ON THE STOCHASTIC MODEL OF RILL DEVELOPMENT IN SLOPE SYSTEM
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The process of rill development on unvegetated slope is classified into four stages as follows; i) sheet flow (ensemble of countless small streaks) ii) formation of streaks iii) occurrence of initial rills (unit stream) iv) formation of steady rills Rills transform themselves remarkably by meandering, branching and joining in the transition from iii) to iv), which leads us to the following hypotheses in the investigation of the rill developing process in the definite region, A) joining ratio of rills increases as the number of rills increases, B) branching ratio of rills increases as the relative width (width/depth) of rills increases. Now, we will start with the number of initial rills N at a specified region with definite width (cf. Figs. 1 and 2). If at an instant t, the number of rills has become k, the probability that one rill will be joined with another rill in the section Δx in the interval (t, t+Δt) is equal to Qk=α(k-1)F(x)ΔxΔt (1-1) and the probability that the rills will branch is equal to Rk=β_??_G(x)ΔxΔt (2-1) where Qk, Rk are the probabilities, α the joining coefficient, β the branching coefficient, Wi the width of i-th rill, w0 the width of unit stream, Di the depth of i-th rill, and F(x and G(x) the functions of stream length x (cf. Figs. 2 and 3). Finally, the probabilities that a rill will branch into more than two rills, and that more than two rills will join during (t, t+Δt) are assumed to be O(Δt). We denote by Pk(t) the probability that the number of rills will be k at the instant t. The probability that there are k rills in the region at time (t+Δt) i. e. Pk(t+Δt) is the sum of three probabilities; 1) k-1 at time t, and that a branching has occurred during (t, t+Δt), 2) k+1 at time t, and that a joining has occurred during (t, t+Δt), 3) k at time t, and that no branching or joining has occurred during (t, t+Δt); _??_(3-1) Considering the idealized slope shown in Fig. 1, F(x), G(x) and Δx can be assumed to be constants. Therefore, replacing αF(x)Δx by α' βG (x)Δxwo by β' and wi/w0 by wi', Eq. (3-1) become _??_(3-2) Then, subtracting Pk(t) from both sides, dividing them by Δt and letting Δt→O, we obtain the system of differential equations; _??_ Next, we investigate the width of rills to obtain convenient approximation for the above equations._??_i.e. Let us assume that total width of initial rills _??_ i. e. WN is equal to w0N. If the number of rills is k, replacing WNJWO by WN' and Di/DO by D'i, the relative total width (Wk/w0-Wk') is _??_ with _??_ where Ni is the:number of unit streams in i-th rill, and Do the depth of unit stream. If we assume _??_becomes, replacing β'/D0 by β'', _??_ Therefore, using Eq. (2-2), we can transform Eqs. (4-1), (5-1) and (6-1) into more convenient forms.
- The Association of Japanese Geographersの論文
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