the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Oct 2023
Numerical stabilization methods for level-set-based ice front migration
Mathieu Morlighem
G. Hilmar Gudmundsson
Abstract. Numerical modeling of ice sheet dynamics is a critical tool for projecting future sea-level rise. Among all the processes responsible for the loss of mass of the ice sheets, enhanced ice discharge triggered by the retreat of marine terminating glaciers is one of the key drivers. Numerical models of ice sheet flow are therefore required to include ice front migration in order to reproduce today's mass loss and be able to predict their future. However, the discontinuous nature of calving poses a significant numerical challenge for accurately capturing the motion of the ice front. In this study, we explore different stabilization techniques combined with varying reinitialization strategies to enhance the numerical stability and accuracy of solving the level-set function, which tracks the position of the ice front. Through rigorous testing on an idealized domain with a semicircular and a straight-line ice front, including scenarios with diverse front velocities, we assess the performance of these techniques. The findings contribute to advancing our ability to model ice sheet dynamics, specifically calving processes, and provide valuable insights into the most effective strategies for simulating and tracking the motion of the ice front.
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Gong Cheng et al.
Status: final response (author comments only)
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CC1: 'Comment on gmd-2023-194', Michael Wolovick, 13 Nov 2023
Thanks for this interesting study! I have a question that I might want your input on.
During my own studies using ISSM to simulate the Filchner-Ronne sector of Antarctica, I found that small values of the reinitialization interval tended to produce a bias towards front retreat. In your analysis of error magnitude (Figures 4 and 5) you show the absolute misfit area, but not the sign of the error. In Figure 2, the maps of ice front error seem to show a bias towards retreat with n_R=1. Am I interpreting that figure correctly? While you are analyzing periodic forcing in this paper, a systematic bias towards either advance or retreat is obviously very important for projecting the response of the real ice sheet to a changing climate. Would you say that your results also support the conclusion that very small values of n_R bias the model towards retreat? Did you see any systematic trend in the sign of the error in addition to the differences in overall error magnitude presented in your results?
Thanks in advance for your answer,
Mike Wolovick
Citation: https://doi.org/10.5194/gmd-2023-194-CC1 -
AC1: 'Reply on CC1', Gong Cheng, 13 Nov 2023
Hi Mike,
Thank you for your insightful question. Following the convention in ISSM, the sign distance function indicates negativity for the ice-covered region. Thus, the negative segment of d(\phi1, \phi2) in Figure 2 implies that \phi1 (numerical solution) is ostensibly more 'advanced' than \phi2 (true solution). This discrepancy arises from the strong diffusion introduced by n_R=1 to the 0-level set contour. Depending on the ice front's shape (convex or concave) and the velocity profile, this effect can introduce a bias towards either advance or retreat.
However, it's crucial to emphasize that one of the primary focus of this paper is to underscore the inadequacy of choosing n_R=1. This choice introduces a significantly higher level of numerical errors compared to other sources. Consequently, we strongly advise to avoid using n_R=1 in any case.Â
The bias associated with other choices of n_R is generally contained within the scale of the mesh, rendering it negligible when compared to errors originating from other sources, such as model inaccuracies and data discrepancies.
Cheng Gong
Citation: https://doi.org/10.5194/gmd-2023-194-AC1 -
CC2: 'Reply on AC1', Michael Wolovick, 14 Nov 2023
Thanks for your response!
I see I misinterpreted the color scale in Figure 2. I had already concluded that using n_R=1 was a bad idea, so there's no need to worry about that issue. But it's interesting to know that the bias towards retreat that I saw in my domain was a consequence of the particular details of front geometry and velocity profile, not anything fundamental to the algorithm.Â
In the case of spatially refined meshes, what do you expect to be the effect of gradients in mesh size near the front? If the numerical errors depend on mesh size, how do you think that will affect front geometry and migration rate, if the front is migrating through an area where mesh size is changing?
Citation: https://doi.org/10.5194/gmd-2023-194-CC2 -
AC2: 'Reply on CC2', Gong Cheng, 14 Nov 2023
Thank you for bringing attention to this aspect. We systematically examined diverse mesh resolutions, and the outcomes align closely with the principles of finite elements and corresponding stabilization techniques. Consequently, we opted to present results solely at a resolution of 100m, the typical finest mesh resolution in ice modeling.
In the context of spatially refined meshes, we do expect the mesh size will influence the level-set function. However, our insight is that this influence, while present, is overshadowed by the more dominant effect from the velocity solution. The ice velocity has a direct influence on the frontal migration rate, therefore, the front geometry as well.Â
The primary message of our paper centers on the pivotal role of thoughtful choices in stabilization and reinitialization. Our findings affirm that, with careful selection of these parameters, we will not introduce higher numerical errors beyond those inherent to the system.
Citation: https://doi.org/10.5194/gmd-2023-194-AC2
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AC2: 'Reply on CC2', Gong Cheng, 14 Nov 2023
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CC2: 'Reply on AC1', Michael Wolovick, 14 Nov 2023
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AC1: 'Reply on CC1', Gong Cheng, 13 Nov 2023
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RC1: 'Comment on gmd-2023-194', Matt Trevers, 17 Nov 2023
Thank you for the opportunity to review this manuscript. Please find my full referee comment in the attached document.
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AC3: 'Reply on RC1', Gong Cheng, 28 Nov 2023
The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2023-194/gmd-2023-194-AC3-supplement.pdf
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AC3: 'Reply on RC1', Gong Cheng, 28 Nov 2023
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RC2: 'Comment on gmd-2023-194', Anonymous Referee #2, 20 Nov 2023
The paper explores different stabilization methods for level-set equations and the impact of reinitialization on the accuracy of the solution.
The methods are demonstrated on an idealized geometry (union of a rectangle and a semidisk) to mimic a fjord with a semi-circular ice front. The front velocity is prescribed. The authors nicely present how the different stabilization approaches and the frequency of the reinitialization affect the accuracy of the position of the level set.
The paper addresses a very important topic in ice-sheet modeling and it is easy to read. However, I have major concerns which prevent me from recommending the paper for publication in its present form.
- My main concern is that despite the title and the presentation of the work, there is little about ice front migration in this manuscript. In fact, the geometry and the prescribed velocity are too simplified to be representative of an ice front migration problem. In addition to the very simplified description of the fjord, the prescribed front velocity is aligned with the fjord axis, which is at odds with the fact that the calving component of the front velocity is typically assumed to be orthogonal to the ice front.Â
The standalone advection level-set equations have been extensively studied in the literature, and this paper adds little to what is already available. On the contrary, I would have found the paper very valuable if the authors targeted a more realistic ice sheet problem as well, where the level-set velocity was computed using ice flow equations (e.g., the Shallow shelf Approximation) for the ice velocity and at least one of the calving laws typically used in the literature. - Another concern I have is that the authors do not explain what reinitialization method they are using, despite the fact that the effect of reinitialization is one of the main topics of the paper. When they introduce the reinitialization they reference two papers they co-authored but I could not find any detail there either. Further, plots in figure 2 show a loss of symmetry, which is likely due to the reinitialization procedure, but the authors do not offer any explanation of why that is happening. I worry that there might be an issue with the reinitialization procedure which would affect the results and possibly the paper conclusions.
- Finally, the forward and backward diffusion stabilization considered in this paper aims at keeping the level-set function close to the distance function, so that no reinitialization is needed. This is qualitatively confirmed by their results. However, the authors miss this point in the discussion of the results. Also the authors do not provide any reference for this stabilization method.
Citation: https://doi.org/10.5194/gmd-2023-194-RC2 -
AC4: 'Reply on RC2', Gong Cheng, 28 Nov 2023
The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2023-194/gmd-2023-194-AC4-supplement.pdf
- My main concern is that despite the title and the presentation of the work, there is little about ice front migration in this manuscript. In fact, the geometry and the prescribed velocity are too simplified to be representative of an ice front migration problem. In addition to the very simplified description of the fjord, the prescribed front velocity is aligned with the fjord axis, which is at odds with the fact that the calving component of the front velocity is typically assumed to be orthogonal to the ice front.Â
Gong Cheng et al.
Video supplement
Animations of the evolution of misfits Gong Cheng https://doi.org/10.5281/zenodo.8400628
Gong Cheng et al.
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