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That resulted in this data:
Figure 1: From Plot from Mukund Gupta, top graph is Global Volcanic aerosol forcing, and bottom is global temperature.
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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
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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:
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