## 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.