This hydrological model is designed to simulate the routing and accumulation of surface water over a given topography. Its core principle is that water flows across the landscape until it encounters depressions where it can be stored, allowing for the formation of lakes or oceans. A key feature of the model is that these lakes/oceans are not fixed boundaries: they can completely evaporate under arid conditions and reappear during wetter periods. When a lake reaches its maximum storage capacity, the model transfers the overflow to downstream depressions. This mechanism enables the transfer of water volumes between reservoirs provides a way to trace drainage pathways and reconstruct watercourses. To simulate these dynamics efficiently, the model makes several simplifying assumptions. In principle, surface-water flow is governed by the conservation of mass and momentum equations integrated overflow depth. However, solving these equations at high resolution and over large domains is computationally prohibitive. Instead, our approach assumes that water follows the steepest topographic slope until it reaches an unfilled depression.

The hydrological model is based on the existing frameworks of and . This framework is specifically designed to handle complex terrains with numerous depressions, like craters for instance (Fig. a). It uses a binary tree data structure, known as the depression hierarchy, where each node represents either a watershed of a depression (“leaf” depression) or a meta-depression formed by the merging of two filled depressions (Fig. b). The depressions are interconnected in a network that defines the water flow paths within the binary tree (Fig. c). This structure significantly enhances computational efficiency, as the connections between depressions are only determined once during the pre-processing stage. Additionally, pre-computed functions link the lake elevation, water volume, and lake area for each depression, enabling quick estimation of lake surface or elevation from water volume. All of this information is gathered in a hydrological database that will be described in Sect. . The implementation of the hydrological model is described in Sect. .

A word of caution: the model is originally designed for watershed-scale simulations, and a key limitation for its planet-scale applicability is that we assume that water transfer between depressions occurs instantaneously. This simplification may lead to a slight underestimation of evaporation during transfer and neglects potential transient storage effects. Furthermore, each node in the binary tree represents the watershed of a depression with its own storage capacity, meaning that intra-watershed water routing is not explicitly simulated. Infiltration and subsurface flow processes are also not included at this stage of model development. Some of these limitations can be mitigated through post-processing, as described in Sect. .

The hydrological database aims to store all the information derived from the high resolution map that needs to be computed only once. This approach ensures computational efficiency, as the model only needs to load and apply the pre-computed database. To build this hydrological database, we used the Depression Hierarchy algorithm to calculate the binary tree (Fig. c, Sect. ) and we pre-computed the depression information (Sect. ) and functions that establish the link between volume lake, elevation lake and surface area lake for each depression (Sect. ). All the information saved in the hydrologic database are listed in the Table .

The hydrological model governs the redistribution of excess water along the depression hierarchy graph. Using the pre-computed functions stored in the hydrological database, the model reconstructs and updates the hierarchical structure at runtime based on the parameters summarized in Table . For each depression i (represented by a node in the binary tree), the variables describing its hydrological state: water volume Vi, outgoing discharge Qiout, active state ai, lake surface area Ai, and elevation Zi are continuously updated during the simulation. The active state ai is a flag that indicates the highest level of the depression that contains water, i.e. the leaf or meta-depression that have surface water in contact with the atmosphere. A filled depression with a filled parent depression is considered inactive (ai=0) since its water volume is not in contact with the atmosphere. On the contrary, a filled depression with an empty parent depression is considered active (ai=1) since the water is in contact with the atmosphere. Only leaf depressions can be active even if they are empty.

The simulation outputs are saved in a NetCDF file and can be saved at each iteration, depending on the frequency specified by the user, or only at the end of the simulation. The saved data include the lake surface elevation Zi, the area covered by the lake Ai, the volume of water stored in the depression Vi, the active state ai (indicating whether it is active or inactive), and the water overflow Qiout for each depression. These outputs enable the reconstruction of water reservoir distributions and the analysis of hydrological flows for comparison with observations. The hydrological model outputs are then post-processed to analyze the water volume distribution, water depth and accumulated outflow in each depression. To compute the water depth, the model uses the lake elevation Zi, the ID label map and the elevation map. The ID label map is used to identify the area covered by the depressions associated. The water depth map is then computed by subtracting the elevation of each grid cell of the depression i from the lake elevation. The water depth map provides insights into the water distribution across the planet and the topographic features of the depressions. The volume distribution map can be obtained by multiplying the water depth map by the area of each cell. The volume distribution map provides information on the amount of water stored in each depression. The accumulated outflow map is generated by mapping the overflow discharge Qiout for each depression, either normalized by the lake surface area or distributed across the contributing watershed area. While the model outputs the accumulated overflow Qiout for each depression, it does not explicitly resolve the flow pathways between depressions. To overcome this limitation and reconstruct a continuous river network, a post-processing procedure is applied. The first step consists in modifying the original topography by assigning the lake water levels Zi to the corresponding depressions. This creates a new topographic surface where water can flow continuously across lake basins which overflow. Flat zones introduced by the filled lakes are then corrected using the “resolve_flats” function from the Pyshed library , allowing proper flow routing. The flow direction is computed using the D8 algorithm , and flow accumulation is then calculated following the method of . Based on this accumulation map, the main drainage paths are extracted with the “extract_river_network” function of Pyshed. Finally, the overflow discharges from the hydrological model are assigned to the extracted river channels, producing a spatially explicit reconstruction of surface water pathways. This step provides a physically consistent representation of surface runoff and facilitates comparison with observed valley networks on Mars.

We used the topographic map from MOLA Mars Orbiter Laser Altimeter, which covers all the Mars' surface with a fine resolution of 463 mat the equator (1/128° per pixel, Fig. a). This dataset provides a detailed digital elevation model (DEM) of Mars which represents approximately 1 billion cells (23 040 × 46 080 cells). This high-resolution topographic data is crucial for accurately identifying depressions and their watersheds. It is clear that the MOLA topographic map does not fully represent the topography of Mars during the periods when liquid water was stable on the surface. However, this current topographic map provides, on several preserved areas, mainly in the Southern Hemisphere, a good approximation of the past topography during the Noachian periods (4.1 to 3.7 billion years ago) when most of the fluvial features were formed . While the current MOLA topographic map accurately represents present-day Mars, it does not provide a global representation of the ancient surface topography required to study early Mars hydrology. Then, we also applied our model on a reconstructed topographic map of Mars built by which remove the True Polar Wander (TPW) to better represent the global hydrology on early Mars. However, this reconstructed topographic map is only available at a lower resolution of 1 pixel per degree, which limits the ability to capture fine-scale topographic features and depressions.

Despite its limitations, the topographic analysis of MOLA data has provided significant insights into the hydrological activity on Mars, revealing a variety of geomorphological features indicative of past water flow (Fig. a). Valley networks, such as those identified by and , suggest the presence of fluvial processes that shaped the Martian surface, with drainage densities and morphologies pointing to sustained precipitation and surface runoff. Additionally, the identification of open- and closed-basin lakes, as documented by and , highlights the role of topographic depressions in capturing and storing water. These lakes often exhibit inlet and outlet valleys, providing evidence of hydrological connectivity. Furthermore, deltas, such as those observed by , mark the confluence of sediment-laden streams with standing bodies of water, offering clues about sediment transport and deposition processes. Finally, potential shorelines, including the Arabia and Deuteronilus levels described by , suggest the existence of ancient oceans in the northern lowlands.

Additionally, the recent crater database of 2020 contains more than 385 000 craters with a diameter larger than 1 km. The density of craters is not uniform across the planet, with a higher density in the Southern Hemisphere and a lower density in the Northern Hemisphere (Fig. a). This gives insights into the minimum number of leaf depressions that can be contained in the hydrological database.

To highlight the application of our model, we first present results at the global scale, then focus on three well-studied systems: Nirgal Vallis, Jezero Crater, and Gale Crater (Fig. b–d). Nirgal Vallis (Fig. b) is one of the longest valley networks on Mars, extending over 700 km. Its morphology makes it an ideal case study for understanding fluvial processes and the potential for past water flow on Mars. High-resolution MOLA data enable detailed hydrological simulations within this catchment. Jezero Crater (Fig. c) and Gale Crater (Fig. d) are of particular interest because they are the landing sites of the Perseverance and Curiosity rovers, respectively. Both craters host well-preserved deltas and sedimentary deposits such as conglomerates and stratified layers, offering strong evidence that they once contained standing bodies of water. By focusing our model to these sites, we can evaluate its ability to reproduce the formation of paleolakes.

The simulations presented here should be interpreted as an idealized reference case, in which precipitation and evaporation are prescribed as spatially uniform over the planetary surface. In this framework, the resulting steady states do not aim to reproduce a specific early-Mars climate scenario, but rather describe water distributions that are topographically reachable under globally distributed water input. The exploration of the parameter space allows to evaluate the sensitivity of the model to different climate conditions and initial states, and to identify the range of conditions that can lead to the formation of lakes and river networks. The homogeneous precipitation/evaporation pattern allows the identification, at the global scale, of water flow pathways and the locations of possible lakes.

This section defines the range of parameters explored in this conceptual application, including the evaporation rate E and the GEL. The evaporation rate E can be estimated using the following bulk aerodynamic formula: 9E=ρairCdvwindqsat(Tsurf)-qair where ρair (kg m⁻³) and vwind (m s⁻¹) are respectively the volumetric mass of air and the wind velocity, Cd is the aerodynamic coefficient (unitless), qsat(Tsurf) is the water vapor mass mixing ratio at saturation at the surface which depends on the surface temperature Tsurf, and qair is the mixing ratio in the atmospheric layer. Assuming Cd=2.5×10-3, ρair=0.02 kg m⁻³, qsat=0.01 kg kg⁻¹, neglecting qair and by varying the wind velocity vwind between 1 and 10 m s⁻¹ , the evaporation rate E ranges from approximately 0.01 to 1 m yr⁻¹. This conceptual study proposes to test three evaporation rates: 0.01, 0.1 and 1 m yr⁻¹. Additionally, to test the robustness of this model, a non-realistic evaporation rate of 10 m yr⁻¹ is also tested. Such value is much higher than the maximum evaporation rate observed on Earth, which is around 3 m yr⁻¹ . This additional evaporation rate will allow to evaluate the model behaviour under extreme conditions and assess its ability to handle large volumes of water.

Moreover, several GEL values are investigated. Present-day Mars is estimated to have a GEL of approximately 34 m , while the late Hesperian era is associated with a GEL of 64 m . Other studies suggest higher values, such as 137 m , 550 m , and even up to 1500 m and 1970 m . To encompass this variability, four representative GEL values are selected for this conceptual study: 1, 10, 100, and 1000 m.

To evaluate the sensitivity of the hydrological model to initial conditions, this GEL values are distributed in three distinct initial water distributions: a homogeneous distribution across the planet (Fig. a), in the northern lowlands (75° N, 60° W, Fig. b), and in the Hellas crater (40° N, 60° E, Fig. c). These configurations allow us to assess the model response to varying initial states. Considering the tested evaporation rates, GEL values and initial states, a total of 48 simulations were performed to explore the parameter space.

By using the MOLA topography map (Fig. ), the hydrological database was built on computing cluster (32 cores 2 AMD EPYC 7302 16-Core, 16 GB of memory per core) in less than 2 d of computation time (41 h equivalent to 1312 CPU hours). The database contains 5 967 453 leaf depressions, 5 967 452 meta-depressions, i.e. a total of 11 934 905 depressions. The map of ID watershed and the watershed area Aiw were used to build the Fig. a. It shows the extent of the watersheds of leaf depressions near to the Gale crater (9777 leaf depressions on this figure). This figure illustrates the high resolution of the database, higher than crater scale, which provides detailed information on the depressions and their associated watersheds. By looking at the craters on this map (circles contour plot, black lines), several depressions can be identified into them, including Gale crater, which is localized by the white star. The presence of depressions in the craters is due to local pits that are present in the craters.

As mentioned in the Sect. , the hydrological functions are pre-computed for each depression. The hydrological functions of a random depression (i=4281753, white point in the map Fig. a) which is close to the Gale crater, are shown in Fig. b. The pre-computed lake volume Vil and lake area Ail are plotted as functions of the lake elevation Zil. As expected, the lake volume monotonically increases with the lake elevation, while the lake area can have a flat or increasing trend. The lake area function can not strictly increase because the lake area can remain constant with the lake elevation when a lake is bordered by vertical cliffs for instance. The hydrological functions provide insights into the behavior of the depressions and their associated lakes, enabling the model to accurately simulate the water redistribution process.

Simulations were conducted on a supercomputer (32 cores 2 AMD Genoa EPYC 9654, 4 GB of memory per core), achieving a computational speed of approximately 22 000 iterations per day of simulation. A total of 48 simulations were performed to test the model sensitivity to the initial state (Fig. a–c), the amount of water present on the planet and the fixed evaporation rate (Fig. ). The simulations were run for a maximum of 100 000 iterations. As shown in Eqs. () and (), the precipitation rate is directly linked to the lake area. A constant precipitation rate (i.e. a constant P/E ratio) indicates that the lakes are stable and that the model has reached a steady state (Fig. d–g).

The timescale to reach equilibrium (Fig. d–g) in our model depends primarily on the evaporation rate (E), the adaptive time step Δt and, to a lesser extent, on the Global Equivalent Layer (GEL) of water. The timescale varies linearly with the evaporation rate, as this parameter controls the rate of water transfer between depressions. Regarding the GEL, the timescale generally increases with GEL because a larger GEL implies that more water must be redistributed before equilibrium is reached. However, a high GEL also corresponds to a larger cumulative lake area, which increases the evaporated volume and thereby accelerates the redistribution of water. This explains why GEL has a secondary effect on the timescale to reach equilibrium. In our simulations, the timescale ranges from approximately 200 years for a GEL of 1 m and an evaporation rate of 1 m yr⁻¹, to over 100 000 years for a GEL of 1000 m and an evaporation rate of 0.01 m yr⁻¹. These timescales are consistent with geological constraints on valley network formation on early Mars, which suggest that they could have persisted for 105 to 107 years .

The results also demonstrate that the model is not sensitive here to the initial state, as all simulations converge to the same P/E ratio for a fixed GEL. The Fig. d–g illustrates, for homogeneous precipitation, that the final water distribution on the planet depend only on GEL value, s demonstrated in Appendix . For a fixed GEL and homogeneous precipitation, the simulations confirm that the model is able of redistributing water across the planet and converging to a unique steady state. As expected, with this homogeneous climatic pattern, the converged P/E ratio depends only on the total lake surface area which is directly related to the GEL. It is also observed that the higher the GEL, the higher the final P/E ratio, as the water surface has a larger exchange area with the atmosphere. The precipitation pattern also controls the final water distribution and the resulting P/E ratio, as shown by simulations with a latitudinally varying precipitation pattern (Appendix ).

The main limitation is the model dependence on the topography map which is a crucial input for the construction of the hydrological database, as it defines the watershed of depressions and their spillover point. The resolution of the topography map can significantly impact the model ability to accurately represent the hydrological features of the planet. We tested the model with the MOLA topography map degraded to 1 pixel per degree. The model was run with a GEL of 100 m and an evaporation rate of 1 m yr⁻¹. The model is compared with results using the high resolution topography map (128 pixel per degree). The Fig. b–c show the water distribution according to longitude and latitude. As Fig. , the histogram shows the relative water volume as a function of latitude/longitude degree and the solid line shows the cumulative relative water volume. While the distributions of water reservoirs are relatively similar according to longitude, the distributions according to latitude are significantly different, notably in the Northern Hemisphere. The high resolution model transfers more water to the Northern lowlands, while the low resolution model shows a more uniform distribution of water across the area between 90° N and 0°. This difference can be explained by the fact that the smoothing of the topography map can increase the elevation of the spillover point and consequently increase the storage capacity of the depressions. The same work was done with a pre-True Polar Wander (TPW) topography map (Fig. a). This topography map contains significantly less depressions (craters) than the current MOLA topographic map and the Tharsis region is not represented. As expected, the pre-TPW model shows a significant difference in the distribution of water reservoirs compared to the model base on MOLA topography. If the pre-TPW topography accumulates less water in the Northern Hemisphere, around 45 % of the total water volume is stored in lower latitudes, between -15 and 60° N, placing one of the main reservoirs closest to the equator than in the current topography (Fig. c). In the Southern Hemisphere, the water is mainly concentrated between 0 and 45° in the Hellas and Argyre basins.

According to the longitude (Fig. b), we can observe an absence of water accumulation in the area between 30° W and 30° E, characterized by the flat curve of the cumulative relative water volume. In the Southern Hemisphere, this area corresponds to the Noachian highlands, which are preserved in the MOLA topography. The previous simulations with the MOLA topography show that these areas are favorable for water storage, as they are heavily cratered and contain many depressions. The absence of water accumulation in the area between 30° W and 30° E in the pre-TPW topography, and the reduced water storage in the northern lowlands compared to MOLA (whether high or low resolution), suggests that the differences arise from fundamental topographic changes rather than resolution effects alone. The degraded MOLA topography (1 pixel per degree) maintains water pooling in the -30 to 30° E longitude band and in the northern lowlands, demonstrating that coarse resolution alone does not eliminate these features. The pre-TPW topography, by contrast, shows a qualitatively different hydrological configuration: the vast interconnected northern ocean present in both MOLA versions is largely absent in the pre-TPW case. This indicates that the True Polar Wander rotation, crater relaxation, and volcanic infilling that occurred between the pre-TPW and present-day have substantially modified the topographic depressions. This result suggests that using the current topography map brings some challenges to understand early Mars. However, models of water surface flow on early Mars are limited by our ability to reconstruct the ancient surface topography.

To fully leverage the potential of such a hydrological model, it would be essential to use a high-resolution topography map that accurately represents the surface of early Mars. This would involve reconstructing the topography by removing the effects of the True Polar Wander (TPW) event . Additionally, the exclusion of younger craters that formed after the Noachian period would be crucial, as their presence introduces additional reservoirs that were not part of the ancient Martian landscape . This reconstructed map would provide a more accurate representation of the ancient Martian surface, enabling better simulations of early hydrological processes and facilitating comparisons with geological and geomorphological observations. An additional improvement for reconstructed ancient topography maps would be to artificially “recraterize” younger terrains. This could be achieved by statistically adding impact craters using a realistic size-frequency distribution, consistent with the expected crater population for the Noachian or Hesperian epochs. Such an approach would help restore the hydrological storage capacity and drainage patterns that would have existed prior to resurfacing events, thereby improving the fidelity of hydrological simulations on early Mars.

In these results, we assume no infiltration (β=1) and neglect subsurface flows. However, incorporating infiltration and groundwater flow processes could significantly improve the model ability to simulate the Martian hydrological cycle, offering valuable insights into the interactions between surface water and the Martian subsurface. The process of infiltration and groundwater flow can significantly influence the distribution and retention of surface water on the planet. By adding these processes to our hydrological model, we would expect a redistribution of surface water driven by subsurface flow, with contrasting effects depending on topographic setting. In particular, surface water retention would likely be overestimated in high-elevation depressions, where infiltration would induce lake volume losses through recharge of the subsurface. Conversely, surface water retention would likely be underestimated in low-elevation depressions, which would act as groundwater convergence zones, receiving additional water through subsurface flow and thus sustaining larger or more persistent surface water bodies.

This implementation would allow the simulation of processes such as aquifer recharge, subsurface flow pathways and groundwater discharge/sapping. Developing a global groundwater flow model, inspired by terrestrial hydrological frameworks and previous Martian studies , could further enhance the integration of surface and subsurface hydrological processes to compare, for instance, the model outputs with the U- and V-shaped valley networks profiles which result from the groundwater sapping and surface runoff, respectively.

In this conceptual study, the precipitation rate P, which depends on the evaporation rate E and the lake surface area, is considered homogeneous. This simplification allows the model to simulate a steady state, where the hydrological system reaches equilibrium. The steady-state results can be analyzed using metrics such as the X ratio or the Aridity Index (AI) . The X ratio, defined as the ratio of evaporated water to precipitated water, provides a quantitative measure of the hydrological balance in a region. Similarly, the Aridity Index (AI), calculated as the ratio of lake area to the total watershed area, offers insights into the spatial distribution of water and the relative humidity of different regions. These metrics also enable comparisons with previous works.

By coupling the hydrological model with a GCM, it becomes possible to explore the interactions between atmospheric circulation, spatial and temporal precipitation/evaporation variations. This coupling would provide a more comprehensive understanding of Mars' water cycle, enabling the study of dynamic processes such as the formation and disappearance of waterbodies according to the seasons. Additionally, the model could be extended to study transient hydrological states, such as the formation of large valleys in the Martian highlands caused by catastrophic paleolake overflows . These events, driven by sudden breaches in crater rims or other topographic barriers, could have significantly reshaped the Martian surface. By simulating such transient events, the model could help identify the conditions under which these features formed, linking them to specific climatic and hydrological scenarios.

The next goal of our work is to integrate this hydrological model into the Planetary Evolution Model (PEM) , a new modelling framework based on an asynchronous coupling with a GCM. The PEM can simulate long-term climate dynamics which allow us to study the impact of orbital-scale climate variations on the water cycle. In particular, we could investigate long-term accumulation/depletion of liquid water in lakes, including their cycle due to temperature and pressure changes, runoff after precipitation, the transport of water by rivers, the diffusion of water into the deep subsurface, etc. Moreover, the PEM can be used to set initial states for other simulations by determining steady-state realistic distribution of water for given orbital parameters.

A quantitative comparison between simulated water distributions and observed geomorphological features on Mars (Fig. ) would strengthen the validation of our model. However, such a comparison is constrained by two key limitations in the current work. First, the assumption of spatially uniform precipitation is a strong constraint that may not reflect the realistic precipitation patterns predicted by Global Climate Models for early Mars . Second, the use of present-day MOLA topography, rather than a reconstructed ancient surface, introduces systematic biases in the spatial distribution of depressions and their connectivity. Despite these limitations, our results provide useful insights: the 100 m GEL simulation broadly reproduces the northern ocean extent between the Arabia and Deuteronilus shoreline levels, and the model successfully identifies the major topographic basins (Hellas, Argyre, northern lowlands) as primary water repositories.

By coupling this model with a GCM, we will be able to compare simulation results with geological and geomorphological observations. For instance, we can compare the simulations with existing databases on open- and closed-basin lakes , identified deltas , potential shorelines and valley networks . Additionally, the results can be compared with detected hydrated minerals and stratified sedimentary deposits , using lake levels and river discharges to estimate sedimentary deposits. Finally, the simulated river discharges can be compared with estimates of Martian fluvial discharges , thereby validating and refining hypotheses on Mars' hydrological and climate evolution.