Formation of glacier tables caused by differential ice melting

Glacier tables are structures frequently encountered on temperate glaciers. They consist of a rock supported by a narrow ice foot which forms through differential melting of the ice. In this article, we investigate their formation by following their dynamics on the Mer de glace glacier (Alps, France). We report field measurements of four specific glacier tables over the course of several days, as well as snapshot measurements of a field of 80 tables performed on one given day. We develop a 5 simple analytical 1-D heat conduction model accounting for the various mechanisms of the heat transfer on the glacier using local meteoreorological data, and which displays excellent agreement with the field measurements. We show that the formation of glacier tables results from a competition between two effects: on the one hand, a geometrical amplification of the heat flux received by the rock and transmitted to the ice underneath on the one hand, and on the other hand, a shielding effect resulting from the warmer temperature of the rocks compared to that of the ice (reducing the sensible and net infrared incident fluxes). 10 The latter is greatly amplified by direct solar irradiation, which induces a strong temperature gradient across the thickness of the rock.

On a temperate glacier, the ice ablation rate is influenced by the presence of debris on its surface. Indeed, a dense debris layer covering an ice surface can, when thin enough (less than 0.5 cm), enhance the ice melting compared to a bare ice surface or, if thick enough, act as an insulation layer and reduce the melting rate (Östrem, 1959). The insulation effect is well-captured by complex energy balance models which use meteorological data as input parameters (Reid and Brock, 2010;Collier et al., 2014), and more recently using enhanced temperature index models (Carenzo et al., 2016;Moeller et al., 2016). The melt 30 amplification effect for thin debris layers was explained by its patchiness (Reid and Brock, 2010) or by its porosity to air flow (Evatt et al., 2015). At a more local scale, patches of dirt or ashes on a glacier are known to lead to the formation of ice structures known as dirt cones, which consists of an ice cone covered with a centimetric layer of dirt (Swithinbank, 1950;Krenek, 1958;Drewry, 1972;Betterton, 2001).
Glacier tables (see fig. 1c) are structures frequently encountered on the leveled part of temperate glaciers (Agassiz, 1840). 35 They typically consist of a meter-size rock supported by a column (or foot) of ice. They form over the course of a few days to a few weeks at the end of the spring, from May to June and may progressively disappear during the summer. Very large tables however, whose size can reach 10 m, can last for several years (Bouillette, 1933(Bouillette, , 1934. Glacier tables form because the ablation rate of the ice is lower under the rock than at the air-ice interface. When the ice foot becomes too thin, the table falls, usually on its south side. The ice foot progressively disappears and the rock can potentially form another table, if not too late in the 40 season. Similarly to what is know for debris layers, smaller rocks tend not to form tables but can instead increase the melting rate and gradually sink into the ice, creating narrow and deep a hole which can reach up to 15 cm in depth (McIntyre, 1984). In a previous work (Hénot et al., 2021), artificial glacier tables were reproduced experimentally in a lab controlled environment (constant temperature, humidity, absence of wind), at a centimetric scale. The study focused on the initial behavior of pattern formation using cylindrical "rocks" of various sizes, aspect ratio and materials, initially resting on a flat ice surface. Although 45 this small scale study under controlled conditions allowed one to understand the physical mechanisms that could play a role glacier table formation, it did not encompass the complexity of the energy balance on a natural glacier, in particular the effects of the direct solar irradiation and of the wind.
In this article, we report field observations made on the Mer de glace glacier (French Alps), of the formation dynamics of glacier tables monitored over the course of a few days, as well as a systematic measurement of already-formed tables on a given 50 day. The Mer de Glace is a temperate glacier, meaning that the ice temperature is always given by the melting temperature of water, T ice = 273 K. We use local meteorological data to fit the ice ablation rate and characterize the heat transfer mechanisms a. b.
c. at the surface of the glacier. We then develop a 1D conduction model taking into account the effect of the solar irradiation as well as sensible and longwave heat fluxes, which is in excellent agreement with the field measurements and which illustrates the synergistic effect between solar irradiation and sensible flux responsible for glacier table formation.

Location and definitions
In this article, we report two sets of observations made on the Mer de glace glacier: a time-resolved camera recording of the formation and evolution of four glacier tables and a field observation of 80 glacier tables.
The lower part of the Mer de Glace (below a 2000 m altitude) is largely covered with granitic debris, with sizes ranging from 60 submillimetric up to several meters. In the following, their dimensions are characterized by their thickness e and their larger and smaller widths d 1 and d 2 (see fig. 2b). We define an effective width of the rock d eff = 2d 1 d 2 /(d 1 + d 2 ) whose expression will be justified in part 3.2 and the aspect ratio β = e/d eff . conductivity of granite was measured in a previous work (Hénot et al., 2021) as λ = 2.8 W·m −1 ·K −1 , which is in agreement with the literature (Cho et al., 2009).

Time evolution of four large tables
Time-lapse images were obtained using an autonomous solar powered camera (Enlaps Tikee) placed on three 1.5 m-long wood rods set into the ice. Pictures (4608 px × 3456 px) were taken every 30 min between 5 am and 10 pm during 5 to 7 days and e. The position of the ice surface was followed using a scaling rod, embedded into the ice and located approximately 2 m away from the rocks.
The surface temperature of rock 4 was measured every 5 min during period B using thermocouples and a homemade battery July 2021, we also measured the air temperature at location 2, 3 m above the glacier surface (see suppl. mat.) which was systematically 2.5 • C higher than the AWS data. In the rest of the article, we use the AWS data to which an offset was added:

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T air = T air AWS + 2.5 • C. We do not expect the measured solar flux and the wind speed to be significantly affected by the distance from the measurement site. The meteorological data are displayed in fig. 4b-d and f-h.

An 800-m 2 field comprising 80 glacier tables
On June 3 rd 2021, we systematically measured, in an area of 10 m×80 m located at point 1 in fig A detailed analysis and a modeling of these observations is provided below, but a first observation shows that there exists a critical size above which rocks tend to form glacier tables. The larger the rock, the greater its ability to protect the ice underneath from melting and, assuming all tables started their formation at the same time, the higher the ice foot. Below this critical size, however, the rocks have a tendency to sink into the ice surface. The transition between this two regimes seems This is reminiscent of the results obtained in a controlled environment in the absence of solar radiation, for which the transition occurred for a critical diameter d crit,Φ=0 = 4λ/(ηh eff ) of the cylinders, independently of their aspect ratio (Hénot et al., 2021).  The lowering of the ice surface of the glacier, due to melting, is shown with blue markers in fig. 4a and e. The daily ablation rate on clear-sky days was of the order of 8 cm·day −1 on period A (3/8 clear-sky days) and 5.5 cm·day −1 on period B (5/5 clear-sky days). In the following, we characterize the heat flux balance on the glacier surface by linking the ablated ice thickness to local meteorological data. The net incident heat flux causing the glacier surface to melt can, in general, be expressed as: where Q SW and Q LW are the net shortwave and longwave radiative fluxes, Q H and Q L are the turbulent heat fluxes corresponding respectively to sensible and latent heat and Q R is the flux associated with rain. In the following, we make simplification assumptions in order to extract the main physical mechanisms of the heat transfer on the glacier. The shortwave radiation (when neglecting the diffuse radiation) is given by where α ice is the ice albedo and Φ is the incoming direct solar radiative flux. The net longwave flux comes from a balance between the radiation emitted by the ice, 110 at temperature T ice , and received from the atmosphere, at temperature T air , (both emissivities are taken equal to 1) leading to  Q LW = Fσ(T 4 air − T 4 ice ), where σ is the Stefan-Boltzmann constant and F = 0.7 is a cloud factor whose value corresponds to clear sky conditions (Hock, 2005). As it will prove useful in the glacier table formation model, we linearize this expression around T ice : Q LW ≈ 4FσT 3 ice (T air − T ice ) since T air − T ice T ice (with temperatures in kelvins). The turbulent sensible flux is assumed to be proportional to the wind velocity: Q H = Cu a (T air − T ice ) where C is a constant depending on the surface 115 rugosity (Brock et al., 2006;van der Veen, 2013). We neglect turbulent latent heat flux as it usually corresponds to less than 10 % of the total flux (Hock, 2005). There were no precipitation and thus no associated heat flux on the glacier during the duration of our study. From this, the net total incident heat flux at the ice surface is: where h eff = 4FσT 3 ice + Cu a is an effective heat exchange coefficient accounting for sensible and longwave radiative fluxes.

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The incoming heat flux directly causes the ice to melt, and the vertical position of the ice surface is thus given by: where L fus = 303 MJ·m −3 is the volumetric enthalpy of fusion for ice. This model has two adjustable parameters: C and α ice .  fig. 4a and e correspond to the application of eq. 2 and 3 using the values of the wind speed, air temperature and solar flux measured at the Requin AWS and displayed in fig. 4b-d and f-h, with parameters α ice = 0.33 and 125 C = 2.3 J·K −1 ·m −3 adjusted on the data of period A (see suppl. mat. for a discussion on the sensibility).
The agreement between our prediction and the field measurements is excellent, which validates the assumptions made. Note that the integration is reset every day from the last experimental value of the previous day in order to avoid the integration of a small error over several day. Moreover, the value of the adjusted ice albedo corresponds to what is reported in the literature for aged surface glacier ice (Brock et al., 2000;van der Veen, 2013). The value of the turbulent convection coefficient C also falls 130 in the range (1.9 to 4.9 J·K −1 ·m −3 ) previously reported (Nakawo and Young, 1981;Oerlemans and Klok, 2002). The effective heat exchange coefficient shows important variations between the two time periods going from h eff, A = 18.2±8.4 W·K −1 ·m −2 (mean ± SD) to h eff, B = 5.6±1.9 W·K −1 ·m −2 . This is due to the drastic difference in mean wind speed, going from 6.5 m·s −1 in period A to 1.0 m·s −1 in period B, which in turn explains the variation in daily ablation rate between the two periods.

Table formation
135 Rocks 1 and 2 formed glacier tables over the course of approximately a week (see fig. 4a and e), after which they felt on their side. Rock 4 was initially sitting on an ice foot, which grew bigger, while rock 3, although exhibiting dimensions close to those of rock 2 went down almost at the same rate than the bare ice surface and did not form any significant glacier table (see suppl. mat. for pictures).
In (Hénot et al., 2021), an analytical model was developed in order to explain the formation of artificial glacier tables made 140 of cylindrical cap in a controlled environment, in which the heat transfer (natural convection of air and infrared radiation from the enclosure) was modelled through an effective heat exchange coefficient. In the following, we adapt this model to the non-symmetrical geometry of the rocks and we include the effect of solar irradiation and wind. Assuming that rocks are cuboids (see fig. 2b), the initial area of contact with the ice is A base = d 1 d 2 while the external area in contact with air is For the sake of simplicity this area will be considered constant, although it evidently varies over 145 time. Yet, as discussed in Suppl. Mat., this has little effect on the heat transfer between the rock and the ice beneath.
The mean effective area on which the solar irradiation is received by a rock is denoted as A sun . This area should depend on the shape and orientation of the rock with respect to the course of the sun, but for sake of simplicity in the following it is simply assumed to be equal to the rock base surface area A sun = d 1 d 2 (the validity of this assumption is discussed in the suppl. mat.). In a quasi-static regime, the energy balance of a rock can be written as: where α rock = 0.18 is the albedo of the granite rock (measured by (Watson, 1971), see suppl. mat.) and T rock is the rock surface temperature, assumed homogeneous. Note that the heat flux received from the air and transmitted to the ice is amplified by a geometrical factor: A ext /A base = 1 + 2e d1+d2 d1d2 = 1 + 4e/d eff = 1 + 4β. The heat flux transmitted from the rock to the ice underneath can be estimated using a 1D conduction model as: where λ is the thermal conductivity 155 of the rock and d 1D is a length scale for the conduction process in the rock. It was postulated, and verified experimentally (see  2021)), that this length could be estimated by: where V rock is the volume of the rock and η = 2.5 is a numerical prefactor adjusted from the lab experiments. From this and eq. 4, the temperature of the rock surface is: and the heat flux transmitted to the ice can be expressed as: The integration of the flux given in eq. 7 is shown in solid lines in fig. 4a and e. The agreement between the measurements and the model prediction is over all very good for all four rocks, which validates the simple assumptions of the model. In our model 165 under the quasi-static approximation, the heat flux received by the rock is assumed to be instantaneously transmitted to the ice underneath. Fig. 5 shows the measured temperature of the top surface of rock 4 (solid line) as well as the value predicted by eq. 6 using the meteorological data for T air , u a and Φ.
As this seems to be well verified in the case of rock 4 whose temporal changes in surface temperature are well captured by the model, this is justified only if the rocks are not too thick compared to the thermal skin depth δ diff = T day D rock /π = 0.2 m 170 where D rock = 1.4×10 −6 m 2 ·s −1 is the thermal diffusivity of the rock and T day = 24 h is the period of variation of the external heat flux (i.e. Φ(t)). This condition is indeed well verified for rock 4, and acceptable for rocks 1 and 2 but not in the case of the 1.7 m thick rock 3 whose daily evolution of surface temperature can therefore not be captured by eq. 6. However, due to the linearity of the heat exchange process with respect to the temperature, the daily average energy balance is expected to remain valid, which explains the good agreement between the data and the model for rock 3 in fig. 4e. and using the average value for T air : where Φ = Φ /(h eff (T air − T ice )) is a dimensionless number characterizing the effect of solar radiation. Note that this expression does not depend on the hourly meteorological data but rather on averaged values over the time period of interest.
On both periods A and B, T air ≈ 7 • C and Φ ≈ 340 W·m −1 on clear-sky days.
Equation 8 predicts the linear behaviour of H vs. H ice and allows one to compute their ratio, which evidently depends on the 185 tickness of the rock, e, which is plotted in fig. 6b for periods A (dashed lines) and B (solid lines) for each rock aspect ratio β.
Here also the agreement between the field measurements and the model is excellent and illustrates the sensibility of the table formation dynamics to the effective heat exchange coefficient and thus to the mean wind speed.

Analysis of the 80 glacier tables field
Having successfully modelled the time evolution of the 4 larger glacier tables, the model can be applied to the data obtained 190 on a given day for a field comprising 80 rocks.
H ice H ice Table formation No   The rocks standing on taller ice feet probably started to form glacier tables before May 26. They correspond to bigger rocks that may have emerge from the snow layer early on. As for the rocks that sunk into the ice surface (H < 0, in blue markers), the expression of eq. 8 is not expected to remain valid after the onset of the formation as the geometrical amplification effect disappears if the rock is set into the ice. We can therefore conclude that using the previously determined values of α ice and C allows on to satisfyingly predict the height of glaciers tables above the critical diameter.

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That critical diameter corresponding to the transition between the two regimes (the rock sink into the ice or form a table), is set by H/H ice = 1, and can be expressed as: Fig. 7b shows a diagram of the ability for a rock to form a table as a function of its effective width d eff (which in practice can be replaced by the mean width) and the effective heat exchange coefficient h eff for various values of the rock aspect ratio.

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The four monitored rocks as well as the field data from the 80 rocks found in the 800-m 2 field are plotted in this diagram. The distribution of aspect ratio β of rocks (46 of them) close to the transition (defined by |H| < 0.2 m) is plotted in fig.7c. The bounding values (β = 0.16 and 0.66) as well as the value corresponding to the maximum of the distribution (β = 0.27) are used to plot in fig. 7b the results of the model given by eq. 9, which shows an excellent agreement with the transition visible in the field data. This diagram illustrates the sensibility of the table formation to the effective heat exchange coefficient: the 215 critical size above which a rock can form a table differs by a factor 2 between the two periods A (windy) and B (no wind).
The dotted line corresponds to the model prediction in the absence of solar radiation and shows that the sun has the effect of lowering the critical size despite the lower albedo of the rocks compared to the ice.

Physical discussion
The ability for a rock to form a table appears to be controlled by two opposing effects. On the one hand, the heat flux received 220 by the rock from the environment on all its surface (sensible heat flux and longwave radiation) and transmitted to the ice is amplified by a geometrical factor related to the aspect ratio of the rock. This effect dominates for small rocks and can cause them to sink into the ice surface. The lower albedo of the rock compared to the ice also lead to an amplification of the direct solar radiation. On the other hand, there is an attenuation effect of a portion of the heat flux received by the ice under the rock compared to the one received far away from it. The heat flux that depends on the surface temperature of the body receiving it 225 (sensible heat flux related to thermal convection and net longwave radiative flux) are smaller for rocks than for ice as they are warmer. This effect is controlled by a dimensionless parameter: h eff e/λ. The higher this number, the stronger the attenuation effect, which is thus dominant for large rocks. This effect was already identified and discussed in (Hénot et al., 2021). In natural conditions, this effect amplified by the solar irradiation which imposes a thermal flux going from the rock top surface to the ice and induces a temperature gradient inside the rock. This rises the surface temperature of the rock which amplifies the cooling 230 effect due to the wind and the infrared emission. Overall, the effect or solar irradiation lowers by a factor 2 to 4 the critical size for which a rock is able to form a glacier table.

Conclusions
We studied the formation of four glacier tables over the course of a week and we measured the characteristics of 80 glacier tables on the Mer de Glace glacier (Alps). We found out that the size of the rocks is a determinant factor governing table