W. E . H . Culling*

Steady State Distributions in the Measurement of Soil Creep

If a probe or any other rigid or semirigid body is placed in the soil in order to measure mass transport rates, two difficulties arise. It may sect the flow itself, causing soil particles to concentrate on the upslope side, or, on the other hand, particles may slide round the obstacle and their movement go unrecorded. It may prove possible to measure these effects and to turn them to advantage in the measurement of soil creep. The problem then is to model the “pileup” and “slip- round of soil particles upon encountering a rigid barrier.

The disadvantages of existing methods from the viewpoint of the soil as a partic- ulate medium are cataloged in Culling (1981). Direct methods of measurement relying on radioactive tracers appear to be beyond present technology at the accuracy required bearing in mind safety, to say nothing of public reaction. Being indirect, the methods proposed here depend more than is usual in geomorphology upon a theoretical model for their motivation, although ultimately this is true of all observations.

In this paper we detail one of the methods outlined in Culling (1981) and extend its scope to a more general case. It is dependent upon a diffusion type model of soil creep, where a random component is superimposed upon a downslope drift term. As we are concerned with local conditions about the barrier, the method can be extended to drift terms however complicated on the global (landform) scale, provided we can assume constancy over the local configuration. Although constructed with a stochastic theory of soil creep in mind, the general principle of using a steady state distribution to measure flow is applicable in principle to alternative models. Steady state methods possess one great advantage-they re- quire just one observation, which may therefore be destructive.

A stochastic theory of soil creep is given in Culling (1963, 1965) and its relation- ship to other transport processes on slopes is given in Kirkby (1971) and Hirano (1975) or in the relevant chapters of Carson and Kirkby (1972) or Young (1972). A recent summary of measurement techniques is Statham (1981), to which the pro- posed technique is complementary.

*This work was supported by a N.E.R.C. Research Grant, which is gratefully acknowledged. Help and encouragement has been received from Professor J. B. Thornes of Bedford College and Dr. N. J. Cox of Durham; Dr. M . Knott of the London School of Economics and Dr. M . Green of City Polytechnic. The figures were drawn by Mrs. Jane Pugh and Clare Wastie.

W. E . H . Culling is Esmee Fairburn Research Fellow, Kings College, Rogate Field Centre.

GeographicalAnalysis, Vol. 15, No. 3 (July 1983) 0 1983 Ohio State University Press Submitted 5/82. Revised version accepted 12/82.

W. E . H . Cu&ng 1 213

FINITE SOURCE MODELS

If a finite amount of a recognizable component is introduced into the soil aggre- gate, its behavior in general will exhibit three modes. First, dispersion from the point of introduction dominated by diffusive movements. Then, the superposition on the diffusion of a drift downslope governed by an external force (downslope component of gravity) activated by the individual particle motions. Finally, if a barrier is interposed, a steady state may exist. In all three modes the presence of the diffusion component implies particle movements counter to the direction of drift. Such movements have been noted by field observers (Statham 1981).

The problem has a well-known solution. The movement of particles subject to fluctuating forces and under the influence of an external directed force field is governed by Smoluchowski's equation (Chandrasekhar 1943, p. 41).

aw - = div(D grad W - KW) , at

where K is the external force vector, W the concentration and D the diffusion coefficient. One of the first applications was made by Smoluchowski himself to the sedimentation of Brownian particles.

For well-established motion of particles on a plane, slope equilibrium condi- tions are assumed in the y and z directions, and we consider the one-dimensional case of a homogeneous and isotropic layer of particles parallel to but at a small depth below a plane surface and for which D and c, the diffusion and drift coeffi- cients respectively, are regarded as constants. For this case (1) takes the form

a2w aw at ax2 ax

= D- + c - . aW -

If the particle is initially sited at x = xo,

W -+ 6(x - xo) as t -+ 0

and with the barrier at the origin,

D - + c W = O a s t > O a t x = O , (;:)

(3)

(4)

then the solution of (2) satisfying (3) and (4) is (Carslaw and Jaeger, 1959, p. 358)

+ e - ( ~ + ~ & 4 D t + - C e - ~ ~ / D

2 0 (5)

For large xo and sufficiently large t, the error function term and the second Gaussian expression may be neglected, to give for the second mode

W = e - ( ~ - x , - ~ t ) 2 / 4 D t

2(mDt)li2

214 I Geographical Analysis

&om which we may derive

IJ. = xo - ct

u2 = 4 D t .

Finally,

to give the steady state distribution on the positive side of a reflective barrier. The three distinct modes of behavior are illustrated in Figure 1.

Field observations on soils will need to extend over considerable time before injected material can be expected to show a measurable amount of drift or diffu- sion. This disadvantage could be offset if a site could be found where it was known that at a specified time in the past a finite amount of recognizable material was incorporated into the soil at a precise site. In the event of the insertion being more complex than a simple instantaneous and localized type, then solutions for such conditions as finite duration or periodic entry can be arrived at by superposition.

If the site includes a barrier perpendicular to the direction of drift and of sufficient extent to reduce net flow to negligible proportions in directions other than the x-axis, we can seek for the steady state distribution. This will give a measure of the ratio cID so at least one of the coefficients must be determined by other methods. There are other problems connected with the steady state method besides the purely practical. That there will be loss of material is not a great problem provided it is not severe for we need to measure the slope of the distri- bution not its absolute magnitude. More serious problems are related to the effect of compaction and reconsolidation upon the values of the coefficients; a topic to which we return.

The use of finite source solutions to measure flow rates is much more fruitful in the laboratory. Two samples of a soil, to one of which has been added a recogniz- able component, can be placed in juxtaposition, the separation being as close to a vertical plane as experimental conditions allow. The samples can then be sub- jected to a variety of treatments-heating, freezing, wetting, compacting, tilting, vibrating-r just left alone. If diffusive movements occur, they will blur the

2.0 I I

FIG. 1. Diffusion, Drift, and Steady State Distribution of a Finite Amount of Material

W. E. H. Culling 1 215

boundary. Particles should move in both directions across the dividing plane, the concentration varying according to (5) for small values of time. Similar methods have been used to measure solid diffusion in metals (Hevesy 1938). Measurements of drift by this method will require protracted times unless the process can be speeded up under controlled and theoretically justified conditions.

A recent application of rate process theory to particulate transport processes has supplied a quantifiable relationship between geomorphic soil creep, rheologic creep as studied by engineers, and the extreme cases of liquefaction and fluidiza- tion. Finite source solutions are ideal for measuring these, much faster, processes. The parallel between injecting marker particles into a sloping fluidized bed and the celebrated experiments of Haynes and Shockley (1949) and Shockley (1950, p. 55) on the flow of holes injected into a semiconductor is complete. Under the influence of an applied electric field, the holes diffuse and drift in the direction of the field, acting like a positive electron gas, according to

(10) aP - = div(qpEp) - 9 DV2(p) , at where p is the hole density, 9 the charge on one hole, p the hole mobility, E the electric field strength, and D the diffusion constant. This is of the same form as (l), and, apart from the steady state distribution, Figure 1 will appear on the oscilloscope.

CONTINUOUS SOURCE SOLUTIONS

Circulur Cylindrical Barrier

If, instead of an instantaneous and finite source, the supply of marker material to the soil were continuous, as will more generally be the case in natural settings, then the imposition of a barrier will cause buildup until the barrier is overtopped. Mathematically, a continuous source under the conditions (2)-(4) possesses no steady state solution. However, where soil escapes round the barrier, there exists the possibility of a steady state solution. We begin with the circular cylinder perpendicular to the plane of the slope. Besides being the easiest case to deal with, it is of great practical importance as most probes and allied hardware placed in the soil are of circular cross-section.

We require the steady state solution f(r, 0) and, if possible, the complete solu- tionf(r, 0, t) to the diffusion equation in the xy plane in the region r > a, outside a circular boundary, radius r = a, in a medium of initial concentration v = V,, and subject to a uniform drift velocity U in the direction (downslope) of the negative x-axis. The steady state solution and an illustration are given in Culling (1981), where, unfortunately, a cosine term has been omitted. The correct expres- sion for the steady state distribution of concentration of material about the cylin- drical barrier is

Ur 2 0

U --case

D v = v, + v, - cos 0 e

216 I Geographical Analysis

c - U \ - I +

FIG. 2. Steady State Solution for the Diffusion Equation with Drift for the Region Outside a Circular Cylinder

where E, = 1, E, = 2, n L 1; and h = (Ul2D) cos 8. The derivation of (11) is given in Appendix 1 and illustrated in Figure 2.

This expression is not as complicated as it looks. It is of Fourier-Bessel form

v = C a,, K, - cos no (:;) and strongly convergent, being dominated by the first term. The modified K-type Bessel functions are strongly “concave-to-the-sky’’ and are radially disposed about the circle of the barrier. The characteristic asymmetry of the distribution is im- posed by the cosine factor

Ur - - cos e

cos 8e Z D

which is symmetrical about the x-axis but antisymmetrical about the y-axis. The exponential function in (13) tends to compress the effect on the upslope and to prolong it on the downslope. At 8 = d 2 , 37~12, the series terms vanish and the expression reduces to V,, the background concentration.

The time variable solution is much more complicated and has to be left in integral form. For those interested it is given in Appendix 1. Series expressions for small values of time are available but cumbersome to use.

Elliptical Cylindrical Barrier

Apart from the case of a horizontal ground slope (which is trivial), it is not normally the case that a cylinder inserted into the soil is perpendicular to the

W. E . H . Culling 1 217

plane of the slope. Care is usually taken to place it vertically, and therefore the cross-section in the plane of the slope is no longer a circle. This may be a fine point at present levels of observation and measurement, but there are other advantages to the more general elliptical case. It can provide a wider range and a better approximation to natural obstacles. Degenerate cases include the flat plate as well as the circular cylinder.

The solution of the diffusion equation for elliptical regions is expressed in Mathieu functions. These are more difficult to work with and are much less well known than Bessel functions. As for the circular cylinder, we merely quote the steady state solutions; the derivations are given in Appendix 2.

If the major axis of the elliptical cross-section of the barrier is parallel to the (downslope) x-direction, then the steady state solution for the same background conditions as for the circular cylinder is described by

U - cosh 5 cos q

D v = Vo - V, - h sinh Eo cos q e 2D

* * E, (- 1)" I , (g cosh to) 0

r = O

r = O

where g = (UhIZD) sinh to cos q and a2 = (Uh/2D)2. In Cartesians, x = h cosh ,$ cos q; y = h sinh 5 sin q, where 2h is the interfocal

distance. Variation in 6 gives a set of confocal ellipses, with 6 = to marking the barrier boundary and variation in q a set of confocal hyperbolae orthogonal to the set of ellipses.

The steady state solution is seen to be of the same form as (11), particularly when it is remembered that the Fek Mathieu functions are defined by a series of K-type Bessel functions, and the cosine-elliptic function ce, is defined by a series of cosine functions. The Fek functions are "concave-to-the-sky" like the K-type Bessel functions; the ce2, functions are symmetrical about both axes; the ce,,, functions about the major axis only. However, once again it is a factor

- 2 cosh 5 cos q h sinh to cos q e eD

corresponding to (13) that imposes the general pattern of the distribution (Fig. 3). As the eccentricity of the ellipse tends to zero and the barrier cross-section

tends to a circle, (15) degenerates to (13) and each term of the Mathieu function series tends, in a one-to-one fashion, to correspond to the Bessel function term in (11) (see App. 2).

Natural obstacles are unlikely to be so obliging as to always align along the downslope direction. We refrain from quoting here the steady state distribution for drift in the direction of the minor axis (it is given in App. 2). We do quote the

218 I Geographical Analysis

I FIG. 3. Steady State Solution for the Diffusion Equation with Drift for the Region Outside an

Elliptical Cylinder, the Drift Parallel to the Major Axis

important factor that governs the general form of the steady state solution corre- sponding to (13) and (W), namely,

- 2 sinh 5 sin q h cosh to sin q e 2D

The steady state distribution is illustrated in Figure 4(Z.). As before, if the eccentricity e --t 0, we arrive at (11). More interesting is the

degenerate case for e -+ 1, while to --f 0. The ellipse degenerates to the interfocal line, representing a plane strip placed perpendicular to the direction of drift (Fig. 4 ~ ) . Finally, for arbitrary alignment, the steady state solution comprises compo- nents of the two previous solutions with proportions according to the x and y components of the drift direction.

Before we move on to discuss some of the problems that beset a field verifica- tion, we note one important conclusion. The presence of the ratio UID, where U

FIG. 4. Steady State Solutions for the Diffusion Equation with Drift for the Region Outside an Elliptical Cylinder, the Drift Parallel to the Minor Axis (Z.), and Degenerate Solution for a Planar Strip Perpendicular to Drift (K)

W. E . H. Culling 1 219

is the drift velocity and D the diffusion coefficient, in the exponential function of the important factor (13), (15), and (16), implies that U and D are of the same order of magnitude. The best available evidence puts U - 1-2 mm/year (Young 1972, p. 56). If D were larger by a magnitude, the effect on the soil would be too obvious; if it were smaller by a like amount, the effect would be too small to detect in the field. This latter possibility presents the greatest practical challenge to field verification.

FIELD VERIFICATION

The steady state solutions are to be regarded as first approximations giving the general form to be sought in the field and the laboratory. They presuppose a constant value for the diffusion coefficient as if we were dealing with a problem in heat conduction. The pileup of material on the upslope side of a barrier will be accompanied by an increase in the normal stress across interparticle contacts, and, probably just as important, it will witness a reduction in the volume of pore space. Both these effects will reduce the mobility of the particles. We are thereby trans- lated into the much more advanced problem of variable (concentration dependent) diffusion. This would have the overall effect of decreasing the levels of density at all radial distances below those predicted by the constant value model but con- serving the general form of the distribution. The next development is therefore to seek numerical solutions appropriate to concentration dependence.

Upon the imposition of a barrier, compressive stress will build up elastically and therefore relatively rapidly through the network of particle contacts. A pressure bulb will develop and a Boussinesq or alternative formulation will be used as a starter solution. From physical theory the effect of increased pressure upon the value of the diffusion coefficient in liquids is of an exponential nature. It will be reasonable then to begin with an exponential relationship in a series of simulations for selected parameter values and then compare these with the exact solutions for a constant value coefficient.

Reconsolidation processes, interpreted in the widest sense, will have two effects upon the distributions we have been discussing. Vertical movements will tend to convert density variations into variations of surface elevation. Secondly, due to the particulate nature of the soil, such movements will be accompanied by random motions. These will supplement those otherwise present in tending to diffuse away any increase in concentration about the barrier. We are thus brought back to the importance of the ratio between the drift velocity and the diffusion coefficient, UID.

All the steady state distributions given include a factor of the form

where r measures the distance from the origin along a radius or hyperbola, along which a is a constant. Although, in any computation, the Bessel or Mathieu function terms cannot be neglected, their general form is such that they decrease away from the origin in an “exponential manner” and therefore merely enhance the effect of the factor so that we may simplify the discussion by recounting the effect of the ratio U/D on the distribution solely in terms of the factor.

An increase in U relative to D increases the value linearly at r = 0, but the exponential cuts off more rapidly; on the other hand, a relative increase in D promotes the opposite. Provided we stay within the range wherein U and D are within one order of magnitude of each other, then the effect of both is detectable

220 1 Geographical Analysis

in the distribution. Outside the “window” wherein UID ranges from - 0.01 to - 1.0, one or the other can be regarded as predominant. If it is the drift velocity, then diffusive movements become negligible and the motion becomes laminar, particles flowing round a barrier with or without a boundary layer. The pileup of material on the upslope side of a barrier assumes a different form being cusp- shaped about the stagnation point, as is the case for fluvial or aeolian examples (Prandtl 1953, p. 45). If diffusion predominates, then no buildup of concentration will be detectable, but, by the same token, the diffusive movements will be large enough to be measurable by other means; for example, finite source methods.

So far we have envisaged drift velocities of - 1-2 mmtyear possibly rising to - 10 mmlyear, but the expanding boundary about a root crop such as turnips moves at a rate of several cms a growing season. The entailed rearrangement of soil should be detectable as a zone of compaction or, if it is not present, then the rate of diffusion is such that it should be measurable. Natural or artificial expand- ing or advancing boundaries provide a further means of measuring soil creep.

SUMMARY

Although the idea of using a steady state distribution to measure transport rates is of wide application, the distributions developed in this paper stem from a particular hypothesis as to the nature of geomorphic soil creep. This hypothesis views the soil mass as essentially particulate, subject to drift downslope, activated by individual particle movements brought about by geomorphic agents, either including or wholly comprised by a randomly oriented component. This compo- nent will induce diffusive behavior in the soil mass upon which is superimposed the downslope drift. It is one of a series of qualitatively different flow patterns to be found in particulate matter as the Reynolds number rises from the very low values of condensed granular media to the high values of turbulent transfer.

Thus the steady state distributions supply a crucial test of the relative signifi- cance of the diffusive movements. Alternatively, the transport process could be dominated by the drift, the flow pattern describable as hydrodynamic, with indi- vidual particles executing relatively smooth flow paths round barriers rather than the Brownian trajectories characteristic of diffusion. The density distribution is also qualitatively different, being bulbous for diffusive flow and cusp-shaped for laminar hydrodynamic flow. If the mean (drift) velocity is further increased, thereby raising the value of the Reynolds number, the density and flow patterns on the downslope side of a cylindrical barrier will undergo a further change and exhibit the characteristic bifurcation forms and periodic structures of the transition zone (Marsden and McCracken 1976, p. 14), until finally, at a sufficiently high Reynolds number, turbulence sets in with a return to diffusive (chaotic) move- ments (Ruelle and Takens 1971).

However, the first task is to search for suitable sites to see if anything approach- ing the exact steady state distributions exist and whether the steady state method can be used to study the microscopic behavior of soil particles. The correct strat- egy appears to be to concentrate effort, in the first instance, on predominantly sandy soils and this is now under way.

APPENDIX 1

Circular Cylindrical Barrier

The solution is best approached using the Laplace transform methods devel- oped by Carslaw and Jaeger (1940) and Jaeger (1942), though the investigation of such problems by classical methods goes back to Nicholson (1921). We follow and refer to Carslaw and Jaeger (1959, particularly p. 390).

W. E . H . Culling I 221

An infinite medium, initially at constant concentration Vo, is moving with con- stant velocity - U along the x-axis. For t > 0, the cylinder r = a acts as a reflective barrier. The equation describing the flow is

a% a% u a v 1 av - + - 2 + - - =-- a 2 ay D ax D at '

The boundary condition at r = a is

av u - + - c o s 0 v = o . ar D

Introducing the transformation

we derive, from (Al),

Pansforming to cylindrical polar coordinates,

Taking Laplace transforms, the subsidiary equation is

where q2 = (p1D) + (v2/4D2), and the bar denotes a Laplace transform. We seek a solution of the general form

* { C , cos n0 + D, sin ne} , 647)

We require the solution to be finite as r + m and are not concerned with the value at r = 0, so A, is zero. Furthermore, the trigonometrical terms need to be appropriate to values that share the sign of the x-axis to be representative of the physical conditions, so D, is also zero. The appropriate solution of (A6) is therefore

ii = 2 a, K ( q r ) cos n0 . (A8)

The boundary condition at r = a becomes, upon the two transformations,

222 I Geographical Analysis

or

where h = Yz(UID) cos 8. Then, from (A8),

vo cos 0 a* = - -- P D

and, upon substitution,

where we have taken the opportunity to employ

From the Fourier-Mellin inversion theorem,

y + i m

1 27ri

u = - I 5 e " d A . y - i m

The integrand has a branch point at the origin and we employ the contour of Carslaw and Jaeger (1941, p. 92), where a cut along the negative axis joins a small circle about the origin to a semicircular arc of radius (Fig. 5). It is known (Watson 1944, p. 511) that &(z) has no zeroes within the contour, and from Cauchy's theorem the value of the integral around the contour is zero. It can be shown that the integrals on the semicircular arcs vanish as r --* 03, and so the value of the line integral is given by the values of the integrals along the cut plus the contribution of the small circle about the origin.

This latter is

where a' = Ual2D and r' = Url2D. For the integral along the lower margin of the cut, put A = Du2 ein - (u2/4D),

then q = iu. Along the upper margin put A = Du2 e-in - (vI4D), giving 9 = -iu, and the contribution to the complete integral from the upper margin is minus the conjugate of the contribution from the lower (Carslaw and faeger 1959, p. 335).

W. E . H . Culling I 223

FIG. 5. Contour for Inversion of Laplace Transforms

Combining the two contributions and using

the contribution along the cut is

30

U vo - cos 8 I c. En Z"(U') cos ne e-Du2t - ("2*'4D)

=D 0

U

* (u2 + (v"/402)) d" '

where for convenience we put rand a for ur and ua respectively and have denoted Un(ur) Yn+I(ua) - Yn(ur) Jn+l(ua)] by Cn,n+l(r,a) and similarly so for the other terms in the numerator.

If we now apply (A3) to both (A14) and (A15), we have the complete solution, which is seen to be of the same form as that of Carslaw and Jaeger (1959, p. 390, eq. 31), with the added complication due to the more demanding boundary conditions.

That the integral term vanishes as t --* ~0 can be shown directly or much more simply by employing the theorem (Carslaw and Jaeger 1941, p. 255) to the effect that

lim x( t ) = lim p F(p) , t+m P-0

224 / Geographical Analysis

so that the steady state solution consists of (A14) after the transformation (A3) and is quoted in full in (11). If only the steady state solution is required, the use of this method saves a great deal of complicated analysis.

Solutions for small values of time can be derived from (A13) by expanding the Bessel functions in asymptotic series, a method pioneered by Goldstein (1932) but which goes back to essentials to Heaviside. The result in this case is enormously complicated and will not be given, but it is available if ever required for the interpretation of field or laboratory data.

APPENDIX 2

Elliptical Cylindrical Barrier

Solutions of the equation of conduction of heat for elliptical regions are not covered by Carslaw and Jaeger (1959). The standard work on Mathieu functions in English is McLachlan (1951) but this does not deal with the region outside an elliptical boundary; this case being considered by Tranter (1951), but for the constant boundary condition only. We refer to McLachlan (1951) as M, giving where necessary page and equation.

The diffusion equation in the x y plane is transformed as before by (A3) and Laplace transforms are taken to give (A4). Then, in elliptical polar coordinates (5, q), the equation for solution becomes (M, p. 170)

where q = (h2/4) [ ( p / D ) + (@/4D2)], 2h is the interfocal distance, and 5 = to marks the elliptical boundary.

The solution of (Al8) must satisfy

(i) as 5 -+ a, ii + 0

(ii) no flow at 5 = E0

Solutions of Mathieu’s equation (A18) comprise the product of two Mathieu func- tions selected to suit the conditions of the problem. To satisfy (i), one must be nonperiodic and of these we choose the alternative (Bessel function product) form, which possesses certain advantages (M, p. 257), namely, Fek,(E, -q), defined (M, p. 248, eqq. (5) and (6)), the K signifying the K-type modified Bessel function. The second Mathieu function must correspond to the symmetry of the problem (M, p. 296), and we require a function symmetric about the major axis, namely, the cosine-elliptic function ce,(q, - 4).

We therefore seek a solution of the form

For the condition of no flow at 5 = 50,

av U h sinh E0 cos q - + - v = o , an D 4

where

1, = h(cosh2 5 - C O S ~ q)% .

W. E . H . Culling 1 225

Transforming as for (Al), we derive

Uh - cash 50 cos 7 vo u ; (0 < q < 2 ~ ) , (A22) aii - + g ii = - - - h sinh to cos q eZD dt P D

where

Uh 2 0

g = - sinh to cos q .

Then, from (A19) and (A22),

Uh - cash 50 cos q - _ - - - vo ' h sinh to cos qe2D P D

We find that if we use the orthogonality argument of McLachlan (M, p. 22) or Tranter (1951, p. 462) to evaluate C,, we reach a solution that, upon the degener- ation of the ellipse to a circle, does not give the whole of the solution for the circular barrier (A14) and (A16) but only the first terms. Instead we put

Uh - cosh to cos q eZD = f: E, I,(% cosh to) cos nq ,

n =O

where e0 = 1, en = 2, n 2 l., and employ (M, p. 21)

Then, substituting in (A19) and (A24),

m

- VO u u = - - - sinh to cos q E, I , P D n = O

where we have used the convention that En denotes and l/[Zr,O ( - 1)r BLF!l')] for odd terms and where a2 = (Uh/2D)'.

( - 1)' AkT)] for even

The Fourier-Mellin inversion theorem (A13) and the theorem quoted (A17),

226 I Geographical Analysis

together with the transformation (A3), give the steady state solution quoted in (14).

Transition to the Circular Cylinder (M, pp. 367-79)

As the eccentricity e -+ 0, 5 -+ + 00 and h -+ 0, in such a manner that Yzh eg + r and ?h h eb -+ a (the radius of the circle). Then h sinh to and h cosh &, --+ a, and it follows that

Uh U 2 0 2 0

g = -sinhEocosq-+-cos8 = h

for the confocal hyperbolae become radii, with q = 8. The factor outside the Mathieu function series becomes

Ur W Vo - a cos 8 e

D n=O

All the Aim) in the series for ce,(q, - 9) tend to zero except that the A t ) -+ 1 and

From (M, p. 369), as e -+ 0 the Fek functions tend to K-type Bessel functions: so ce,(q, - 9) -+ cos n 8. For n = 0, A$') = 2-' and co(q, -4) -+ 2-'.

and similarly so for odd functions. derived from the comparison of the dominant terms

in the asymptotic expansions of corresponding Mathieu and Bessel functions, cancel out on substitution and need not detain us. Now

The multipliers pH,,,

to give the steady state solution. So

T 2 0

If we remember that

substitution gives

Ur U - ~ C o S 9

V = Vo + V, - cos 8 e D

W, E. H . Culling / 227

to agree with the steady state solution (11).

Variable Solution We employ the same contour as before. For the small circle about the origin,

we derive the steady state solution (14), in like manner to the circular case. On the lower margin of the cut, put A = D u ~ ~ " " - ( " " ~ ~ ) and q = - (h2u2)/4 = - P2, and the integral along the lower margin becomes

U - Du't - (dtI4d) du . (u2 + (U?4D2$

On the upper margin of the cut, put A = Du2 e-im-(u2'4D), and the integral is minus the conjugate of (A34).

From (M, p. la),

Substituting, combining the two integrals and rearranging the variable solution is given by

m 2u v = Vo - Vo - h sinh So cos q 2 E,( - 1)" I ,

7FD n = O

m

e - D ~ % - (Vt/4D) du , U

* cen(q' P2) (u2 + (v2/4D2)) (-436)

From P2 = ( l ~ U / 2 ) ~ and k, = [(p/D) + (v2/4D2)]46 = u, together with the transi-

228 I Geographical Analysis

tions listed for the steady state case, substitution and use of the cylinder function relationship

gives finally the variable solution for the circular cylinder (A16).

Drift Velocity Parallel to the Minor Axis

The equation to solve is now

Putting

and taking Laplace transforms, the equation in elliptical coordinates is identical to (AH). The boundary conditions become

(i) E + O a s c - + m

(ii) no flow at 6 = to ,

which in this case entails

av U h cosh to sin q - + - v = o , an D 11

where l1 is given by (A21).

seek a solution of the form The conditions of the problem require symmetry about the minor axis and we

The cosine-elliptic function is symmetric about both axes; the sine-elliptic func- tion, seZn+,, is symmetric about the minor axis but antisymmetric about the major.

From (A41),

vo u 9 sinh 50 sin s\ E! + g' E = - - - h cosh to sin q e2D at P D

where g' = (Uhl2D) cosh to sin q. Substituting

('443)

Uh - sinh &,sin q eZD = n=O E, ( - 1)" { I , , (g sinh to) cos 2nq

W. E. H. Culling 1 229

+ Z2n+l (g sinh to) sin(2n + l)q } (A44)

and proceeding as before, we can derive the steady state solution

U - !!! sinh 5 sin q v = Vo - Vo - h cosh to sin q e 2D

D

where En, E,, and a2 have the same connotation as before.

Transition to the Flat Plate Boundary

(M, p. 171), and g‘ + 0. While Zo(z) -+ 1, Z,(z) -+ 0; n f 0. As the eccentricity, e -+ 1, a + h and to -+ 0; sinh to + 0 and cosh to + 1

Substituting

,=O

LITERATURE CITED

Carslaw, H. S., and J. C. Jaeger (1940). “Some Two-Dimensional Problems in Conduction of Heat

. (1941). Operational Methods in Applied Mathematics. Second edition. London: Oxford Uni-

. (1959). Conduction of Heat in Solids. Second edition. London: Oxford University Press.

with Circular Symmetry. ’ Proceedings of the London Mathemutical Society, 42, 361-88.

versity Press.

Carson, M. A,, and M . J. Kirkby (1972). Hillslope Form and Process. London: Cambridge University

Chandrasekhar, S. (1943). “Stochastic Problems in Physics and Astronomy.” Reoiew of Modern Phys-

Culling, W. E. H. (1963). “Soil Creep and the Development of Hillside Slopes.”Journal of Geology,

. (1965). “Theory of Erosion on Soil-Covered Slopes.” journal of Geology, 73, 23G54.

. (f981). “New Methods of Measurement of Slow Particularate Transport Processes on Hillside Erosion and Sediment Transport Measurement (Proc. Florence Symp.) 1981. I.A.H.S.

Goldstein, S. (1932). “Some Two-Dimensional Diffusion Problems with Circular Symmetry.” Proceed-

Haynes, J. R., and W. Shockley (1949). “Investigations of Hole Injection in Transistor Action.” Physics

Hevesy, G. von. (1938). “Self-Diffusion in Solids.” Trans. Farad. Soc., 34.2, 84145. Hirano, M. (1975). “Simulation of Developmental Process of Interfluvial Slopes with Reference to

Jaeger, J. C. (1942). “Heat Flow in the Region Bounded Internally by a Circular Cylinder.” Proceed-

Press.

ics, 15, 1-89.

71, 127-61.

Slopes. Publ. 123, pp. 267-74.

ings of the London Mathematical Society, 34, 5148.

Review, 75, 691.

Graded Form.”]ournal of Geology, 83, 113-23.

ings of the Royal Society of Edinburgh, A. 61, 223-28.

230 I Geographical Analysis

Kirkby, M. J. (1971). “Hillslope Process-Response ‘Models Based on the Continuity Equation.” In Slopes: Form and Process, edited by D. Brunsden. Institute of British Geographers, Special Publ. 3.

McLachlan, N. W. (1951). Theory and Application of Mathieu Functions. London: Oxford University Press.

Marsden, J. E., and M. McCracken (1976). The HopfB#urcation and Its Applications. New York: Springer.

Nicholson, J. W. (1921). “A Problem in the Theory of Heat Conduction.” Proceedings of the Royal Society, A. 100, 226-40.

Prandtl, L. (1953). Essentials of Fluid Dynamics. London: Blackie. Ruelle, D., and F. Takens (1971). “On the Nature of Turbulence.” Communications in Mathematical

Shockley, W. (1950). Electrons and Holes in Semi-Conductors. New York: Van Nostrand. Statham, I. (1981). “Slope Processes.” In Geomurphological Techniques, edited by A. Goudie. Lon-

don: Allen & Unwin. Tranter, C. J. (1951). “Heat Conduction in the Re ion Bounded Internally by an Elliptical C linder

and an Analogous Problem in Atmospheric Ddusion. Quart. Journ. Mech. and AppZieJMath, 4.4., 461-65.

Watson, G. N. (1944). Treatise on Bessel Functions. Second edition. London: Cambridge University Press.

Young, A. (1972). Slopes. Edinburgh: Oliver & Boyd.

Physics, 20, 167-92; 23,343-44.