Strahler 1952 Dynamic Basis of Geomorphology Geology

Geological Society of America Bulletin

doi: 10.1130/0016-7606(1952)63[923:DBOG]2.0.CO;2 1952;63, no. 9;923-938Geological Society of America Bulletin

ARTHUR N STRAHLER DYNAMIC BASIS OF GEOMORPHOLOGY

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BULLETIN OF THE GEOLOGICAL SOCIETY OF AMERICAVOL. 63. PP. 923-938 SEPTEMBER 1952

DYNAMIC BASIS OF GEOMORPHOLOGY

BY ARTHUR N. STRAHLER

ABSTRACT

To place geomorphology upon sound foundations for quantitative research into fundamental principles,it is proposed that geomorphic processes be treated as gravitational or molecular shear stresses actingupon elastic, plastic, or fluid earth materials to produce the characteristic varieties of strain, or failure,that constitute weathering, erosion, transportation and deposition.

Shear stresses affecting earth materials are here divided into two major categories: gravitational andmolecular. Gravitational stresses activate all downslope movements of matter, hence include all massmovements, all fluvial and glacial processes. Indirect gravitational stresses activate wave- and tide- inducedcurrents and winds. Phenomena of gravitational shear stresses are subdivided according to behavior ofrock, soil, ice, water, and air as elastic or plastic solids and viscous fluids. The order of classification isgenerally that of decreasing internal resistance to shear and, secondarily, of laminar to turbulent flow.

Molecular stresses are those induced by temperature changes, crystallization and melting, absorptionand desiccation, or osmosis. These stresses act in random or unrelated directions with respect to gravity.Surficial creep results from combination of gravitational and molecular stresses on a slope. Chemicalprocesses of solution and acid reaction are considered separately.

A fully dynamic approach requires analysis of geomorphic processes in terms of clearly denned opensystems which tend to achieve steady states of operation and are self-regulatory to a large degree. Formula-tion of mathematical models, both by rational deduction and empirical analysis of observational data, torelate energy, mass, and time is the ultimate goal of the dynamic approach.

CONTENTSPage Page

Introduction 923 Mathematical models in geomorphology 935Basis of a dynamic approach 925 References cited 937

Type of stress 925Type of material 926Type of strain 927

Gravitational stress phenomena 928 ILLUSTRATIONSMolecular stress phenomena 932Surficial creep phenomena 933 Flgure Page

Chemical processes 934 1. Plastic and fluid flow 926Tectonic and volcanic stresses 934 2. Stream profile as a function of time andDynamic open systems and the steady state.. 934 distance 937

INTRODUCTION transportation and deposition. The concepts setforth in this paper have been established as

The aim of this paper is to outline a system guiding principles underlying the quantitativeof geomorphology grounded in basic principles investigation of erosional landforms by theof mechanics and fluid dynamics, that will writer and his associates under Contract N6enable geomorphic processes to be treated as ONR 271, Task Order 30, Project No. NR.manifestations of various types of shear stresses, 089-042, Office of Naval Research, Geographyboth gravitational and molecular, acting upon Branch. The writer is grateful to Professor W.any type of earth material to produce the C. Krumbein of Northwestern University andvarieties of strain, or failure, which we recognize Professors Sidney Paige and Donald Burmisteras the manifold processes of weathering, erosion, of Columbia University for their kindness in

923

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924 A. N. STRAHLER—GEOMORPHOLOGY

reading the manuscript and suggesting variousimprovements, and to Mr. Samuel Katz of theLamont Geological Observatory for developingthe mathematical analysis of river profiles.

For more than half a century, the study oflandforms in North America was dominated byan explanatory-descriptive method of studyused by W. M. Davis and his students. Davishimself maintained that the aims of his methodwere geographic; that the consideration ofprocess was introduced merely to permit anorderly genetic system of landform classifica-tion. The weakness in understanding of geo-morphic processes (and hence also a weaknessin the understanding of the origin of landforms)has not been confined to the American conti-nent. The geomorphologists of France andEngland, closely attached to schools and de-partments of geography, have also tended togive much attention to descriptive, deductivestudies of landform development and to regionalgeomorphological treatments. Even in sodistinctively different a treatment as WaltherPenck's morphological analysis, processes andforms are analyzed by a deductive methodsafely removed from the reality of existinglandforms and without mention of basic princi-ples of soil mechanics and fluid dynamics.

If geomorphology is to achieve full stature asa branch of geology operating upon the frontierof research into fundamental principles and lawsof earth science, it must turn to the physicaland engineering sciences and mathematics forvitality which it now lacks. The geomorphicprocesses that we observe are, after all, basicallythe various forms of shear, or failure, of ma-terials which may be classified as fluid, plastic,or elastic substances, responding to stresseswhich are most commonly gravitational, butmay also be molecular.

Unless the fundamental nature of materialsis understood, we are in a poor position to addanything worthwhile to what is already largelyself-evident concerning the behavior of streams,landslides, glaciers, or wave-induced currents.We cannot hope for anything better than asuperficial knowledge of the form and motion ofa sand dune unless we can interpret the dunein terms of aerodynamic principles; or of astream profile unless we understand the princi-ples of fluid dynamics and the transportationof sediment; or of the moulding of a drumlin

unless we study the flowage and fracturing ofice; or of the production of an off-shore barunless we know something of the dynamics ofwaves and the transport of sediment by oscil-lating or pulsating wave-induced currents; orof the causes of a great earth-flow unless wecan appreciate the principles of plastic flowage.

To delegate to the civil engineer all funda-mental research on geomorphic processes andforms has certain disadvantages. With his at-tention focused upon problems dealing withman-imposed modifications of the natural land-scape, the engineer may have neither the timenor the inclination to investigate a broad rangeof natural phenomena where they are bestdisplayed. Furthermore he is likely to haveonly a limited acquaintance with geologic ma-terials and forms, whereas the geomorphologist,trained as a geologist, has built up a life-timestore of information and experience, much of itrelating to theoretical and historical aspects ofgeology.

Although the study of fundamental principlesof geomorphology by engineers is to be wel-comed and encouraged, there is a real dangerthat the engineer will find it necessary to takeover an increasingly greater proportion of geo-morphic research and thereby cut it off fromits most logical parent, the field of theoreticalgeology. A specific example is the field oflandslides and related gravity movements. Thegeomorphologists now owe their most pene-trating analysis of the fundamental principlesof these phenomena to research by specialists insoil physics and soil mechanics. Karl Terzaghi's(1950) work is outstanding in this respect. Farfrom being a supplier of basic theoretical knowl-edge to civil engineers in this field, as he shouldbe, the geomorphologist has been receiving thisinformation from them.

It is appropriate in this paper, which isphilosophical in nature, to clarify the functionof time in geomorphology. Two quite differentviewpoints are used in dynamic (analytical)geomorphology and in historical (regional) geo-morphology. The student of processes and formsper se is continually asking "What happens?";the historical student keeps raising the question"What happened?". Bucher (1941) has aptlylabeled the two types of geological informationas timeless and timebound knowledge re-spectively. It is largely with the timeless knowl-



INTRODUCTION 925

edge that the field of dynamic geomorphologydeals. The principles are usually most easilydiscovered by a study of contemporary proc-esses and existing forms, but the dynamicgeomorphologist will refer to any part of thegeologic record for evidence which will increasehis understanding. He is not, however, primarilyconcerned with the actual series of eventswithin a particular geologic period and in aparticular geographical location, as is the his-torical geomorphologist. Although manyhistorical-regional geomorphic investigationshave been ably conducted with minimum refer-ence to geomorphic processes, it is only rea-sonable to suppose that a better knowledge ofhow processes operate and normal forms evolvewill increase the effectiveness of historicalstudies and reduce the likelihood of drawingerroneous inferences of past events.

On the other hand, studies of dynamic geo-morphology based on existing landforms cannotbe prosecuted in ignorance or disregard of pastchanges of climate, and hence, of relative ratesand importance of various processes dependenton climate, during the long period required todevelop the forms. One cannot, for example,extrapolate rainfall intensity-frequency-dura-tion statistics of the past 50 years back over aperiod of 50,000 years in correlating drainagebasin forms with rainfall characteristics. Thedynamic geomorphologist must be alert toevidences of important differences in processesoperating during earlier stages of developmentof the forms he is studying. Thus historical-regional geomorphic treatment cannot bedivorced from dynamic investigations. The dif-ference in the two types of study lies inproportion of each one involved, for neither cansuccessfully be pursued independently of theother.

BASIS OF A DYNAMIC APPROACH

Type of Stress

Table 1 outlines the organization of a dy-namic treatment of geomorphology. The firstbasic subdivision is made according to thenature of the stresses involved: (A) gravitationalstresses and (B) molecular stresses. The gravita-tional stresses act upon all earth materials.Where these have a sloping surface, compo-nents of the gravitational stress tend to produce

movement of matter downslope. Under thisheading, therefore, are placed both the massgravity movements and the erosional-trans-portational fluids: water, ice, and air. Energytransformed during the operation of these proc-esses is for the most part the potential energyof position or elevation. Earth materials raisedby erogenic or epeirogenic crustal movementsare gradually moved to lower elevations, withan accompanying transformation of potentialenergy to kinetic energy of heat or motion.Water raised from sea level to divide regionsby expenditure of solar energy in atmosphericheating and turbulence makes its way to lowerelevations, at the same time expending a smallfraction of the total transformed energy inovercoming the cohesion or bonding of rockand soil particles and in transporting them tolower levels.

The work of wind is a somewhat differentform of response to gravitational stress in thatsolar heating has set up differences in air massdensities which tend to be equalized by airflow from regions of higher to lower pressure.Some winds (katabatic) are actually a form ofdownslope flowage and are thus akin to streamsand glaciers. Shore currents may be wave-induced, in which case the generation of wavesmay be traced back to gravitational stressesthrough the winds which produced them, ortide-induced, in which case gravity flow re-sponding to differences in water level occurs.The general arrangement of materials in Table1A is in order of decreasing resistance to shearstresses.

The molecular stresses (Table IB) may act inany direction with respect to gravity, and in ahomogeneous or isotropic soil or rock materialmay be distributed at random in all possibledirections through a given point. The charac-teristic movement is that of dilatation of amass. Downslope movement is not essential,but creep to lower levels is inevitable on anyslope, because a component of the gravitationalstress then adds its vector to the otherwiserandom stress distributions.

Table 1C lists chemical processes, which arenot themselves stress-producing, although ofgreat importance in geomorphology. Listed hereare chemical reactions and simple solution.Such chemical processes as hydrolysis and oxi-dation, which would produce expansive stresses




by increasing volume and developing adsorptiveproperties, have been placed under molecularstresses.

Type of Material

Three fundamental types of materials can berecognized, although the distinction in a given

mass. A law of plastic deformation may bestated as follows: (Burmister, 1948, p. 93)

ino>

:T

0 Rate of shear du/dyFIGURE 1.—RELATION OF RATE OF SHEAR TO

SHEAR STRESS IN PLASTIC SOLIDS ANDViscous FLUIDS

case may be difficult to make: (1) rigid, orelastic, solids, (2) plastic solids, and (3) fluids.

An elastic solid is a mass of rock or soilwhich possesses elasticity and yields by elasticstrain. Ideally this strain follows Hooke's Law,in which strain is proportional to stress. Stressesof greater magnitude cause failure by rupturealong discrete planes and may take the form ofsliding movement on shear planes or pullingapart on widening tension fractures. We willassume for the moment that there is little orno region of slow, permanent deformation(creep) between the development of elasticstrain and rupture. Our definition of elasticsolid is here broadened to include homogeneoussoil or incoherent bedrock, which, in a denselypacked state without excessive moisture, mayact as an elastic continuum (Krynine, 1947,P- 92).

A plastic solid is a material that deformsby distributed intermolecular or intergranularshear—that undergoes flowage in the mannerof a fluid, if the shear stresses exceed a limitset by the internal cohesion or friction of the

(F -/)dy

where du/dy = rate of shear (rate of change ofvelocity, u, with respect todepth, y)

F = shear stress/ = yield limit17 = consistency of the material.

Figure 1 illustrates the principles of plasticflow. In region A, increasing shear stress iswithin the limits of internal resistance of thematerial and no shear occurs. In region B.flowage of slow but increasingly rapid rate setsin. This is the region of plastic creep. In regionC, flowage is essentially that of a true fluid inthat rate of shear is directly proportional toapplied stress. Bingham (1922) has describeda plastic material as one which "has a propertyof permanently supporting a shearing stressless than a certain critical yield value withonly slight total deformation, but for greatershearing stresses plastic flow takes place at acontinuous uniform rate, as for a highly vis-cous fluid."

A fluid is a substance that offers little re-sistance to change in form and is incapable ofany internal adjustment that will enable it tomaintain equilibrium at rest while subjectedto shear stress, however small (Dodge andThompson, 1937, p. 1). Fluids include bothgases and liquids, hence the study of geomor-phic processes involving flow of water or windis governed by principles of fluid flow. Whereasliquids are largely incompressible and deformby changing shape, gases change readily bothin form and in volume.

A perfect fluid, one which offers no resistanceto deformation, does not exist. All fluids possessviscosity, a quality of resisting shear stress.Energy is required to overcome this resistance,hence stress must be applied continually tomaintain shear in a fluid, and the rate of shearis proportional to the applied stress, as statedby the following equation:

dudy

F_u



BULL. OEOL. SOC. AM., VOL. 63 STRAHLER, TABLE 1

Table 1. DYNAMIC BASIS OF 6EOMORPHOLO6Y

A. 6ravitational Stresses

1.

I.

1

4.

5.

6.

7.

8.

9.

MATERIAL INVOLVEDIN MOTION

Crystalline rocks,arenaceous rocks,limestones ; dry so/Is

Glacial Ice ,near surface.

Argillaceousrocks and soils

Unconsolidafed rock,soil + water

Glacial ice underheavy load.

Wafer film onsloping Surface.

Water in permeablerock, soil . (Nosurface slope)

Water layer onsloping surface

Water in slopinglinear channel

PROPERTIES OFMATERIAL

Rigid , elastic solid or"elastic continuum"

Obeys Hooke's Law:Strain oc stress

Plastic solid in regionof -elastic behavior.

Plastic solid in regionof slow creep.

Plastic solid in regionof flowaqe. Obeys Binaham's: r~) ^1 c Ylaw: Kate ot shearproportional to stress, abovea yield limit.

Plastic solid with highyield value. Non-linearincrease of shear rate with

increase of stress.

Newtonian fluid. Moyield value. Linear increaseof shear rate with stress.Subject to capillary influences.

Newtonian fluid,subject to capillaryinfluences.

Newtonian fluid.

A/ewton/an fluid.

TYPE OF FAILURE(STRAIN)

Sudden rupture alongshear surfaces or

tension fractures.

Sudden rupture alongshear surfaces ortens/on fractures.

Continuous, slow laminar•flow (distributed shear)

Plastic flow (shear betweengrains) when stress exceeds

yield limit. Movementceases below yield value.

Flow laminar or turbulent.

Laminar plastic flowabove yield limit. Belowyield limit, returns tobrittle elastic solid.

Laminar- flow, ceasingwhen water thins belowcapillary control limit.

a). Silts : Laminar flowfollowing Darcy's Law.

b). Sands : Mixed laminarand turbulent flow,

c). Gravels •• Turbulent flow.

Sheet runoff in turbulentor mixed turbulent- laminarflow.

Stream flow, turbulentexcept in bed layer.

GEOMORPHIC PROCESSESAND FORMS

Landslides: Slump, slide(compressiona/ stress);

rock- fa II (tensional stress).

Crevassmg , overthrusting ,calving of glaciers.

Large-scale creep phenomena.Superficial deformation of clays.

Solifluction , earth flow,mudflow, highly turbidstreams, turbidity currents.

Continental and alpineglaciers,. Erosion forms dueto ice abrasion. Deposifionalforms moulded by ice flow.

Sheet runoff on slopes androck surfaces. Slope reductionby removal of ions, Colloids,clays. Fluting, grooving of limestone.

Infiltration of precipitation, carryingdown of ions, colloids, clays, Silts( illuviation). General slopereduction. Karstic forms inhighly soluble rocks.

Slope erosion, transportation.Slope forms of fluvial drainagebasins.

Stream erosion, transportation,deposition. Drainage systems.All fluvial landforms.

Indirect Responses to Gravitational Stresses

iO

11.

Standing waterbodies - Oceans,lakes.

Air

Newtonian fluid.

A gas : compressiblefluid of extremelylow viscosity.

Turbulent flow asa). Pulsating or oscillating

currents caused by waves.t>). Tide -induced currents.

Turbulent flow inducedby pressure gradients,

(gravitational stress on air mosses)

Shoreline processes of erosion,transportation, deposition.Shoreline landforms: cliffs,benches, beaches, bars, spits.

Wind erosion, transportation,deposition. Deflational andahrasional -forms. Dunes, loess.

4.

6.

7.

1.

2.

B. Molecular Stresses

MATERIALSINVOLVED

Rock :si rang, hardcrystalline,glassy or

crystal aggregate.

a). Permeable rack+ water

b). Clay Soils

Permeable rockor soil + water

and salts

Rock or soil+ colloidsand water-

Rock or soil+ capillary wafer

Rock or soil+ plant roots

Strong, hardmonolithic

bedrock

PROPERTIESOF MATERIAL

a). Elastic Solid,non- homo-

geneous.

b). Elastic solid,homogeneous

a). Elastic solid,

b). Plastic solid.

Elastic solidor elasticcontinuum

Elastic orplastic solid



Elastic solid

STRESS AND CAUSE

Shear stress due tonan- uniform expansion ~

contraction in cyclictemperature changes.

Shear stress sef upby thermal gradient•from surface heating.

a). Shear stress set up byinters-filial ice crystalgrowth,

b). Stress -from growth ofice lenses, wedges.

Shear stress sef up byinterstitial growth ofsalt crystals.

Shear stress Set up bydictation accompanyingwater adsorption and drying.

Shear stress set up bydilitation accompanyingchanges in capillary filmtension.

Shear stress set up byswelling of rootlets under

osmotic pressure.

Shear stresses of tectonicorigin stored as elastic

strain at depth.

KIND OF FAILURE

Rupture by shear ortension fractures betweengrains, along cleavages,joints , bedding planes.

Rupture betweenlayers paralleling rockSurface.

a). Rupture between grains,cleavage pieces , jointblocks, beds,

b). Plastic deformation ofclays adjacent to ice.

Rupture between grains,cleavage pieces, joint blocksor beds.

a). Rupture between grains.b). Plastic deformation of

clays during Swelling.

Rupture between grainsor masses of soil.

Rupture between grains,cleavage pieces,, joint blocksor beds.

Rupture of rock on planesparalleling surfaces after

release of confiningpressure.

WEATHERIN6 PROCESSAND FORM

Granular or blackydisintegration of rocks,esp. coarse-grainedcrystalline rocks.

Exfoliation of rockby fire, lightning ; solaror atmospheric heating-cooling.

a). Frost disintegration ofrocks. Felsenmeer.

b). Heaving of clay soiJs,frost mounds, polygons.

Effloresence , granulardisintegration in dryclimates. Caliche heaving.

Exfoliation of basaltic,granitic rock upon alteration

of silicates. Slaking ofshales, argillaceous ss.

Disintegration of granularpermeable rocks. Heavingor subsidence of clays,silts .

Disintegration of rock byprying of roots. Deforma-tion of soils.

Exfoliation of domes,slabs, shells. Quarryrupture, rock-burst.

C. Chemical Processes

MATERIALS

Soil, rock +acids , water

Soil, rock +water

PROCESS

Reaction between acidions and mineral surfaces.Removal of products insolution.

Simple solution (ion/cation)of susceptible minerals.

FORMS PRODUCED

Lowering of rock and soil surfaces.Pitting, cavitation of rocks, esp. carbonates.Cavern and karst forms (see Table A, no. 7).Weakening of bonds between mineral grains.

Cavitation of soluble salt formations.Slow attrition of exposed mineral surfaces.



BASIS OF A DYNAMIC APPROACH 927

where y is depth above bottom«is velocityF is shear stress\i is viscosity.

Under this type of flow, termed Newtonian vis-cous flow, the relationship of shear stress torate of shear is linear, the slope of the straightline being proportional to viscosity. Note inFigure 1 that the line passes through the origin,signifying that no matter how small the appliedstress some deformation by shear will occur..

Type of Strain

The type of failure, or strain, that may beexpected as a result of the application of gravi-tational or molecular stresses on elastic, plastic,or fluid earth materials determines the geomor-phic process and form. To some extent types offailure have already been noted in defining thefundamental properties of the materials.

Rock or soil behaving as an elastic solid failsby one of two possible forms: (1) A shear frac-ture may develop as a slip plane formed in thezone where shear stresses first exceed resistanceto shear. Differential movement of one mass orblock over another by sliding along the shearsurface constitutes the type of movement andis characteristic of slump types of landslides orof the surface zone of glaciers where overthrust-ing occurs. (2) A widening tension fracture maydevelop in material subjected to tensile stressestending to produce elongation of the mass. Ex-amples are particularly striking on glaciers andlandslides where gaping crevasses and fissuresare formed. Where molecular stresses are activein a rock or soil having elastic properties, theshear fractures and tension fractures may havethe extremely small dimensions of grain con-tacts or cleavage planes. Thus weathering phe-nomena involving disintegration of bedrock arelargely of this type of failure.

In fluids, and in plastic solids above the yieldlimits, flowage occurs in two forms: laminarflow and turbulent flow, both of greatest im-portance in geomorphic processes. In laminarflow, parallel layers of the fluid are in motionover one another without the existence of crosscurrents of motion. Each layer of the fluidmoves at a higher velocity than the layer ad-jacent to it on the one side but lower than the

layer on the other side. The intermolecularforces are strong enough to prevent the de-velopment of cross-threads of particle move-ment.

Laminar flow can be maintained in deepwater only at very low velocities. Thus evenextremely slow flow in sluggish streams andtidal currents may be turbulent. Erosion andtransportation of coarse-grained materials can-not occur in laminar flow; instead the settle-ment of suspended sediment is permitted..Somewhat more rapid flow of laminar nature ispossible in thin films of water, such as insheetflow over soil or rock surfaces or while in-filtration is occurring through permeable rockor soil. The removal of ions, colloids, extremelyfine clays, or silts may occur by this processand it is therefore of considerable geomorphicimportance in denudation.

Laminar flow is possible at higher velocitiesin fluids of high viscosity than in those of lowviscosity; glacier ice, which behaves essentiallyas a fluid above its elastic limit, flows exclusivelyby laminar principles. Air, by contrast, hassuch low viscosity that for all practical purposesany air currents of sufficient velocity to movefine sand or dust can be considered as in turbu-lent flow.

Turbulent flow sets in when flow velocityexceeds a critical velocity determined by factorsof depth, viscosity, and roughness of the bound-ing surface (Hjulstrom, 1935, p. 238). Turbulentflow is characterized by components of velocityin many directions within the moving fluid.These have been thought of as innumerablecylindroids of rotation continually forming anddissolving in the fluid. Although an averageforward velocity prevails at any one particularpoint in the stream of flow, the instantaneousvelocities are distributed at random, both indirection and magnitude (Einstein, 19SO, p.13). The result is vertical components of flowwhich can sustain solid particles; hence thetransportation of rock materials in suspensionis possible by flowing air and water at thevelocities normally prevailing in streams, wave-induced currents, and winds.

The work of running water, waves, and windcannot be understood without a thorough ap-preciation of the principles of laminar and




turbulent flow. Fluid dynamics is, therefore, acornerstone of geomorphology.

In the plastic earth materials, such as earth-flows and mudflows, our definition of "flow" isbroadened to include shear more or less uni-formly distributed throughout the mass andoccurring by the rotation or slippage of grainswith respect to one another. As the particlesize involved diminishes, the deformation simu-lates more and more perfectly fluid flow inwhich shear is among molecules of the liquidor gas. On the other extreme is the flow of agreat landslide mass in which huge boulders ofrock are incorporated. Considered in detail,such movement would not follow the laws offluid flow, but if we think of the mass as awhole and if the landslide is very large in rela-tion to the particles of which it is composed, theprinciples will apply. In any case, the type ofdeformation is distinctly different from the slid-ing movements in which a large mass movesas a unit on a single shear surface.

Before leaving the topic of general principlesof types of failure in response to shear stresses, aword of comment upon the composite behaviorof certain earth materials is in order. Ice andsome types of argillaceous sediments responddifferently to stresses depending upon the dura-tion of the stress. A weak but continuously ap-plied stress will cause continuous deformation(creep) by flowage, whereas a severe stress ap-plied suddenly will be met by an elastic re-sponse.

GRAVITATIONAL STRESS PHENOMENA

Table 1A outlines the various gravitationalstress phenomena, listed in a general way ac-cording to the property of the material in-volved. Beginning with rock, soil, or ice be-having as elastic solids, the table proceedsthrough the plastic earth materials to thefluids, taken in order from higher to lowerviscosity—first water, then air—with laminarflow being placed ahead of turbulent flow.

Most of the processes covered by this tableinvolve the downslope movement of material.Here gravitational stress tending to produceshear is some fraction of the total gravitationalforce. The slope function, which is the sine ofthe angle of slope, determines the intensity oferosional and transportational activities. In the

case of water percolating through a permeablerock or soil, the surface slope function is absent,as the process occurs in a subsurface location.Subsurface flow lines are functions of perme-ability and excess hydrostatic head and mayfollow various directions and velocities. Thisdistinction is worth noting because, in regionsof very low surface slopes, the removal ofmineral matter by percolation of ground watermight achieve quantitative importance equalto that of surface runoff and mass-movementprocesses.

A brief explanatory statement follows foreach tier of Table 1A with mention of a fewrepresentative published works concerning each:

1. Rock and soil which is relatively dry. Thismaterial possesses a high degree of internalfriction or cohesion but lacks a tendency toplastic deformation; it behaves under near-surface conditions of temperature and con-fining pressure as an elastic solid or elasticcontinuum, hence yields slightly by elasticdeformation until the limit of internal cohesivestrength or intergranular friction is exceeded.Because this occurs along a localized surface ofmaximum shear stresses, the rupture is in theform of a sliding plane, as in slump blocks ofbedrock or soil where an up-concave surface ofsliding is formed and the mass rotates to a posi-tion where forces tending to produce shear aremore nearly balanced throughout the shearzone. In other cases, particularly where a plas-tic substratum is in flowage, the rigid upperpart of a landslide mass ruptures in nearlyvertical tension fractures. The break off ofoverhanging rock masses on a cliff is a formof rupture of an elastic solid, but the plane ofbreakage is usually predetermined by joint orbedding planes along which the cohesion ismuch less than in the solid rock.

Yielding of elastic rock materials by ruptureis characteristically followed by rapid motionof the mass, often accelerating at the rate of afreely falling body. This is because the gravita-tional shear stress built up before rupture isvastly greater than the stress required to movethe detached body once the bonds along thefracture surface have been broken. (For furtherdiscussion of dynamics of slumping and land-sliding see Terzaghi, 1943, 1950; Terzaghi andPeck, 1948; Krynine, 1947; and Hubbert, 1951.)Engineers have long been familiar with the



GRAVITATIONAL STRESS PHENOMENA 929

principles of stress distribution in slumps butgeomorphic studies of naturally occurringslumps have not thus far used the quantita-tive-dynamic approach.

2. Glacial ice in the surface zone where, toall visible evidences, the behavior is that of anelastic solid. Perutz (1950) has estimated, fromdeformation of a pipe embedded in a glacier,that to a depth of about SO meters no plasticflowage is observable. Crevassing, the tensionaltype of rupture associated with surface ice, isapproximately limited to this depth. Ruptureby overthrust faulting occurs in the ablationalzone of a glacier where ice thickness is less thanthe limit required for flowage, but where stressestransmitted from points upstream exceed theelastic limit. Calving, the break-off of ice blocksfrom the front of a tide-water glacier, may beconsidered as rupture resulting from undercut-ting by melting or by differential verticalstresses caused by rise and fall of tide level.

3. Argillaceous rocks and clays subject toexceedingly slow, continuous creep in a down-slope direction (Terzaghi, 1950; Hollingworthet al, 1944). These materials are character-ized by a high cohesion of particle surfacesbut a relatively low coefficient of friction amongthe particles. A coarse sand or gravel, with nocohesion but high internal friction, is totallylacking in such plastic properties. Valleywardflowage results in development of superficialfold and thrust structures resembling large-scale tectonic structures. Thrust sheets movinglong distances under gravitational stress havebeen postulated by geologists, and these occur-rences may be thought of as intermediate inscale between tectonic and surficial forms. Thematerials involved in continuous slow creepwould normally rupture as elastic solids undersevere, suddenly-applied stresses.

4. Unconsolidated rock or soil with liberalamounts of water, such that the material be-haves as a plastic solid. When shear stress ex-ceeds the yield limit, flowage sets in at a rategoverned by the intensity of the shear stressand the consistency of the material. Flowageconsists of intergranular shear distributed moreor less uniformly. Geomorphic phenomena in-cluded in this type of flowage are solifluction,earthflow, and mudflow, in order of decreasingconsistency and increasing velocity. To thislist might be added turbidity currents which,

through the presence of suspended sediment,may perhaps be more exactly classed as plasticflow than fluid flow.

A definitive feature of the plastic flow phe-nomena is the cessation of movement while thematerial is still on a valley-side slope or in asloping channel. Even though gravitationalshear stress is till acting, the material maycease to move because either the stress hasfallen below the yield limit (because of the massreaching a lower slope) or the yield limit hasincreased (by desiccation of the mass) abovethe previous yield limit. Unlike a fluid, whichwill continue to flow until a water-level sur-face is attained, the plastic solid can remainstable on a slope. Plastic flowage in soils hasbeen discussed by Burmister (1948, p. 91-101),but the principles have not yet been fully ap-plied to the study of naturally occurring flow-age movements on slopes.

5. Glacial ice under load. Characterization ofglacier movement as plastic flow is based uponthe recognition of a surface zone, some 30 to50 meters thick, in which no flowage is observedin the ice, but rather rupture along tensionfractures or thrust planes. Perutz (1950) calcu-lated from observations of deformation of asteel tube sunk into the Jungfraufirn that therigid zone extended to a depth of 50 meters,at which level a yield stress of 0.1 kg. per cm.2

existed. Observable deformation set in fromthis point downward. Rate of increase of shearwith respect to shear stress was not, however,Newtonian linear increase, but increased morerapidly with depth. It is, of course, possiblethat slight flowage does occur in ice of lessthickness, but it must be so slow as to be un-important in relation to the time span of theexistence of the ice in the surface zone whereablation is in progress.

Glacier flow is laminar because of its ex-tremely high viscosity (consistency) and greatdepth. Treated in this light, glaciers would bejudged incapable of lifting debris through tur-bulence, but must obtain their englacial loadfrom overthrusting or from the surface. Ifglacier ice is regarded as a plastic solid, wehave an explanation of the absence of flowagein embryonic or relict glaciers, even thoughthey rest on a considerable slope.

Landforms included under the category ofplastic glacial ice are, of course, the glaciers




themselves, both alpine and continental, andall erosional and accumulational landforms pro-duced by the flowage of ice over ground sur-faces. Rdches moutonnees, grooves, striae,troughs and other ice-carved forms in bed-rock are included, as are drumlins and othertill forms moulded of debris by ice flowage.Further progress in understanding the originof these features must be based upon funda-mental principles of ice flowage, now beingstudied intensively by physicists.

6. Flow of thin films of water on slopingsurfaces. The water is treated as a Newtonianfluid without a yield value but controlled inits behavior by capillary forces. Water heavilycharged with fine matter in suspension mightbe more accurately treated as a plastic solidof very low consistency. Water in thin filmson soil or rock surfaces moves largely or en-tirely by laminar flow. This is likely to occuron smooth rock surfaces where the water isheld by capillarity against the surface; on mostrough soil surfaces, flow is normally mixedlaminar and turbulent (Horton, 1945, p. 312).

The geomorphic activity of laminar waterfilms is removal of ions, colloids, and fine clayparticles from slopes. Fluting and grooving oflimestones and basalts are the striking formswhich may be attributed in part to this process,but unseen increments of slope reduction bythis process may be universal in distribution;ind quantitatively of great importance.

7. Downward percolation of water throughpermeable soil or bedrock (Burmister, 1948,p. 111-115). In fine-grained materials of siltsixes, the flow is entirely laminar and followsDarcv's iaw:

where » = average velocity of flowK = a constant11 = headL = length of soil column.

In sands, flow may be mixed laminar and tur-bulent; in gravels with large void spaces, flowis turbulent, above a critical velocity. Becauseof great variations in the cross-section of thepaths of flow, the velocity is unsteady, alter-nating between laminar flow in the constricted

passages between grains and turbulent flow inthe wider voids.

The geomorphic importance of percolationmay be considerably greater than indicated bytextbooks and published geomorphic papers,which give this phenomenon little or no men-tion. Depending upon the permeability of thesoil or rock, ions, colloids, clays, or fine siltsmay be carried down from the surface layerto lower depths or to stream channels. Accom-panying compaction under direct gravitationalstress lowers the ground surface. Where rain-fall is heavy, vegetative cover strong, and slopeslow, this process could be the major agent ofslope reduction. Its relative importance will in-crease as the surface flow of runoff decreases inerosional intensity. Where dominant, percola-tion removal would be expected to modify ordestroy normal fluvial-erosional topography,producing an irregular surface as in manykarstic terrains.

8. Water flowing on sloping soil or rocksurfaces where depth and velocity are suchthat turbulent flow results. The hydrodynamiccharacteristics of surface runoff have been stud-ied intensively by Horton (1945), Little (1940),Schiff and Yoder (1941), and others engaged insoil investigations. Flow is mixed laminar andturbulent in some cases where depth is slightand vegetative cover creates constrictions inflow paths, but with discharges typical of tor-rential rainfall the flow is wholly turbulent.

The geomorphic importance of surface run-off is paramount. The development of drainagebasin slopes is largely attributable to erosionand transportation by sheet flow. Because avery large percentage of the earth's landsurfaceis composed of fluvial erosional topography,this process may perhaps be the most importantsingle process in forming the landscape. Geo-morphologists have neglected the intensive,quantitative-dynamic study of slope develop-ment by surface runoff. This represents a glar-ing deficiency in a field of earth science whereeven adequate understanding of any process orform is a rarity.

9. Water flowing in a linear channel, i.e. asa stream. All stream flow of sufficient velocityto transport sediment and perform significantmorphological work may be regarded as turbu-lent. The development of drainage systems,



GRAVITATIONAL STRESS PHENOMENA 931

valleys, and all of the depositional features as-sociated with floodplains, deltas, terraces, al-luvial fans, and glacial meltwaters are includedin the work of streams.

Fortunately streams have been studied fromthe dynamic-quantitative standpoint by compe-tent scientists over a long period of years.Among the important geological contributorsare G. K. Gilbert (1914), W. W. Rubey (1933;1938), John Leighly (1932; 1934), and FilipHjulstrom (1935). Fundamental work on dy-namics of streams with particular reference tothe transportation of bed load and suspendedload has been published in recent papers byG. H. Matthes (1941; 1949), H. A. Einstein(1950), Samuel Shulits, (1936; 1941), V. A.Vanoni (1947; 1948), Hunter Rouse (1939),A. A. Kalinske (1947), and J. F. Friedkin(1945), to list but a few examples. Futuregeomorphic studies of stream-formed featuresand of the Pleistocene and earlier history ofriver systems under the influences of changingconditions of discharge, load, gradient, andbase level must be based upon sound principlesof stream action as they are being discoveredtoday through the experimental and quanti-tative field studies of the hydraulic engineer.

10. Shore-line processes, which we may re-gard as water currents or pulsations generatedby the impingement of water waves upon ashoaling bottom or by differences of level ac-companying tides. Wind-generated waves can-not be regarded as direct responses to gravi-tational shear stresses acting on a slope, butthe winds from which the energy is derivedby shear of air over water owe their existenceto differences in air density. One might, ofcourse, make a separate stress group out ofpropagated water waves (tsunami) generatedby crustal movements. But the pure deep-waterwave does not directly accomplish significanterosion or transportation at the shore line. Onlywhen the energy of orbital motion is trans-formed into mass movements of water such asthe swash (uprush), backwash (backrush), orlongshore current, does the wave become amajor agent. In these cases, gravitational stressdue to differences in water level or to slope ofthe beach is important in producing or modi-fying the flow. The backwash is simply thereturn gravitational flow, while some longshore

currents may be hydraulically induced by theraising of water level near shore by shorewarddrift of water under strong onshore winds.

Wave- and tide-induced currents! of sufficientvelocity to transport beach detritus are in tur-bulent flow. Unlike the more or less steady.unidirectional flow of streams, the wave-in-duced currents take the form of short-periodpulsations or oscillations.

Landforms associated with shore currentsinclude all features of shore lines, erosionaland depositional. Beaches, bars and spits, wave-cut cliffs, and abrasion platforms all requirefor their study an understanding of the funda-mental principles of pulsating and oscillatingwater currents. Although the qualitative-de-scriptive phases of shore processes and formshave been extensively treated by geomorphol-ogists, the quantitative-dynamic aspects havebeen given little attention by these same work-ers. In recent years, studies by laboratorypersonnel of the Beach Erosion Board of theU. S. Army Corps of Engineers and of suchhydraulic laboratories as the Fluid MechanicsLaboratory of the University of California atBerkeley, have brought forward much newquantitative information correlating wave char-acteristics with beach forms and with theattainment of the steady state in the shoreprofile. (See J. W. Johnson, 1949; Krumbein,1950; Keulegan and Krumbein, 1949; and Wil-lard Bascom, 1951.)

11. Air, the fluid of lowest viscosity amongthe geomorphic agents, with its content of dustor sand in transport. With the exception ofkatabatic winds, which are simple downslopeflows of cold air formed by inversion near theground or local upslope winds of thermal origin,winds are not directly related to topographicslopes. They may, however, be traced to gravi-tational stresses acting upon air masses of dif-fering density, or air pressure. In this respect,the transportational work of wind is unique,because material can be moved upslope or insome direction other than the downslope linesfollowed by water and mass movements of soilor rock.

Air has extremely low viscosity in comparisonwith water, hence air engages in turbulent flowat very low velocities. Excepting the thinboundary layer of laminar flow, turbulent flow




may be assumed in air for all practical purposes.Transport of particles of the clay and fine siltsizes takes place by suspension in the turbulentflow; coarse silt and sand move close to theground in a form of saltation by elastic reboundthat is unique among the geomorphic trans-portational processes.

Landforms and geomorphic processes of tur-bulent air flow include abrasional and defla-tional activities, movement of fine-grained par-ticles as dust storms with the eventualproduction of loess or volcanic ash accumula-tions, and all dune forms. Geomorphic literatureabounds with descriptions of dune forms andwith attempts to explain the variations of duneforms with respect to wind strength and direc-tion, vegetative cover, and sand source, butthe approach has been descriptive and non-quantitative. To R. A. Bagnold (1941) we owea great advance in the application of principlesof aerodynamics to the study of the movementof sand and the development of drift and duneforms. In a single publication Bagnold has laidthe foundations for quantitative and dynamicstudies of the erosional, transportational, anddepositional work of wind.

MOLECULAR STRESS PHENOMENA

Table IB outlines the molecular stress phe-nomena, more conventionally termed the"weathering processes." Molecular stress isprincipally stress set up by changes of tempera-ture or physical-chemical changes. The impor-tant point is that the direction in which thestress acts is independent of gravity and maybe distributed in a random manner throughoutthe rock or soil. Because the source of energyis solar in most of the phenomena listed, theprocesses are limited to surface or near-surfacelocations.

1. Direct thermal stresses set up by heatingand cooling of rocks. Although all crystallinesolids are affected by temperature changes, itis assumed here that unconsolidated granularmaterials are merely agitated with respect toone another and that actual rupture to formnew surfaces of breakage takes place only inhard, crystalline rocks in which mineral grainsare tightly knitted by cementation or originalbonding of crystal intergrowth. The material is

therefore described as an elastic crystallinesolid.

Shear stress by temperature changes can bedivided into two categories: (a) Nonuniformexpansion and contraction by heating and cool-ing gives rise to shear stresses between adjacentgrains of unlike orientation or physical proper-ties. Assuming for the present that the stressesare capable of exceeding the elastic limits ofthe rock, some granular disintegration, espe-cially of the coarse-grained crystalline rocks,may be attributed to this cause, (b) As a resultof cyclic temperature changes at the surface,whether daily, cyclonic, or seasonal, thermalgradients are set up in the rock. Range oftemperature at depth for a given heating andcooling cycle at the surface is a negative ex-ponential function of depth. Where surfacechanges are great and of short period, con-siderable shear stresses may be set up in therock, tending to cause rupture in planes ap-proximately parallel with the surface. Exfolia-tion of hard, fresh rock surfaces is known tooccur during intense heating by forest andbrush fires and it is possible that other causesof exfoliation may be supplemented by normalsolar and atmospheric heating and cooling.Griggs (1936) has applied principles of experi-mental physics to this problem.

2. Stresses developed during the growth ofice crystals in interstices of rock or of ice lensesand wedges in soil. Open systems are assumedin which atmospheric confining pressures aremaintained during crystal growth. Ice crystalsof needle-like form, their axes perpendicular tothe walls of the opening, exert pressure as watermolecules derived from water films are fittedinto place at the ends of the crystals. (SeeTaber, 1930; Grawe, 1936.) Bedrock may beshattered by growth of crystals between grains,cleavage fragments, joint blocks, or beddingsurfaces. Soil is heaved upward by the growthof ice needles or lenses parallel with the groundsurface. The geomorphic effects of rupture ofhard rocks by freezing water are especially con-spicuous above timber line on mountain sum-mits where talus, felsenmeer, or rock glaciersabound.

3. Stresses exerted by growing salt crystals.This process is essentially the same mechani-cally as the growth of ice crystals but proceeds



MOLECULAR STRESS PHENOMENA 933

under quite different hydrologic conditions. Inarid climates and climates with a long, hot dryseason, excessive surface evaporation results inthe surfaceward capillary movement of soil andground water. Upon evaporation, salts, usuallycarbonates and sulphates, crystallize in theinterstices of the soil or rock, exerting pressuresupon confining walls. Efflorescence of buildingstones and brick, granular disintegration ofsandstones, and growth of caliche layers andnodules in dry climates are expressions of thisprocess.

4. Shear stresses set up by the adsorptionof water by colloids in the rock and soil. Alter-nating periods of drought and rainfall will causelosses and increases of soil moisture, which inturn cause shrinkage or swelling of clay soilsor argillaceous sedimentary rocks. Rupture mayoccur between grains of the rock or soil, andthere is the further possibility of exfoliation ofrock shells where moisture penetrates a surfacelayer, causing it to develop shear stresses alongsurfaces parallel with the outer surface. Shalesand sandstones, as well as igneous and meta-morphic rocks, might suffer exfoliation fromthis cause. In the case of rocks containing feld-spars and ferromagnesian minerals, the mineralalteration products, being hydrous clay miner-als, would be subject to swelling and contrac-tion, whereas unaltered rock would not. Thephenomenon of slaking of shales and bentonitesupon exposure is a manifestation of water ad-sorption.

5. Shear stress set up by contraction of capil-lary water films between grains of a soil orgranular rock. Where discrete capillary filmsoccupy the contact areas of adjacent grains, ashrinkage of the film through evaporation in-creases the capillary tension of the curved filmsurface. This in turn causes the grains to bepressed more tightly together and, as a result,the entire mass tends to shrink and to increasein density and strength (Burmister, 1948, p.80-81). The stresses thus set up may causeshrinkage cracks to form in fine-grained soilsin the dry state. In a sandstone, the tensionmight be sufficient to rupture the rock. Wherea mud film on a rock surface dries out, theshrinkage might loosen fragments from therock surface to which the mud has tightly ad-hered.

6. Shear stress set up by the growth of plantrootlets in soil or rock. Rupture between grains,joint blocks, cleavage fragments, or beddinglayers may result from osmotic pressures ofroot growth.

7. Expansion of rock upon release of confin-ing pressure, as a result of preexisting elasticstrain within the rock. Whether the strain isof diastrophic origin, a relict of mountain-building processes, or a result of solidificationof magma under high confining pressures is notfully decided, but the phenomenon of rockburstin quarries, tunnels, and mine shafts is wellknown and the expansion of slabs upon releasefrom confining rock walls is measurable (Bain,1931). Large exfoliation domes, as well as wide-spread development in igneous and metamor-phic rocks of exfoliation planes parallel withthe hill slopes, have been attributed to expan-sion accompanying release from confining pres-sure, or unloading, by erosional removal.

SURFICIAL CREEP PHENOMENA

Not listed in the table but clearly a resultof simultaneous action of both gravitationaland molecular stresses is the slow surficial creepof soil, weathered rock, or weak bedrock downa slope. Sharpe (1938) discussed the mechanismsof soil creep, but the fundamental law of creepwas stated much earlier by Gilbert (1909, p.345) who wrote: "Whatever disturbs the ar-rangement of particles, permitting any motionamong them . . . promotes flow, because gravityis a factor in the rearrangement and its tendencyis down the slope." All types of rupture or inter-granular shear of rock or soil treated under themolecular stresses are forms of disturbance ofsurficial materials in which the directions ofstress are either oriented in a random mannerthroughout the material, or oriented with re-spect to the rock surface and independent ofgravity. Unless there is a slope to the groundsurface, no systematic aggregate movement inone direction will occur, but should a slopeexist, however faint, a component of gravita-tional stress will be added to the molecularstresses in the down-hill direction, subtractedin the up-hill direction. Downslope movementis thus cumulative throughout countless minutemovements among the grains of the soil. Sur-




ficial soil creep differs from continuous creep,in which flowage deformation progresses be-cause of the plastic nature of the rock. Surficialcreep can occur in dry, nonplastic materialswith a high degree of internal friction, whichwould not otherwise flow under gravitationalstress alone.

The related phenomenon of downslope creepby rain-drop impact may be noted in connec-tion with surficial creep of soil. Heavy raindropsstriking a bare soil surface produce cratersfrom which soil particles are thrown into theair (Ellison, 1950). With perpendicular fall ofdrops on a horizontal surface, no aggregatemovement of soil in any one direction wouldbe expected, but, where a slope exists, thegravitational stress differential comes into play.The trajectories of particles aimed upslope areshorter; those aimed downslope are longer andimpart downslope movement to the particlesstruck. A downslope transfer of particles is thusaffected. Unlike the other creep mechanisms,however, gravitational stress on the falling drop,rather than molecular stresses, disturbs theparticles and drives this process.

CHEMICAL PROCESSES

Chemical processes (Table 1C) are set apartbecause they do not directly produce shearstresses, yet are of great importance in land-form development. Distinction is made be-tween (1) chemical reaction in which acids ionsin soil water or surface water react with mineralsurfaces and (2) simple solution or ionizationof unusually soluble minerals, such as halite.Removal of the ions in the circulating groundwater and in surface runoff constitutes a formof mass reduction that might otherwise be im-possible because of resistance of the hard crys-talline or glassy minerals to corrosion or toother forms of mechanical reduction.

Chemical reaction or simple solution causesloosening of cementing bonds between grainsof a sedimentary rock and selective removal ofmineral grains in igneous and metamorphicrocks, thereby weakening the rock and reduc-ing its resistance to both gravitational andmolecular shear stresses. Minor forms such aspitting and grooving of rock surfaces are usuallydescribed in connection with dissolution ofrocks, but the general lowering of land surfacesby removal of mass throughout the soil and

weathered rock layer may be of vastly greatersignificance. In tropical rainy climates, wherelarge quantities of water percolate through thesoil and temperatures are most favorable tochemical reaction and solution, the major shareof landmass denudation might conceivably becarried out by solution-removal processes. Lowslopes or absence of surface slope would notprevent this form of denudation. (See discus-sion of the flow of water downward throughpermeable materials, Table 1A, No. 7.) Thekarst landscape features, typified by disruptionof normal surface drainage systems, would beassigned to the chemical processes in combina-tion with subsurface percolating or channel flow,

TECTONIC AND VOLCANIC STRESSES

Although landforms of various types are pro-duced directly by tectonic stresses or by vol-canic extrusion, these processes have not beenincluded in this discussion, which takes as itsscope the processes of weathering, mass wast-ing, erosion, transportation, and depositionpowered or triggered by solar energy. Whereasthese exogene or external processes affect onlya thin surficial zone of the landmasses, the tec-tonic and volcanic processes are deep seated;they involve structural and petrographicchanges of enormous masses, along with thetransformation of enormous amounts of energywhose source we may regard as internal. Suchlandforms as fault scarps, initial domes or anti-clines, volcanoes, and lava flows are superficialfeatures in terms of the deep-seated changes ofstate which they accompany. A complete treat-ment of stresses and behavior of these materialsat depth would require inclusion of all of struc-tural geology, petrology, and geophysics intogeomorphology. Moreover, initial landformsproduced by these deep-seated processes areeasily identified and understood as landforms,hence cause no special concern to the studentof geomorphology. He considers crustal changesas changes of potential energy of mass withrespect to base level, the quantitative valuesfor which are readily calculable by multiplyingmass times elevation.

DYNAMIC OPEN SYSTEMS AND THESTEADY STATE

Geomorphology will achieve its fullest de-velopment only when the forms and processes



DYNAMIC OPEN SYSTEMS AND STEADY STATE

are related in terms of dynamic systems andthe transformations of mass and energy areconsidered as functions of time (von Ber-talanffy, 1950a; 19SOb). Walling himself off insacrosanct confines of his geological societiesand journals, the geomorphologist has paidlittle attention to the development of thermo-dynamic principles and their steady infiltrationfrom pure physics and chemistry into sciencesof biology, economics, psychology, and politicalscience. True, the Davisian concept of cyclerelated changes of form to changes of time, butthe treatment is not based upon mathematicallaw, and no thought is given to energy rela-tionships, despite the inviting opportunities forsuch treatment.

Many of the geomorphic processes operatein clearly defined systems that can be isolatedfor analysis. A drainage system—whether ofwater or ice—-within the geographical confinesof a watershed represents such a dynamic sys-tem. A cross-sectional belt of unit width acrossa shore line or sand dune, or down a givenslope from divide to stream channel, wouldconstitute another, more limited, type of dy-namic system.

Two major types of thermodynamic systemsmay be recognized (von Bertalanffy, 1950a):(1) the dosed system which has a clearly de-fined boundary through which neither materialsnor energy are exchanged, and (2) the opensystem which exchanges either material orenergy (or both) with outside environments.The closed system tends to establish an equil-ibrium in which entropy attains the maximum,available free energy the minimum. An examplemay be found in the state of water vapor in theair standing above a water surface in a sealedjar. If no heat flows into or out of the jar, thewater vapor will attain a certain concentration,maintained without further change of tempera-ture or pressure. The only activity in this sys-tem will be exchange of a few molecules betweenthe gas and water. Such a system obviouslydoes not describe a stream or glacier wheremotion continues with time and material iscontinually entering and leaving the system.

Form and composition of the open systemdepend upon the continuous import and exportof materials and energy. Normally a time-inde-pendent steady state is achieved in which theform remains unchanged but the activity con-

tinues. In the case of a segment of a stream inuniform flow, a steady state ensues when theenergy developed through the descent of thewater is entirely dissipated in overcoming re-sistance to shear within the fluid and againstthe channel boundary and to the movement ofbed load. The discharge is constant throughoutthis stretch of stream, and there is no accelera-tion of motion except what is accountable tochanges of channel form, roughness, or slope.In this steady state, the form of the stream isunchanging with time, but should the importof water be cut off the form will be destroyedat once. Open systems require energy from out-side to maintain a steady state; equilibrium ofthe closed system requires none.

Open systems such as streams or glaciers, orcells of living matter, are able to adjust in-ternally to changes in supplies of material orenergy from outside. The open system is, inother words, a self-regulating mechanism (vonBertalanffy, 1950a). When a stream is graded,it is in a steady state. If the bed-load supply isreduced or increased, the stream changes itsslope so as to readjust to a new steady state.A shore segment whose beach possesses the"profile of equilibrium" is in a steady statewith respect to the energy supplied by breakingwaves. When the wave characteristics change,the beach profile is altered in slope until a newprofile, independent of time, is established.

MATHEMATICAL MODELS IN GEOMORPHOLOGY

In attempting to quantify his statements ofgeomorphic process and form, the geomorpholo-gist has, in general, two types of mathematicalprocedure open to him, both fruitful. He may,by statistical analysis of experimental andsample field data, derive empirical equationsthat best state the observed interrelationshipsbetween two variables. In its first form, thissort of empirical equation states the degree ofcorrelation between two form elements, both ofwhich may be products of a third and unknownindependent variable. For example one mightrelate drainage density to length of overlandflow in a statistical correlation. Although nei-ther quality of terrain is a cause of the other,the degree of correlation is very close becauseboth qualities are controlled by a third, inde-pendent factor. Aside from predicting the mag-




nitude of one form element when the other isknown, this mathematical statement is oflimited value for it does not improve under-standing of the genesis of the landform.

In a higher form, the empirical equation maydescribe a regression in which the independentvariable is a force, or time itself, whereas thedependent variable is a landform element. Forexample, if drainage density were plottedagainst surface resistivity, a close but inverserelationship would be found. Here resistivity,or resistive force, is a cause; drainage densityan effect. Analysis of this type is found ingeneral engineering practice; it is, in fact, theonly way in which carefully observed sets ofvalues can be impartially and objectively re-lated.

As a second general procedure, the geo-morphologist may formulate, through a type ofinvention or intuition based upon the sumtotal of his experience, a relatively simplemathematical model (Rafferty, 1950) which isa quantitative statement of some point ofimportant general theory otherwise definableonly in words, qualitatively. The establishmentof such mathematical models may be regardedas the highest form of scientific achievementbecause the models are precise statements offundamental truths. The two methods—empirical and rational—would tend to convergeas time goes on and the fund of informationgrows. The statistical analyst cannot hope toderive quantitative relationships of general ap-plication from small samples because of theirinherent variability, but, as his sample dataincrease and the influences of variables areisolated, his empirical equations tend to ap-proach the status of general laws. New knowl-edge of the observed influences of variables inturn results in keener deduction on the part ofthe analyst who is formulating his generalmathematical laws by intuitive, deductive men-tal processes.

To illustrate the formulation of mathematicalmodels in geomorphology, a specific example isoffered on the subject of the longitudinal profileof the graded stream. Mr. Samuel Katz of theLamont Geological Observatory of ColumbiaUniversity has very kindly worked out thefollowing steps at the suggestion of the author:

We are interested in deriving a time relation-

ship between the elevation of a given point ona graded stream and the horizontal distance ofthe point from the head of the stream, makingonly the minimum number of a priori assump-tions. If y is the elevation, * the horizontaldistance measured from the head of the stream,and t the time, we shall take y to be a functionof the two independent variables x and ;, andshall look for a quantitative expression of thisfunction.

From an analysis of many graded streamprofiles (Shulits, 1941; Krumbein, 1937), wefeel confident that the relationship between theelevation y and the horizontal distance x of agraded stream is given at a particular time /oby an exponential function of the form

(1) y, = Ate~

(The constant A is determined by the value ofy when x = 0, and the constant k\ by the valueof y when x = 1/k.)

The overall reduction of relief and valley-wall slope steepness will cause a steady re-duction of load and a diminishing supply ofpotential energy in a drainage system. Conse-quently, the regrading of the master streambecomes increasingly slower. We thereforepostulate that, at any given time, the rate atwhich the stream profile is lowered at a givenpoint is proportional to the slope at that point.Expressed analytically,

From the basic definition of the differentialof a function of two variables y(x, t),

(3) = dx+ Mdt.\«A

Substituting (2) into (3), we eliminate thepartial derivative with respect to the time andobtain

(4) dy = (dx+ kidt)[ —

Equation (4) holds for all values of t, in par-ticular for t = t0. Differentiating (1),

(5)>* /«



MATHEMATICAL MODELS IN GEOMORPHOLOGY 937

Substituting (5) in (4),

dy = (dx + ksdt)(-kiy) or

(6) Q = -M*- hktdt.y

Integrating and using the initial conditionsthat for x = 0 (the head of the stream) and

TimeFIGURE 2.—STREAM PROFILE AS A FUNCTION OF

DISTANCE AND TIMEA. Profile at a given time, /.B. Variation in elevation at a point, x, with

time.

t = /„, y = An (the elevation at the head of thestream), we obtain

In y — In AO = — k\x — kikit or

(7) y = 4

Equation (7) gives the longitudinal profile ofthe graded stream both as a function of thetime and of the distance from stream head. Ithas been obtained with no explicit assumptionabout the variation of slope profile with time.

Figure 2A is a schematic plot of equation (7)for a given value of t = to, and thus shows thestream profile at a given moment. Figure 2Bis a similar plot for a given value of x = *0,and shows the reduction of elevation with timeat a point (x<s) on the profile. The exponentialrelationship of distance to elevation is readilychecked by existing profile data, but the rateof reduction of elevation with time will beextremely difficult to verify with field evidenceextending back into geologic time. Perhapscontrolled model experiments would be requiredin the latter case.

In summary, the proposed program for futuredevelopment of geomorphology on a dynamic-quantitative basis requires the following steps:(1) study of geomorphic processes and land-forms as various kinds of responses to gravita-tional and molecular shear stresses acting uponmaterials behaving characteristically as elasticor plastic solids, or viscous fluids; (2) quantita-tive determinations of landform characteristicsand causative factors; (3) formulation of em-pirical equations by methods of mathematicalstatistics, (4) building of the concept of opendynamic systems and steady states for allphases of geomorphic processes, and finally (5)the deduction of general mathematical modelsto serve as quantitative natural laws. Theprogram is vast and qualified investigators few,but we are already a half-century behind ifdevelopment is to be measured againstchemistry, physics, and the biological sciences.The need for rapid dynamic-quantitative ad-vances is, therefore, all the more pressing.

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DEPARTMENT or GEOLOGY, COLUMBIA UNIVERSITY,NEW YORK, N. Y.

MANUSCRIPT RECEIVED BY THE SECRETARY or THESOCIETY, MARCH 18, 1952



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Strahler 1952 Dynamic Basis of Geomorphology Geology