# Sea Level Rise due to Thermal Expansion

This is a simple model of sea level heights that may be appropriate for very introductory level students studying climate change.

For this model I will assume that the ocean consists of two parts: the surface ocean and the deep ocean. The surface ocean is uniform in depth, temperature, and salinity. The depth of the surface ocean is 500 meters. The average initial temperature of the upper ocean is 14C. The deep ocean is everything else, and is assumed to not change.

The volume of water in the ocean is given by the equation: V=A*d, where A is the surface area of the ocean and d is the depth of the ocean. We also know that the mass of an object is equal to its volume multiplied by its density; m=V*ρ. We can solve these equations for d, the depth of the ocean.

d = m/(ρ*A)

And since we’re interested in the change in d, or Δd, that’s equal to

Δd=d-d0, where d0 is the initial height of the ocean, 500m.

In our equation above, we have the change in depth as a function of density, and we’re assuming that the mass of the ocean and its surface area do not change. However, that’s not really interesting, so let’s find how the density of sea water changes with temperature. As stated above, the salinity, or the saltiness, of the ocean will be held constant. So the density is only dependent upon the temperature. We could go through the calculations here to figure out the density as a function of temperature, or we could cheat and use this handy water density calculator. I’ve reproduced some values in the table below:

We now have all we need to figure out how much sea levels will rise do to thermal expansion.

This simplistic model of the surface ocean shows that if the average sea surface temperature rises by 1.4C, sea levels will rise by about 6 inches.

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