My name is John Bond, and I took 12.307 in Spring of 2019. For my Dig Deeper project I investigated the global cooling effects of large volcanic eruptions. When a volcano erupts, there are two major side effects. First is the ash cloud, which largely stays in the troposphere, and can cause local issues, like agricultural failures and the clogging of rivers with the ashfall, not to mention high concentrations of air pollutants that are harmful for people to breathe. The second part of an eruption, which consists of the lighter sulfate particles that are thrown into the stratosphere, remain there for years. If the eruption is sufficiently large, then the amount of solar radiation reaching the earth is severely reduced, leading to cooler temperatures worldwide. Furthering the work of Mukund Gupta, I took the data he used in his project on the Little Ice Age (from around 1250-1850). This data takes solar radiation at the Earth's surface, which is gathered from observing ice cores and tracking layers to observe timing and magnitude of volcanic eruptions. This radiation data is put up alongside global temperature variation, which is observed from tree rings, taking their width through the years, as tree bark grows thicker in higher temperatures and better seasons.
That resulted in this data:
Figure 1: Plot from Mukund Gupta, top graph is Global Volcanic aerosol forcing, and bottom is global temperature.
In order to analyze this data, I plotted it via a MATLAB script, a large portion of which I obtained from Mukund Gupta (though I made personal edits to the script with his assistance), and zoomed in on various points, like this one circa 540 CE:
Figure 2: Zoomed in view of same dataset, circa 540. Observe the temperature dip followed by a relaxation period that coincides with the volcanic eruption
The particular pattern I was looking for generally consisted of a sharp negative temperature spike, followed by an exponential relaxation toward zero in the following decades. The ideal pattern looks like this:
Figure 3: Idealized version of temperature fluctuation curve
After analyzing several potential spikes, I compiled them all into a matrix. There were six in total.
Figure 4: Aggregate plot of all 6 observed temperature events, marked by year of occurence
It should be noted that not all of these temperature tracks follow the desired exponential pattern. Because there are many factors that influence global temperature, these signals are all incredibly noisy. However, after averaging these curves together, we end up with an aggregate curve that is striking.
Figure 5: Averaged curve resulting from taking the mean of all curves in Fig. 4. Offset from a pure average by 0.4848 degrees C.
One brief note, this curve is shifted upward by about a half degree. If you refer to the first graph, you can see that for the vast majority of the past two millenia, the average temperature has been significantly lower than the assigned zero point, which was taken in 1950, after the effect of the industrial revolution significantly dragged up the global average temperature. So I took the average of the temperature from 0 to 1800, and subtracted that from the averaged curve to get the above figure. After this, an exponential regression was completed to observe the average behavior of the temperature curve after an eruption.
Figure 6: The average of the six temperature plots is in blue, and the fitted curve is in red. The fit curve's formula is below
This fit has the equation:
The main analysis done in trying to determine the impact of this equation has to do with a simple model of the earth's thermodynamic behavior, which basically treats the earth like a giant ball of water. Since it is mostly water and the thermal impact of that is far greater than the impact of land, this model is mostly accurate. This assumption results in the model below:
Figure 7: Photo and formulae credit: Mukund Gupta. F is the forcing of the atmosphere, which in this case results from the volcano. rho is the density of water, cp is the specific heat, and T is the global average temperature. H is the water depth, which is taken as 100 m, as this is the depth of the ocean's mixing layer, over which it can be assumed that the ocean's temperature is relatively uniform as the water is well mixed by the wind. Lambda is the "climate relaxation factor" This factor, in W/m^2, represents the speed at which the planet returns to normal after a large temperature disturbance. After some manipulation of the above formula, a global temperature equation is reached:
Tau is the time constant, and Tc is the maximum distance from average, which in this case, is the initial sharp temperature decrease following the eruption. Applying the exponential regression we achieved earlier to the above equations, keeping in mind the the time for tau needs to be converted from years to seconds, we arrive at:
Given that the range for lambda was expected to be somewhere between 0.5 and 3, this value lines up neatly with the expected results. This project was quite fun, and it was interesting to see how some volcanic eruptions could pretty clearly be tied to large temperature variations from the same time frame. However, it also was clear from observation that this was not always the case. For example, in 1257 there was a massive eruption recorded, but there was no visible temperature change beyond a fluctuation during the one year.
Figure 8: Look in the very middle of the graph, just before 1260, to see the aforementioned eruption and observe the temperature surrounding that time. The temperature is low at that time, but there is no observable downward spike or subsequent relaxation. Clearly, there are more factors at play. But it is quite interesting to be able to use data to take a look back in time and see how volcanic eruptions effected our world. Given that volcanic eruptions will be a part of our planet for the foreseeable future, we must learn what we can from these events in order to better prepare for what lies ahead. Also, volcanoes are just plain cool, no?
