The manuscript by Gailleton et al. presents a suite of algorithms designed to solve the shallow water equation over gridded topographic data, utilizing graph theory-related data structures and associated tree scanning algorithms. By doing so, the authors leverage advancements and overcome some challenges of two existing approaches for hydrologic and landscape evolution calculations. Specifically, the authors combine (1) recent advances in applying graph theory-related algorithms in large-scale Landscape Evolution Models, which speed up computations of drainage area and consequently erosion/sedimentation rates (e.g., Braun and Willett, 2013) but commonly assume that flow occurs as 1D lines, and (2) shallow water equation solvers that more realistically represent flow routing over complex topography but commonly suffer from high computational costs, limiting their usage to small length scales and short time scales.

The presented numerical advances are impressive and substantial. The manuscript makes convincing arguments that their utility could be fundamental for specific hydrologic and landscape evolution problems. However, the manuscript remains in the realm of model development and, as such, doesn't provide new insights into natural processes (including what models' emergent dynamics teach us about how nature could work). Consequently, the manuscript reads more like (an impressive) technical report. If expanding the analysis toward more physical insights is desired, the current manuscript holds some threads that could be pulled to produce new, insightful understandings.

For example, the analysis presented in figure 7 could potentially be utilized to produce preliminary (and possibly case-specific) insights about the relationship of flow width as a function of bedforms and local relief (and precipitation input, of course). There are more opportunities like this in the manuscript.

Specific comments:

• Lines 53-54 list several studies demonstrating the advantages of integrating 2D hydrodynamics to inform the study of landforms. What is missing in these studies?
• Fig 1. What are the arrows? How are they scaled?
• Line 70. What does "remains hampered by the physics behind which explicitly simulates wave propagation" mean?
• Line 120. The text about xmax​ is out of context.
• Line 146. Unsure if z was defined?
• Line 164. Since the hydraulic surface is a crucial concept, readers might benefit from a formal definition and possibly a demonstration (relative to topography).
• Line 182. "The magnitude of Qout flux is the same for MFD and SFD schemes." Is this an assumption (represented by the correction factor) or a fact? Is it also valid for the transient case? Since this is such a central corollary, perhaps the authors can expand on this issue.
• Line 207. What does "while in reach mode, given entry nodes receive an arbitrary Qin​" mean?
• Fig 2. Can't understand the comparison. Which of the panels are a and c?
• Line 215. What does "(then determined in respect to CFL conditions)" mean?
• Fig. 3. It is possible to guess from the context, but readers will benefit from using different symbols for the stationary and transient solutions. I would also be interested in the actual time represented by these simulations (perhaps multiple time axes are needed). It is interesting to understand the time necessary for the transient solution to converge to the analytic solution, as this represents (gives a hunch for) the time scale over which the transient solution is important. Additionally, consider adding a graphical representation of the rectangular channel.
• Fig. 8. Wrong caption. Duplication of fig 7.
• Fig. 9. S−a_s. The interpretation of the colluvial sections is surprising. Are the white pixels (channels) in the left-hand side, lower relief landscape truly colluvial? The different types of slope break from colluvial to fluvial (lower slope on the LHS and higher slope on the RHS) could indicate that the interpretation should be more complex. Additionally, the slope increase at the high S – s_a end is surprising. While mentioned in line 327, its source is still not clear.
• Line 412. Q_{out}​ and Q_{in}
• Consider reducing the use of acronyms. Acronyms are useful for the authors but less for casual readers (specifically those not from the same field).

Please see the attached file for review comments.