glew, ford (1980) - a simulation study of the development of rillenkarren

8/3/2019 Glew, Ford (1980) - A Simulation Study of the Development of Rillenkarren

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EARTH SURFACE PROCESSES, VOL. 5, 25-36 (1980)

A SIMULATION STUDY OF THE DEVELOPMENTOF RILLENKARREN

J . R. GLEW A N D D. . FORDDepartment of Geography, McMaster University, Hamilton, On tario, Can ada L8S 4K 1Received 17 July 1978Revised 2 January 1979

SUMMARYRillenkarren a re patterns of tightly packed , small solution rills found upon bar e, sloping surfaces of soluble rocks in allclimates. They head at the crest of a slope and are replaced dow nslope by a planar solution surface, the ausgleichfliiche.Development has been simulated successfully using a rainfall simulator and plaster of paris blocks. Ten principalexperiments were completed with blocks at inclinations ranging 22$-60, temperature and rainfall intensity beingconstant. Results suggest that rillenkarren develop within a hydrodynam ic zone of rim effectwhere the depth of threadsor sheets of runoff is insufficient to preve nt direct ra ind rop impact up on th e underlying soluble solid. Where depth ofrunoff becomes sufficient rills are rep laced by the ausgleichflache. Between upp er a nd low er limits, rill length isproportional to slope in a log linear manne r. Rill cross-sections approximate t he parabola, the most effective sha pe forfocussing raindrop e rosion in th e axis of the trough: this explains the tight packing characteristic. Ausgleichflache andrill troughs evolve by parallel retreat at the original slope angle, the erosion rate being greatest at abo ut 45".KE Y WORDS Karst Karren Solutiona l erosion Rim effect Parabolic focussing

INTRODUCTIONRillenkarren are landforms of solutional origin belonging to the karren family of small-scale karst features.They are a type of free karren (Bogli (1960)), i.e. developed upon rock surfaces free of any soil or vegetalcover. Examples are shown in Figures 1 A and B. At their simplest, rillenkarren a re straight solutionalchannels displaying remarkable regularity of form and dimension. They head at the very crest of a slope,extend directly down its fall line and are extinguished downslope where they are replaced by a planarsolution surface, termed an ausgleichjluche by Bogli. The channels are tightly packed together, interfluvesbeing limited to mere partitions tapering to razor-sharp edges.Inmany instances rillenkarren are less regular than this. Adjacent individuals vary greatly in length. They

may be discontinuous. Rills curve or are sinuous, and lack parallel alignments. Such irregularities can berelated usually to variation of the crestlines and/or curvilinear form of the host rock surface. Frequently,they reflect the fact that rills are secondary colonists upon the characteristically irregular surfaces of subsoilkarren that have been denuded of their soil.

Rillenkarren are reported on all three dasses of comparatively soluble rocks-the carbonates, sulphatesand halides-and from every type of climatic region. They are of general interest in geomorphology, notonly for their regularity of form and universality of distribution, but because they may be seen as theconverse of conventional First Order channel distribution on a planar slope as this was envisaged by R. E.Horton (1945). The rillenkarren channels propagate at the crest and are succeeded downslope by a 'belt ofno (channelled) erosion', They are uniformly packed together whereas Horton's initial channels areuniformly separated.

Early explanations of rillenkarren development invoked wind action or flowing ice, as well as runningwater or combinations of erosion processes. These have long been rejected in favour of the development0360-1269/80/0105-0025$01.00@ 1980by John Wiley& Sons,Ltd.

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26 J. J. GLEW AND D. . FORD

Figure 1A.Natural rillenkarren and au sgleichflacheon a Palliser lim estone block in a landslide pile at Surprise Valley, Jasper NationalPark, Alberta. Scale is in inches. B . Simulated rillenkarren on a plaster of paris block at 50, McMaster laborator y. Scale in cms.

under rainfall that is clearly suggested by the wide climatic range of the features. But there has been littleconsideration of precisely how they are formed, nor are there published quantitative accounts of forms thatwould appear to be well suited to simple field measurement and statistical analysis. The principal study is byBogli (1960) who attributes the aqueous solution of rillenkarren and ausgleichflache surfaces on limestonesto the first phase of a four-phase division of carbonate solution processes. Following Plummer and Wigley(1976) this may be written:

H ~ O ++OH- (14CaC03+H C O H - eCa+++HCO; +OH- (1b)

i.e. solution is by simple, almost instantaneous, formation of bicarbonate rather than by reactions involvingdissolved and complexed COz as in the genesis of most larger karstic features. That this attribution isbroadly correct (though, no doubt, there is some reaction with COz at some times when it is raining onrillenkarren) is indicated by the occurrence of apparently identical forms on gypsum and salt rocks also,where molecular dissociation is the only solution process normally occurring, e.g. :

Bogli states that the length of rill channels tends to increase as the gradient of the host surface increases, butoffers no supporting data. Rate of molecular dissociation is temperature-dependent in a linear manner andBogli cites the opinion of H. Lehmann that rillenkarren are longer in warmer climates. This issupported by


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DEVELOPMENT OF RILLENKARREN 27Wilford and Wall (1965), Jennings (1971) and Sweeting (1972), but no measured comparative data areoffered. There appear to be no published studies of the packing characteristic of rillenkarren. Lundberg(1977) analysed the cross-sectional form of more than 1,000 rills at Chillagoe, Australia. No clearrelationships between width, depth, radius, derived parameters, could be established. There was muchscatter of data. None of these variables displayed significant correlation with the slope of the host surface.

In 1967, Ford (unpublished) measured rillenkarren on fifty sample blocks of crystalline limestone in alarge landslide pile in Surprise Valley, Rocky Mountains of Canada (see Luckman (1976, p. 189) forlocation). This was an ideal random sampling site because the host rock surfaces offered every variation ofgradient and aspect, and of orientation with reference to lithological grain in the rock. Rather sharp lowerand upper limiting gradients for the rillenkarren development were established. They were absent whereslopes were less than 123-15"and where greater than 75-80", rills were attenuated by development ofshallowly scalloped ('cockled') surfaces that are known to be a consequence of the introduction of boundarylayer discontinuities into the fluid flow (Goodchild and Ford (1971),Blumberg and Curl (1974)).However,inside the range 12$-75", no statistically significant relationship between gradient and rill length could beestablished from a sample of more than 1,400 individuals, although both aspect and orientation with regardto lithology could reasonably be excluded as interfering variables: an explanation of this failure is offeredbelow. Packing of rills along the crests of the 50 measured blocks was remarkably uniform, the arithmeticmean frequency being 24 individuals per 30 cm of crest length, with a standard deviation of only 0.50.

Failure to establish a significant relationship between rill length and surface gradient from a large sampleat an ideal field site led us to suppose that the problem of rillenkarren formation was best tackled by physicalsimulation under controlled conditions. Lundberg's similar experience with the cross-sections of rillsunderscores the point. This paper reports the results of a series of successful simulation experiments in thelaboratory.

EXPERIMENTAL METHODSThe basic equipment comprised a rainfall simulator delivering continuous precipitation at constant rate tocast plaster of paris surfaces oriented at fixed gradients. Experiments were conducted at a temperature of20f C.

Following Yalin (197l ) , satisfactory hydraulic modelling by physical simulation requires that eithergeometric or dynamic or kinematic similarity with the natural situation be maintained. Ideally, all three willbe maintained, or nearly so. Natural rillenkarren are small so that in our experiments it was possible toobtain exact geometric equivalence, i.e. to work at full scale. Accurate modelling of the dynamic andkinematic relationships reduced to considerations of the eroding material, of raindrop size and sizedistributioh, fall velocity and rainfall intensity.

Plaster of paris (artificial gypsum) is a suitable substitute for a rock because it dissolves by moleculardissociation in the presence of water, is inexpensive and easy to handle and gives comparatively rapidresults. Both reagent and commercial grades of plaster were used, without causing significant variation inresult in this particular set of experiments. Working surfaces measured 60 x 36 cm, with a depth of 8 cm.Variations in natural rock porosity, permeability and solubility attributable to deposition, lithification andsecondary fissuring were not reproduced in the plaster models so that these should be considered simplifiedsoluble substitutes having physical attributes similar to those of limestone, gypsum, etc.

The rainfall simulator was of the spraying type, being an enlarged and modified version of a machineoriginally built at the Universityof Alberta to study soil erodibility (Bryan (1970)).The McMaster machinehas a fall height of 2.6 metres and was able to deliver simulated rainfall uniformly to all parts of the plastersurfaces for periods as long as 800 hours. It was operated at rainfall intensities of 35 or 40 mm/hour. Fulltechnical details are supplied in Glew (1976), pp. 75-84.

Properties of the simulated rain were determined by measuring droplets retained in an oil bath placed inthe fall area. Their kinetic energy was then calculated from data of Gunn and Kinzer (1949).Results areshown in Figure 2. The simulated rain gave a slightly smaller droplet size range than has been measured innatural rains of the same intensity (Figure 2A) and terminal velocity could not be attained for droplets


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28 J. J. GLEW AND D. C. FORD16 I

14

12

E 10E 8a-B0-c0 6

8 42

LcQ

1 2 3 4 5DROPLET DIAMETER Im.mJ

100

902,U.---e 80-0.-Ec 70c0c

60Lp"

5 0 0 I 2 3 4 5DROPLET DIAMETER I m m J

Figure 2A. Comparison between raindrop diameters produced in th e M cMaster rainfall simulator and tho se of a natural storm ofsimilar intensity reported by Gunn and Kinzer (1949).B . Relationship between raindrop diameter and percent terminal velocityobtained in the McMaster rainfall simulator

greater than 1mm diameter (Figure 2B ) because of a height limitation in the laborato ry. The refo re, precisedynamic and kinematic similarity were n ot attained in the modelling but, both intuitively and fr om t heexperiments, t he deviation in physical process is regarded as on e of degree only, not kind. Its effect waspossibly to slow down the rates of rill formation and surface reduction. This is indicated by the results,rillenkarren that appe ared identical to natural co un terp arts being produced in all experim ents. It was anunusually successful simulation in this respect.Ancillary techniques were used t o measure the evolving rillenkarren an d properties of th e fluid flow upo nthe experimental surfaces. The re were direct m echanical measurements of rill length, width and depthduring and after each experiment. From a special camera m ounting, stereographic pairs an d triple lineprojections were tak en a t fixed time intervals. To tal reduction (lowering) was measured at 25 positions oneach exp erimen tal surface every 100hours by a micrometer accurate to 1/1000 nch and referenced to fourbrass pins cast into the plaster. A ccurate measurement of th e tiny, pulsing thr ead s of flow in individual rillsproved infeasible but rill flow was observed in a general fashion by injecting dye into th e simulator.


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DEVELOPMENT OF RILLENKARREN 29Thickness of the film of water flowing on th e ausgleichflache was det erm ine d indirectly by th e simultaneou scomparison of colour saturation of dyed liquid on th e surface with that passing through a calibrated wedgeapertu re. With stand ard camera equipment, this permitted estimates to within f .05 mm. D etails of the setechniques ar e given by Glew (1976), pp. 85-98.A tota l of 18experim ents with fresh blocks were conducted . Thr ee preliminary ex perimen ts with plasterblock at 45 " were run for more than 650 hours each. These established that the simulation w orked well,giving reproducible results as indicated in Figure 3 and tha t rillenkarren atta ined equilibrium length an dstable width after 300-350 hours. Ten principal experiments with plaster we re then run in which gradientwas varied for different experiments between 2 2 g and 60",an d the ancillary measures we re employed.Each experiment was operated for 500+ hours: measures of rill length, etc., quoted here are frommeasurements m ade at termination when all rills had long attain ed equilibrium. In the last two experimentsin this series inferences concerning critical water film flow thickness for rill and ausgleichflache develop-ment were tested by introducing flow fences of coppe r sheet i nto th e working surfaces. Finally, five briefexperim ents were con ducted with blocks of crystalline comm on salt cut from comm ercial salt licks. With th esame rainfall intensities an d range of gradients as were used for the plaster m odels, rillenkarren of greaterwidth and de pth rapidly developed upon the salt. Length of rillenkarren could no t be com pared with thoseon plaster because t he salt blocks were too narrow to eliminate edg e effects described below.

EXPERIMENTAL RESULTS AND DISCUSSIONCollectively the experimental data support the hypothesis that rillenkarren are rainfall-inducedphenomena. In all cases closely packed rills were propagated from the crests of the inclined workingsurfaces and termin ated in ausgleichflache. T he re was no significant change of slope angle during the courseof each experim ent, i.e. taken a s a whole, the surfaces displayed simple parallel retrea t.In all experim ents with plaster sh ort, very shallow rills could be discerned at th e crest after a few tens ofhours of rainfall. T hese lengthened and d eepe ned steadily for approximately 200 hours after which th e rateof lengthening declined, ceasing altogether between 300 and 400 hours. R ate of d eepe ning also declinedbut some deepening m ay still have been occurring when t he experiments were terminated at 500 hours.Initial rills varied substantially in width; greater uniformity was prod uced by a process of lateral coalescencein which wider rills consumed ad joining, narrow er one s. This process was rarely fully completed: at 500hours m ost surfaces retained a few narrow, partly consum ed rills towards the ir crests. This characteristic isto be se en on mo st natural rillenk arren surfaces as well. It is a principal cause of variance in th e statisticsrepo rted below.I I I I AI

0

Rill length In centimetresFigure 3. Histogramsof rillenkarren length obtaine d on plaster at 45" in three separate experiments


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30 J. J. GLEW AND D.C. FORDIn all experim ents, rills closest to th e sides (lateral edges) of the w orking surfaces were much longer thanth e others. This is an edge effect attributable to loss of water over th e sides of th e models an d identical in itsconsequences to the rim effect discussed below. To reduce it, no rillenkarren within 10 cm of e ithe r edgewere included in the statistical analyses. Extend ed rillenkarren a t edges are comm on on natural exposures.It is most probab le that failure to discount these adequ ately explains the absence of a significant relatio nsh ipbetween gradient and rill length at the S urprise Valley field site described a bove.

Slope gradient and rillenkarren lengthTh e experime nts indicate tha t ther e is a stropg log linear relationship be tween th e gradient of a surfaceand th e length of rills develop ed upon it, after due allowance is made fo r edge effects. Th e relationship isillustrated in Tab le I and Figure 4B. he m easure of separation of the th ree different histograms in Figure

4A is com paratively poo r because of inclusion of partly consum ed initial rills and because, de spite th e 10 cmexclusion bands, it is suspected that minor e dge effects remain. W e believe tha t if t he ex perime nts wererepeated with wider models run for more than 1,000 hou rs each to eliminate these effects, much s harpe rseparation would be obtained. Such experiments would be less realistic because in natural cases localvariations of gradient and su rface roughness will always cause so me dispersion of rill iength .Th e equation of the line of Figure 4 B isE = 1.27 exp [Om52 S] (3 )

where S = gradient in degrees. The lower limiting gradient for rillenkarren development, indicated byextrapolation , is 8but it will be noted tha t this has not been tested experimentally. Th e Surprise Valley dataindicated a higher value for limestone under natural conditions, which may be a consequence of greatervariation of all variables in the field. Th e upp er limiting gradient for rillenkarren could not b e specificallyinvestigated because of height limitations in the simulator. But rills propagated from the working crestsdown th e reverse slopes of the plaster models suggest that reduction of length may commence arou nd 70.For natural rillenkarren on solub le rocks, Equation 3 may be generalizedL = T u ( I ,U,) xp [ b S ] (4)

where T = empera ture , u = u ( T , Ir, Ut)= Tu ( I , ,Ut),Ir = intensity an d /o r d roplet size of rain fall an d U,= a term for the textural hom ogeneity of the rock. Thetem peratu re term remains to be evaluated but, following autho rs cited above, is believed to be significant.Th e term fo r rainfall intensity and droplet size is included for the s ake of theoretical com pleteness but, given

Table I. Rillenkarren lengths for eight principal experiments withplaster of paris blocks

No. Gradient(degrees)45355530455060

224

No. ofmeasuredrillenkarren ff rill length(cm) (+ rill length(45656455150574824

12.57.423.46.14.112.522.325.0

4.52424.51.71.14.93.82.6

t Because of the anticipated length of rills, the 60 X 36 cm block was re-oriented for experiment number 8. Rills developed from a 36 cm crest widthinstead of 60 cm as in previous experiments. This explains the smaller numberof rills created.


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DEVELOPMENT OF RILLENKARREN

-31

35 Degrees

i-'I 145 Degrees

r - - 1 II 7-11 1 1

20

15

10

5

0

A

55 DegreesLI I I I II I I I355 10 15 20 25 30

LENGTH OF RlLL IN CENTIMETRfS

BS L O P ED E G

22.5

30

35

45

50

5560

L E N G T HCM

4.1

6 .?

7.4

1 25

23.3

23.4250

20 30 40 50 60SURFACE SLO PE IN DEGREES

Figure 4A. Histograms of rillenkarren length obtained on plaster in experimen ts at 35",45" and 55". B. The relationship betweensurface slope an d rillenkarren length ob tained in the principal experiments with plaster

the extreme variability of these quantites in and between storms at any natural site, it is probablyintractable. Th e term may not be significant except a t a lower limit where light rains of small drop let size willnot produc e rillenkaren, o r at the very initiation of the r i ll pat terns. In E quation 4 t may appea r that a termfor t he solubility of a principal mineral is desirable but because t he do minan t solution process is effectivelyinstantaneous the absolute mineral solubility is unlikely to be important as a determinant of length.Textural homogeneity or inhomogeneity of the host rock (size and roughness of soluble grains, size andfrequency of insolubles and voids, etc) certainly is an important determ inan t. It m ay not b e easily quantifiedfor, in our field experience, most soluble rocks fall into one of two extreme categories: either they aresufficiently homogeneous to perm it rillenkarren extension without impediment or they are so inhomo-geneous that rillenkarren ar e precluded. Interm ediate cases ar e to be foun d, particularly o n dolomites, but


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32 J . J . GLEW AND D. C. FORDTable II. Mean rillenkarren width (mm) on experimentalplaster of paris surfaces, measured at percentage meanlength stations for each block

26 per cent 43 per cent 87 per centGradient from crest from crest from crest- 7.5

30" 6.6 6.6 8 - 535" 6.0 7 - 2 8.545 " 6.2 6.8 8.05 0" 6.3 6.8 8.260" 5 . 5 7.0 7.9

22 9 -

they are comparatively rare. Recognizing difficulties encountered with the terms for rainfall and rockhomogeneity, an approximate but practical equation for rillenkarren-bearing rocks is

L = Tatexp [ b S ] ( 5 )Rill width and cross-section

In all experiments patterns of rills propagated to occupy all available space in the crest zone of theworking surfaces, interfluves being reduced t o the sharp intercepts of adjacent channels. Rill width did notvary with gradient (see Table 11)but there was a tendency for it to increase downslope as a consequence ofthe more complete elimination of narrow initial rills. Rill width isdetermined by a characteristic of the hostmaterial. Figure 5 compares widths from typical experimental plaster and salt blocks with natural exampleson Palliser limestone (Devonian) in Surprise Valley. Th e controlling characteristic is probably grain size:the experimental plaster was invariably fine-grained and the salt coarse-grained. The Palliser limestone ismedium-grained: qualitatively, the widths of its rillenkarren are similar to those we have observed on manyother crystalline limestones. Natural salt tends to be coarse-grained like th e experimental salt and in rilledexamples viewed in the geological collection of the University of British Columbia, rill width appeared to beas great as or greater than that produced in our experiments with the artificial solid.

The evolution of rill cross-sections on the plaster models was monitored carefully by means of heightgauges, casts, photography and line projection. Figure 6A, comparing a mature experimental cross-section

15

LQ

0

t @ S i m u l a t e d

I--0

plaster surface

One millimefre class intervalFigure 5. Histograms of rill width o n typical experim ental plaster and salt blocks, and upon a natural limestone block from SurpriseValley, Rocky Mountains of Canada


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DEVELOPMENT OF RILLENKARREN 33

A

NATURAL LIMESTONE0 10 20 30 40 50--m

SIMULATE0 (PLASTER!

Figure 6A . Cross-sections of rillenkarren from an experimental plaster block a t 60" and from a natural limestone block at 70" inSurprise Valley, R ocky Mountains of Canada. B . Evolution of plaster surface at 45", 120-500 hours after beginning the experimen t.Parabolas of varying value of n are shown for comparison

a t 60" with a natural cross-section at 70 " on Palliser limestone, indicates the high degree of geometricsimilarity obtained in the exp eriments. Figure 6B illustrates evolution of a portion of a 45" model between120 and 500 hours. From many profiles such as those in this figure it was determined that rillenkarrencross-sections appro ach a parabolic fo rm . Because rill width is fixed by comp etition with neighbouring rills,deepening by solution as time progresses produces a cross-section that conforms closely to the equat ion

- -

where d =wid th .Fo r the experimental rills on plaster and salt values of n ranged fro n approximately 10 n th e early stagesof development to 3 for rills approaching an equilibrium state. Natural rillenkarren o n the Palliserlimestone sample of Figure 6A have n values within this rang e. Physical implications of th e parabo lic formare discussed in the concluding section of this paper .


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34 J. J . G L E W AND D. C. FORDSolutional reduction of the experimental surfaces

As determ ined by 25-point m icrometry on th e 60 x 36 cm blocks solution reduces rill troughs and allparts of th e ausgleichflache in parallel a t the original gradient, implying that solution rates ar e constanteverywhere on the surface except the tiny rill partitions. This finding is supported by the grossly planarcharacteristic of most simple rillenkarren surfaces in nat ure, although it does not ap ply where rillenkarrenare secon dary colonists on the sides of larger solutional feat ure s such as kamenitzas (solution pits).Ra te of red uction of the exp erimental surfaces was fou nd to dep en d directly o n gradient and rainfallintensity, not surprisingly: these two factors determ ine the amo unt of wate r intercep ted and its residencetime on a surface, therefore regulating the amo unt of solution tha t can tak e place. Figure 7 depicts the gr aphof reduction of the surfaces (500 hours) against gradient, rainfall intensity invariate. The optimumcombination of collecting area an d renewal of exposure to denud ation occurs at 45" for this simp lest case oferosion by aqu eou s solution in the presen ce of a saturate d fluid boundary layer. On natu ral surfaces it mustbe supposed tha t prevailing winds, etc., may cause so me local distortions of the function. F or com parison,the Renn er (19 36) and H orto n (1 945) graphs for mechanical erosion by she et, ri ll and gully flow o n larger,mantled slopes are superimposed o n the graph, though it should be noted that the ordinate scales aredifferent.

Genesis of rillenkarren and ausgleichflacheMaintenance of a uniform rate of solution on surfaces which develop tw o distinct morpho logical zones,rilled and planar, u nd er sustained rainfall indicates that whilst the solvent capacity of th e water is similar atall points, two unlike eroding processes are localized. It is suggested that this differentiation is acons equ ence of a critical chan ge in the thicknes s of th e film of flowing wa ter and of th e deg ree of turbulencepropagated at the base of that film by impacting droplets. An alternative hypothesis, tha t the differentiationis a respon se to changing aggressivity of the w ater as it moves downslope (L ehm ann (1927), Bogli (196 1))must be rejected because of the constancy of erosion ra te th at has be en d emo nstrated: this suggestion is also

80

60c2w0LlaaZ 40u)0aw0

20

00 10 20 30 40 50 60 70 80

SLOPE ANGLE DEGREESFigure 7 . The relationship between mean surface lowering of the exp erimental plaster blocks and their gradient. For comparison, the'percent erosion' curves for mechanical erosion published by R enner (1936)and Horton (1945) are given. N.B. The ordinate scalesare different


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DEVELOPMENT OF RILLENKARREN 35unlikely to be true because of efficient replenishment and mixing of the flowing film by raindrops strikingeverywhere on the surface.Any element of an inclined surface receives direct rainfall plus a volume of run-off from upslope. Undersustained rainfall impermeable surfaces have deeper water on the lower slopes because acceleration of flowis retarded by surface roughness. At 45 nclination and 35 mm/hour rainfall water film thickness was foundto increase from 0.15f .05 mm to 0.20 f .05 mm down the experimental ausgleichflache. Therefore, thecritical value of film thickness may be set at =z0.15 mm. Upslope surfaces and lateral edges where this valueis not attained may be described as subject to rim effect and rillenkarren as phenomena of a rim effect zo ne .Length of the zone (length of rillenkarren) is not simply predetermined by the combination of initialgradient and rainfall intensity. The experiments revealed that developing rillenkarren extended the zoneinto the ausgleichflache until equilibrium length was attained.Rim effect was first described by Smith amd Albritton (1941) and Hoffmeister and Ladd (1945)whenstudying the raised edges found o n some weathered clints (flat, segregated limestone surfaces). It is evidentfrom our experiments and those of Purdy (1974) that the effect is self-perpetuating, partitions and edgesincreasing in relief to an equilibrium value determined by properties of the rock, probably grain size andbonding.In the case of rillenkarren, in the hydrodynamic zone of rim effect and under the random fall of raindrops,initially planar surfaces are transformed into sets of close-packed rills of straight plan form and approxi-mately parabolic cross-section. Two factors explain the transformation. The first is the unidirectional andstrong effect of gravity. On planar slopes rills and partitions a re always oriented directly down the fall linebecause gravity is effective in directing falling droplets, splash and film flow. The second factor relates to aunique property of the parabola, that it deflects parallel forces through a focal point. In the experiments andin nature, most raindrops are falling in parallel motion as they approach the surface. The rillenkarrencross-section in the mature form ( n < 6 in Equation 6) may be regarded as a most efficient shape thatminimizes droplet impact upon channel sides, thereby reducing dissipation of energy across rill partitionsand allowing only a narrow central portion of the rill trough to receive impact directly. Direct and deflecteddroplets reduce the trough at a rate closely commensurate with that prevailing upon the ausgleichflache.Rillenkarren are therefore the expected, stable landform in zones of rim effect on soluble rocks, given thatrainfall intensity, surface gradient and rock homogeneity are sufficient.The rillenkarren channel is radically different from flow-formed conventional stream channels of cor-rasional or other origin. First Order rills, as Horton (1945)referred to them, can only be established wherethe film flow of an ausgleichflache or belt of no (channelled) erosion reaches an unstable thickness andbreaks down to turbulent flow. The karstic equivalent of First Order corrasional rills are rinnenkarren(Bogli (1960))which head at the base of ausgleichflache, are separated by broad interfluves and are an orderof magnitude larger than rillenkarren. Occasionally rillen, ausgleichflache and rinnen occur in association inone field assemblage but this is uncommon.

ACKNOWLEDGEMENTSWe are indebted to Dr. M. F. Goodchild and two anonymous referees (one of whom suggested themodification of Equation 6 adopted here) for review of this paper and to R. Bignell, R . Zablocki and J .Mayer for technical assistance in the laboratory. The National Research Council of Canada providedfinancial support.

REFERENCESBogli, A. (19 60) . Kalklosung und K arrenbildung, Zeirschrifr fur Geomorphologie,Supplementband 2, 4-2 1.BogIi, A. (196 1). Karrentische, ein Beitrag tu r Karstmorphologie, Zeitschrifr f i i r Geomorphologie,5 , No. 3, 185-193.Blumberg,P. N. , and Curl, R. L. (1974 ). Experimentaland theoretical studies of dissolution roughn ess,Journal of Fluid Mechanics,Bryan, R . B. ( 197 0). A n improved rainfall simulator for use in erosion research, Canadian Journalof Earth Sciences, 7,1525-1531.Glew, J . R. (1976). The simulation of rillenkarren,unpub lished McMaster University M.Sc. thesis, 116 p.Goodchild, M. F., and Ford, D . C. (197 1). An analysis of scallop patterns,Journal of Geology,79, No. 1, 52-62 .Gum, R ., and K inzer, G .D. (19 49) . Terminal velocity of fall for water dro plets in stagna nt air, Journal ofMereorology,6,243-248.

6 5 , 7 3 5 .


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36 J . J . GLEW AND D. . ORDHoffmeister, J. E., and Ladd L. S. (1945). Solution effects on elevated limestone terraces, Bulletin, Geological SocietyofAmerica, 56,Horton, R. E. (1945). Erosional development of streams and their drainage basins; hydrophysical approach to drainage basins andJennings, J. N. (1971).Karst, Cambridge, Mass., and London, The M.I.T., Press.Lehmann, 0. (1927). Das Tote Gebirge als Hochkarst, Mitteilungen der geographische Gesellschaft, W ien ,70, 201-242.Luckman, B. H. (1976).Rockfall and Rockfall Inventory Data: Some Observations from Surprise Valley, Jasper National Park,Lundberg, J. (1977).An analysis of the form of rillenkarren from the tower karst o f Chillagoe, Austr alia, Proceedings, 7th InternationalPlummer, L. N. , and Wigley, T. M. L., (1976).The dissolution of calcite in C02-saturated solutions at 25C and 1 atmosphere totalPurdy, E. G. (1974). Reef Configurations: Cause and Effect, in Laporte, L. F. (Ed.), Reefs in Time and Space, Society of EconomicRenner, F.G. (1936).Conditions nfluencing erosion on theBoise River W atershed,U.S. Department of Agriculture, Technical BulletinSmith, J. F., and Albritton, C. C. (1941). Solution effects o n limestone as a function of slope, Bulletin, Geological Society ofAm erica,Sweeting, M. M. (1972). Karsr Landforms, London, Macmillan.Wilford, G. E., and Wall, J. R. D. (1965).Karst topography in Sarawak, Journal of Trop ical Geography, 21,44-70.Yalin, M. (1971). Theory of Hydraulic Models, London, Macmillan.

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glew, ford (1980) - a simulation study of the development of rillenkarren

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