A revised sea level budget equation to accurately represent physical processes driving sea level rise
 ^{1}University of Bristol, School of Geographical Sciences, Bristol Glaciology Center, Bristol, United Kingdom of Great Britain and Northern Ireland (bd.vishwakarma@bristol.ac.uk)
 ^{2}Delft University of Technology, Department of Geoscience and Remote Sensing, Delft, The Netherlands
The sea level budget (SLB) equates changes in sea surface height (SSH) to the sum of various geophysical processes that contribute to sea level change. Currently, it is a common practice to explain a change in SSH as a sum of ocean mass and steric change, assuming that solidEarth motion is corrected for and completely explained by secular viscoelastic relaxation of mantle, due to the process of glacial isostatic adjustment. Yet, since the Solid Earth also responds elastically to changes in present day mass load near the surface of the Earth, we can expect the ocean bottom to respond to ongoing ocean mass changes. This elastic ocean bottom deformation (OBD) has been ignored until very recently because the contribution of ocean mass to sea level rise was thought to be smaller than the steric contribution and the resulting OBD was within observation system uncertainties. However, ocean mass change has increased rapidly in the last 2 decades. Therefore, OBD is no longer negligible and recent studies have shown that its magnitude is similar to that of the deep steric sea level contribution: a global mean of about 0.1 mm/yr but regional changes at some places can be more than 10 times the global mean. Although now an important part of the SLB, especially for regional sea level, OBD is considered by only a few budget studies and they treat it as a spatially uniform correction. This is due to lack of a mathematical framework that defines the contribution of OBD to the SLB. Here, we use a massvolume framework to derive, for the first time, a SLB equation that partitions SSH change into its component parts accurately and it includes OBD as a physical response of the Earth system. This updated SLB equation is important for various disciplines of Earth Sciences that use the SLB equation: as a constraint to assess the quality of observational timeseries; as a means to quantify the importance of each component of sea level change; and, to adequately include all processes in global and regional sea level projections. We recommend using the updated SLB equation for sea level budget studies. We also revisit the contemporary SLB with the updated SLB equation using satellite altimetry data, GRACE data, and ARGO data.
How to cite: Vishwakarma, B. D., Royston, S., Riva, R. E. M., Westaway, R. M., and Bamber, J. L.: A revised sea level budget equation to accurately represent physical processes driving sea level rise, EGU General Assembly 2020, Online, 4–8 May 2020, EGU20208808, https://doi.org/10.5194/egusphereegu20208808, 2020
Comments on the presentation
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CC1:
What about the elastic response of the ocean column itselff?, Aslak Grinsted, 04 May 2020

AC2:
Reply to CC1, Bramha Dutt Vishwakarma, 04 May 2020
Hi Aslak,
I am assuming that the compression will affect the density through heights.
In the theoretical faremwork we allow for a change in density through depth. The final experssion for sea level budget components have surface ocean water density and ocean density near the ocean's bottom.
However, if such effects are not included in the data products (such as steric), then application of the budget equation with available data will not account for compressibility of water column.
I hope this answers your query. please let me know if you have more questions/comments.
Thanks,
Bramha

AC4:
Reply to CC1, Samantha Royston, 04 May 2020
As Bramha said, theoretically you could include it, but it is generally not accounted for by observational products.
My understanding is, if you were comparing two `full depth' profiles taken at two points in time, then using the TEOS10 equation of state you effectively account for p because the equations use the pressure level that the measurements are taken at (i.e. the coefficients account for p). But for the steric data products, using mostly data to 2000 m depth, this isn't going to be accounted for; neither in reanalysis models. In terms of order of magnitude, yes I think the volume reduction is about 2% for a 4 km column of water. So at present rates, a 30 mm increase in EWH over 10 years we should expect 0.3 mm reduction in the water column height due to compression. So roughly an order of magnitude smaller than the OBD and far smaller than the uncertainties in the sea level budget!

CC2:
Reply to AC4, Benjamin D. Gutknecht, 04 May 2020
Greetings!
I do agree that volumetric compression _at 4 km depth_ would be in the order of 0.018 in total (dimensionless). Compressibility: ~ 4.5·10^{10} Pa^{1}, which, due to water internal structure, increases with lower temperature near 0 °C, increases with freshness, and decreases with depth (pressure). For a more advanced approach, one would want to use a compressability integral function. But: this percentage cannot simply be taken for _additional_ mass in time; you would have to put another ocean on top for that, I suppose. If we assume compressibility constant with regard to salinity etc. and we add 30 mm ewh of water mass, it results in a relative volume change in the order of only 10^{7 }from (additional) compression. Thus, even more insignificant than in your approximation, I would say. It is an interesting thought, but I alos do not see a necessity for integrating compressibility in the budget (e.g. in view of uncertainties as already mentioned). We have not integrated it in the esa SLBC project.
ben

CC3:
Reply to CC2, Aslak Grinsted, 05 May 2020
I dont think the 1e7 is correct. Here's how I got 2 cm: I have a 4km water column which is elastically in equilibrium with respect to its own load.
 I now add 1m on top of that.
 Pressure everywhere increases by 10 kPa
 Every part of the column will compress by 10e3Pa/2e9Pa (Bulk Modulus = 2e9Pa)
 This leads to a further compression of 4000m*10e3Pa/2e9Pa = 2cm.The OBD amplitude looks like ~5% of the Mass amplitude in your results plot. This elastic water column effect would be 2%, so slightly less than half of the OBD amplitude. I agree that this is small compared to uncertainties. But on the other hand it seems trivial to apply this correction, so why not try to get people to do it? It would be a different and probably simpler study than your nice OBD one, but I think it is worthwile. Taken together with OBD then this becomes a sizable fraction of the budget.

AC5:
Reply to CC3, Bramha Dutt Vishwakarma, 05 May 2020
Hi Aslak,
Thanks for sharing the step by step process for reaching to that value of 2cm. I guess there is a confusion because we are talking about change and not absolute values. Here is my take:
At time t1, for an ocean 4000 m deep, we would already have such compression, denoted by h_c(t1) = 2 cm. Thus the ocean depth ideally should be (4000e3 2) cm.
At time t2 let's assume the water mass increases by 1 m, the corresponding value h_c(t2) would be (as per the same relation) 4001*(10000/2*e9) = 2.0005 cm. Thus we can conclude that, corresponding to 1 m increase in sea level, change in h_c is 0.0005 cm. If the change in sea level is of the order of mm (which is the case), the correspodning correction would be of the order of 1e7, as pointed out by Benjamin.
If we were not considering rate of change, but absolute change (filling oceans from an empty state), this compression correction becomes significant and must be included. I hope i am making sense. If I have missed anything, please let me know.Thanks,
Bramha

CC4:
Reply to CC3, Benjamin D. Gutknecht, 05 May 2020
The ∼2 cm are the total compression at depth of the entire water column down to 4 km. To my understanding, we all are interested in the change, after adding e.g. 1 m on top, which is the difference of compression of the 4000 m column with respect to a 4001 m column. Adding 1 m water does not cause 2 cm compression by any means. A simplified approximation:
Comp_{4000} = 1025 · 9.81 · 4000 Pa · 4.5e10 Pa^{1} = 0.018 ≈ 2% (dimensionless)
ΔComp_{40014000} = 1025 · 9.81 · (40014000) Pa · 4.5e10 Pa^{1}, i.e. just the compression effect of 1 meter.
Hence, the total effect of 2% is notable, but the change is less than marginal.

AC5:
Reply to CC3, Bramha Dutt Vishwakarma, 05 May 2020

CC3:
Reply to CC2, Aslak Grinsted, 05 May 2020

CC2:
Reply to AC4, Benjamin D. Gutknecht, 04 May 2020

AC2:
Reply to CC1, Bramha Dutt Vishwakarma, 04 May 2020

AC1:
Comment on EGU20208808, Jonathan Bamber, 04 May 2020
Hi Aslak, that's a good question, which I don't really know the answer to. It won't be included in any steric calcs because these don't have any change in mass. I doubt it's included in any SLB calcs as far as I am aware.

AC3:
Reply to AC1, Bramha Dutt Vishwakarma, 04 May 2020
Thanks Jonathan. The theoretical farmework allows us to incorporate such effects, but the available datasets do not. So I am sure that compressibility of water column is missing.

AC3:
Reply to AC1, Bramha Dutt Vishwakarma, 04 May 2020
Your study made me wonder if the water column also will compress. We water is usually treated as incompressible, and it nearly is, but not completely. The bulk modulus is such that a 4km deep column of water will compress by 2cm if you add 1m to sea level (by mass). Atleast that is my naive estimate. Not alot, but not neglible either.
Question: Do you know if this effect of pressure on the density of water is taken into account in any steric estimates? Should it be accounted for separately in the budget equation?