Abstract

Introduction

In the earliest stage, quantitative models were mostly focussed on an empirical approach, relating soil information to soil processes, such as erosion, soil organic matter production, mineral dissolution etc. The need for better understanding of the processes, has lead to the development of mechanistic models.

Little or no work has been done in soil science in order to formulate a general mechanistic model for the soil production and equilibrium in the landscape. In the fields of geology and geomorphology, integrated soil production and slope development formulations have been developed, however. Heimsath et al. (1997) proposed a model for soil production in a landscape and have verified it with field data from Tennessee Valley. They also illustrate how this model can be used to determine whether a landscape is in equilibrium or not.

A simple mechanistic model for soil production at the catena scale is introduced and its implementation in a spreadshhet program is presented.

Theory

Consider a landscape with surface elevation z, soil with thickness h and soil-bedrock interface e along a horizontal x-axis (Figure 1). The change in soil thickness over time depends on two major processes: the production of soil from weathering of bedrock and the transport of soil by erosion process, or in simple mathematical form:

Soil production depends on the rate of breakdown or weathering of the underlying parent materials under physical, chemical and biological processes, which will result in the lowering of soil-bedrock interface. Considering the simple nature of the model, these factors are not analyzed individually.

The rate of weathering or lowering of bedrock surface (-) is usually represented as an exponential decline with thickening of soil (Cox, 1980; Ahnert, 1988):

     (2)

where P₀ [L T^-1] is the potential weathering rate of bedrock at h = 0 and b [L^-1] is an empirical constant.

Soil erosion depends on elevation z, which act as a potential driving the process. The first derivative of z with respect to space () is the slope or gradient. Gradient is the maximum rate of change of elevation and is directly proportional to the rate of sediment flows. The second derivative of z with respect to space () is the curvature (or profile curvature), defined as the rate of change in gradient. A positive curvature reflects a convergent slope (valley), and negative one is a divergent slope (hill).

The process of erosion or sediment flow may occur by overland flow and mass wasting. The transport rate (q) of material (soil) can be defined similarly to Darcy’s law for water transport in soil:

     (3)

where q is the flux [LT^-1], volume of material transported across unit area per unit time [L³L^-2T^-1] and K_e is the erosivity of the material [LT^-1] .

The movement of materials in the landscape is usually expressed as diffusive transport (Scheidegger, 1990), where the flux is defined as:

     (4)

in which q_s is the sediment flux [L²T^-1], volume of material that flows across a slope profile width per unit time [L³L^-1T^-1], and D is the erosive diffusivity of the material [L²T^-1]. The erosive diffusivity depends on the erosion factors of the soil, eg. vegetation cover, soil physical properties, and weather.

We can formulate the continuity equation for the soil thickness over time at equilibrium as: the change in soil thickness over time and the production of soil (lowering of e) is equal to the transport of soil by erosion. The equation is:

     (5)

where C_s is the concentration of soil (mass over the volume of soil) at h [ML^-3], C_r is the concentration of rock at e. In diffusion terms,

     (6)

where r_s is the density of soil and r_r is the density of rock. It should be noted that this equation does not take into account chemical weathering.

If we combine equation (3) and (5) we obtain the general equation for one-dimensional soil production:

     (7)

Likewise, by combining equation (4) and (6) and assuming D and r_s constant over space, then the soil production function is:

     (8)

The above equation described that the production of soil (thickening of h) over time depends on the rate of breakdown of rock (lowering of e) and the transport of soil through erosion, which is dependent on the diffusivity (D) and curvature of the landscape ().

Given appropriate initial and boundary conditions, the above partial differential equation can be solved numerically using the finite-difference approach. The formulation using explicit approximation on a rectangular grid over space (x) and time (t):

     (9)

under the following initial and boundary conditions:

h = h_i, z = z_i at t = 0 , 
at t > 0 , 
at t > 0 , 

The initial condition describes the soil thickness and elevation at h_i and z_i. The second condition states that the soil production rate is assumed to be an exponential decay function of the soil thickness. And the third condition defines the lower and upper boundary for the finite difference space grid. The numerical solution of Equation (9) is stable under the condition where the stability ratio  should be 1.0 (Press et al., 1992).

Implementation

To illustrate the application of the model, a hypothetical landscape with initial elevation representing a series of hills and valleys are simulated with initial soil thickness h_i. The density of rock and soil, and diffusivity D is assumed to be spatially and temporally constant. The weathering (bedrock lowering) rate is exponentially decreasing.

A spreadsheet program (MS Excel) called Pedogenesis was developed. Using the rows and columns as time and space grids, the solutions for soil thickness and land surface elevation is calculated and an animated graph is presented to illustrate the landscape development.

An example of the program output is shown below,
 

Obtaining the spreadsheet
 
 

References