and we know that it is a much more potent greenhouse gas than CO2

Recall methane is a feedback enhancement, not a primary, mechanism. The reason is the timescale... methane content is more or less a function of other parameters due to its short equilibration time. The reason CO2 is so significant is its longer time scale. See the intro articles on
RealClimate for more detail.

"Methane is transient in the atmosphere, so if the anthropogenic source stays constant, the methane concentration stays constant. This is different from CO2, which accumulates. Methane is well-mixed in the troposphere. Gets oxidized by OH in the troposphere and zapped by UV light in the stratosphere. Gases don't sink out very much in the atmosphere because it circulates so quickly. You can measure gravitational settling of gases in stagnant columns of air like in firn above ice cores, but in the atmosphere, gases don't really settle out. The atmospheric measurements are straightforward, replicated, reliable."

--David Archer on RealClimate

ciao,
Bruce

Natural methane may be a feedback mechanism, but rice paddies, landfills and pipeline leaks are not.  If we can reduce atmospheric methane by turning landfill gas into electricity and feeding our livestock differently, that's changing a human greenhouse contribution.

Methane is transient in the atmosphere, so if the anthropogenic source stays constant, the methane concentration stays constant.

On the other hand, if the anthropogenic source were to lessen, then the concentration would lessen, wouldn't it?

That's very true... unlike with CO2 there is no significant delay hence methane doesn't accumulate in the way CO2 does. I don't remember the residence time for CO2 in the atmosphere, but ...(google)... it's about a century for CO2 and about a decade for CH4. The latter figure is long enough for CH4 to be well mixed and short enough that even with increased input the effective sinks keep the concentration in equilibrium. The response (eq conc as a fn of forcing) is nonlinear, but in this case the nonlinearity is stabilising (i.e., you don't get explosive growth without a catastrophic event).

Those interested should google on these things a bit, there is lots of interesting stuff on the net. It is easy to stick to scientific (or just multiple) sources.

ciao,
Bruce