Though it is clear that much complicated conditions rule over the actual process of slope evolution, it is also the fact that a series of slope forms of geomorphological interest can be derived from the mathematical model of diffusion type for two-dimensional slopes, namely, _??_ where u denotes elevation of land surface, t is time and x represents horizontal distance from the divide, respectively. The model is justified as a sort of the first approximation to the basic equation describ the phenomenon even viewed from the physical background of the mathematical expression to the various processes operating on the slopes.
A theory of graded slope can also be developed starting from the model mentioned above. The model on this theory maintains that the graded slope is produced as a time-independent form (steady-state solution of the equation) under the definite balance between erosion and such material supply as achieved by crustal movement whose effect is introduced by the term f (x, t).
Under the condition in which graded slopes appear, the relation ∂u/∂t=0 is satisfied. The same is also the case when graded form appears in connection with the river down-cutting at a constant rate, except that the former relation must be replaced by ∂(u-u_r)/∂t=∂t=0, where u_r means the elevation of the river-bed. Such proportionality between rate of down-cutting and ground slope as assumed by W. Penck is satisfied with good accuracy so far as steady-state solution is concerned.
When the agent that brought the occurrence of the graded slope is disappearred, the slope thereafter changes its form irreversiblly with time as illustrated by Davis and declines itself successively and gradually. Though two types of slope forms distinguishable in the course of slope disintegration are controversial in the discussion by Davis against Penck, they are both acceptable in relation to difference of relative magnitude of the coefficient a to b in the model, when slopes of the same dimension are compared each other.
Measurement of the coefficients a and b is possible using the chracteristics of the graded slope and this is especially easy for the slopes accompanied with the antecedent river. Application of the present theory to morphometric data on the valley wall of the river Kiso dissecting the Toki surface shows that a=0.02-0.05m²/year and that b=1.7-0.75mm/year, under the appropriate assumption on the age of formation of the surface, The value of the recessional coefficient bü shows good coincidence in order with that obtained from morphometric analysis on the fault scarp of the Yôrô mountain range, Central Japan.