See attached file.

This manuscript presents a new model to explain the functional relationships between the width of a valley W and several parameters such as drainage area and tectonic uplift. It is very well written, with very broad implications for understanding landscape dynamics over geological time. The widening of valleys is still poorly understood, and such a model provides a framework that could enable this element of the landscape to be interpreted quantitatively in terms of climate and tectonics. The fit between the data and the model is remarkable, even if some of this fit is due to the adjustment of certain parameters. This suggests that the scaling relationships between W and the various ingredients of the model are correct. The model is based on a number of simplifying assumptions, starting with the assumption that valley widening reaches a limit through time. This model assumes and applies to a stationary state of W. To derive the model, J. Turowski and colleagues follow an original approach, starting with a simple definition of W and gradually integrating the ingredients that lead to the final equation linking W with the other parameters. However, I did not always fully understand the derivation of these equations and I have some doubts about certain assumptions. The rest of my report is more a discussion of these misunderstandings than a challenge to the model. These comments can be used to improve the presentation of the model.

I'm not sure I fully understand Equation (1) linking valley width W with lateral migration velocity V. If I understand correctly, this equation assumes that the width W is a constant width, obtained after a time Deltat. If I still understand correctly, this time is defined as the average reoccupation time of the same site by the channel, in particular the edge of the valley. However, a little above, it is stated that the width is set by the average time during which the channel migrates in the same direction. How are these two times related? What is the underlying vision: a channel that migrates for a certain time in one direction and then abruptly changes position (through avulsion or some other process)? In other words, why is the maximum bound in the integral of Equation (3) a mean time and not an infinite time, since we are looking for a stationary solution? Why is an integral needed here rather than writing directly that the width is determined by the product of the speed of migration in one direction and the average time Deltat during which the channel is in contact with the edge of the valley (=V.Deltat) ?

Does it make a difference if the total water discharge splits between multiple channels? Croissant et al (2018 JGRES) show that continuity of transport capacity as the valley widens requires a change from a single channel to multiple channels. Does the number of channels in the valley influence the likelihood of widening the valley?

Are there arguments to justify a Poissonian distribution of waiting times? Would another type of distribution (from one valley to another) dramatically change the model's predictions?

In Equation (5) the characteristic shift time is proportional to the ratio between the lateral transport capacity and the cross-sectional volume of a channel. I would have rather written that it is proportional to the ratio between the longitudinal flux Qs of  sediment passing through the channel and the volume of the channel (e.g. Sun et al., 2002, WRR). This flux can be related to the upstream uplift rate but also the drainage area and could therefore change the scaling relationship between width W, the  steepness index  and the drainage area. Similarly, avulsion frequency increases with sediment flux Qs (e.g. Bryant et al., 1995, Geology). If Lambda depended on Qs, would this dramatically change the model's predictions?

In equation (15), two different definitions of the probability of a channel "touching" the edge of the valley seem to be combined. The first corresponds to a Poissonian process (Equations (2)-(11)) and the second to a homogeneous probability which only depends on the ratio between the width of the channel and the width of the valley. Is there not a conflict between these two visions and a contradiction in combining them in equation (15)?

Would it be possible to test the validity of the model by analysing the sudden changes in valley width when a river moves from a confined to an unconfined situation as it leaves a mountain range?

The model predicts functional relationships between valley width and several geomorphological parameters, but not lateral erosion rates. Some data from Zavala et al (GRL 2021) seem consistent with this model. For example, Zavala et al. found a very low valley-side erosion rate for the valley located upstream of a knick-zone in the Tana valley in Chile. The valley is not very wide and is not deeply incised into the pampas (this point is an outlier in the erosion rate versus W graphs). This seems consistent with the fact that the valley has reached a state of equilibrium between qL and qH which determines Wc, and for which lateral erosion is necessarily low.

Specific comments

Line 59, "rock type" and weathering.

Lines 71-72 These two sentences about the climate seem to me to contradict each other.

Line 215. What is Wu (not used afterwards)? Is it Wc?

Equations (19) and (20). Is the dependence of Wc on A taken into account in the model? And if not, wouldn't this change the model's predictions about the scaling relationship between W and A?

Line 557. I'm not sure I understand. Do you mean that the width increases so slowly in the EW model that this amounts to fixing the width of the valley?

Line 591: "thereby slowing the lateral back-and-forth movement of the river and narrowing the valley ": On the contrary, it could be argued that increasing Qs favours lateral mobility (Bryant et al., 1995) and widening (Baynes et al., 2020, ESPL).

Line 592: "absence of lithological control on the steady state valley width in a regional perspective". I am not sure that the model calibration exercise in figure 5C really demonstrates this. Insofar as the fitted data is an average per mobility number value class, there can only be one value of Wc for the whole data set. Shouldn't you separate the data into different datasets by lithology category to check that W does not depend on the lithology globally?

I wish you good luck for the revision.

Sébastien Carretier