is biased to smaller sizes in Antarctica than elsewhere (Harvey, 2003). By extension, this process may explain the restricted altitude range of productive Antarctic meteorite stranding zones, the number of new discoveries on revisiting a blue ice area, and the absence of meteorite finds from the Greenland Ice Sheet (Harvey, 2003; Haack et al., 2007)

In this study we have undertaken a detailed investigation of shortwave radiative transfer in optically thick ice where englacial bubbles cause scattering, using a newly developed Monte Carlo model that also includes consideration of inclusions. Our results are primarily applicable to blue ice areas where the surface can be considered horizontal and the ice is relatively compact and snow-free. We have applied an idealised geometry, deriving and using optical properties that are representative of blue ice areas, whilst retaining information about their likely range. This study was motivated by the need for a more extensive investigation into the vertical motion of meteorites within Antarctic blue ice areas. However, the general conclusions are expected to be relevant for optically thick glacial ice where scattering dominates, noting that in specific cases the internal radiative processes can be complicated by the presence of snow and firn and their microscopic geometries (see also Haussener et al., 2012, who investigated this point for snow), as well as the macroscopic surface geometry. Notwithstanding this simplification, the general results that follow are expected to have wider importance.

First we find that there can be an appreciable enhancement of the subsurface downwelling flux, above the level of the incident irradiance. This was previously observed by Jiang et al. (2005) for a single wavelength of 550 nm but has been explored here both spectrally and for a solar-integrated quantity. For the normalised units of irradiance that we have used throughout, the integrated solar downwelling flux within the ice can be up to 1.31 times the incident irradiance. There is a corresponding enhancement in the upwelling flux (thus conserving energy). This enhancement is a result of the refractive air–ice boundary overlaying a material volume where scattering dominates, resulting in multiple scattering and internal reflections. For specific wavelengths where absorption is minimal, the enhancement can be as high as 1.735, just less than the theoretical maximum of 1.743 for ice at solar wavelengths in the absence of absorption.

Considering the albedo, our calculations produce the expected wavelength
dependence that is interpreted by our visual system as “blue” (Warren,
2019), with lower-porosity ice producing more saturated colours. Thus the
perceived saturation of blue ice could be used in the field as a visual
proxy for the porosity of ice, its density, and the degree of scattering
present, and moreover it could be used to coarsely assess the penetration of solar
shortwave radiation into the ice. The calculated broadband albedos agree
favourably with previous observations, but with a narrower range of 0.585 to
0.621 – though we note that even small changes in broadband albedo can
have considerable global consequences over geological timescales (e.g.
Pierrehumbert et al., 2011). Considering the spectral behaviour, the effect
is largest at 820 nm where we see a 79 % change when

Our results are predicated on a diffuse incident field under particular
assumptions of the surface environment. However, permitting the solar zenith
angle to vary, we find a reduced dependence once the incident field is
partitioned between diffuse and direct incident components: broadband
albedos range between 0.597 and 0.618 for the most representative solar
zenith angles (49 to 69

For absorbing inclusions embedded within the ice and its internal radiation field, self-shadowing reduces the irradiance incident on the surface of the inclusion. This is a geometric effect, with the irradiance decreasing for larger inclusions. Conversely smaller inclusions whose dimensions are less than the mean free path for scattering absorb a greater fraction of the available radiation; for all inclusion sizes downwelling and upwelling fluxes lie between the available irradiance and the single-scattering component, with the inclusion acting as a local sink for photons. Interpreting the inclusion instead as an optical detector, we conclude that both self-shadowing and the introduction of a lower refractive interface between the detector and blue ice must be taken into account when assessing measurements. If they are not taken into account, then our results suggest that measurements of the irradiance within the ice column would be an underestimate of the actual irradiance present.

Next we assess the Monte Carlo results for ice without an inclusion as a
whole and formulate an empirical expression describing the typical behaviour
of the net downwelling irradiance in optically thick blue ice. The broadband
albedo is separated out, thus leaving the normalised depth-dependent aspects
to be numerically fitted. The wide range of absorption coefficients
exhibited at UV and blue wavelengths, in contrast to those at IR
wavelengths, result in a depth dependence that is inadequately modelled as a
single exponential. Instead our result takes the form of a triple-exponential function; the two fast decaying components represent the
absorption of longer IR wavelengths at shallow depths, whilst the remaining
one exhibits a decay constant of 1.93 m

Finally in Sect. 6 the results of the solar shortwave Monte Carlo modelling
are applied to a 3-D finite difference implementation of the full heat
equation to explore the vertical motion of meteorites in Antarctic blue ice
areas. Here we find the process is more nuanced than predicted by Evatt et
al. (2016), with iron meteorites typically residing at shallower depths
(

In order to study the passage of solar radiation through bubbled ice, it is
necessary to calculate the solar spectrum incident on that surface,
considering that many aspects of the transfer are wavelength dependent.
Here, the incident spectral irradiance at the ice surface is calculated
using the libRadtran radiative transfer model (Mayer and Kylling, 2005) with
relevant atmospheric inputs. These inputs are broadly similar to those in
Evatt et al. (2016) used to calculate a climatology of integrated shortwave
fluxes, but for completeness we describe them in some more detail here.
Internally we use the sdisort radiative transfer solver with
pseudo-spherical approximation and the reptran molecular absorption
parameterisation to calculate surface spectral irradiance between 250
and 2800 nm at 1 nm intervals. The extra-terrestrial solar spectrum is from
Kurucz (1994). The output altitude is set at 2.04 km with a nominal albedo
of 0.62, suitable for the BIA located near Frontier Mountain range in
Antarctica (72.95

The clear-sky atmosphere profile used is the subarctic summer profile (Anderson et al., 1986) with a climatological total ozone column of 300 DU (Diaz et al., 2004). The spring–summer aerosol profile is taken from Shettle (1989) with the aerosol optical depth (AOD) at 550 nm scaled to 0.028, the mean of the high- and low-elevation AOD estimates given by Tomasi et al. (2007). In order to include the effect of clouds, the model is run three times: once with no clouds with the parameters above, once with the addition of a high-altitude glaciated cloud, and once with a lower-level cloud. The three spectra are then combined linearly in proportion to the relative occurrence estimates of clear skies and high-level and lower-level clouds to form a single irradiance spectrum. Cloud heights, depths, occurrence frequencies, and bulk microphysical properties are taken from Adhikari et al. (2012).

Care has been taken to select representative parameter values for the specific BIA locality. However it is anticipated that the resultant spectral shape, though not necessarily the absolute total irradiance, should also be generally applicable to high-altitude polar regions (see further in Sect. S2 in the Supplement).

Summary of bulk ice bubble data contributing to mean parameter
set. Dadic BIA caliper density parameters select only samples directly
measured by calipers (including samples designated as having cracks). Dadic
BIA combined mCT and caliper parameters incorporate a full set of samples,
with data homogenised as described in the text of Appendix B. Dadic BIA
mCT density parameters select only samples with micro-CT density
measurements (chosen to avoid regions with cracks) selected as representing a
realistic upper bound for density. Mellor and Swithinbank (1989) parameters
are constructed from an estimate that BIAs have a porosity of 6 %,
combined with the bubble radii–density regression noted in the accompanying
text. The Bintanja (1999) (850 kg m

Summary of additional bulk ice data bubble data not contributing to the mean parameter set in Table B1. Dadic BIA unadjusted (no-crack) parameter set calculated from samples classed as not including cracks and relying on unadjusted micro-CT density measurements. Dadic BIA mCT density (SSA adjusted) is derived from Dadic BIA mCT in Table B1, but adjusted for SSA attributed to crack vs. no-crack proportions. Dadic BIA crack regions select crack-only regions, with a consequently low bulk density.

Once the incoming solar radiation reaches the ice surface, the key
moderators of radiation through blue ice are the number concentration and
radii of bubbles. As noted in the main text there is a paucity of relevant
in situ data and thus we principally rely on a homogenisation of bubble
number concentration and radii measurements carried out by Dadic et al. (2013) near the Transantarctic Mountains, Antarctica. The samples collected by Dadic et al. (2013)
cover a range of ice conditions along two BIA transects with sample depths
down to 0.94 m and are analysed by a combination of micro-CT analysis,
estimates derived from specific surface area (SSA), and caliper-and-scale
measurements. As the authors state, the bulk ice densities from the caliper
measurements are expected to be underestimates of the density, whilst those
from micro-CT analysis are expected to be overestimates. To reconcile these
differences and formulate a best estimate of bubble radius and number
concentration, we first calculate the mean ratio of micro-CT and caliper
densities for samples where both methods have been employed, which allows us
to calculate an adjusted density estimate for all samples. In a similar
fashion a simple regression is found between measured bubble radii and bulk
ice density for the subset of samples undergoing both analysis methods, to
give an estimate for samples lacking radii measurements. In this way we
formulate a self-consistent set of densities and radii for the 27 samples
available, which together have a mean density of

To ensure we are not overly reliant on a single set of field campaign data, we apply the density–radii regression and the usual density–porosity–number concentration relations to subsamples of the data of Dadic et al. (2013) and estimates from the wider literature. In this way we aim to represent the range of environments present in BIAs, and we are able to calculate an overall mean parameter set and choose example parameter sets corresponding to low and high bulk densities. We additionally select values corresponding to an outer upper bound for blue ice density and referred to as the “no-crack” parameter set. Each of these four parameter sets is self-consistent between its estimates of effective bubble radius and number concentration and in line with observed bulk densities and porosity values. The parameter sets contributing to the overall mean are listed in Table B1, with additional data being noted in Table B2.

We extract dimensions from 94 recently discovered samples detailed in ANSMET
newsletters (ANSMET, 2017); all meteorite data are combined to give a mean
length

This defines three macroscopic ellipsoidal inclusions with upward-facing
surface areas of 4.00, 13.4, and 40.9 cm

Based on these ellipsoids, five planar and five spherical targets are also defined with linear dimensions and radii chosen so that their upward-facing surface areas match those of the set of ellipsoids.

Copies of the model routines are available from the corresponding author on request.

The supplement related to this article is available online at:

ARDS led the writing of the manuscript and was responsible for the Monte Carlo calculations. GWE provided overall scientific input and direction. AM was responsible for the 3-D and dynamical modelling aspects, and EH carried out preliminary analysis on blue ice bubble data. All authors were involved in discussions and contributed to the preparation of the manuscript.

The authors declare that they have no conflict of interest.

The authors would like to thank both reviewers for their time and constructive comments and the editorial team for their assistance during the preparation of this manuscript.

This research has been supported by the The Leverhulme Trust (grant no. RPG-2016-349), the EPSRC MAPLE Platform (grant no. EP/I01912X/1), and the Paneth Meteorite Trust (Summer Bursary).

This paper was edited by Benjamin Smith and reviewed by Stephen Warren and Ruzica Dadic.