**Effects of biomass burning on climate, accounting for heat and moisture fluxes, black and brown carbon, and cloud absorption effects (Journal of Geophysical Research, Atmospheres, 2014) (pdf)**

**Recent shift from forest to savanna burning in the Amazon Basin observed by satellite (Env. Res. Lett., 2012) (pdf)**

**The short-term cooling but long-term global warming due to biomass burning (Journal of Climate, 2004) (pdf)**

Correction to Equation 3 of the 2004 paper. This correction has no effect on any figure or result in the paper. Figure 1b of the paper, which refers to Equation 3 of the original paper, is the correct figure for the conditions given in the figure caption when applied to the corrected Equation 3, below.

Equation 3 gives the time-dependent change of anthropogenic CO2 in the atmosphere due to periodic burning of vegetation or crops followed by regrowth. The corrected equation applies both to savannah (the original application) and to any original vegetation replaced by a crop.

The equation demonstrates analytically that biofuel (e.g., ethanol or biodiesel) burning, considered a "renewable" energy source, is only partially renewable even ignoring energy inputs due to growing, refining, and transporting the resulting fuel. Considering only combustion and regrowth, biofuel burning always elevates atmospheric CO2 until the burning is stopped.

Equation 3 should be rewritten as follows

X(t) = X(0) + Bs,max + Bs * (exp(-MOD(t,Ns) / TAUs) - 1) / (1 - exp(-Ns/TAUs) (3)

where

X = atmospheric quantity of carbon

Bs,max = quantity of carbon burned or decayed from the land initially to clear
the land.

Bs = annual carbon emission rate from the regrown vegetation or from an annual
crop that

replaces the original vegetation

TAUs = e-folding lifetime (years) of the vegetation/crop against its full regrowth
(= 1/regrowth rate).

t = current time (years) past initial (t=0) clearing of land

Ns = time (years, between burning)

Please note that Equation 4 of the paper (Bs,max = Bs / (1 - exp(-Ns/TAUs)) applies only when the vegetation burned annually is the same as the vegetation originally cleared from the land. When a crop, for example, replaces the original vegetation, Bs,max is independent of Bs).

Figure 1b of the paper shows an application of revised Equation 3, above, to savannah burning (when Bs,max is related to Bs by Equation 4).

Below are some example uses of the equation when a crop replaces the original vegetation:

Example A: If land at time t=0 containing Bs,max = 3.5 units of carbon is cleared and replaced with a crop that has a regrowth rate of TAUs = 0.3 years (96.5% regrowth after 1 year), and the crop is burned at the end of each year with an annual emission rate of Bs=1 unit of carbon, the net accumulation of carbon at the end of 1 year from Equation 3 ranges from

X(t-->infinity) = 2.5 to 3.5, with an average of approximately 3.

Example B: In the case of agriculture replacing tropical rainforest land, where Bs,max = 20 or more,

X(t-->infinity) = 19 to 20 with an average of approximately 19.5

Example C: In the case of Bs,max = Bs = 1 (agriculture replaces equivalent vegetation),

X(t-->infinity) = 0 to 1 with an average of approximately 0.5

For comparison, the accumulation of carbon in the atmosphere due to fossil fuel combustion is

X(t-->infinity) = Bf * TAUa (3a)

(obtained by setting the derivative to zero in Equation 1 of "Updates to the 'Control of fossil fuel' paper at

http://www.stanford.edu/group/efmh/fossil/fossil.html

In Equation 3a, above,

Bf = annual emission rate of fossil-fuel carbon

TAUa = mean e-folding lifetime against removal of CO2 by all loss rates = 40
years

(with a range from 30-95 years - from the "Updates..." paper)

Thus, for Bf = 1,

X(t-->infinity) = 40

Thus, from Example A, 7.5% (3/40) of biofuel carbon emitted annually is not renewable in steady state relative to the same annual emission of fossil-fuel carbon.

In Example B, 49% (19.5/40) is not renewable.

In Example C, 1.25% is not renewable.

Note that Bs,max in Equation 3 is not attenuated by TAUa because removing the crops and allowing the natural vegetation to regrow will always allow Bs,max-Bs carbon to be removed from the atmosphere and stored on the land, so the loss of Bs,max-Bs to the air is constant so long as crops replace the natural vegetation.

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