3.10: Appendix
- Page ID
- 25014
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Calculations for Table 3.3 and Figure 3.7 Using a Simple Equilibrium Model
The amount of organic matter in soils is a result of the balance between the gains and losses of organic materials. Let’s use the abbreviation SOM as shorthand for soil organic matter. Then the change in soil organic matter during one year (the SOM change) can be represented as follows:
SOM change = (gains) – (losses) [equation 1]
If gains are greater than losses, organic matter accumulates and the SOM change is positive. When gains are less than losses, organic matter decreases and SOM change is negative. Remember that gains refer not to the amount of residues added to the soil each year but rather to the amount of residue added to the more resistant pool that remains at the end of the year. This is the fraction (F) of the fresh residues added that do not decompose during the year multiplied by the amount of fresh residues added (A), or gains = (F) x (A). For purposes of calculating the SOM percentage estimates in Table 3.3 we have assumed that 20% of annual residue additions remain at the end of the year in the form of slowly decomposing residue.
If you follow the same cropping and residue or manure addition pattern for a long time, a steady-state situation usually develops in which gains and losses are the same and SOM change = 0. Losses consist of the percentage of organic matter that’s mineralized, or decomposed, in a given year (let’s call that K) multiplied by the amount of organic matter (SOM) in the surface 6 inches of soil. Another way of writing that is losses = (K) x (SOM). The amount of organic matter that will remain in a soil under steady-state conditions can then be estimated as follows:
SOM change = 0 = (gains) – (K) x (SOM) [equation 2]
Because in steady-state situations gains = losses, then gains = (K) x (SOM), or
SOM = (gains) / (K) [equation 3]
A large increase in soil organic matter can occur when you supply very high rates of crop residues, manures and composts or grow cover crops on soils in which organic matter has a very low rate of decomposition (K). Under steady-state conditions, the effects of residue addition and the rate of mineralization can be calculated using equation 3 as follows.
If K = 3% and 2.5 tons of fresh residue are added annually, 20% of which remains as slowly degradable following one year, then the gains at the end of one year = (5,000 pounds per acre) x 0.2 = 1,000 pounds per acre.
Assuming that gains and losses are happening only in the surface 6 inches of soil, then the amount of SOM after many years when the soil is at equilibrium = (gains) / (K) = 1,000 pounds / 0.03 = 33,333 pounds of organic matter in an acre to 6 inches.
The percent SOM = 100 (33,000 pounds of organic matter / 2,000,000 pounds of soil).
The percent SOM = 1.7%.