# A Differential Equation for the Concentration of Methane in the Atmosphere: Estimating Parameters

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By Matthew Fulkerson, just a former physicist turned applied mathematician.

Previously, we verified a solution to a simple differential equation for the concentration of methane in the atmosphere under a constant emissions scenario. In the document “Differential Equations for the Concentration of Methane in the Atmosphere,” we go through this case, and explore more general scenarios such as a linearly increasing emissions rate.

In this article, we further explore the constant emissions scenario. We change up our notation a bit to follow Gilbert Strang in his book Differential Equations and Linear Algebra. Let a(t) be the rate of decay of methane in the atmosphere and q(t) be the emissions rate. This is Strang’s notation, except we have a minus sign difference in a(t) to explicitly account for our interest in exponential decay rather than exponential growth.

The general differential equation for arbitrary a(t) and constant q(t) is then:

Here we will solve this equation for constant a(t) = a_0 and constant q(t) = q_0. Making these substitutions, we obtain:

It is not immediately obvious why we have factored the equation in this way. We have done so because the equation is now separable. That is, we can get all of the terms involving y on one side of the equation and all the terms involving t on the other.

Now we can integrate both sides of the equation and obtain:

This is the same solution we verified previously, but this time we derived it rather than simply verified it.

Now we want to estimate a_0 and q_0 from real-world data. Since the concentration of methane is measurable, for example on Mount Mauna Loa in Hawaii, we can get the ratio q_0/a_0 from measurement. From Wikipedia and NOAA, we have the following plot of monthly averages of methane measurements:

Image credit: NOAA

We see that methane concentration is rapidly approaching 1900 parts per billion in the atmosphere. So q_0/a_0 is 1900e-9.

Next we can compute a_0 from the half life of methane in the atmosphere (denoted tau_CH4), which according to the Wikipedia article linked above is 9.1 years:

So then we have a_0=0.0762 and q_0 = 1900e-9*0.0762 = 1.45e-7 in units of inverse years.

Now we are finally getting to the punchline. We can understand what is meant by methane having 84 times more greenhouse gas forcing than carbon dioxide over 20 years. In the linked google doc, we work out the following for the average forcing over a given period of time.

It turns out that Kappa is the immediate forcing relative to carbon dioxide. That is, methane is initially 164 times worse than carbon dioxide as a greenhouse gas. Over a 20 year period, methane is on average 84 times worse than carbon dioxide. And over 100 years, methane is about 21 times worse. This is all depicted in the following chart:

A big takeaway is that we now have an estimate as to how many times worse than carbon dioxide methane is immediately. 164 times worse! A second takeaway is that methane may be 21 times worse than carbon dioxide over 100 years, but after 100 years it is basically gone and has no effects besides its byproducts (carbon dioxide and water).

Stay tuned for a future article, in which we will verify Strang’s general solution to the differential equation, and explore the case where emissions are linearly increasing.

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