1. A simple continuous model of soil organic matter transformations
S. I. Bartsev1,2 A. A. Pochekutov1
1) Institute of Biophysics SB RAS, Federal Research Center “Krasnoyarsk Science Center SB RAS”,
2) Siberian Federal University
Krasnoyarsk, Russia

2. SOIL AS A CARBON RESERVOIR
More than 50% of carbon involved in global carbon cycle is estimated to be in the form of soil organic matter (SOM).
Therefore any model of global carbon cycle have to include a module capable of describing the soil response (release or accumulation of carbon) to global climate change.
Unfortunately it is not completely clear in what chemical forms SOM exists, what are the factors protecting SOM from decomposition, and what are the mechanisms of SOM transformation.
In this situation of uncertainty the most effective way seems to be creation of phenomenological model representing coinciding general provisions of existing concepts.

3. Analyses of existing concepts (Essington, 2004; von Lützow et al
Analyses of existing concepts (Essington, 2004; von Lützow et al., 2006; Berg, McClaughrety, 2008) showed that all of them contains notions of gradual increase of SOM stability toward transformation and concurrently occurring partial decomposition of SOM.
Phenomenological representation of the most general notions about the nature and direction of SOM transformation process does not require details of internal mechanisms of it, which are different for existing concepts.
In this case the model can and has to be constructed in accordance with the imperative “make it As Simple As Possible”. 3

4. Derivation of the model in the scope of humification concept
litter
Organic matter mineralization, k(h)
humus
detritus
Organic matter humification, v(h)
The big arrow represents a scale for h – “constant” of organic matter humification rate. Then the humification process can be described as the motion of small portion of organic matter along this scale.
The state of organic matter involved into the process can be described as C(h, t) - the concentration of organic matter having humification rate “constant” inside the small interval (h, h+h) at time moment t. 4

5. Derivation of the model in the scope of humification concept
h'+h' h
h+h
The amount of matter in the range (h, h+h) is
After a time t the amount of material has displaced along the coordinate h to the point h'. In this case the same amount of material is located in the shorter interval (h’, h’+h’). 5

6. IMPORTANT NOTE
Note that the range of h becomes shorter and SOM concentration increases with time. 6

7. Derivation of the model in the scope of humification concept:3
Then from the matter conservation law we can write:
After usual manipulation corresponding equation can be obtained:
where v(h) is the velocity of the motion of the matter along the h-scale, k(h) is mineralization rate coefficient, and D(h, t) is litter input rate distribution over h. 7

8. v(h) - ?
v(h) is the velocity of the motion of the matter along the h-scale.
It seems reasonable to suppose the rate of transformation is proportional to a variable which is a measure of matter decomposability (Carpenter, 1981). Here this measure is the rate of humification - h. Thus,
−v(h) = dh/dt = −r · h.
But r should be equal to h too, since it is a coefficient of the rate of the reaction that causes further movement of the matter along h-scale. Thus h is simultaneously the changing value and the coefficient of the rate of change h itself. Then
and 8

9. THE BASIC EQUATION OF SOM TRANSFORMATION
So dynamics of the distribution is described by typical transport equation:
We assume k(h) as a simple nonlinear empirical function:
where b and p are constants corresponding to different ecosystems.
is litter input rate distribution over h,
where δ is the Dirac delta function, index i numbers plant litter components characterized by their annual average amount D0i and initial transformation rate h0i.

10. ANALYTICAL SOLUTION OF THE BASIC MODEL EQUATION
The basic model equation has the following analytical solution:
where θ is Heaviside step function.

11. Stationary solution of the basic model equation
SOM distributions in formed soils are described by stationary solution of the equation written for one (i-th) litter component
litter

12. RELATION BETWEEN THE SCALES H AND Z (DEPTH)‏
While the model allows to get analytical solution, the variable h is very difficult for measuring. The SOM distribution along the depth of soil is much more convenient for observation.
We again choose the simplest variant of relation between SOM stability (associated with h) and vertical SOM transport velocity w(h), which allows to map scale h onto the scale of depth - z:
The function h(z) necessary for further computations can be obtained by numerical (unfortunately) solving this transcendental equation.

13. One-to-one mapped stationary distributions of SOM
along the transformation rate, h along the depth, z

14. THE SIMPLEST PATTERNS OF VERTICAL SOM DISTRIBUTION.
Even in this simplest case these patterns cover wide variety of soil types.
Pattern 1α: cambisols, luvisols, ferralsols, acrisols;
Pattern 1β: umbrisols, gleysols, andosols;
Pattern 1γ: brown earths. 14

15. Sometimes abrupt changes of SOM permeability between different horizons may occur
This auxiliary function a(h) is approximately equal to constant aj in the ranges between hj and hj+1. The hj are correspondent to the depths of horizon bounadies. Parameter εj indicates a sharpness of change of intensity of transportation. Index j is the number of soil horizon boundary. 15

16. PATTERNS OF VERTICAL SOM DISTRIBUTIONS FOR INHOMOGENEOU PERMEABILITY CONDITIONS
Pattern 2α – two horizons, vertical SOM transport is intensive in the upper horizon and weak in the lower one – solonetz;
Pattern 2β – two horizons, vertical SOM transport is weak in the upper horizon and intensive in the lower one – kastanozems, chenozems;
Pattern 3 – three horizons, vertical SOM transport is weak in the upper and lower horizons, and intensive in the middle one – podzols. 16

17. ACTIVE AND INACTIVE SOM
Coming to quantitative description of data and accounting that essential part of SOM can be into mineral bound state we subdivide SOM into:
- undergoing transformation active SOM, c(h, t).
- bounded inactive SOM, s(h, t).
Then the total SOM distribution is: C(h, t)=c(h, t)+s(h, t).
Organic matter mineralization, k(h)
detritus
litter
stable forms
v(h)
α(h)
(h)
s(h, t)
c(h, t) 17

18. ACTIVE AND INACTIVE SOM
Let's again assume the simplest form of α(h) and β(h) :
α(h)=qh and β(h)=rh.
In steady state , and thus:
Again assuming the simplest form of the function w: w(h)=ah, we obtain completely analytical expression of vertical SOM distribution:

19. Root litter
Since for some biomes essential part of organics comes with root litter we have to take into account the depth-distributed input of it.
The stationary distribution of the products of root litter transformation is described by equation:
where zm is the maximum depth where roots are found; is
distribution function of root organic income.

20. Quantitative comparison of model predictions and field data
Model calculations of vertical SOM distribution were fitted to averaged field data for three different biomes – boreal forest, temperate grassland, and evergreen tropical forest (Jobbágy and Jackson, 2000).
Plant litter was divided into three components – leaf, root, and wood litter. Parameters of them were taken from (Bontti et al., 2009; Chambers et al., 2000; Gholz et al., 2000; Rodin and Bazilevich, 1967; Vedrova 1995, 2005).
The vertical distributions of root litter were obtained by piecewise linear interpolation of the literature data (Jobbágy and Jackson, 2000; Rodin and Bazilevich, 1967).

21. Calculated curves of SOM vertical distribution curves and field data

22. Used litter characteristics and fitted values of model parameters22

23. CONCLUSIONS
A simple (may be the simplest) mathematical model having rather small number of parameters can provide adequate quantitative (and always qualitative) description of vertical organic matter distributions observed in real soils.
SOME ORDER (ADVICE) TO PEDOLOGISTS
To determine parameters of so simple model three experimental values (the stock of soil organic carbon, the rate of plant litter input, and initial rates of litter decomposition) are required.
Thus, there is a minimum set of parameters that should be measured in any study of soil dynamics. 23

24. Our publications on the presented model
Bartsev, S.I., Pochekutov, A.A., A continual model of soil organic matter transformations based on a scale of transformation rate. Ecol. Model. 302, 25–28.
Bartsev, S.I., Pochekutov, A.A., The vertical distribution of soil organic matter predicted by a simple continuous model of soil organic matter transformations. Ecol. Model. 328, 95–98.
Bartsev, S.I., Pochekutov, A.A., Quantitative description of vertical organic matter distribution in real soil profiles by means a simple continuous model. Ecol. Model. 360, 219–222.

25. Thank you!