A Simple Sea Level Rise Model

... because curve-fitting is so much fun.

Background

Mark Bofill raises an interesting question over at Lucia's:

SLR is evidence warming is occurring. It doesn’t put the A in AGW though, FWIW. It’s always seemed to me that sea level rise started a trifle early for CO2 increase to be the original cause.

There being a number of SLR reconstructions floating about, I asked which one(s) he's been looking at.  He proposed I we have a look at Jevrejeva et al. (2008), Recent global sea level acceleration started over 200 years ago?

Abstract: We present a reconstruction of global sea level (GSL) since 1700 calculated from tide gauge records and analyse the evolution of global sea level acceleration during the past 300 years.  We provide observational evidence that sea level acceleration up to the present has been about 0.01 mm/yr 2 and appears to have started at the end of the 18th century.  Sea level rose by 6 cm during the 19th century and 19 cm in the 20th century.  Superimposed on the long-term acceleration are quasi-periodic fluctuations with a period of about 60 years.  If the conditions that established the acceleration continue, then sea level will rise 34 cm over the 21st century.  Long time constants in oceanic heat content and increased ice sheet melting imply that the latest Intergovernmental Panel on Climate Change (IPCC) estimates of sea level are probably too low.

A Quick Discussion of the Paper

I typically look at the pretty pictures first, here is the first one most relevant to this discussion:

There are other versions out there without the polynomial fit, which is what I'm used to seeing.  And indeed, the link Mark provided for me didn't have the trend line either.  Here's what I had to say as an initial response:

… the trend looks to start going up around 1780, is wiggly until about 1850, and basically linear from that point on. That about right?

That worked for him.  Here's one more figure from the paper itself:

I like this better than a simple polynomial fit.  The 30 year trend windows does bring out the 60 year quasi-periodic rate fluctuation mentioned in the abstract.  And finally, the upward-sloping rate plot indicates an accelerating trend.

A 60-year cycle smells like AMO to me.  I may not be too far off; from the paper:

A global pattern of 60-year variability is supported by comparison of the GSL and North East Atlantic variability (Figure 3), where a similar pattern of variability is seen, though with differences in amplitude and timing of prior to 1950, which are suggestive of an Atlantic driving mechanism. This may be related to an underlying variability in the thermohaline circulation [Delworth and Mann, 2000], perhaps through advection of density anomalies or combinations of gyre and overturning advection [Dijkstra and Ghil, 2005]. However, direct observational evidence on these long cycles in thermohaline circulation is very limited and modelling using coupled Global Circulation Models (GCMs) show rather ill-defined power on these timescales [Knight et al., 2005].

Mann (2009) presents an AMO reconstruction going back to 500 CE.  Had it already been published it would have been interesting to see what Jevrejeva & Co. might have done with it, if anything.  My first instinct was to not use Mann's reconstruction because it's not de-trended, and we're more interested in CO2's putative influence on SLR than what might be manifest in Atlantic SSTs.  That didn't mean I wished to ignore other plausible mechanisms which might have contributed to the "early" SLR and/or some of the wiggles.

A Dirt-simple Multiple Regression Model

Total solar irradiance (TSI) is one obvious candidate.  The paper mentions volcanic activity -- I myself was already thinking Tambora in 1815, and Krakatau in 1883.  But I only have volcanic aerosol data back to 1850 my fingertips.

There is one not so obvious candidate; length of day anomaly (LOD), data for which I have all the way back to 1623 CE.

CO2 data are of course easy to find going back hundreds of thousands of years.

So, without further ado, here's the quite elementary multiple linear regression model of SLR I managed to kluge together:

Figure 1 - SLR regression model.  120-month trailing means for all data series except TSI (132-months).  CO2 and TSI are lagged 360 months.

The fit is actually fairly good for CO2 only, but the TSI and LOD combine to slight rise starting around 1790, the flat trend from 1820-1880, and the seemingly early upward trend beginning around the turn of the 20th century.

I have some other interesting tidbits to add from work done by peer-reviewed professionals, which I plan to add to this note as updates.  For now, I'm going to rush this to press so Mark and others can have a looksee.

Update 4/15/2016

In comments, Mark Bofill refers us to a slide presentation by Dr. Trenberth.  The text of slide 12 reads:

Where does energy go?
• An imbalance at TOA of 1 W m-2 is 3.2x10 7 J/yr m-2 = 1.6x10 22 J/yr globally
• To melt 10 6 km 2 ice 1 m thick (2007) to 10 C = 0.8x10 20 J
• To produce 1 mm rise in sea level requires melting 360 Gt ice or 1.2x10 20 J Plus 12.5% to warm melted waters to ambient 1.35x10 20 J
• To produce 1 mm rise in sea level by warming the ocean (thermosteric) depends greatly on where energy is placed
• Fresh water has a maximum in density at 4 C, but not so for sea water.
• Coefficient of expansion varies with temperature and pressure by factor of 6 from 0 C to 20 C
• For warming over top 700 m to give 1 mm can take from 50 to 75x10 20 J, or below 700 m 110x10 20 J
• Hence melting ice vs warming ocean is a factor of about 40 to 70 more effective in raising sea level (if in top 700m) or 90 (if below 700 m)
• 1 W m-2 gives sea level rise of 93 mm (melting ice) vs 3 to 1.5 mm (thermal expansion)
• Need to distinguish eustatic vs thermosteric sea level rise wrt energy

From these figures, Mark argues:

1. It's much cheaper energy-wise to get SLR from melting ice than thermal expansion of sea water.
2. Even so, Dr. T gets only 93 mm for each W/m^2 forcing increase,
3. Which means about 372 mm for a doubling of CO2 I think? 4 W/m^2?

So the problem is that the model, although it fits bee-a-utifuly, doesn't appear to be physically correct. There's not enough energy for that much SLR for that much CO2 increase.

Which is a nice way of saying my simple linear regression model above is busted.  If it's not immediately apparent why, note carefully the title of my Figure 1 above, which puts CO2's contribution to SLR at 1.23 meters for a doubling of CO2.  The canonical radiative forcing calculation for CO2 doubling is:

5.35 W/m^2 * ln(2) = 3.71 W/m^2

Multiplying by 93 mm m^2/W gives 344 mm of SLR per CO2 doubling.  This implies that my regression model is magnifying CO2's effect on SLR by a factor of 3.6.

Not so fast.  CO2 is the dominant anthropogenic forcing since at least 1950, but it is not the only one (note that there are some negative ones as well).

And then there are feedbacks.  The Met Office provides an enumeration with short descriptions, though no numbers.

The dominant positive feedback is water vapor.  Trying to track it down in AR5 is a pain, but to my knowledge the AR4 estimate is still valid (plus it's in handy HTML format instead of the abhorrently bulky slow-loading .pdf documents of AR5):

In the stratosphere, there are potentially important radiative impacts due to anthropogenic sources of water vapour, such as from methane oxidation (see Section 2.3.7). In the troposphere, the radiative forcing due to direct anthropogenic sources of water vapour (mainly from irrigation) is negligible (see Section 2.5.6). Rather, it is the response of tropospheric water vapour to warming itself – the water vapour feedback – that matters for climate change. In GCMs, water vapour provides the largest positive radiative feedback (see Section 8.6.2.3): alone, it roughly doubles the warming in response to forcing (such as from greenhouse gas increases). There are also possible stratospheric water vapour feedback effects due to tropical tropopause temperature changes and/or changes in deep convection (see Sections 3.4.2 and 8.6.3.1.1).

See Footnote [1] for why this confuses me.  Ignoring my confusion; taking the above paragraph as written, water vapor amplifies 3.71 W/m^2 forcing per CO2 doubling by a factor of 2 to 7.42 W/m^2, implying 688 mm of SLR in that scenario.

There are of course other feedbacks.  Rather than try to look them all up and net them out as a function CO2 change, I will "cheat" by simply grabbing some LW flux variables from the CMIP5 RCP6.0 model ensemble.  Since Dr. Trenberth is talking about radiative imbalance at TOA, one's first instinct might be to grab outbound LW flux at TOA from the models.  It does not make sense to do so for my simple linear regression model approach:

Figure 2 - Modelled outbound LW at TOA from CMIP5 historical/RCP6.0 ensemble.  Source: KNMI Climate Explorer.

For this exercise, I thought it made more sense to go after either the up or down LW flux variable, here's what downwelling LW flux at the surface looks like:

Figure 3 - Modelled inbound LW at surface from CMIP5 historical/RCP6.0 ensemble.  Source: KNMI Climate Explorer.

That makes more sense.  Regressing that against the RCP6.0 CO2 scenario over the entire 1861-2100 interval gives 17.44 W/m^2/2xCO2.  Divide by 3.71, and we get a factor of 4.7 over the expected forcing of a CO2 doubling alone.  This seems promising.  However ...

Don't quote that figure.  Reason being, it's inclusive of other GHG increases, changes to aerosols and land use, as well as all as the net of all feedbacks.  I only use it here to allow CO2, being the dominant driver of AGW, as kind of a proxy for roughly guesstimating what SLR might look like under the RCP6.0 CO2 emissions scenario:

Figure 4 - Projected SLR using RCP6.0 Surface Downwelling LW only.  All series trailing 120 month means, no lags.

This model gives 45 mm SLR for every 1 W/m^2, Trenberth is calculating 93.  So by all rights, I should be projecting on the order of 1.4 m of SLR by 2100.  As well, over the first part of the 21st Century, I'm projecting GMSL well higher than present observational estimates, on the order of 20 cm.  Why?

As BBD points out in comments below:

Hot off the presses at Nature Climate Change we have Slangen et al. (2016) Anthropogenic forcing dominates global mean sea-level rise since 1970.

 Going to the paper, we read:

Under CMIP5 control run forcing, most contributions show little variability, and no significant trend on a centennial timescale (Fig. 1a). However, if the glacier model is initiated to its 1850 state and then forced with control run variability, there is a contribution of 30 ± 13 mm for 1900–2005 (cyan) as a result of the continued retreat of glaciers to higher altitudes after the Little Ice Age (LIA relaxation) 5,11, as glaciers typically take decades to centuries to establish a new equilibrium after climate changes.

In my first model, Figure 1 above, I used a 30-year lag for CO2 and solar forcing.  If in the Figure 4 model, we mentally slide the projected curve forward 30 years, the 2100 projection works out to about 50 cm of SLR over the 1986-2005 baseline mean for RCP6.0.  Let's see what the IPCC actually projected:

Figure 5 - GMSL projections for all RCPs.  After AR5 SPM Figure 9, credit: Aslak Grinsted.

For RCP6.0, the central estimate for the 2081-2100 mean is ... 50 cm, about where my model would put it if I laid in a 30-year lag in forcing.

Damn I'm good.  Or maybe just lucky.

This doesn't really answer Mark Bofill's original question, at least not directly.  At the very least I better understand the arguments for the IPCC SLR projections being conservative ... here I make it by about half.  Exploring that would be interesting, and I will perhaps dig into that in a future article.

Footnotes

[1] If we first consider Table 3 from Kiehl and Trenberth (1997), water vapor accounts for about 75 W/m^2 and CO2 about 32 W/m^2 of the "greenhouse effect" in clear sky conditions, 51 and 24 W/m^2 respectively under cloudy conditions.  That gives a water:CO2 ratio of about 2.2:1.

For sake of argument, it seems reasonable to assume a linear relationship for a CO2 doubling: 1 + 2.2 = 3.2.  We are looking to explain an apparent discrepancy in my simple model of a factor of 3.6, so invoking water vapor feedback gets us pretty close, with other net feedbacks making up the difference.  Yet AR4 tells us to only expect an amplification factor of 2.

So I'm confused.  Perhaps some literati out there can help a buddy out here.