Representing subgrid-scale heterogeneity in land-surface models requires balancing realism with computational complexity and -demand as well as with the availability of data needed to describe interactions between surface elements. In this first T-REX implementation, this trade-off is addressed by adopting a tiling approach that focuses on a small number of generalized landscape elements whose interactions follow a broadly applicable physical logic. Consequently, it is most naturally suited to simplified and idealized tile configurations and not necessarily designed to capture every aspect of spatial heterogeneity that may be important for a specific region. Instead, it targets landscape elements that are expected to play a key role across a wide range of environments, and whose interactions follow a consistent structural logic that can be applied throughout the model domain. This approach allows T-REX to start from simple configurations that are straightforward to set up for idealized cases (see Sect.  and ) while providing a framework that can be extended to increase the degree of realism in future simulations (see Sect. ).

The main requirement of T-REX is that the relationships between any two tiles must be describable by a set of characteristic connectivities that reflect their interactions reasonably well. For example, surface runoff generated on elevated terrain will generally flow toward surrounding lowland areas via slopes, suggesting that any grid-cell can be subdivided into elevations, slopes and lowlands, with elevations being hydrologically connected to the slopes which in turn are connected to the lowlands. These characteristic connectivities are used to calculate the spatial relationships between two tiles (see Sect. ) as well as the time-lag factors (see Sect. ) that govern the lateral transport (note that all key variables are summarized in Appendix ). In the present implementation, they are treated as inherent properties of the pair of tiles, invariant in time and independent of the location and composition of a grid cell. Here, the characteristic connectivities merely describe the general connection between two types of surface clusters, based on the internal logic underlying the tile definition of a given setup. However, the characteristic connectivities do not necessarily constitute the degree to which two tiles are connected within a specific grid cell. Since in most cases the inter-tile exchange ultimately depends on the grid-cell composition, the actual connectivities also account for the fractional cover of the tiles within a given grid cell (see below). The characteristic connectivities Λpt,s between a tile t and its siblings s need to be provided to the model in the form of a n×n connection matrix, where n is the number of tiles considered in the setup. T-REX uses several matrices to describe the lateral interactions between tiles, since the connectives depend on the process p that is being considered (Table ).

The most basic of these matrices provides the characteristic connectivities due to direct contact between two tiles (Λctc). Here, the matrix elements (Λctct,s) constitute the relative (geometrical) contact lengths, corresponding to the fraction of the boundary of a surface patch of tile t that interfaces with a patch of given sibling tile s. These connectivities are used to determine the spatial relationships between a tile and its siblings and all those lateral fluxes that are directly proportional to the inter-tile gradient in the corresponding state variables – that is the conductive and diffusive fluxes of heat and water – as well as the redistribution of snow. The matrix elements need to be provided in such a way that (i) if a connectivity Λctct,s>0.0 is assumed between t and s, a connectivity Λctcs,t>0.0 may not be given for the relation between s and t, since this would result in certain fluxes being computed twice per time-step for the same connection. Here, the order should be in such a way that (ii) the connectivity is provided for t, with t being the tile that is encompassed by sibling s or in a way that t is the tile with the smaller characteristic area (see Sect. ). (iii) The sum of all connectivities that are defined for tile t (∑s=1nctΛctct,s, with nct being the number of connected tiles) may not exceed 1., while the sum of all its connectivities, including those that have been defined for the tile's siblings (∑s=1nctΛctct,s+Λctcs,t) may be smaller than 1, implying that a given fraction of the patch's boundary interfaces with non-interacting surface patches.

As stated above, the characteristic connectivities Λctct,s alone may not be sufficient to accurately describe the effective connectivity between a tile and a sibling within a specific grid cell, particulalrly for tiles with multiple connections. Here, an important factor that is being neglected is the relative abundance of the connected siblings within the cell which, for many factors, can be regarded as a reasonable proxy for the likelihood that a larger fraction the tile is connected to a specific sibling rather than to the other siblings. Thus, neglecting the grid-cell composition can produce highly implausible fluxes – for example, a sibling occupying 70 % of the area may receive only half of the runoff generated on the tile, while a second sibling also receives half of the runoff but occupies only 1 % of the grid cell. To avoid such situations, the effective connectivity is assumed to be proportional to the relative abundance of the connected sibling within the cell. In the standard setup, this is implemented by weighting the characteristic connectivities by the cover fraction of each connected sibling (cs), while preserving the overall geometric assumptions of the original connectivity matrix: 1Λctc*t,s=Λctct,s⋅cs∑k=1nctΛctct,k⋅ck⋅∑k=1nctΛctct,k. This weighting is optional and only relevant when a tile interacts with multiple siblings and, if the connectivity between tiles is inherently independent of the grid cell composition, Λctc*t,s can simply be set equal to Λctct,s.

With respect to surface- and subsurface runoff and the corresponding advective heat fluxes, it is assumed, that the movement of water does not happen uniformly across the surface or through the soil, but through dominant hydrological flow paths. Consequently, the runoff fluxes from a tile to its connected siblings may not be directly proportional to the contact lengths. A good example for this are tiles that are connected to two siblings with different surface elevations. Here, the relative contact length may be the same with respect to the two siblings but water at the surface would only run off to the sibling that has a lower surface elevation than the tile. The connectivities are distinct for the micro- and macro-scale fluxes as well as for the surface and subsurface flow. Consequently, the present T-REX setup considers four additional matrices, representing micro-scale connectivities at the surface (Λhfp,micro,srft,s) and below ground (Λhfp,micro,blgt,s) as well as macro-scale connectivities at the surface (Λhfp,macro,srft,s) and below ground (Λhfp,macro,blgt,s). On the macro-scale, connections may even exist for two tiles that do not interface directly, but where the hydrological flow paths are assumed to pass the area of another tile without interacting with the latter. This simplification was required to be able to represent runoff across hillslopes - where water may initially pool in micro-scale depressions – with only a small number of tiles. Hillslope runoff may best be described by a depressional fill–spill cascade, in which a fraction of the runoff initially pools in small scale depressions until it overflows and moves down-slope across the even surface of the slope to again pool in a lower lying depression . For many applications, it may, however, not be feasible to resolve the lateral fluxes representing individual height bands, as proposed by e.g. , and the entire hillslope is subdivided into two tiles, representing the relatively even slope surface – that is the areas where water may not pool – and small scale depressions. In such a setup, the model can not represent hill-slope flow cascades and the inclusion of a flux from the depression tile back to the tile representing the even surface fraction results in a circular connection that has the potential to retain water on the slope indefinitely. To avoid such closed loops, the runoff from the depression tile can be routed to a connected lowland tile directly, bypassing the tile representing the relatively even surface areas of the slope.

On the macro scale, all spatial relationships between connected tiles are merely implicitly included in the parametrizations of the model, via the slopes and the time-lag factors used to determine the macro-scale runoff (see Sect. ). On the micro scale, however, the lateral exchange between a tile and its connected sibling tiles is in large parts determined by spatial relationships that the model determines explicitly, more specifically by differences in surface elevation, the distance between the centers of the patches that the tiles represent and the characteristic interface length of the connection. These are determined based on the characteristic surface elevation, the characteristic area covered by a patch of a given tile, the fraction of the area of the patch that is typically connected to a patch of a specific sibling-tile and by the cover fractions of the tiles, all of which need to be provided as input parameters. A concise overview of the key variables used to describe the spatial relationship between a given pair of tiles is provided in Table .

The lateral hydrological fluxes accounted for in T-REX encompass the diffusive (Sect. ) and gravity-driven fluxes of water (Sect. ) and the micro-scale redistribution of snow (Sect. ), with a concise overview of the key variables used to describe the inter-tile exchange given in Table .

Concerning the lateral heat fluxes, T-REX distinguishes between conductive heat transport (Sect. ) – the heat transfer through direct molecular diffusion – and advective heat transport (Sect. ) – the heat transfer via bulk water movement – with a concise overview of the key variables used to describe the inter-tile exchange given in Table .

For the first example application, we aim to investigate the effect of micro- and macro-scale lateral transport processes on the land's capacity to retain water and the resulting impacts on the turbulent land-atmosphere exchange and surface temperatures. On the macro scale, we separate the grid cell into those parts that generate runoff, the uplands, and those that receive the generated runoff, i.e. the lowlands. With respect to the micro scale, the setup distinguishes between those grid-cell areas where water (including snow) may pool on top of the surface, i.e. local depressions, and those where excess water runs of immediately, i.e. relative elevations. Here, the logic of the tile definition provides the following characteristic connectivities, assuming that the depressions, both on the uplands and lowlands (DU, DL), are fully surrounded by and receive the runoff from relative elevations (EU, EL) and that the surface runoff from the uplands occours via surface channels that are primarily connected to lowland depressions, while the below-ground runoff similarly constitutes inflow to the lowland elevations and depressions (with the actual connectivity being determined by the cover fractions of the two lowland tiles): Λctc=EUDUELDLEU0000DU1000EL0000DL0010Λhfp,micro,srf=EUDUELDLEU0100DU0000EL0001DL0000Λhfp,micro,blg=EUDUELDLEU0100DU0000EL0001DL0000Λhfp,macro,srf=EUDUELDLEU0001DU0001EL0000DL0000Λhfp,macro,blg=EUDUELDLEU000.50.5DU000.50.5EL0000DL0000. The respective fractions are determined by combining the 30 m resolution version of the Copernicus DEM together with the CEH CTI dataset at 15^′′ resolution as follows: In a first step, we use the Whitebox Tools Open Core to fill all local depressions in the DEM data. The difference between this and the original data provides us with the depression depth. Next, we aggregate this data to a 15^′′ resolution thereby also computing the depression fraction as grid-cell area belonging to local depressions relative to the total cell area. Furthermore, we use the Connected-Components implementation of the OpenCV Toolbox to label all grid cells belonging to the same depression. Thus, we can divide the overall depression area by the number of different depressions to gain an estimate of the mean depression size for each 15^′′ cell. These depression characteristics are then cross-referenced with the CTI data. Wherever the CTI ≤ 5.7 we consider the cell to be an upland and therefore assign the depression fraction, depth and average size to the upland region and vice versa for lowlands with a CTI > 5.7. Here, we chose a threshold value of 5.7, because it is the median of the global dataset and because it roughly corresponds to the value commonly used separate lowlands from uplands and slopes . Finally, the data is further aggregated onto the R2B4 grid (≈160 km resolution) which is used for our global simulations. Grid cells with missing values are set to the average of their respective quantities. We conduct ICON-Land-standalone simulations driven by climate forcing from the Global Soil Wetness Project Phase 3 (GSWP3, ). The simulations were performed for the period 1979–2018, with the analysis using the model output from the period 2009–2018. In the following, we compare simulations in which all subgrid-scale lateral transport of water is disabled to simulation in which either micro-scale or macro-scale transport processes are accounted for as well as a simulation in which transport processes on both scales are enabled in the model.

Water is transported laterally mainly in those regions of the world where the water availability at the land surface exceeds the atmospheric moisture demand, at least temporarily. Thus, the model simulates only small lateral subgrid-scale fluxes in the arid and semi-arid regions, while they can exceed 500 mm yr⁻¹ in the humid areas (Fig. a). However, also the humid regions in the tropics feature extensive areas with comparatively little lateral water movement, such as the Amazon basin. Here, it is most often the topography that does not favor large quantities of water being moved laterally through the soil or at the surface. Comparatively small slopes in the contributing areas generate only little surface- and lateral subsurface runoff and, in the model, most of the water drains at the bottom of the soil column, contributing to the channel flow as base-flow below the bedrock boundary (not shown). In contrast, the flat areas in the cold regions often feature larger lateral fluxes, since here the ground is predominantly frozen, at least during the snowmelt season, and water can not percolate deep into the soil but is forced to run off laterally within a comparatively shallow layer close to the surface. In case of the micro-scale fluxes another minor effect stems from the redistribution of snow, which already shifts water from relative elevations to the depressions during the cold season (see Sect. ).

Lateral transport processes enable low-lying parts of the gridcell to store the runoff from upslope areas and allow a redistribution of water between those areas with a large depression storage and those that may not retain water at the surface. Thus, they almost exclusively increases the terrestrial water storage (Fig. d), with marked increases in the total soil water content and in the wetland area (not shown). Additionally, snow redistribution further increases total water storage, as the preferential accumulation of snow in small-scale depressions delays the moment when the grid cell becomes completely snow-free. There are, however, a few grid cells, mainly in the (sub-) polar regions, where the total terrestrial water, in particular the soil water content, is reduced. Here, the lateral transport initially increases the moisture content in the near-surface layers after snowmelt, which increases the heat conductivity of the ground (not shown). This, in turn, raises the ground heat flux and allows a higher fraction of the soil ice to melt. And with more water being liquid, the resulting increase in evapotranspiration and drainage during summer and fall causes an overall reduction in the annual mean soil water content. Substantial increases in the terrestrial water storage are mainly limited to the temperate zone and the (sub) polar region, where the additional retention capacity allows to partially compensate the differences in the seasonality of water availability and (atmospheric) moisture demand. Depressions, in particular, can store parts of the spring snowmelt, with the water subsequently diffusing into the surrounding areas increasing the water availability during spring and summer when potential evaporation rates are high. Here, the effect on the simulated wetland extent is especially pronounced and increases in the inundated area – relative to the potentially inundated area – of up to 70 % across large parts of the temperate and most of the polar and subpolar regions show that lateral exchange processes are essential for maintaining standing water at the surface (not shown). Pronounced effects on the total soil water storage are, however, limited to those regions that feature a deep bedrock boundary, where increases in total soil water of up to 0.2 m often correspond to a relative increase in the soil water content of about 20 % (not shown). In regions where the bedrock boundary is comparatively shallow, such as most of eastern Siberia, there are only small increases in the soil water content, despite large lateral fluxes. In contrast, some large effects are observable in regions where the lateral fluxes are comparatively small, such as the West Siberian Lowlands, where a large depression capacity, deep soils and low potential evaporation rates facilitate the water storage over longer periods allowing even comparatively small lateral fluxes to have a notable impact on the soil moisture and depression storage.

Within the subtropics and the tropics, the effects on the terrestrial water storage are largely negligible (Fig. d). Here, most precipitation transpires and evaporates locally and the model simulates only minor lateral fluxes. Additionally, the temporal offset between water availability and demand is much smaller than in the temperate and polar region and most of the water that is redistributed laterally evaporates or is transpired swiftly rather than being stored in the soil or in depressions over longer periods. Consequently, there are areas in the tropics and subtropics with only a minor impact on the terrestrial water storage but with a notable effect on the latent heat flux (Fig. g). In general, the inclusion of the lateral exchange fluxes increases the latent heat flux notably, mainly in those areas that also feature large changes in the terrestrial water storage where respective effects can correspond to relative increases of up to 10 %. And since the increase in latent heat flux is mainly balanced by a decrease in the sensible heat flux, the Bowen ratio changes by as much as 20 %. Yet even a 20-percent change in the Bowen ratio does not drastically alter the overall surface energy balance on the grid-cell level. Consequently, the effects on the (grid cell) average surface temperatures are comparatively small and the cooling due to the inclusion of the lateral water transport rarely exceeds -0.2 K (not shown). However, the spatial subgrid-scale temperature variability is extremely sensitive to the inclusion of the lateral transport processes, since the local effects on the Bowen ratio in the lowlands and depressions can far exceed the grid-cell mean value. Here, the temperature differences between the warmest and the coldest grid-cell fraction can increase by as much as 3 K (Fig. j), corresponding to an order of magnitude increase across most of North America and Eastern Siberia.

A comparison of the micro- (Fig. b) and the macro-scale fluxes (Fig. c) reveals a large similarity with respect to the spatial patterns and there are only few areas in which the model simulates relatively large micro-scale fluxes but only relatively small macro-scale fluxes (and vice versa). This suggests that the global patterns are mainly controlled by scale-independent factors, such as the state of the atmosphere – which determines precipitation and potential evaporation but also the general thermal state of the soil (frozen or unfrozen) – and the soil properties, and less by the specific topography within a given grid cell. In contrast the overall magnitude is quite different, with the lateral fluxes being predominantly larger for the micro-scale exchange – on average by about 40 %. With respect to the total terrestrial water storage (Fig. e and f), the effect of the micro-scale transport is about 25 % larger than that of the macro-scale transport, with the impact on the total inundated area being almost 50 % larger. In contrast, the increase in latent heat flux is actually about 15 % smaller for the micro- than for the macro-scale lateral fluxes, due to the different ways the water is redistributed within the grid-cell system (Fig. h and i). Here, macro-scale runoff increases the soil moisture in a comparatively large area, i.e. the lowland region, while the micro-scale transport processes predominantly lead to a moisture convergence in a comparatively small part of the grid cell, i.e. the depressions. And since evapotranspiration rates have a non-linear dependency on the water availability, most importantly they are limited by the available energy, a small increase in the soil moisture within a large region has a stronger effect on the latent heat flux than a large increase in soil moisture in a small region. The above, however, also means, that the effect on the maximum (spatial) temperature gradient is notably larger – on average by about 35 % – for the micro-scale fluxes (Fig. k and l). Due to the moisture convergence in a comparatively small part of the grid cell, the local temperatures within the depressions predominantly show a stronger reduction due to micro-scale fluxes, than the temperatures in the lowland fraction due to the macro-scale fluxes. Finally, it should be noted that the effects in the simulation that enable both micro- and macro-scale transport processes are not necessarily the same as combining the effects from the micro-scale-flux-enabling and the macro-scale-flux-enabling simulations. The lateral-movement of water in the fully-enabled simulation is very close to the sum of the effects in the two simulations that only enable one flux component (Fig. a–c). However, in case of the terrestrial water storage the former is about 25 % smaller than the sum of the effects (Fig. d–f) and for the effect on the latent heat flux it only amounts to about 60 % (Fig. g–i). This discrepancy arises because the terrestrial water storage and the latent heat flux are not only limited by the available water, with former additionally being limited by the pore volume and the potential depression storage and the latter by the available energy. Thus, an increase in the laterally transported water does not necessarily entail an increase in the water storage or in evapotranspiration.

In the second application, we aim to investigate the effect of the lateral heat transport on the subgrid-scale temperature variability. Here, we set up the model to represent typical patterned-ground structures – e.g. non-sorted circles, ice-wedge polygons – that are often found in the arctic tundra. The presence of such patterns is limited to periglacial regions and, while the coverage of environmentally suitable spaces has been estimated at around 3 million km² for ice-wedge polygons , other studies conclude the actual extent of the polygonal tundra to be merely around 0.3 million km² . However, these structures present an ideal test case for our model, since soil organic matter concentrations, and with that hydrological and thermal soil properties, can vary dramatically on the sub-meter scale . For this application, we assume circular structures, that consists of an organic-rich center (C), with a volumetric organic matter fraction of 95 % at the surface – i.e. the uppermost 0.1 m of the soil, encompassed by the rim section (R), which has a near-surface soil organic matter fraction of 35 %. These two are in turn surrounded by an outer section (O) with mainly mineral soil throughout the below ground column, i.e. an organic matter fraction of 5 % at the surface, resulting in the following connectivity matrix: Λctc=CROC010R001O000. We assume the radius of these circles to be evenly subdivided into the center, rim and outer section, with water draining from the circle either at the soil bedrock interface or laterally through the outer section, with the former only being possible if the active layer is sufficiently deep. For simplicity reasons, we assume these circles to be omnipresent in the permafrost region, allowing us to conduct our analysis for the pan-Arctic average rather than focusing on a specific grid cell. We conduct ICON-Land-standalone simulations driven by climate forcing from the Global Soil Wetness Project Phase 3 (GSWP3, ). The simulations were performed for the period 1979–2018, with the analysis using the model output from the period 2009–2018. For different setups we compare the spatial, below-ground temperature variability, i.e. the temperature differences between center, rim and outer section, between simulations in which the lateral heat transport is disabled and simulations with the heat transport active.

In a first step, we assume a radius of one meter, with center, rim and outer section each covering 0.33 m. Additionally, we assume the circle to be level so that there is no vertical offset between the surface of the sections. The latter may not be the most common configuration, since the circular structures often have a distinct micro-topography, e.g. an elevated center in case of a hummock-type structure, but it makes the analysis much more straight-forward, since effects due to the lateral water transport and a lateral coupling with a vertical surface offset are (largely) absent. This provides the following connectivity matrices for connections due to preferential hydrological flow paths, assuming that they are the same for above- and below-ground fluxes (note that macro-scale fluxes are not present in this particular setup): Λhfp,micro=CROC010R001O000. More importantly, this setup allows us to compare our simulations with those of the Dynamic Soil Model DynSoM, a pedon-scale soil model, which was developed to explore the movement of energy specifically in permafrost-affected soils. DynSoM simulates the key physical variables, such as soil temperature, and soil water and ice content, as well as the surface and below-ground heat fluxes both in the vertical and horizontal direction (note that the lateral movement of water is not represented in the present DynSom setup). Here, DynSoM, in contrast to ICON-Land, is a true 2D model which explicitly resolves the soil in one horizontal direction. For the present setup, the model uses a 10 cm horizontal grid-spacing and a variable spacing in the vertical direction, with the uppermost meter of the soil column being resolved in 10 cm steps. The simulations with DynSoM are based a 1 m soil transect of a non-sorted circle in the Chersky region (; Supplements profile CH-E [note that in the original transect the inner circle – i.e. the circle center – features the low organic matter concentration and the inter circle area – i.e. the outer section – the high concentrations, but the order can simply be reversed without affecting the results of the DynSoM simulation]) and are forced with CRUNCEP data for the respective region. The setup of the DynSoM soil transect differs to the synthetic ICON-Land setup in that the organic rich region (surface organic matter fraction of 95 %) and the organic poor region (organic matter fraction of 5 %) both only cover 0.2 m, while the intermediary region (surface organic matter fraction of 35 %) covers 0.6 m. Furthermore, the mineral soil below the top layer also shows differences along the transect, while the present ICON-Land setup assumes spatially homogeneous mineral soil properties within one grid cell. However, since we do not attempt a site-level validation of our model but merely aim to compare its general behavior to a model that explicitly resolves the horizontal exchange on the pedon-scale, we assume that respective setups of the models are sufficiently similar.

In DynSoM, soil organic matter predominantly lowers the soil heat conductivity, which impedes the ground heat fluxes during the snow-free season (not shown). When neglecting the lateral exchange, this entails notably lower soil temperatures in the organic rich center than in the rim and the outer section, with differences at the surface starting to emerge after the snowmelt in early summer (Fig. a and b). The signal subsequently propagates downwards and the peak temperature differences in depth > 2 m are being reached the following spring. At the same time, the lower heat conductivity in the center also results in a more pronounced heating of the surface during late summer and a less pronounced cooling during early fall, with the near-surface temperatures exceeding those in the rim and the outer section between August and October. However, as soon as the soil is insulted by an increasingly thick snow cover in fall, also the near surface temperatures in the center fall below those of the rim and the outer section. During the cold season, the lower heat conductivity of the organic matter plays a subordinate role mainly because the soil is insulted by the snow cover. Additionally, ice has a much higher heat conductivity than liquid water and the relative differences in the near-surface heat conductivity between the sections are much smaller when the soils are frozen. Consequently, the cooling of the center, relative to the rim and the outer section, during summer is not fully balanced by a relative warming during the cold season. A temperature convergence during the cold season is observable in some soil layers, however, the temperatures in the center remain below those in the rim and the outer section even in these layers. Here, the temperature differences between center and rim and between center and outer section are largely similar. However, the signal is slightly more pronounced for the center-rim comparison, despite the near-surface soil organic matter concentrations in the center actually being closer to those in the rim than those in the outer section. The reason for this lays in the fact that, at depth > 5 m, the rim has a higher heat conductivity than the outer section, which has a notable effect on the magnitude of the bottom heat flux (not shown). Here, DySoM does not employ a zero-flux condition, but a prescribed temperature. With 0 °C, this temperature is higher than the simulated soil temperatures in the superjacent layers resulting in a positive bottom heat flux. Owing to the higher conductivities, the rim exhibits a larger warming at the bottom of the soil column, hence, higher temperatures than the outer section. Thus, the temperature differences to the cooler center are also more pronounced for the rim.

In the ICON-Land simulation, the temperature differences between the sections are equally driven by differences in the top- and f). Consequently, the overall effects are quite similar between the two models, especially given the fact that we are comparing simulations of a specific site (DynSoM) to the pan-Arctic average based on a synthetic setup (ICON-Land). However, there are also important differences, most notably in that large temperature differences between the sections are confined to the uppermost meter of the column and that the near-surface temperatures during the cold season are actually higher in the circle center. Both of these differences are related to the hydraulic properties of the soil organic matter – which can differ quite strongly to those of the mineral soil fractions – and to the assumed drainage pathways of the circle. Contrary to the DySoM simulation, the soil in the ICON-Land simulation is not necessarily water saturated and the degree of saturation of the soil in a given circle section depends on the overall water holding capacity, the infiltration rates and the position on the drainage path. Here, organic matter has a higher porosity than the mineral soil fractions – hence can hold more water – and also a higher hydraulic conductivity – hence allows for higher infiltration rates. Furthermore, the circle center drains laterally through the rim and the outer section and without vertical offset between the three sections, lateral fluxes mainly occur if the rim and the outer section are less saturated than the circle center. As a result, the soil column is more water saturated in the organic rich center than in the rim and the outer section, which entails higher heat conductivities below the organic layer (not shown). During the snow-free season, when the soil warms from the top, the combination of lower heat conductivities at the surface and higher heat conductivities below does not only reduce the overall heat uptake, but also enhances the downward transport, resulting in a reduced vertical temperature gradient. This difference in heat distribution causes the cooling of the center (relative to the rim and the outer section) to be more pronounced closer to the surface and less pronounced deeper within the soil. Similarly, the higher heat conductivities also lead to a reduced vertical temperature gradient during the cold season. However, during this period, the column cools from the top and the reduced temperature gradient corresponds to an enhanced upwards heat transport in the center and, consequently, higher near-surface temperatures than in the rim and the outer section. Here, the respective effects are more pronounced in the center-outer-section than in the center-rim comparison, i.e. larger positive and negative temperature differences close to the surface and a slightly less pronounced relative cooling below depths > 1 m. This is because the near-surface soil organic matter concentrations in the center are closer to those in the rim than those in the outer section and because the soil column in the latter, as the outermost circle section, is predominantly less water saturated than in the rim.

As discussed above, there are some differences in the general effect of a heterogeneous soil organic matter distribution between the models. Nonetheless, they agree well on the impact of the lateral heat fluxes on the spatial temperature variability. In both simulations, the temperature differences between the circle sections are strongly reduced and any notable spatial variability is largely limited to the uppermost meter of the column as well as to the snow-free season (Fig. c, d, g and h). Here, the fact that the lateral heat fluxes have a stronger influence on the temperature differences at greater depths has two main causes. On the one hand, the temperature differences deeper within the soil are not only balanced by the lateral heat fluxes on the respective layers, but they are also reduced since the vertical heat exchange with the superjacent layers is spatially more homogeneous due to the lateral heat exchange in the layers above. On the other hand, the temporal temperature variability is strongly dampened with increasing depth. Thus, while large near-surface temperature differences between the sections may not be fully balanced on the short term, the signal is attenuated while propagated downwards to the extent that temperature differences in the deeper soil layers only emerge if the longer-term mean temperature near the surface is notably different.

Overall, the above results suggest that, at the pedon-scale, the lateral heat fluxes are sufficiently strong to largely balance the below-ground temperature differences resulting from the spatial variability in the surface heat fluxes. Furthermore, it appears that these effects can be captured reasonably well by LSMs employing a tiling-approach. The latter allows us to use ICON-Land simulations to investigate the extent to which the ability of the lateral heat exchange to the equilibrate soil temperatures depends on the horizontal distances considered. For a radius of 10 m, with the circle sections each covering 3.33 m, we find that the temperature differences between center, rim and outer section above 1 m, appear to be hardly affected by the lateral heat exchange (Fig.i–l). This strongly suggest that at these length scales the lateral heat exchange is too weak to balance the short-term effects resulting from the differences in the seasonal variability of the surface heat fluxes. However, the near-surface temperatures in the center do in fact show a slight longer-term increase due to the lateral fluxes (e.g. for the center-outer-section comparison this is visible in the [positive] temperature differences during the cold season being slightly larger) and the resulting annual mean temperature is very close to that in the rim and the outer section. Consequently, in those soil layers that are less sensitive to the seasonal variability in the surface heat flux, the spatial temperature variability is reduced substantially. At depths > 1 m, there are no marked temperature differences neither between the center and rim nor between center and the outer section. Here, it should be noted that the absence of large temperature differences comparatively close to the surface is partly caused by the rather coarse vertical resolution of the model, since the damping of the temporal temperature variability depends on the latter. With a thickness of almost 3 m, the first soil layer below a depth of 1 m possesses a substantial thermal inertia and the temporal variability of the vertical heat fluxes only has a small effect on the temperatures of this layer. Thus it is highly plausible that, in reality or with a higher vertical resolution, the spatial temperature variability extends much further downward than in our simulation. Increasing the radius to 100 m, with the sections each covering 33.33 m, further reduces the lateral heat fluxes. As a result, the latter are not only too weak to balance the short-term effects due to the variability of the surface heat fluxes, but also the longer-term mean state of the near surface layers is substantially different. Consequently, notable differences in the below-ground temperatures persist even at depths > 4 m and, for these length scales, the impact of the lateral heat fluxes appears to become negligible (Fig. m–p).

As stated above, a level surface of the circle may not be the most common configuration as in reality we often find high-centered hummock-type structures, or low-centered hollow-type structures. On the one hand, the vertical offset between the sections of the circle affects the soil thermophysical properties, since it determines the flow of water, hence, the saturation of the soil and, thusly, the heat conductivity. On the other hand, the ground features a strong vertical temperature gradient during most of the year, with the vertical offset between two sections determining the distance to the surface of any two interfacing points (Fig. ). Thus, in the final example we investigate the extend to which the micro-scale topography of the circle effects the spatial temperature variability. First we focus on a low-centered circle with a 1 m radius, in which the surface of the center and outer section are both located 0.1 m below the surface of the rim, with the connectivities for hydrological flow paths being set in the following way: Λhfp,micro=CROC000R0.500.5O000. For, this configuration we find notable differences between the temperatures of the sections despite the small horizontal extent of the circle (Fig. q–t). The near surface soil layers in the center are cooler during summer and warmer during winter, with the differences being larger in the center-rim than in the center-outer-section comparison. Here, the temperature differences between center and rim are less determined by the differences in heat conductivities and mainly reflect the vertical temperature gradient across the uppermost 0.1 m of the soil. The lateral heat fluxes are driven by the temperature differences of two points with the same absolute vertical position, hence the surface temperatures of the center are essentially adjusting to the temperatures at 0.1 m depth in the rim. During the summer the (downwards negative) vertical temperature gradient near the surface reaches a maximum of about 20 K m⁻¹ and during winter roughly half that value (downwards positive), meaning that temperature differences of up to +2 and -1 K, are fully explainable by the vertical offset between the two tiles. Furthermore, since the vertical temperature gradient averaged over longer-periods is close to zero, there are no notable differences in the temperatures in the deeper soil layers. The temperature differences between center and the outer section (which have the same vertical offset to the rim) show, however, that there is also a notable variability in the temperatures of points with the same absolute vertical position, i.e. that the surface temperatures in the rim and the outer section do not fully adjust to the temperatures at 0.1 m depth in the rim. This variability is much larger for the low-centered than for the level circle (Fig. h and t), indicating that the lateral heat fluxes are smaller in case of the former. The reason for the reduced lateral exchange lies in the different levels of saturation of the rim. Due to its raised position – in case of the low-centered circle – the rim drains more readily into the other sections, resulting in overall drying of the soil column. This in turn leads to a lower heat conductivity and, consequently, reduced lateral fluxes. For a high-centered structure, in which the surface of the circle center is located 0.1 m above the surface of the rim, which in turn is located 0.1 m above the surface of the outer section – resulting in the following connectivity: Λhfp,micro=CROC010R001O000 we find that the lateral heat fluxes have essentially the same effect as for a low-centered circle. However, since the center is elevated relative to the rim and the outer section, it features higher near surface temperatures during summer and colder temperatures during winter (Fig. u–x), with the magnitude of the effects increasing with the vertical offset. As a result, the temperature differences, resulting from the lateral heat fluxes actually oppose the ones in the simulation in which the lateral fluxes are being neglected.