Atmospheric methane per head of cattle

Rough and ready order of magnitude calculation: Cows burp 100kg methane each year, which lasts 10 years before decaying in the atmosphere, and has about 100x the warming potential of carbon dioxide while it's up there.

Cows (and other ruminants such as sheep) continuously burp CH$_4$, a powerful greenhouse gas.  In fact, gram for gram methane warms the planet 87 times more than carbon dioxide.  However, CH$_4$ reacts in the atmosphere so that the carbon atom eventually finds itself in a CO$_2$ molecule.  We can think of it as having an atmospheric half life of about 8.6 years.  For this reason, the Global Warming Potential (GWP) of methane is sometimes averaged over 100 years to give a GWP100 figure of 27-30, i.e. twenty seven to thirty times as powerful (over 100 years) as carbon dioxide.

Methane enters the atmosphere from a number of sources.  Two significant ones are biogenic sources such as cows, and fossil sources, such as when unwanted methane is flared by oil rigs, or when fossil methane leaks from the distribution grid.  Fossil sources add new carbon into the atmosphere from geological reservoirs where it's been safely locked away for millions of years.  For this reason, a steady flow of fossil methane into the atmosphere will gradually cause atmospheric carbon to increase, along with global temperatures.  Biogenic sources, on the other hand, don't add any carbon to the atmosphere-biosphere, they just form part of a cycle.  This means that a steady flow of cow burps and farts would not increase carbon levels or temperatures.  Unfortunately the flow has not been steady, instead it has grown enormously over the last century as meat consumption has gone through the roof!

The number of cattle in the world has increased from about 450 million in 1900 to about 1.5 billion today.  The figure is still going up, but if it were to stop increasing, then at some point in the future the atmospheric CH$_4$ they cause would level out.  But to what value?

Let's call $M$ the total mass of methane in the atmosphere due to cattle.  We know that the rate at which CH$_4$ turns to CO$_2$ is proportional to $M$, so let's call that $M/T$, where $T$ is some, as yet unknown, time constant.  Let $P$ be the population of cattle.  We know that each cow produces methane at some rate $r$ so the rate at which cattle produce methane is $rP$.  This gives rise to a differential equation:

$$
\frac{dM}{dt} = rP -\frac{M}{T}
$$

We can solve the above equation for a steady state $dM/dt = 0$ to get a solution for $M$.  This is the level of methane that the world will converge towards if we finally stop increasing the cattle population $P$, and it is given by

$$
M = r T P
$$
Let's work out what $r$ and $T$ are in the above solution.  Each year an average cow produces 220 pounds, or 100kg, of methane.  So $r = 100 kg/$year.  What about $T$?  Well, if we imagine that $P=0$ the equation becomes $dM/dt = -M/T$ which is solved by $M=M_0 e^{-t/T}$.  We know the half life of atmospheric methane is 8.6 years so we need to solve $e^{-8.6/T} = 0.5$ which gives us $T=-8.6/ln(0.5)$ i.e. $T = 12.4$ years.  So if the population of cattle stays at 1.5 billion the atmospheric CH$_4$ it causes will eventually level out at 1.86 billion tons.  This has the same warming potential as 162 billion tons of CO$_2$ since methane is 87 times as powerful as carbon dioxide.

Now, the 2018 figure for the atmospheric carbon concentration was 410 ppm (up from a pre-industrial average of 280).  This corresponded to 3210 billion tons of CO$_2$, of which 1018 billion tons was anthropogenic.  That means that, if the cattle population remains fixed at today's levels, when the methane levels equilibrate they will be responsible for about 16% of anthropogenic warming!

Let's try to find the general solution for constant $P$.  We already have one solution to

$$
\frac{dM}{dt} + \frac{M}{T}= rP
$$
namely the steady state solution $M = r T P$.  This is called a "particular solution".  All we now need to do is solve what's called the "homogenous equation"

$$
\frac{dM}{dt} + \frac{M}{T}= 0
$$

This is because if $M$ is a particular solution and $M'$ a solution to the homegenous equation then $M+M'$ is another particular solution, and all particular solutions can be found this way.  Luckily the homogenous equation is easy to solve and its solution is $M = Ae^{-t/T}$ where $A$ is an arbitrary constant.  This gives a general solution of

$$
M = r T P + Ae^{-t/T}
$$

So this shows the mass of methane converging towards the fixed rate solution $rTP$ with a time constant of $T$.

In real life the population of cattle is increasing at some rate $R$ and we can model $P$ as equal to $Rt$.  In this case the particular solution becomes $M = rT(Rt - RT)$ and the general solution has the same decay term as before.  Another way of saying this is that the cow-genic methane concentration we have today is the equilibrium level for the cattle population of 12.4 years ago!

There are a number of conclusions we can draw from all this.  The first, obvious, conclusion is that we should stop increasing the amount of cattle.  The second is that if we do so, the remaining cattle will not cause atmospheric carbon levels to keep increasing indefinitely.  This is totally different to the case with fossil methane emissions which must stop entirely to halt the rise in atmospheric carbon.  Another way of looking at it is as an opportunity.  Although reducing the cattle population is strictly speaking optional, doing so would actually bring down methane concentrations, not just emissions.  Every cow eliminated offers a one off reduction of atmospheric methane of $rT$, or 1.24 tons CH$_4$.  That's 107 ton CO$_2$e removed from the atmosphere per cow gone.

 

So, removing a cow is equivalent to growing four large northern red oaks from saplings, and maintaining them.  There is a difference though: it only takes 12.4 years for the atmosphere to respond to a missing cow, but 100 years for an oak to grow to maturity.  Another difference is that planting trees uses land, whereas removing cows frees it up.  And if the freed-up land is reforested the positive impact on the climate will be even greater than estimated above!