We are interested in the flow of soil on a hill slope, in the area of an obstruction such as a telephone pole. On a sufficiently small scale, we can ignore production terms and have the simple conservation low

where is the mass flux

since we are assuming that flux is a function of (x, y, t) not of depth.

The boundary conditions are as and at the boundary of the pole.

We wish to compare two models of the flux. The linear model is a Fick’s law model

(2)

and the other is a modification of this

(3)

where is a critical slope (NB: the limit of equation 3 as gets large is equation 2)

Consider the steady state problem with the nonlinear .

(4)

Expanding we get

multiplying by the denominator gives

(6)

Now let , the square of the ratio of the background slope to our critical slope, be .

This gives (7)

Assuming is small, we look for a solution in the form . This gives

(8)

Solving the zeroth order first we have (9)

which can be solved as (10)

This is the solution to the linear model.

The order equation then is (11)

The first term on the right is 0. Since

so

. (13)

This gives

. (14)

Using and

gives

(15)

or

(16)

This is already in a Fourier Series, so we can write

and get

(17)

with boundary condition

Solving gives

so

and

.

Now consider

. This gives

Thus

Height without pole = cR

Height due to linear model = +2cR

Height due to nonlinear model

Thus, if P= the actual amount of soil piled up beyond the background slope,

i.e.

then

.