Nonlinear weighting across the grounding zone

[JanJereczek]

Left panel: time series of the tidal sea-surface height perturbation from Rignot et al. (2024).
Right panel: resulting probability density function (pdf, "how frequent is a sea-surface height perturbation"?) and cumulative density function (cdf, "how frequent are tides below a given value" - e.g. there is a frequency of about 0.85 that the tidal anomaly is 50 cm or less). Two fits of the cdf are shown: one uses the cdf of a normal distribution (relies on error function erf) and the other uses a reference polynomial of degree 5. The former one should be used, unless we want to get experimental (the polynomial can be tuned to do whatever we want)...

Conceptually, we want to associate the lowest tides with gz_Hg0 and the highest ones with gz_Hg1. To do this, we apply a linear map that is then passed to the cdf fit. The thus obtained weight has following statistical meaning: "how often is a point with gz_Hg0 < H_af < gz_Hg1 grounded?". The result is shown below for gz_Hg0=0m and gz_Hg1=100m :

For the relatively coarse resolutions we are running, I don't expect this to give very different results compared to the linear version of PMPT that was implemented so far. I think this can however become crucial on higher resolutions (melt is larger for cells close to gz_Hg0 and smaller for cells close to gz_Hg1).

I am waiting for the data of Chen et al. (2023) to confirm that the nonlinearities that are implemented here are adequate regardless of the region treated.

Diving into the code:

• topography.f90 l. 1590: case distinction between "pmpt-lin", "pmpt-gausscdf" and "pmpt-polyref", which now need to be specified in the namelist.
• l 1798: define parameters of gaussian cdf and the linear map between tides and gz_Hg.
• l 1845: case distinction (internally).

If this PR is accepted, the according namelist entry of yelmox shoud be changed to:

    bmb_gl_method       = "pmp"             ! "fcmp": flotation; "fmp": full melt; "pmp": partial melt;
                                            ! "nmp": no melt; "pmpt-lin" partial melt with linear tide;
                                            ! "pmpt-gausscdf" partial melt with realistic tide;
                                            ! "pmpt-plyref" partial melt with tunable tides (experimental)

[alex-robinson]

Hi Jan. This looks like a good approximation to tidal frequency, which is fine. One thing is to check the code and see if it does what you propose, which I am confident it would.

But another thing is still the question of whether this tidal frequency is the dominant control on duration of floating versus grounded fraction within the GZ. This goes back to Marisa's question of, tides or intrusions?

For example, do we have a clear understanding of how the SSH anomaly of 100cm translates to a grounding zone spanning possibly 100 m of equivalent ice thickness? It cannot be due to the change in SSH alone, which means basal hydrology under the ice sheet plays a role, allowing seawater to penetrate farther upstream, perhaps as proposed by Robel et al (2022) - see Fig 1. Furthermore, Fig. 1 (or Fig. 5) of Robel et al (2022) also would suggest that the height of the seawater intrusion diminishes as it moves upstream, giving it less melting capacity. Maybe it points to incorporating some information about H_w into the parameterization, although that complicates things, especially if one wants to run simulations without interactive hydrology. Then the question is does the timescale of that intrusion match with the timescale of SSH changes, and does that also lead to a good weighting which should also account for that diminishing melting power inland? For this, actually it looks like a lot of the modeling presented by Robel et al (2022) would be relevant. In fact, I think my parameterization was inspired by their Fig. 5, recognizing that L was not a very practical parameter, but H_grnd would be. But I didn't do any of the hard work of thinking about these questions.

So, I feel like a certain amount of literature review and clarification of how we conceptualize these questions above within the parameterization is needed to bring more complexity into the approach. That said, we could include your technical changes and internally keep using pmpt-lin until those questions are resolved.

[JanJereczek]

Yes, I agree that there is a underlying assumption here (a tide of 100m implies reaching gz_Hg1) which we are not sure about. This PR is really just about having a technique to make things a bit better without getting into more processes (e.g. basal hydrology). Assuming a linear weighting is equivalent to assuming a constant pdf, which is worse since we know that extremes are not as likely as intermediate values (see for instance Kim et al. (2024)).

[marisamontoya]

To me the question is indeed what Alex was pointing out above: how the tidal fluctuations translate into long intrusions. The SSH + hydrostatic equilibrium alone won't make it. Chen et al (2023) explicitly say that long intrusions are not in phase with the tides - see their Figure 2d. Their timescales are also different, see Figure 2e. On the other hand they are ultimately driven by the tides, but affected by other features. The most clear relationship is with the slope of the bed (Figure 3a). The relation is exponential. So at this point I was truly wondering what PMPT exactly captures - will it capture this exponential relationship? Another issue is how is all this related to the layered seawater intrusion of Robel et al (2022) among others (which is what I think motivated our parameterisation).

[JanJereczek]

Ok, it seems that you both agree on the fact that this is probably not a robust (albeit minor) improvement of how realistic our parametrization is. I have the feeling that the pdf above matches to some extent what is observed by Kim et al. (2024) in terms of grounding line position but it's a bit difficult to make a one-to-one comparison... For now, let's leave this open but we can close it (potentially without merging) after the summer.

[alex-robinson]

Ok, yes, I see now in Kim et al. (2024), you are referring basically to this picture:

And the time series of GL-position.

Indeed in this case, it looks like the timing correlates quite well and so the time can be considered analogous to the tides.

But I think we should study a bit more and revisit this after the summer. It looks like the datasets we have that are sparse are also not too easy to interpret every time. If we get more confidence that this behavior is roughly universal, and consistent with Robel et al. (2022) and Gadi et al. (2023) modeling, then I'd feel better about it.

For example, a cursory look at Gadi et al. (2023), which models Petermann glacier, it looks like the opening and closing of the cavity happens on tidal timescales (see caption of Fig. 2 (snapshots at time=18h and time=24h):

This would support your approach. Maybe we could request some time series from them too.

It would be nice to have an example from Antarctica too.

[marisamontoya]

Alex do you mean like doing a histogram with this? Here it seems the centre would be less populated?

[alex-robinson]

@marisamontoya , yes maybe, though it's hard to figure out what is going on at all in that figure.

Ok, I have taken some time now to read a few things.

In fact, the paper Jan first cited (Rignot et al., 2024) is pretty clear that the flexure and tidal intrusions operate on the timescale of the tides too. See the scatterplot:

And then the text is also quite clear on this point. For example,

We find a strong correlation between SSH at spring tide and the length of intrusion

which must then also correlate in time. And

2.3. Rushing of Seawater. Ice flexing in the main trunk has a range of ±70 cm vs. a subsidence/uplift of O(10 cm) upstream, i.e., seawater intrusions are thin. For seawater to rush multiple kilometers in half the diurnal cycle, or 6 h, requires speeds O(10 cm/s). For a 6-km intrusion in 6 h, the water speed is 28 cm/s. At the upper part of the tidal cycle, we detect intrusions
of 12 km inland, which imply a water speed of 56 cm/s. In the “shallow water” assumption, the wave speed will be gHw, where g is the acceleration of gravity and Hw is the water thickness, or O(1 m/s) for Hw O(10 cm). During a transition to low tide, seawater extrusions may be limited by the resistance to water flow in the cavity (34) or be trapped in bed pockets (35).

In addition, like Kim et al. (2024), this figure from Kim et al. (2025), now for Helheim glacier also would hint at this higher frequency at the center:

And they write:

Over a span of 6 hr, the grounding line migrates by 500–800 m, that is, a rate of ice cavity opening of 2.3–3.7 cm/s, which is not a high water speed, but still potentially able to carry a significant amount of ocean heat beneath the glacier. The area of intrusion on the southern flank grows to 3.85 square km in 6 hours, leading to a seawater flux of 89–142
m^3/s.

So this again, does seem to indicate that the timescale of the intrusion is correlated with the tidal timescale.

At least in terms of timescale then, this modification proposed by Jan is probably a reasonable addition.

At this point, I think this and other considerations require an offline meeting and a decision about how to investigate this more rigorously and efficiently. This thread does not seem like the way to do it.

I also want to note: one additional thing that seems important is that the ability of seawater to move upstream is predicated on the presence of a layer of basal freshwater. Right now, our parameterization assumes that freshwater is present everywhere at the GL, which may or may not be reasonable.