Sunday, February 28, 2010

Thermal Expansion

In Sea Level Rise - Part 2, I had estimated the magnitude of sea level rise (SLR) due to ice melt alone, since the last glacial maximum (LGM). It was 97 meters. This was based on an ice coverage and ice topography reconstruction made available by Peltier (1993).

Total SLR since the LGM is about 130 meters according to Fleming et al. (1998). The IPCC figure is 120 meters (4AR WGI FAQ 5.1.) The Red Sea reconstruction provided by Siddall et al. (2003) also indicates it's in the order of 120 meters.

So we're missing at least 23 meters of SLR. Could this difference be the result of thermal expansion? As it turns out, no.

I will estimate a ballpark figure for SLR since the LGM due to thermal expansion alone. For this, I will use a simplified model of the ocean: A rectangular pool of water with 3.5% salinity, a surface area of 361·106 km2, and a depth of 3,600 meters. The temperature at the surface of the pool has increased from 12C to 16C. The temperature at a given depth is calculated using the ocean water temperature profile model I derived in the last post. I will also assume that water pressure does not affect our calculations significantly.

In order to carry out the calculation, I divided our simplified ocean in 36 layers of 100 meters each. For each layer, I calculated current temperature and LGM temperature.

You can also calculate the density of water given its temperature and salinity. The following graph illustrates the relationship between temperature and the inverse of density (or volume of 1 kg of water in liters.)

The density of water with 3.5% salinity is, thus, well approximated by:

T is the temperature in degrees Celsius. The units for density are kilograms per liter.

With this equation, we can calculate the density of each layer of our simplified ocean, at present and during the LGM. The volume of each layer at present is 3.61·107 km3. The volume of each layer in the LGM can be calculated by multiplying 3.61·107 km3 by the current density and dividing it by the LGM density.

The total volumetric difference turns out to be 9.5·104 km3. This translates to 0.26 meters of SLR, or 26 cm.

This should not be taken as a precise figure. The point is that it's nowhere near 23 meters. Clearly, the change in ice volume I calculated from the Peltier (1993) reconstruction must be an underestimate.

Ice melt is obviously a much more significant problem, in the long term – meaning thousands of years – than thermal expansion. However, if the temperature is rising rapidly, thermal expansion can temporarily contribute more SLR than ice melt. Presumably, the effects of thermal expansion are immediate. Ice melt is slow.

Friday, February 26, 2010

The Temperature of Ocean Water at a Given Depth

In order to corroborate the results of the last post on sea level rise, I would like to estimate the impact of thermal expansion alone since the last glacial maximum. (I anticipate I will need to post a correction, but first things first.) In order to do this properly, I need to know how the temperature of ocean water varies with depth. I've tried to look for a straightforward formula that solves this problem, to no avail. (I did find a graph that was illustrative.)

So I went ahead and derived such a formula, based on ocean water characteristic data from the Argo database. More specifically, I used the ensemble mean grid made available by Asia-Pacific Data-Research Center of the University of Hawaii.

For each depth file, I calculated temperature means for 3 different latitude groups, which I've called 0S, 30S, and 60S. The 0S group includes latitudes -15S to 15N. The 30S group includes -45S to -15S. The last group includes -75S to -45S. Data is not available for all latitudes, so the mean values obtained should not be considered latitudinal averages; that's not the purpose of this exercise.

Let's start by looking at a graph of mean temperature at a given depth for each of the groups.

It doesn't look very easy, does it? Clearly, the temperature of ocean water must depend on variables other than the sea surface temperature and depth. But I was able to come up with a model that fits the data quite well (R2=0.988.) The formulas follow.

Where:
• S is the sea surface temperature plus 0.338, in degrees Celsius.
• D is the depth in meters.
• T(D) is the temperature at a given depth D, in degrees Celsius.

Derivation

The model was empirically derived. The first thing I noticed is that the depth D is approximately inversely proportional to T(D). In fact, the following formula is a rough approximation of temperature at a given depth.

The problem with this formula is that it doesn't work very well at shallow depths. You might have noticed in the figure that the temperature is roughly stable at a depth of 30 meters or less. In order to fix this problem, the approach I came up with is to transform D into D', where D' tends to zero when D is small, but tends to D when D is big.

If we multiply D by a logistic function, we obtain something close to the desired result for D'.

Once I had the general form of the equation, all I had to do is figure out the 4 coefficients involved. I used genetic programming to solve this.

Monday, February 22, 2010

The Greenland Canard

A favorite argument of AGW "sceptics" has to do with Greenland and what is known as the medieval warm period (MWP). The idea is that if the Earth was warmer some time in the past, this would undermine AGW theory. I don't find this line of argumentation to be robust either way, but let's examine it, shall we?

The Vikings colonized Greenland from 986 AD. They were able to farm, fish and raise cattle. The settlements disappeared by the 15th century, presumably because of the little ice age (LIA).

The first thing that needs to be pointed out is that Greenland was still a very cold place during the MWP. It was not a green paradise of any sort. Skeptical Science and A Few Things Ill Considered have the details on that.

Another counter-argument I've encountered is that Greenland might have been considerably warmer than it is now during the MWP, but this was most likely a local or North-Atlantic phenomenon, not a global one. Nearly all climate reconstructions that cover the MWP (and there are many of them, by many different authors, using several different methods) do not show it to be warmer than today.

Is there a way to corroborate that Greenland was indeed unusually warm during the MWP, unlike the rest of the northern hemisphere? Absolutely.

Alley (2000) provides a 50,000 year temperature reconstruction from Central Greenland whose resolution is not bad at all. The reconstruction is ice-core based, and it provides temperatures as absolute values. In order to calculate a "temperature anomaly" for purposes of graphical comparison, I added 31.29 to the temperature values provided by Alley (2000).

Let's start by looking at data from 200 AD to 1850 AD. I will use the CPS Northern-Hemisphere reconstruction from Mann et al. (2008) for comparison.

The difference between the MWP and the LIA in Greenland was around 1.6°C. Interestingly, the MWP temperature peak in Greenland occurs almost exactly at the time of the Viking colonization. Presumably, that's not a coincidence. Additionally, there are already some indications in this graph that the climate of Greenland experiences abrupt changes from time to time.

Not convinced? Let's look at 50,000 years of data. For comparison, I will use the temperature reconstruction from Vostok station, Antarctica made available by Petit et al. (1999). This is also an ice-core based reconstruction.

Here we see that the temperature of Greenland fluctuates in a manner that is not matched by the temperature of Antarctica. In particular, notice the effect of the Younger Dryas stadial on each of the series.

In synthesis, the climate of Greenland is quite peculiar, and as a result, it should not be thought of as a proxy of global or even hemispheric climate.

Wednesday, February 17, 2010

Ice Melt

In the previous post on Sea Level Rise, I argued there is a clear association between the Red Sea sea level reconstruction from Siddall et al. (2003) and the Vostok temperature reconstruction from Petit et al. (2000), with sea level lagging temperatures by about 4,700 years, at least for the last 50,000 years. This is corroborated by other reconstructions, such as the SL reconstruction from Arz et al. (2007) for the time span 83,000 years to 13,000 years before present.

The magnitude of SLR since the last glacial maximum 20,000 years ago is about 130 meters (427 feet), which again, is non-trivial. If we had similar SLR today, essentially all coastal cities around the world would be under water.

Much of that SLR has to be the result of ice melt. The rest, if any, would have to be the result of thermal expansion. Ice that is floating in the ocean should not cause SLR when it melts. What matters in this sense is ice over land.

In this post I'd like to estimate the magnitude of sea level rise due to ice melt alone.

NOAA provides a reconstruction contributed by Peltier (1993), along with applets that let you visualize ice coverage and ice topography in a world map. The resolution is 1,000 years, and the data goes all the way back to the last glacial maximum.

Fortunately, most of the ice coverage is over land, as you can see. So I took the 21K-year-old ice coverage data, and used it as a baseline for the calculation of changes in ice volume. The topography data includes land topography, evidently, but we're interested in ice volume change (not so much absolute ice volume) so it will do for now. In a footnote I will provide the processed data, along with an explanation of how it is obtained. The following graph shows the ice volume data along with the Vostok temperature reconstruction.

So we're talking about a change in ice volume in the order of 3.9·107 km3 in the last 21K years. The density of glacial ice is about 90% that of ocean water. (The tip of an iceberg is about 10% of the iceberg.) So the change in ice volume translates to 3.5·107 km3 of additional ocean water.

Now, the surface area of the ocean is 3.61·108 km2. If sea level rises, this surface area will change in a manner that is negligible, even if some areas are flooded. Does that make sense? Therefore, a good approximation for the change in ocean volume (ΔV) that results from a rise (ΔL) in sea level, is:

`ΔV = 3.61·108·ΔL`

Solving for ΔL, we have that:

`ΔL = 0.097 km`

That is 97 meters. It follows that thermal expansion must be responsible for the remaining 33 meters of SLR since the LGM.

[Correction 2/28/2010: As it turns out, SLR due to thermal expansion can't be anywhere near 33 meters. See Part 3 of the series.]

What if we lost all ice?

Let's assume, conservatively, that only 2.5·107 km3 of ice remain over land. If we managed to melt all of it, the ocean would get an extra 2.24·107 km3 of water. Dividing this volume by the surface area of the ocean, we get 62 meters of SLR.

That scenario is, of course, not something we'll see in our lifetimes, by a long shot. Even if we raised the temperature of Earth enough, it would take thousands of years to be realized, no doubt.

It's not an impossible scenario, however. About 50 million years ago, during the Paleocene-Eocene Thermal Maximum, global sea level was about 200 meters higher than today.

Data
• Summarized Ice Coverage and Ice Volume from Peltier (1993)
This is obtained by processing all ice*.asc and top*.asc files in "ASCII special" format. The area of a grid cell is calculated by dividing the area of a spherical ring 1-degree wide (corresponding to each latitude in the data) by 360. Only grid cells with ice coverage in the 21K-year-old data are considered when calculating land+ice volume.

Thursday, February 11, 2010

Sea Level Rise - Part 1

I have begun to take a closer look at available sea level data. Frankly, I'm a bit surprised. Of course I've heard of sea level rise (SLR) before, but it was like a fuzzy abstraction. I didn't have specific figures to ponder.

In this particular post I will only discuss paleoclimate data. There's some data that I imagine is pretty well known. The figure to your right, for example, comes from Wikipedia. It's a sea level reconstruction from Fleming et al. (1998) of the last 20 thousand years or so. That's roughly the time span since the last glacial maximum (LGM), when the global mean temperature was 5°C lower than today, give or take. As the LGM ended, and the planet entered the current interglacial period, sea level rose by about 130 meters (427 feet.) Of course, much of that is probably the result of ice melt, and there was a lot of ice in the LGM.

Then there's data that is not well known as far as I can tell. For example, there's a 380,000-year reconstruction contributed by Siddall et al. (2003) based on oxygen isotope records from Red Sea sediment cores. The following graph shows this sea level reconstruction along with the Vostok station temperature reconstruction provided by Petit et al. (2000).

If there were any doubts that temperature drives sea level, I believe the graph above is enough to dispel them. This is the basic premise I wanted to establish with this first post.

The data has some features that I wish to explore further, but not today. In particular, notice that sea level lags temperature by several thousand years. (This is not so clear with sea level data older than about 250,000 years. I'm guessing there's some sort of dating error either in the Red Sea data or in the Vostok data once you go that deep.) If I only consider data for the last 50,000 years, I'm estimating that the best lag is about 4,700 years. I'm not sure I can emphasize enough how important this lag is, but I'll certainly try.

[In Part 2 I estimate SLR due to ice melt alone since the LGM.]

Friday, February 5, 2010

The means by which a greenhouse gas (GHG) forces climate change is sometimes called radiative forcing. You can think of it as the additional irradiance necessary to bring the system back into balance after a change in the concentration of the greenhouse gas.

Apparently, the radiative forcing contribution of different greenhouse gases is typically added to come up with the total contribution. For example, if radiative forcing between 1750 and 1998 is 1.46 W/m2 for CO2 and 0.48 W/m2 for methane, then the total for the two gases is 1.94 W/m2. This makes complete sense on the surface, does it not?

But if the effect of a change in the concentration of a GHG is non-linear, is it really correct to linearly add such effects? I was thinking this would be similar to trying to add decibels.

After some digging around, my impression was that the IPCC had addressed this issue in section 2.8.4 of 4AR WGI. If you read the papers cited, nevertheless, it appears that they refer to the equivalence of the addition of forcings and the addition of temperature changes. I don't doubt these operations are roughly equivalent for relatively small effects. But that's not what I'm talking about at all. I'm questioning the validity of either operation, considering that both forcings and temperature shifts are non-linear responses.

In order to illustrate the problem, I've come up with a thought experiment that I've dubbed the twin GHGs paradox.

Imagine there are two GHGs that happen to be identical in their effects. If you have trouble picturing it, suppose one gas is industrially-produced CO2 and the other gas is naturally-produced CO2. Their effects are logarithmic.

Let's say the concentration of both gases has changed from 100 ppmv to 200 ppmv. Then their combined radiative forcing would be given by:

`ΔF = k·ln(200/100) + k·ln(200/100) = 2·k·ln(2)`

Since the gases are identical, the change just described should be equivalent to, say, that of one gas going from 190 ppmv to 390ppmv and the other gas remaining at 10 ppmv, not changing at all (i.e. the combined total is 200 ppmv initially and 400 ppmv later.) Therefore,

`ΔF = k·ln(390/190) + k·ln(10/10) = k·ln(2.05)`

Combining both equations, we have that:

`ln(2.05) = 2·ln(2)`

Evidently, we end up with reductio ad absurdum.

To get more technical...

When you're looking at the absorption bands of two GHGs, it seems clear that it matters whether the bands overlap or not. The net absorption calculation will depend on this.