the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Apr 2026
Convolution Based Techniques for Computing Self Attraction and Loading in MOM6
Abstract. Self Attraction and Loading (SAL), which includes the deformation of the solid Earth under the load of the ocean tide and the self-gravitation of the so-deformed Earth as well as of the ocean tides themselves, is an important term to include in numerical models of the ocean tides. Computing SAL is a challenging problem that is usually tackled using spherical harmonics. The spherical harmonic approach has several drawbacks which limit its accuracy. In this work, we propose an alternative technique based on a spherical convolution. We implement the convolution technique in the Modular Ocean Model, version 6, and demonstrate that it allows for more accurate tides when measured against tidal datasets based upon satellite altimetry. The convolution based SAL reduces the error by reducing spurious oscillations associated with the Gibbs phenomenon. These oscillations are large in coastal regions under the traditional spherical harmonic approach.
Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-654', Richard Ray, 20 May 2026
-
RC2: 'Comment on egusphere-2026-654', Anonymous Referee #2, 23 May 2026
Review of "Convolution Based Techniques for Computing
Self Attraction and Loading in MOM6" for EGU GMD.
Note the preprint from GMD at the url, https://arxiv.org/pdf/2602.01416,
does not include line numbers.
This review is based on the file named SAL_Paper.pdf which
was provided by the editor; it contains lines numbers.This paper is concerned with the representation of ocean self-attraction and
loading (SAL) in global ocean models. In tide-resolving ocean models, the
effect of the SAL is represented as a perturbation to the equilibrium tidal
potential. However, this perturbation depends on the time-dependent
distribution of ocean mass (which can be expressed in terms of the
sea level anomaly), so the SAL cannot simply be computed once as an external
forcing field and incorporated into the forcing. Instead, the sea level anomaly
computed by the ocean model must be incorporated into the SAL forcing
as the model is running.The effect of a localized infinitesmal sea level anomaly on the surrounding
gravitational potential at the surface of the Earth may be expressed by
a convolution of the sea level anomaly field with a Green's function.
Prior to this work, efforts to incorporate the SAL in global ocean models
either took the SAL as a prescribed forcing computed from independent data,
or they approximated the convolution as a truncated spherical harmonic
expansion. The new element of this work is the approximation of the
convolution directly on the model's computational grid using a quadrature
based on the Cubed Sphere Fast Multipole Method.
This seems like an important effort to accurately and efficiently
compute the SAL from first principles in ocean models.The authors implemented their approach in MOM6, and they evaluate the
success of their efforts by comparing the output of the simulation with
independent estimates of the tidal sea level anomaly, based on a TPXO solution.
Unfortunately, I think this approach conflates several possible sources
of error and makes it impossible to evaluate the quality or significance of
their work. Since their focus is on accurately and efficiently computing the
convolution of a function with a Green's function, they should simply
evaluate this in isolation before using an ocean model as the basis for comparison.
Of course, they ought to use the actual model grid and realistic
test fields for \eta in their comparisons. Likewise, since it is essential
to their application, they should evaluate the accuracy of the gradients
of the SAL (the components of acceleration), as they have done nicely here.
But this evaluation should be done in isolation from the other model numerics
which irreversibly fold in (possibly compensatory) errors due to the following:
- time-stepping stability and accuracy (What time-level scheme is used for SAL?)
- errors in the frictional parameterizations (wave drag and bottom friction)
- errors in coastline location and bottom topography
- errors due to the neglect of baroclinicity
- errors in the TPXO comparison dataFor these reasons I suggest "reject" for this manuscript. Nonetheless, I think
this is an important topic, and the authors are capable and careful scientists,
so I hope they will revisit their approach and publish on this topic in the
future.
l11: antecedent of "which" is not clearl12 and throughout: avoid parens within parens
l19: It seems like you mean "3-dimensional" or "depth-resolving"
model, rather than "baroclinic model".l20: should be plural "configurations", unless the 3 authors all used the
same configurationl23: is the phrase, "large scale", in the right place?
l24: mesoscale eddies aren't necessarily smaller scale than the internal tides
l39: Be more specific: "the computation of a convolution" -->
"the convolution of the sea surface height"l60: "tidal forcing" --> "astronomical tidal forcing"
l73: omit "in line" or move it closer to the word it modifies,
"computing"l76: I have not done these computations myself, but don't the Love
numbers decay like 1/n?
I don't find the phrase, "modified Load Love Numbers", used in Ferrell (1972).
It appears that he compares the Greens function integrals with spherical
harmonic expansions. Why aren't you referring to his work in more
detail? I doubt I am expert enough in this subject matter to
review it adequately. Please distinguish between the "primed" and
"bare" Love numbers.l86: How is the computation of an integral "prone to aliasing"?
What do you mean?l88: Doesn't \eta go smoothly to zero at the shoreline? I don't understand
where is the discontinuity that would give rise to the Gibbs
phenomenon.l102: The sea level "jump" occurs over the scale of a grid cell, so it is
hard to imagine scenarios in which it could be significant.
And, doesn't it involve a minuscule mass of ocean, compared to the
deep water, so its effects in the SAL should be negligible?l104: omit comma before "that"
l113: Maybe it would be useful to show some graphical examples
of the Gibbs phenomena and it amelioration for a 1-dimensional example.
I assume you implemeted Cesaro summation? If not, then omit this
section.l125: Shouldn't you go back to Ferrell 1972 here and simply start
with the Green's function? Since the function \eta is originally
defined in physical space, it seems odd to develop these expression s
from the convolution theorem and its spherical harmonics.l143: Capital "G" in the derivative? Or define exactly what is
the lower-case "g_{SAL}" function.l133 and l137: It seems like you should go back to Ferrell 1972
and either use his Green's function and/or explain why your approximation
in equation (14) is justified.l153: How does omitting i=j "remedy" anything? Don't you need to
express the function as a sum of a singular kernel and a remainder,
and then analytically approximate or evaluate the singular part?
See Green and Nycander for an example.l165: Since you are concerned with the accuracy of an approximation,
the representation of a convolution over a spherical shell with its
truncated spherical harmonic expansion (and with its approximation over
the spatial grid), I think you should simply examine this directly
for some candidate \eta fields. It seems like the implementation in MOM6
should come later, once you are convinced that the approximations
are accurate enough.l166: What is the "error computation"? What is compared to what?
l178: While your MOM6 application might benefit from comparison with TPXO9
tides, I think you should separately study the accuracy of the approximations
of the convolution. There are many other sources of error in MOM6 and in
TPXO9 which could obfiscate or cloud the interpretation of your
approach to computing the SAL.I did not read the manuscript beyond this point.
Citation: https://doi.org/10.5194/egusphere-2026-654-RC2
Viewed
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 199 | 0 | 3 | 202 | 0 | 0 |
- HTML: 199
- PDF: 0
- XML: 3
- Total: 202
- BibTeX: 0
- EndNote: 0
Viewed (geographical distribution)
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
It seems that most global ocean modelers who incorporate tidal forcing into their models now finally appreciate the necessity of implementing self-attraction and loading (SAL). But in light of HOW they are doing it, as described in this paper, it seems necessary that changes be made. This paper describes why and how. The paper will likely become influential in the modeling community. It thus deserves to be published.... eventually.
The second half of the paper is what will be convincing to modelers and justifies publication. The problem is with the first half of the paper which does not use, or even acknowledge, a large already existing literature on this topic. The first half needs a good bit of revision, to acknowledge the existence of that prior work, and also the authors should examine whether that prior work necessitates some changes to their own approaches. This prior work is mostly (maybe entirely) in geodetic journals, and it often addresses loading or gravity but not self-attraction, but the math is essentially the same.
For example, the problem with Gibbs ringing is readily appreciated in the geodesy community, and for this and other reasons, nearly all geodetic loading computations use a convolution (Green's function) approach. Comparisons of spherical harmonic and convolution calculations, as presented here, have been done long ago. For example, a useful paper, not cited here, is:
Schrama, "Three algorithms for the computation of tidal loading and their numerical accuracy", J. Geodesy, 2005. doi: 10.1007/s00190-005-0436-3
Schrama actually does also address self-attraction.
There are popular, widely used, software packages that include many of the computations laid out here. I will mention:
- Duncan Agnew's SPOTL package
- Hilary Martens's LoadDef package
(both easily found via google).
These packages include things like computing Green's functions from loading Love numbers in numerically accurate ways. See especially Martens use of Kummer's series transformation. They (and other papers) discuss integrating over the singularity (or near-singularity) of the Green's functions and related matters -- I am pretty sure it does NOT involve skipping the integration at the origin as is done here (authors' statement after Eq (20)). Even Farrell's long 1972 paper does discuss some of this, too.
Many of the geodetic papers are also concerned with how to integrate over the ocean model near coastlines, which are not well mapped, but at least that problem does not arise here! (The border of the model ocean is known.)
In summary, I think the authors should start with Schrama's paper and at least read through the integration discussion in Agnew's manual. Martens's works are also useful and comprehensive. Then the authors can modify (if needed) some of their approaches, while acknowledging this previous literature. At that point, they can warn fellow ocean modelers of the need to account properly for SAL with the examples they give.
A few other items....
In any resubmission, I would appreciate having line numbers.
I don't understand the desire to break the symmetry of the Green's functions by including the gradient in it. The gradient can be done later, since it has to be done anyway for the grad \eta term. Giving up the symmetry in G is a high price to pay when one has to evaluate gradients later anyway. The SAL field is a smooth function (even over the land/ocean boundary) and easily differentiated, and I see no reason not simply to use that field and keep G symmetric. Maybe I'm missing something.
Expressions (14) and the listed constant terms (a1, b0, b1) appear to me to be very poor approximations when compared with Farrell's Table A2 (notwithstanding different earth models). Farrell tabulates k_n'/n, and it is not a constant until n > 1000 or so.
Figure 3. It's somewhat worrying to me that the degree-1 terms differ. These depend of the reference frame (and thus bear on geocenter motion) and I'm surprised the red and blue curves do not match there. It's worth checking for some reference-frame or other inconsistency here.
I was actually very surprised to read that the "standard" procedure is to use spherical harmonics at N=40, which is surely far too low. I had assumed that packages that use fast spherical harmonics were being used (e.g., spherepack3), which (e.g.) for a 0.5° grid would automatically take an expansion to N=360.
I agree that the problem with the "standard" SAL approach for ocean modeling is probably with the anomalous gradients near land. It might just be due to the fact that this is where the major dissipation from bottom friction occurs and so must be done accurately. Perhaps that is more or less equivalent to the "back effect".
Page 2, line 9: This reads as if my 1998 paper advocates using a scalar approximation, when in fact it does the opposite. I'd appreciate it if you move the (Ray,1998) reference to the end of the following sentence [after "different scales"].
In the final paragraph, I think the reference to earth's free oscillations is irrelevant to the present problem. The speed and frequency of free oscillations have nothing (directly) to do with the earth's response to tidal loading. The only anomaly like this that might eventually come into play is from an anelastic earth response, and new GNSS analyses are starting to bound this; e.g., perhaps at the mm level (in amplitude, so far not delay) in vertical deformation, according to Bos et al (2015; http://dx.doi.org/10.1002/2015JB011884 )