Steric SSH changes

[julia-neme]

I've been calculating steric changes here as:

This integrates the departure of density from a reference density. I didn't know what value to assign to the reference density, so I chose it to be the time mean, that is it depends on (x, y, z). I had done it only for the upper and lower 100m, will do for the entire column now.

[julia-neme]

Total steric contribution to SSH in control, up and down:

Up and down anomalies with respect to control:

Looking at Figure 2 it doesn't seem that changes at the end of the simulation come from steric contributions, since they are approx one order of magnitude less... although someone might want to check my script :)

[StephenGriffies]

@julia-neme is the density here the in situ density? That is what it should be for computing steric effects. I ask since some folks mistakenly use potential density.

[julia-neme]

Hi stephen! Yes, its in-situ density calculated using gsw.rho (Gibbs SeaWater toolbox).

[StephenGriffies]

Sounds good.

[julia-neme]

As per our discussion today, in the calculation I replaced the reference density by a constant value (1035) instead of using the time mean for each experiment. I think it is what we wanted:

Should I still calculate it using density from control and dz from the perturbation?

[adele-morrison]

Nice, yes, so changing the reference density just aligned the starting points. We're still off by a factor of 2-4.

Yes I think it would still be good to do the control density with dzt from WIND+ to see how much of that change is coming from dzt vs rho.

[adele-morrison]

@StephenGriffies we have a question for you. We're struggling to reconcile the net sea level changes over the continental shelf with the Ekman effect (i.e. the change from the zonal wind only experiment) plus the steric effect (calculated above from in situ density change).

Is there anything else obvious that we're missing? Could the fact that we have more sea ice formation and export out of the shelf region be contributing to reducing sea level around Antarctica?

[julia-neme]

Dashed blue line in this plot is using density from control and dzt from WIND+. This confirms that all the changes are coming from density changes right?

But true, this is half of what the total changes in SSH show..

[StephenGriffies]

Hi. I will try to look at this asap. Sorry for my delay.

[StephenGriffies]

Hi @julia-neme and @adele157

Two suggestions. One trivial and the other more fundamental.

1/ Does the sea level you are plotting include the inverse barometer effect from sea ice? Using MOM5 lingo: are you plotting "sea_level" or "eta_t"? The sea_level diagnostic removes the inverse barometer, whereas eta_t is the actual position of the model's free surface top interface. For sea level studies such as in CMIP, we want sea_level. But when interested in model development, eta_t can be more important. Note that if "max_ice_thickness = 0.0", then eta_t = sea_level, since in this case the weight of sea ice does not affect the ocean free surface. This might be the setting in ACCESS, but unsure.

2/ When studying regional sea level patterns, it is useful to decompose sea level changes into local steric changes plus mass changes. Steric changes to sea level result from changes to density as integrated over the total depth of a column. As such, steric effects are typically more important for sea level over the deep ocean, where there is more water to integrate over. That then leaves the shelves as regions where mass changes can be more visible, with mass changes revealed by bottom pressure changes. So my guess is that you are missing the bottom pressure term in your diagnostics, and that can play a big role in some shelf regions.

For the physics and maths of what I am talking about, see Appendix B of Griffies et al (2014): http://dx.doi.org/10.1016/j.ocemod.2014.03.004

One modest qualifier: bottom pressure changes in a Boussinesq model can be a bit tricky given that Boussinesq models do not conserve mass but instead they conserve volume. Griffies and Greatbatch (2012) talk about the way to correct for this term in Section D.3.3 of http://dx.doi.org/10.1016/j.ocemod.2012.04.003

My recommendation for your shelf analysis is to ignore this Boussinesq correction for now. I suspect it is not important for your purposes. Rather, it is important for studies of the angular momentum budget as in

Bryan, F. O. (1997). The axial angular momentum balance of a global ocean general circulation model. Dynamics Of Atmospheres And Oceans, 25, 191-216 pp. doi:10.1016/S0377-0265(96)00477-0

I hope this helps.

[julia-neme]

Thanks @StephenGriffies! Lets see if I got this right.. Regarding:

1. I am using sea_level, and I think that max_ice_thickness = 0.0, but maybe @adele157 can confirm that. Therefore we are not including the inverse barometer effect due to sea ice, and we "shouldn't" worry about that?

2. I am not including mass changes. I thought that we could recover mass changes simply by looking at the decrease that would come from X amount of freshwater export from the study region, but that doesn't bring us closer to fully explaining sea level changes at the end of our simulation by steric+mass changes effects.

   Including only steric effects explains about half of the full sea level changes (see Figure 2 here only up to the year 2158.) However, if you look at Figure 8c, and considering that the total area of our study region is ~5e12 m2, that doesn't amount to the ~0.3cm change we need.

I will take a look at the Boussinesq correction, maybe discuss in our next hackathon and come back to this soon. I hope I have interpreted you answer correctly :)

[StephenGriffies]

1. With max_ice_thickness = 0.0 then sea_level = eta_t. So do not concern yourself with the sea ice inverse barometer as it is absent from this simulation. It does not affect the computed sea level in the model.

2. How are you computing the freshwater export? And are you including the net mass input arising from E-P+R through top of the ocean. Note that I am looking at equation (48) in Griffies et al (2014), where you see that bottom pressure changes arise from E-P+R as well as convergence of depth integrated mass transport. That is, are you sure you have a closed mass budget (really a closed volume budget since model is Boussinesq) for the shelf region?

I do not think the Boussinesq correction is needed. Recall the model is Boussinesq, so this correction term is not what the model will include when computing its free surface. Instead, I think the answer sits with the mass budget.

[julia-neme]

I calculated the mass term now, as the average over our study region of bottom pressure divided by g and rho_0 = 1035, following equation (48) in Griffies et al (2014). Here you can see both terms, mass changes and steric changes:

And here you can see their sum:

Almost there but not quite matching our Figure 2, where at the end of the simulation WIND+ shows ~+-.75cm and WIND- ~-1cm.

[adele-morrison]

Thanks Julia! And thanks @StephenGriffies for the references and info. Julia, do you think you could add the raw sea_level anomaly time series to this for comparison? Figure 2 has some complicated things going on, if I remember right, we subtracted the control climatology and applied a 12 month rolling mean.

[julia-neme]

I had a unit conversion error that @adele157 spotted. Corrected figures below:

It's not a perfect match.. but quite close. From what I gather that the mass contribution is more significant than steric changes actually for what we see in the last years of the simulation..

[StephenGriffies]

This is perhaps the point where the Boussinesq approximation starts to matter. That is, equation (48) is for a non-Boussinesq fluid. In contrast, (47) holds whether Boussinesq of non-Boussinesq. If you do not have the time tendency of the bottom pressure saved, then (48) is a sensible approximation, but it is not precisely correct for Boussinesq. I am going in circles trying to describe this. Sorry.

Suffice to say I think you have shown that the bottom pressure (mass term) is nontrivial and you now have an approximately closed sea level budget. Do you agree?

[adele-morrison]

Yes, I think you're right @StephenGriffies - a) this is probably as close as we can hope to get for now, and b) the mass term is non-trivial. Thanks again for your help!

My intuition is that the initial change (i.e. in first 2 years) in the mass term is due to increased southward Ekman transport moving waters onto the continental shelf in the WIND+ case. But then in the later years, this initial Ekman signal is overwhelmed by a negative anomaly in the mass term in WIND+. Any guesses as to what is causing this? Could it be the extra sea ice export continually removing water volume from the shelf? Or...?

[matthew-england-unsw]

Sorry I’m now a bit confused about what is being shown. In a diagram up above steric dominated over mass, and now in the most recent plots it’s the other the way around. What have I missed? Thanks for any tips

[adele-morrison]

The first plots are wrong, there was an error in the mass term. The most recent ones are correct.

On Fri, Apr 8, 2022 at 5:02 PM, Matthew England @.***> wrote:

Sorry I’m now a bit confused about what is being shown. In a diagram up above steric dominated over mass, and now in the most recent plots it’s the other the way around. What have I missed? Thanks for any tips

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[StephenGriffies]

@adele157 comment from 17 Aug 2022 email:

Regarding the sea level change components, ... the initial increase in sea level in the WIND+ case comes from the mass component and the longer term decreasing trend in sea level has a large contribution from the steric component. However, there is also a trend and large noise in the mass term. I don't think we decomposed the steric into halosteric and thermosteric (is that right Julia?); we had assumed that salinity would dominate. We decided that the complexity of the split between mass and steric terms would detract from the main story, which is why we left it out. However, if you have a suggestion for how to include it, we could reconsider.

The time series do seem noisy. But there is a trend, and that trend is the basis for the discussion in Section 3 of the Overleaf draft.

A visible means of supporting the discussion in the text is to compute maps around the shelf of the linear trend over the 10 years of the simulation. We showed linear trends for sea level, steric sea level, halosteric sea level, and thermosteric sea level in Figures 18,19,20, and 21 in this paper

https://www.sciencedirect.com/science/article/abs/pii/S1463500314000407?via%3Dihub

with details for the analysis in the text. I suspect the trend maps would support for the story, which, in fact, is currently written as if we had seen these maps...

[StephenGriffies]

See Sections 5.3 and B.1.2 of the paper

https://www.sciencedirect.com/science/article/abs/pii/S1463500314000407?via%3Dihub

for the decomposition into thermsteric and halosteric.

[julia-neme]

Thanks Stephen, that's great. I guess my doubt is a bit more of a technical one:

The density is calculated using a time varying salinity, but a reference state temperature, in Appendix B1.2 written as $\theta^{T}$. Conversely for the thermosteric contribution. My question is (and maybe @adele157 can answer this as well):

Should I take these $\theta^{T}$ and $S^{T}$ to be some value such as I took the reference density to be 1035, or should I average the 10 years of experiment on the shelf and use that instead? Sorry if the q is not too clear.

[StephenGriffies]

@julia-neme the reference state is typically taken as the first year of the simulation, so that subsequent years are referenced relative to that first year. So I suggest the annual mean of the first year is suited.

[julia-neme]

Reminder here: on-shelf averages of steric ssh changes plus the mass changes term are very close to the actual change in sea level:

Here you can see the steric and mass term separated: The variability is mostly set by the mass term, but there are trends in both terms...

Here is the decomposition of the steric sea level changes into a thermosteric and a halosteric component: Halosteric changes are clearly dominating changes in steric sea level.

So, going to this sentence in our manuscript: We hypothesise that the deviation of the sea level response from the expected Ekman dynamics after the second year arises from steric sea level change driven by changing water masses on the continental shelf.

And I might be wrong here! But don't this figures show that the response in SSH is dominated by the mass term? I'm not sure how to interpret this.

Maps of trends in the making :)

[StephenGriffies]

@julia-neme , very nice decomposition into halo and thermo steric, with clear trend in the halosteric!

It does seem that the mass term is dominating the variability. Does it have a clear trend too? Tough for me to see from the time series. Could be that the sea level trend is all driven by steric, with the mass term just variability on top of that.

[julia-neme]

Mmm actually the trends are important for both: for example, for the WIND- case, they are 0.01069 for the mass term and 0.003087 for the steric ssh (all in cm/yr).

Does this mean that the changes are dominated by mass export of freshwater cause of sea ice advection?

[adele-morrison]

Are those trends off by a factor of 10? The plot above looks like 0.4cm over 10 years for steric, so 0.04 cm/yr trend?

We could try a back of the envelope calculation: Area of the shelf = 2.8e12 m^2. Mass trend in sea level (0.1 cm/yr, is that right Julia?) = 1e-3 m/yr

Extra volume of water being lost from shelf region via mass sea level term = area*mass sea level trend = 2.8e9 m^3/yr

Compare this with the sea ice export anomaly (from Figure 8c of paper) of 1.2e-2 Sv = 1.2e4 m^3/s = 3.8e11 m^3/yr.

So sea ice export is 2 orders of magnitude larger than the mass lost in sea level? What did I do wrong here?

[julia-neme]

Hey, upps yes embarrassing, missed a zero in the conversion from m/month to cm/year (120 instead of 1200).

Your back of the envelope calculation seems fine to me. What other things could be affecting the mass term? DSW export?

[julia-neme]

Maps of trends of steric SSH changes relative to control, and decomposition into thermosteric and halosteric. All trends in cm/yr (this time the conversions are correct :)). Note that there are different colorbars, because the thermosteric is way smaller than the halosteric on the shelf.

A cool thing to note though, is the thermosteric signal in the West Antarctica.

[StephenGriffies]

Great to see these maps so quickly. Thanks @julia-neme !

Presumably the positive thermosteric in West Ant peninsula is the Spence et al 2017 wave mechanism?

The halosteric is consistent around the shelf except in Bel-Amun region where thermosteric signal is largest and opposite. I presume that is sea ice related?

To focus on the shelf, then I recommend masking out values in the open ocean and lowering the range a bit to bring out more features. The current color range is a bit too large for both thermo and halo patterns on the shelf.

Did you compute the trends for the mass component? I believe there is still the question about whether it is the dominant player. Note that steric only moves by advection and diffusion, whereas mass moves around by barotropic waves. So the mass effects are faster and more nonlocal than steric. So when doing a mass budget, one needs to account for mass that might be moving across the shelf break.

[julia-neme]

Lets see masked now:

and the decomposition into thermo/halo:

sorry about the smallish labels.. but it seems to me that the trends are dominated by the mass term on average (like the time series show), but that there are localized regions were the effect of steric changes is larger. For example, at that trough in the Weddell continental shelf. These are also regions of significant halosteric contributions.

I guess this means that our hypothesis applies on those localised regions, but not around the rest of the continental shelf, where the mass term dominates? Sorry, this is kind of a very informal blabber about this figures, not at all sure of what I'm saying.

[julia-neme]

Also, the notebooks are pushed and merged in case anyone wants to take a look at them :)

[StephenGriffies]

Thanks @julia-neme . These are great figures. But I must admit not fully understanding the mass term yet.

It is a clear signal and one that we should be able to understand. Perhaps just a clear and fast Ekman effect? I think that is the only option over this short of time. Is the sign right?

Sorry to ask for more work, but to help with the mass term could you produce it without masking the off-shelf region? I am trying to put this pattern into what we see over longer time scales, as discussed in Section B1 of

http://dx.doi.org/10.1016/j.ocemod.2014.03.004

The longer time patterns might not be what is revealed from our shorter 10 year expts. But still worth having a look.

All this might be an unnecessary rabbit hole, so I apologize, particularly since the sea level story is not central. But I do think we need to have a bit more to support the discussion in section 3...

[julia-neme]

Its an interesting rabbit hole in any case! Ok, I have plotted the unmasked mass term trends and the full SSH trends:

Notice how the SSH trend is dominated on the shelf by the mass term, whilst off shelf it is dominated by the steric term (see the maps a couple of comments above) (and excuse the colormap range differences). I think the argument in your paper was that the shelf and the open ocean are subject to steric changes of different order of magnitude. This would induce a gradient in ssh, that would in turn induce a mass transport on/off the shelf.

So the explanation for Fig. 2 could be: first years of Ekman transport dominated changes. Then steric changes larger off-shelf than on-shelf induce a mass transport. That dominates the later years. How does this sound?

[StephenGriffies]

Your explanation is what I expect to hold. Namely, a steric lowering of SSH on the shelf relative to the open ocean leads to a rush of mass onto the shelf, so a positive mass trend. But for the Wind+ case, there is a lowering of SSH on the shelf AND a lowering of mass on the shelf. It is not a perfect mechanism since there are geostrophic effects that cause currents to not go straight onto the coast. But it is roughly the case.

So the signs seem to be wrong in the above mass figures. Or am I missing something?

[julia-neme]

I think I was comparing the on-shelf/off-shelf steric ssh trends. Take the WIND+ experiment: the off-shelf decrease in sea level is larger than on-shelf. Therefore the rush of mass is off the shelf, leading to that decreasing trend we see on the shelf. I think the signs are correct, since they correspond with the time series. Does this explanation make a bit more sense? Sorry if I'm missing your point.

[adele-morrison]

That makes sense to me Julia. Steric SSH decreases both on the shelf (due to saltier DSW there) and off the shelf (due to colder AABW?). But the off shelf steric decrease is larger, which could lead to a mass movement off the shelf to compensate.

[StephenGriffies]

ok, so I see some regions where off-shelf steric lowers more than on-shelf. But in the Weddell Sea region that is not the case.

One other facet of the mass trend pattern is that mass is propagated around with boundary waves. So the steric/mass compensation mechanism is not so clean everywhere. It is only suggestive.

Anyhow, good to know the signs are right in your figures. I think we have a rough story, largely what Julia and Adele said, but with nuances related to mass movements with boundary waves.

I am unsure how much or how little of this analysis to show in the paper...thoughts?

[julia-neme]

Mmm that's a good question, I'm not sure because we haven't delved into the mass term in the paper. We could expand with a few more lines at the beginning of section 3 saying that on-shelf/off-shelf differential response to steric changes induces a mass flux? Do you think these figures should be added as suplementary?