Natural convection, also known as density-driven convection, occurs when fluid flow is driven by density gradients, unlike forced convection, where an external force (e.g. pressure gradient) drives the flow. In coarse sediments, natural convection can be generated when there is a large enough temperature gradient between the sediments and the external air. As a consequence, a density gradient develops between the lower (warmer) and upper (colder) boundaries, leading to the upward movement of warmer, less dense air and the downward movement of colder, denser air (Fig. ). This pattern forms clockwise and counter-clockwise convection cells side by side, resulting in a bottom-up energy transfer by the convection cells .

The strength of natural convection in a porous medium is quantified by the Rayleigh–Darcy number (Ra) : 1Ra=gβaρacakνaλΔTaH, where g (ms-2) is the gravitational acceleration; βa (K-1) is the volumetric thermal expansion coefficient of the air; ρa (kgm-3), ca (Jkg-1K-1), and νa (m2s-1) are the density, specific heat capacity, and kinematic viscosity of the air, respectively; k (m2) is the medium permeability; λ (Wm-1K-1) is the thermal conductivity of the medium; and ΔTa (K) is the difference in air temperature between two plains separated by a distance H (m). The onset of natural convection is defined by the critical Rayleigh number (Rac), which ranges from 27 to 40 depending on the upper-surface characteristics . The lower limit can be considered to be representative of an open surface (e.g. coarse surface sediments without snow cover), whereas the upper limit can be taken to be representative of an upper surface impermeable to airflow (e.g. coarse surface sediments with thick snow cover).

The Rayleigh number is particularly sensitive to the permeability of the porous medium, which can vary by several orders of magnitude for natural materials depending on the characteristic grain size . Permeability is commonly estimated using the Kozeny–Carman equation: 2k=Ad102n3(1-n)2, where d10 (m) is a characteristic grain diameter such that 10 % of the sediment particles are finer than d10, n is the porosity, and A=0.0056/4.25 is a constant proposed by that improves the estimation of permeabilities for coarse sediments.

Natural convection in coarse sediments has a strong seasonal variability resulting from air temperature fluctuations. In winter, temperatures inside the sediments are higher than the external air temperature. Air convection intensifies the energy exchange with the atmosphere, causing the sediments to lose energy to the air. In summer, the opposite happens; the temperature inside the sediments is lower than the external air. Since the cold air is heavier than the exterior air, no air convection occurs, and energy exchange is controlled by conduction. The small contact areas between the sediments and the low thermal conductivity of the air result in the sediments acting as a thermal insulator, reducing the amount of energy gained by the ground from the atmosphere . Over the year, the imbalance of increased energy losses in winter and reduced energy gains in summer produces a net energy loss, lowering the average ground temperature .

GEOtop is a versatile 3D hydrological model that effectively integrates ground energy and water budgets while accounting for atmospheric energy exchange and a multilayer snowpack . This makes it suitable for modelling permafrost-relevant variables such as snow and ground temperatures. This work refers to GEOtop 3.0 as the standard GEOtop (S-GEOtop). The system of equations for the water flow and energy balance in S-GEOtop is as follows: 3∂Φm∂t+∇⋅Jw+Fw=0,4∂U∂t+∂G∂z=0, where Φm=Mw+MiρwVc is a dimensionless variable representing total water content expressed as liquid water, with Mw and Mi being the mass of liquid water and ice, respectively; ρw (kgm-3) being the density of liquid water; and Vc (m3) being a control volume. Jw=-krwkμw(∇P-ρwg) (ms-1) is the water flux, with krw being the relative permeability of water, μw (Pas) being the dynamic viscosity of liquid water, ∇P (Pam-1) being the pressure gradient, and g (ms-2) being the gravity vector. Fw (s-1) is a sink term representing evapotranspiration, which can act in the first layers of the sediments. U=Cm(T-Tref)+ρwLfθw (Jm-3) is the internal energy, where Cm (Jm-3K-1) is the medium volumetric heat capacity, T (K) is the temperature within Vc, Tref is a reference temperature, Lf (JKg-1) is the latent heat of fusion, and θw is the volumetric liquid water content. G=-λT∂T∂z (Wm-2) represents the vertical component of the conduction flux, where λT (Wm-1K-1) is the effective thermal conductivity within Vc.

Prior research has utilized S-GEOtop in mountain regions to model snow cover , water and energy fluxes , and permafrost distribution . However, as the model does not consider airflow in porous media, it fails to accurately represent the thermal behaviour of coarse sediments, leading to inaccurate representation of permafrost distribution in mountain landforms such as talus slopes, moraines, and rock glaciers.

Simulations using CE-GEOtop were conducted to evaluate the differences between a case considering the classical LTE assumption and a case using the LTNE approach. These simulations evaluated the impact of considering different heat transport processes in the sediment's temperature without considering surface energy fluxes. The LTE simulations were achieved by artificially increasing the heat transfer term (h; Eq. ), thus enforcing the same temperature between the air and the composite medium. Additionally, S-GEOtop simulations were performed to establish a baseline for comparison with CE-GEOtop. Since S-GEOtop exclusively considers conduction as the heat transport mechanism, any agreement between the results of CE-GEOtop and S-GEOtop implies the absence of natural convection.

The numerical experiment involved a synthetic geometry representing a flat square domain with dimensions of 33 m by 33 m and a 3 m layer of coarse sediments with a permeability of 1.0×10-6 m2 (d10∼70 mm). The domain was discretized into 12 vertical layers of 0.25 m each and a grid size of 1.5 m, resulting in 22 grid cells in both the longitudinal and the transversal directions.

To limit the interaction with the atmosphere, the surface energy balance was removed, which included eliminating shortwave and longwave radiation (by setting surface albedo to 1 and surface emissivity to 0), latent heat (by setting sediment water content to θw=0 and using a low air relative humidity), and sensible heat (by setting a low wind velocity). The sediments were initialized with a temperature of 15 °C, and a no-flux lower boundary condition was applied for the heat transfer. On the top boundary, a fixed temperature (Ttop) was applied to the sediments and air throughout the simulation. The simulations were carried out for various values of Ttop, resulting in a range of Rayleigh numbers from 25 to 265. With a lower impermeable boundary and an upper open surface, Rac was estimated to be 27 . The simulations were conducted with constant boundary conditions over a period of 30 d. The results were analyzed in terms of the layer-averaged normalized temperature Tt*(z): 17Tt*(z)=Tt(z)‾-TtopTini-Ttop, where Tt(z)‾ (K) is the temperature average over the layer at time t (s) and Tini (K) is the initial temperature.

All simulations exhibit identical temperature profiles for the lowest Rayleigh number (Ra=25), indicating the absence of natural convection (Fig. ). However, with a slightly higher Rayleigh number (Ra=65), natural convection influences the sediment temperature profiles after approximately 15–20 d in the LTNE simulation. This suggests that although the Rayleigh number exceeds its critical value, natural convection is not strong enough to affect ground temperatures over short periods. Analysis of the results of the higher Ra values reveals that natural convection becomes increasingly significant for Ra>100 as evidenced by the temperature profiles while considering the LTNE approach.

At monthly timescales, the LTE simulations are indistinguishable from the pure conduction simulations in all cases except for Ra=265. In this last case with the highest Rayleigh number, LTE simulations show lower temperatures than the pure conduction case after 20 d and eventually resemble the LTNE simulations after 30 d, indicating that the system reaches local thermal equilibrium. However, significant disparities are observed for shorter times or lower Rayleigh numbers, highlighting the system's non-thermal equilibrium state. This is evident in the air temperature profiles for the LTNE case (dashed cyan lines), where early air temperature indicates lower temperatures than those of the sediment. As a result, the thermal non-equilibrium state leads to a faster decrease in the temperature of the sediments. Therefore, adopting the LTE approach affects the short-term response of the sediments' temperature and hinders natural convection effects.

Topoclimatic factors, such as air temperature and potential incoming solar radiation (PISR), are typically used to delineate permafrost zones. For the Babylon talus slope case study, these variables would suggest that the upper part of the talus is the most favourable zone for the occurrence of permafrost due to the lower air temperatures and PISR. However, the observed permafrost zone is in the middle–bottom part of the talus (Fig. ), similar to observations in previous studies elsewhere e.g.. This thermal anomaly could be caused by air convection in two characteristic regimes: the winter regime, in which cold external air induces the influx of air into the bottom of the slope and warm air outflow at the top (Fig. a), and the summer regime, in which higher external air temperature causes the influx of warm air at the top and the outflow of cold air at the bottom (Fig. b). Similar convective patterns can also occur on a diurnal timescale: surface cooling by longwave radiation at night can drive upslope air movement, while daytime heating by shortwave radiation can reverse this flow as surface temperatures rise.

To assess whether the CE-GEOtop model can represent the air convection fluxes on a talus slop, the summer and winter regimes will be simulated. The CE-GEOtop (with LTNE) and S-GEOtop models were configured using the dimensions of the Babylon talus slope with a constant slope angle of 21° derived from a DEM. The 3D model domain is confined within the boundaries of the geophysical survey , with the talus slope's longitudinal axis measuring 94.5 m and the transversal axis 38.5 m. Figure c shows a cross-section of the model along the longitudinal centerline of the slope. The grid size is set at 3.5 m, resulting in 27 grid cells in the longitudinal direction and 12 in the transversal direction. The bedrock depth, based on , reaches a maximum of 16 m at the middle section of the talus slope, gradually decreasing to 0 m at the top and bottom regions. Consequently, the initial 16 m of the sediment domain is discretized into 32 layers, each one with a thickness of 0.5 m, while the subsequent 12 layers representing the bedrock have varying thicknesses ranging from 0.5 to 2 m, in total reaching to a depth of 33.25 m. A constant geothermal heat flux of 0.03 Wm-2 was set as the model's lower boundary condition .

For each regime, three cases were run; the first, CG-HK, used the CE-GEOtop model with coarse sediments and a high permeability of k=1.5×10-6 m2 representing low-end values for cobble-sized materials based on laboratory measurements . The second is SG-HK, which utilized the standard GEOtop model with coarse sediments (k=1.5×10-6 m2). The last one is CG-LK, employing CE-GEOtop with sand-sized sediments with low permeability (k=1×10-10 m2) instead of coarse sediments. For all cases, bedrock is assumed to be almost impermeable with k=2×10-15 m2 .

Although CE-GEOtop is capable of incorporating fully time-varying meteorological forcings, the summer and winter regime simulations were conducted with constant boundary conditions for 10 d to isolate the direct effects of higher or lower air temperatures on airflow patterns. As in the preceding simulations, all surface energy balance terms were neglected.

This section elaborates on the CE-GEOtop model's capability to predict permafrost distribution within the Babylon talus slope. The model is forced with half-hourly meteorological data from the nearby (∼0.5 km) Opabin weather station , including precipitation, wind speed and direction, relative humidity, air temperature, incoming longwave radiation, and cloud fraction. Due to the high spatial variability in solar radiation resulting from topographic shading (see Sect. 2.3.4), direct use of measured shortwave radiation is not feasible. Instead, the half-hourly cloud fraction is derived from measured incoming solar radiation and combined with the hourly sun position and topographic shading to calculate pixel-specific incoming solar radiation across the model domain. The spatial distribution of precipitation, relative humidity, and air temperature was achieved using lapse rates calculated by from measurements along an elevation transect in the local area.

The model's dimensions, including spatial discretization and the bedrock depth, are consistent with the setup described in the preceding section (Fig. c). However, this analysis incorporates distinct sediment permeability and thermal properties. Based on the findings of , a low-conductivity layer (k=3.75×10-11 m2) was incorporated in the bottom of the talus sediments. Given that the talus is predominantly composed of cobble- to boulder-sized rocks and lacks fine sediments , a uniform permeability of 1.5×10-6 m2 (d10=80mm) was chosen to represent the entire unit, which is likely conservative considering the larger sediments observed in the field. The sediment lithology is assumed to mirror the overlying rockwall, with approximately 65 % quartzite and 35 % limestone . Quartzite possesses thermal conductivity of λT=5.7 Wm-1K-1 , while limestone has thermal conductivity of λT=2.9 Wm-1K-1 . Additionally, the volumetric heat capacity of limestone (2.1×106 Jm-3K-1; ) is slightly higher than that of quartzite (1.9×106 Jm-3K-1; ). Consequently, the thermal properties of the talus sediments were calculated proportionally to the quantities of quartzite and limestone.

The presence of snow has a strong effect on sediments' temperature. To verify the accuracy of CE-GEOtop's simulated snow water equivalent (SWE), it was compared to the snow cover evolution estimated by in the Opabin watershed, which includes the Babylon talus slope. validated the Utah Energy Balance (UEB) model using extensive field campaigns. The Babylon talus slope falls within the regions covered by oblique-angle terrestrial photography used for UEB model validation, so it is aimed to match the snow cover period rather than the exact SWE values.

The surface aerodynamic roughness was set to 20 mm and the rain threshold temperature to 1 °C in CE-GEOtop . While simulations began in mid-April with measured snow water equivalent (SWE) as the starting condition, CE-GEOtop simulations, driven solely by meteorological data, spanned 2 years to mitigate the uncertainty in initial conditions. Figure b compares SWE from CE-GEOtop and UEB. Although there are differences in the magnitude and timing of peak SWE, both models capture similar snow depletion patterns, coinciding in the timing of snow-free ground. This suggests that CE-GEOtop can reasonably reproduce SWE dynamics in the Babylon talus slope, supporting its applicability in permafrost-related studies.

This section examines the effect of permafrost on spring discharges and temperatures by comparing the results of CG and CG-AO cases. The hydrological year of 2007–2008 was simulated because measured discharge and temperature data of the Babylon basin were available for July–September 2008 . Figure  shows the summer 2008 simulated discharge and spring temperature. Since the GEOtop domain is smaller than the Babylon basin, the discharges are normalized by the area and shown in mmd-1. For profiles P3 and P4, sediment temperature, liquid water content, and ice content for four instants during this period are shown in Appendix B (Fig. ).

The simulations for the scenario involving permafrost (CG) showed that the onset of spring flow occurred approximately 2 weeks later compared to the scenario without permafrost (CG-AO). In early May, the CG case exhibited negative temperatures and low water content throughout the talus, impeding infiltration and groundwater movement. On the other hand, the CG-AO scenario had positive temperatures, enabling groundwater to reach the base of the talus, increasing water content in this area, and resulting in earlier discharge at the model outlet.

These differences are attributable to intra-talus groundwater flow processes. The simulated permafrost within the talus slope is discontinuous and characterized by relatively low ice content, which prevents the formation of a continuous impermeable permafrost layer and consequently precludes supra-permafrost runoff. Thus, the hydrological behaviour is dominated primarily by vertical infiltration through more conductive upper talus layers to deeper, less conductive but more saturated zones near the bedrock.

From mid-May to July, both cases displayed similar behaviour. However, slightly negative temperatures persisted in the middle–lower sections of the talus in the case with permafrost, temporarily delaying groundwater flow by partially freezing and retaining infiltrated water. These conditions explain the subtle differences in discharge observed during May and June. Specifically, in the permafrost scenario (CG), discharge minimums tended to be higher, and the discharge maximums lower, reflecting a more buffered hydrological response to snowmelt and rainfall events. This buffering effect arises from water re-freezing and reduced hydraulic conductivity associated with negative temperatures and higher ice content within profiles P3 and P4 during this period.

The Nash–Sutcliffe efficiency (NSE) coefficient was calculated using daily discharge values, yielding an overall NSE of 0.48 for the CG-AO case and 0.55 for the CG case. According to , daily NSE values greater than 0.5 are considered satisfactory, indicating that the model incorporating permafrost (CG case) provided a more accurate representation of basin discharge. Particularly, both cases aligned more closely with observed discharge during the last week of June and the first week of July. However, from mid-July onward, contributions from late-lying snowpack in zones not considered in the model became more significant (present in the upper part of the Babylon talus slope as depicted in Fig. ). As a result, the simulated discharge showed lower magnitudes than the observed values, matching the observed discharges only during precipitation events. Additionally, when comparing the spring temperatures to the measured data, it is evident that the temperatures from the model with permafrost closely resemble the measured values, while the case without permafrost overestimates the spring temperatures. These findings support the idea that the low water temperature during the summer may be related to the presence of coarse sediments and permafrost occurrence.