| 17 Feb 2026
Nonlinear dynamics of time-variable slope circulation
Abstract. Bottom topography strongly constrains ocean circulation in the Arctic, and both theory and numerical modeling suggest that nonlinear flow–topography interactions influence slope-following currents. Yet, how such interactions modify the circulation response to time-variable surface forcing remains poorly understood. Using idealized shallow-water simulations of flow over a corrugated slope in a re-entrant channel, we investigate how nonlinear features arise and evolve under oscillatory forcing. We observe both a persistent prograde flow bias (aligned with topographic Rossby wave propagation) relative to linear estimates, and an asymmetry in the circulation response, with retrograde flow (opposing wave propagation) exhibiting a saturation of flow strength once the flow reaches sufficiently strong velocities. To identify the mechanisms responsible for these behaviors, we evaluate integrated momentum budgets. Which terms appear as dynamically relevant, in addition to linear surface and bottom stresses, depends on the choice of integration path: when integrated along constant-depth contours, the nonlinear dynamics appear as a cross-slope relative vorticity flux, whereas integration along straight transects instead highlights momentum flux convergence and topographic form stress. These perspectives can be unified under quasi-geostrophic scaling as describing a flux of potential vorticity (PV). This PV flux is strongest during retrograde flow and predominantly down-slope, explaining the prograde bias. When retrograde velocities approach the arrest speed of topographic Rossby waves with wavelengths comparable to the corrugation scale, the flux increases sharply, halting further acceleration and producing the observed asymmetry. These results show how flow–topography interactions shape time-variable slope circulation, biasing the flow toward prograde states and limiting retrograde flow strength. Such effects are likely underrepresented in coarse-resolution numerical simulations, and highlight the need for improved representations of unresolved topographic interactions.
[Dear Editor, just when I was about to submit this comment did I see this: "Please note your comment is only made available to the handling editor, not to the authors" at the bottom of the comment box. Sorry I totally ignored your instructions and wrote this review in the same way as for a traditional journal. How should I fix this?]
This study utilizes an idealized barotropic shallow-water model to explore how arrestation of topographic Rossby waves can lead to strong prograde bias in along-shore barotropic flow. I think the science of the paper is excellent. I enjoyed reading it. I have only minor suggestions that might enhance the scientific discussion of the paper. My semi-major concerns are all about the readability of the text. I hope my comments will be helpful in this regard.
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First of all, I'm not familiar with the literature about form stress due to irregular topography and so, I'm not able to judge the novelty of the present study. I trust the authors that their findings are new.
Second, I'm not familiar with the peculiarity of the Arctic Ocean. I expect in the Arctic the currents to be more barotropic and mesoscale eddies to be smaller in diameter, but beyond that, I'm not too familiar with the situation of the Arctic Ocean. I hope my questions and comments may be helpful to increase readability of the text by non-experts.
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I found the introduction confusing. It's not very clear what past studies found, what has been explained, what hasn't, and what the present authors' hypotheses are. The flow of logic is too circuitous and hard to follow. A more straightforward logic is desired.
For example, the paragraph on lines 59–72 forms a nice hypothesis (retrograde currents arresting topographic Rossby waves, resulting in prograde form stress and hence the asymmetry between the prograde and retrograde directions). So I thought that this is the hypothesis the paper is going to test and that this is the end of "the problem statement". But, then a discussion of Holloway's mechanism starts AGAIN . . .
. . . As I read this paragraph, it dawned on me that a discussion of another mechanism of generating prograde form stress HAS JUST STARTED.
The problem in this particular instance is that the authors have cast the problem into "steady forcing vs variable forcing". But this contrast (steady vs variable) is hardly essential. In Holloway's mechanism, you still need eddies, which form variability. The real issue is that there are two mechanisms.
So, the whole discussion of the authors' hypothesis should start from a statement that there are potentially two mechanisms to generate retrograde stress. One is XXX and the other is YYY. This should be the structure of the discussion . . . or at least this is one way to convey the idea smoothly to the reader.
. . . But then, at the very end of Section 5 (Discussion, lines 558–560) it becomes clear that the 2nd hypothesis isn't tested in the present study.
. . . As (I hope) you can see from this difficulty I had, the logic of the discussion becomes clearer, only when you review what you have read and re-organize the ideas in your head.
In general, ideas and arguments should be arranged to form a straightforward narrative, reducing redundancy as much as possible, so that the reader can understand the logic easily just by reading once from top to bottom.
I've inserted comments to the manuscript PDF file, which I'm attaching to this review report, as well as wrote other comments below. They, especially those in the manuscript, haven't been edited much. Some of the questions I ask in earlier comments are resolved later in the text, and also some of my comments include a lot of redundancy. I'm sorry about that but at the same time I hope this helps the authors see potential difficulty some of their future readers will have.
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There are still things that I ended up not understanding:
- Past studies are sometimes alluded for steady-state response, but it's not clear from the text what the past studies have shown about the steady-state response of the system the present authors deal with (barotropic flow along a corrugated continental slope on an f plane). Can the steady state response be regarded as the limit of slowly-varying forcing and the mechanism of arrested Rossby waves remain the same?
- The Neptune effect (barotropic flow induced by mesoscale eddies impinging on bottom slopes) is mentioned as a second potential mechanism to explain the prograde bias, but this hypothesis isn't tested as the barotropic model doesn't produce mesoscale eddies →
→ Much later in the text (lines 558–561) this issue is taken up as a missing feature in the present study. That means, I was misled to think that this was a second hypothesis that the present paper would test. I hope the introduction will be revised so that this misunderstanding wouldn't happen.
- How does the amplitude of corrugation enter the discussion? What determines the minimum amplitude for the arrested Rossby wave mechanism to work? How does the amplitude control nonlinearity? Does an increase in the amplitude monotonically increase form stress?
- What happens in the limit of R → 0 when there is corrugation?
- The along-shore (x direction) structure of the form stress and Rossby waves is not discussed. If the form stress is perfectly periodic at the wavelength of the corrugation, the authors' re-entrant (periodic) domain is perfectly fine, but is this clean structure expected to hold for all realistic parameter range? I hope at least some words will be added about this issue.
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It's not clear to me what the role of the linear part of bottom form stress is: H ∇φ. Under the linear limit (weak forcing τ_0), is the form stress term always negligible even if R is very small? In a stratified ocean, generation of lee waves can exert net stress on the fluid and this is a linear mechanism, I think, but I'm not familiar with the barotropic situation.
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This is definitely a minor point. I'm glad to learn about the new package Oceananigans, but the text needs to tell the reader how the package contributes to the present study. If you know basic finite differencing, you can in principle construct the same numerical model. So . . .
- Do the author *need* the package? For example, does the package use sophisticated numerical methods for higher accuracy, which would be a lot of work to implement?
- Or, do they use the package just for convenience? . . . Since there is already a well-written package, why not just use it? . . . Is this the idea?
- Or, does the package include tools for analysis and that's the main reason for utilizing this package?
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I'm not familiar with quasi-geostrophy for barotropic flow. For a 1.5-layer model, for example, PV is q = (ζ + f)/D, where D = H + h, where H = const is the mean layer thickness and h is the deviation from it. In the QG limit, q ≈ (ζ − f h/ H) / H. This is the so-called QG PV.
For the barotropic case, D = H(x,y) + η. Does that mean that the QG PV is (ζ − f η/H)/H and the rigid-lid approximation (η ≪ H) is equivalent to QG (of course assuming ω ≪ f)? Or, with corrugated bottom, the nonlinear momentum term tends to be large enough to break geostrophy and so that h(x,y) ≪ H_0 is a necessary condition for QG?
But then, why not just use q = (ζ + f) / D ≈ (ζ + f)/H for PV balance? At least for the const-H line integration, this q should work. Also, is the h ≪ H_0 approximation really necessary in the const-y integration to relate PV with bottom form stress?
In any case, I think that a more careful discussion on the necessity of the approximation that h(x,y) ≪ H_0 is necessary.
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Figure 3 compares a nonlinear solution with the prediction from the linear theory. Before this, we need a comparison of a "linear" numerical solution with the theory. We can reduce the amplitude of the wind stress by a factor of 1/1000 or anything very small for the numerical model. This numerical solution should be very "linear". In this situation, the numerical simulation doesn't develop eddies (correct?) and so we don't have to worry about signal-to-noise ratio even with very week winds (because there is little "noise" in the numerical solution). This comparison would give us confidence in the linear theory, in the authors' calculation of various terms (for example, how accurately is the contour length C calculated for the numerical model?), and in the numerical solution itself (for example, because of the finite resolution, there may be some artificial form stress due to the zig-zag nature of the slope), and at the same time it would be a way to assess the impacts of the assumptions that went into the theory (rigid lid approximation, what else?) and the numerical model's deviations from the idealized situation (for example, what's the impact of the artificial boundary at y = L_y in the numerical model? ).
After this, we would be more confident that the deviation of the nonlinear solution from the linear theory is due to nonlinearity.
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The spatial structure of the nonlinear term should be discussed. If it is exactly periodic in the x direction at the periodicity of the bottom corrugation, the width L_x of the domain shouldn't matter as long as it is an integral multiple of the corrugation wavelength.
In this regard, I'm wondering whether the x-width of the domain might be too restrictive. It includes only two bumps (Figure 2). Nonlinearity might "want" to develop a larger structure?
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For color plots, please consider using color-blind friendly pallets. For example, the orange-green contrast in Figure 4 doesn't seem to be particularly friendly to some type of color-blindness. Even though one is dashed and the other is solid, the legend uses only colors. There are resources on the Internet: for example, https://davidmathlogic.com/colorblind/ . Modern scientific plotting tools usually include a selection of color-blind-friendly schemes.
Having said that, it's always a good practice to increase the distinctiveness between the lines by using symbols and various dash-types in addition to colors.
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Please use an expression such as "friction" and "frictional stress" instead of "bottom stress" throughout the paper. The latter expression is confusing because bottom form stress is also a "bottom stress".
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I found it hard to follow the discussion of momentum flux and acceleration near the end of Section 4.2. You need a schematic diagram to follow it. If the discussion were simply about relation between a single spot of flux convergence and acceleration, it would be fine, but the text talks about multiple convergence zones occurring on the "flanks" of peaks of another quantity . . . In an oral presentation, the presenter would point to specific points in the line graphs in Figure 6 . . . In reading the paper, the reader needs a similar visual guide.
Also, I'm not familiar with the relation between Rossby wave generation and form stress. I guess Rossby waves carry prograde momentum and that momentum is imparted by the solid Earth and therefore the form stress must be prograde . . . Is this what's going on?
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I failed to see the value of the PV discussion. The approximate equivalence of PV flux with bottom form stress is mathematically derived and the numerical solution demonstrated the accuracy of the mathematical derivation (Figure 7) . . . What have we learned from this exercise?
Perhaps there has been a debate or confusion about the form-stress view and PV-flux view and the authors want to resolve it?
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It's not clear what the comparison between 16-day and 128-day forcings shows. By design (equation 16) higher forcing frequency leads to smaller amplitude. Is the weakness of nonlinear impacts in the 16-day solution, then, all due to the smaller amplitude? What would happen to a 128-day solution whose τ_0 is reduced so that u_max is exactly the same as that of the 16-day solution at the mid depth? It would still show the weaker depth dependence [ 1 + (ωH/R)^2]^{−1/2} . . . what else?
→ This question will be resolved later in the text, but I suggest including some guide earlier so that the reader doesn't have to carry this question to the very end of the discussion of the main results.
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The construction of Figure 9 is explained solely in words (lines 450–460) and is hard to follow. Also, the reason why this procedure will tease out Rossby waves isn't clear. Perhaps write the procedure and the assumptions both in math.
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This is just a suggestion: Since most of the important conclusions are derived by analyzing the ω = 2π/(128 d) solution, it's a bit distracting that the text keeps comparing it with the ω = 2π/(16 d) solution. The meaning of this comparison becomes clearer only when the systematic amplitude sensitivity is presented in Section 4.5.
So, I think the logical structure of the Results would be simpler if the numerical experiments are ordered this way:
1) ω = 2π/(128 d) only
1.1) "linear" . . . no corrugation (or extremely small τ_0 ?);
1.2) "weakly nonlinear" . . . smallish τ_0 (so that this is similar to (2.1) below);
1.3) "fully nonlinear" . . . τ_0 = standard size.
2) Sensitivity solutions
2.1) same as (1.3) but ω = 2π/(16 d).
2.2) sensitivity to τ_0
2.3) impacts of corrugation.
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Figure 8 is very informative. And this makes me wonder whether resonance is possible or not. Because the computational domain is re-entrant (cyclic in x) or the actual Arctic Ocean has closed-f/H basins, wind forcing should include frequencies and wavenumbers that match the cycle travel times and wavenumbers of topographic Rossby waves.
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It's not clear to me how the prograde-retrograde asymmetry in Figure 3b is explained, where the currents aren't fast enough to arrest Rossby waves. Is it still due to bottom form stress?
I suppose that this lack of discussion on this point is because the mechanism has already been explained by past studies and it's entirely possible that I missed such discussion in the present manuscript. But it would still be nice if the mechanism is briefly explained in the discussion of Figures 3 and 4 before entering the discussion of "saturation" and also in the discussion of Figure 11a, where corrugation is removed. (I suppose the prograde-retrograde asymmetry is gone when there is no corrugation.)