Scientific Basis of Geomorphology

Science

• human creation and thus what qualifies as science changes over time and differs between cultures
• the process of rigorous, systematic discovery; a powerful system for ordering constructs and concepts
  
  

  constructs

  facts, empirical content, specific, used to define or categorize experiences
  
  

  concepts

  lack empirical content; permit manipulation of constructs; ordered (hierarchical) starting with megaconcepts of number and relationships which are the basis of math and logic

  
  
• other ordering systems: theology, esthetic (art), and common sense (intuition)
• scientific thought didn't exist until after the renaissance and the protestant reformation, because previously there were just beliefs about the natural world
• very much an artifact of western culture and commonly expressed in the language of mathematics

Characteristics of a scientific discipline

1. philosophy
   • set of beliefs; cannot be invalidated by logical arguments since it is believed not necessarily logical
   • source of objectives (e.g. geomorphologists study form, or mechanics of processes, or contemporary landscapes)
   
   
2. methodology
   • internal logic or approach to research, stems from theory and influences the way that theory is developed
   • must be logical (internally consistent)
   
   
3. theory
   • a system of abstract statements (laws)
   • laws only exist as a network (theory)
   • in the empirical sciences, theory must be related to observation
   • the domain of theory must be defined
   • statements about the natural world exist on a continuum from fact-like to law-like
   
   
4. techniques
   • tools that permit the examination of theory and thus advance the discipline towards a stated goal

1. fieldwork
   • observation of form, location, internal character, rates of processes
   • including field experiments
     • "an operation designed to discover and establish some principle or natural effect
     • e.g. draining a periglacial lake, rainfall simulation; experimental control either by human interference or statistical methods
   • thus most geomorphological studies are not controlled experiments
   
   
2. lab work
   • experiments: flumes, wind tunnels, hardware models
   • determining physical and chemical properties (e.g. particle size analysis)
   • examining changes in physical and chemical properties that underlie mechanics of processes (e.g. shear tests)
   
   
3. remote sensing and GIS
   • overlap with older techniques of map and air photo analysis
   • now involve use of computers to collect more data than can be collected or processed by other means
   
   
4. quantitative techniques
   • stochastic/statistical: e.g., correlation and regression to establish relationships between process and response (form)
   • deterministic/analytical: specifying relationships from existing theory and deduction of consequences that must result
   • the theory of important geomorphic relationships is poorly understood, thus many assumptions made and statistical data often used

The nature of scientific activity

• three stages
  • the process of discovery
  • generating support for one's conclusions, a social process
  • disseminating results in a format that meets commonly held standards, a social process;
  
  
• most research results are presented in a manner to prove or verify a conclusion
• most of the scientists in a discipline operate under an accepted paradigm (governing code) for lengthy periods until enough dissatisfaction or unanswered questions buildup and then a new paradigm is fairly quickly adopted (scientific revolution)

Scientific reasoning

• inductive
  • facts exist independent of theory, generalization from a body of unordered facts
  • but how is a fact identified and defined in the absence of theory?
  • there must be some preconceived notions that control the ordering and interpretation of facts; existing theory suggests which data are worth recording
  • also, there are no underlying principles to establish an unvarying response, another set of observations could produce different results
  
  
• deductive
  • postulating a new idea (hypothesis) based on examination of existing theory
  • a hypothesis must have the potential to become a law: i.e. it must not be trivial (it must be a substantive question) and stated in a manner that permits falsification
  • however, how can any hypothesis survive falsification under any circumstances?
  • also, theory itself cannot grow and knowledge cannot increase without new observations

Scientific versus non-scientific

• all inquiry must have a theoretical perspective that is identified once that scientist has an objective (philosophy), this perspective is a major part of and governs the methodology
• facts or observations are not important ingredients of science unless they point to relationships or ideas that are combined to form theory (e.g. natural history versus ecology of physical geography)
• to a critical rationalist, that which is falsifiable is scientific
• however in reality good science is based on good questions which can be derived from theory (deductive) or observation (inductive), that is, the method of hypothesis formulation is irrelevant to the status of the hypothesis as scientific or non-scientific
• scientific research is based on a combination of inductive and deductive reasoning; for the reasons given above, neither can advance theory in isolation
• facts and observations are required to test hypothesis derived from theory and thus new theory requires new observations, conversely inductive reasoning produces hypotheses that apply only to a set of observation, but it suggests relationships (laws) which when derived from theory have much broader scope

Relevant Laws And Principles Of Physics

• only parts of the body of theory of physics and chemistry is directly applicable to geomorphology
• for example, whereas quantum mechanics is fundamental to an understanding of the atomic nature of matter, the behavior of matter at human scales involves the use of classical or Newtonian physics

Newtonian physics: work, power, energy, force, stress and strain

force

• moving a mass with a particular acceleration, in SI units one newton (N) is the force required to give one kg and acceleration of one m s^-2 (F = ma)
• when the force is gravity, a = g = 9.81 m ^s-2
• all motion is the result of forces; in nature several forces are involved at one time; when there is no movement the sum of the force vectors is zero
  
  
• Newton's third law
  
  
  • movement occurs when a resultant force (Fs = mg sin ß) exceeds the resisting force acting in the opposite direction
    
    
  • friction (Ff), the resistance to force over a surface is mostly a function of the normal force (Fn); just prior to motion, Fs = Ff
  • coefficient of static friction = Fs/Fn = mg sin ß/ mg cos ß = tan ß
  • once the static friction has been exceeded by the shear force, movement will continue over lower slope angles, because the necessary force over the surface and thus the coefficient of dynamic friction (Fk = tan ßk) are less
  • in hillslope studies, static friction is of greatest interest
  • in studies of erosion by wind and water, dynamic friction is more relevant, because the geomorphic agent is always moving (forces vary a lot)
    
    
  • viscosity
    • the internal friction in a fluid
    • in a river maximum viscosity occurs where there is the largest gradient in velocity (i.e. between the center and the banks and bed)
      
      

      molecular viscosity

      in laminar flow from the interaction between water molecules in adjacent layers
      
      

      eddy viscosity

      in turbulent flow from mixing of molecules from different parts of the flow

      
      
    • Reynolds number
      • a dimensionless parameter to distinguish between laminar (low values) and turbulent flow (high values)
      • Re = velocity X depth / kinematic viscosity
      • kinematic viscosity (the amount of interference per unit time between adjacent layers in the flow) = dynamic or absolute viscosity / density
      • units of absolute viscosity = force X time /area (e.g. newton-seconds/m²), increases with decreasing temperature
  
  
• moment
  • the product of a force and the vertical distance from the axis of rotation
  • the axis of rotation of a pendulum is the bottom of the swing, thus the velocity of the weight is proportional to the displacement from the axis since the gravitational force is increased and so is the vertical distance from the axis of rotation (e.g. a rotational landslide)
  
  
• buoyancy
  • an upward force equal to the force resulting from the displaced fluid (Archimedes' principle)
  • e.g., a 10 kg stone with a density of 2650 kg m^-3 has a volume of 3.77 m^-3; this much water has a mass which is 37.7% of the stones, so the stone will have an apparent mass of 6.23 kg in the stream

stress

• force over an area, more exactly area is the (cross-sectional) area within a solid or liquid over which the forces is acting
• normal stress acts perpendicular to three principal planes within a mass, along the principal stress axes: major, intermediate and minor, in terms of magnitude; it can be either tensile or compressive, depending on whether the force vectors are directed away or towards one another
• shearing or shear stress is perpendicular to normal stress, in that it leads to slippage along a fracture surface
• tensile and especially compressive forces often create both normal and shear stresses perpendicular and along an inclined fracture surface
• peak shear stress occurs at point of failure, whereas the ultimate or residual shear stress is a lower constant value that exists after a solid has failed
  
  
  • this difference is analogous to the difference between static and dynamic friction
  • once cohesion and the interlocking of particles along a sliding plane has been overcome, less shear stress is required to cause further failure
  • an important factor is the reorientation of particles along the shear plane

strength (S)

• resistance to, and same units (kg/m²) as, stress
• directly proportional to the normal stress (Fn)
• also a function of friction (Ø) among particles (shape, size, compaction), cohesion (c; interparticle attraction) and porewater pressure (p)
• S = c + (Fn - p) tan Ø    (Coulomb equation)
• different angles of shearing resistance (internal friction) correspond to peak and residual shear stresses
• positive porewater pressure under saturated conditions (e.g. at water table) reduces the applied stress to the effective normal stress

strain

• relative changes in dimensions or shape in response to stress, e.g., linear strain = change in length/ initial length
• strain generally is resolved like normal stress to 3 independent components
• relationship between normal (compressive or tensile) stress and strain rate defines three modes of solid behavior
  
  
  • elastic: fully recoverable, strain increases in a linear manner with stress
  • plastic: no deformation until yield limit
  • brittle: little or no deformation until fracture
  
  
• viscous fluids have no upper limit to deformation, there is continuous strain as a function of stress
• most natural materials exhibit a combination of behavoirs, e.g. visco-plastic, elasto-plasitc

work

• whereas stress is force over and area, work is force through a distance, i.e. the product of the displacement of a mass and the component of the force in the direction of movement (e.g. the shearing force generated by the force of gravity, or forces applied by turbulent eddies of air or water)
• since the magnitude of force can vary over the distance, work can be expressed as the integration of forces (dW) over x1-x2, where dW = Fdx, and F can be a component expressed as a trigonometric function (e.g. FcosA)
• the unit of work is a joule, the work done when a force of one N acts over a distance of one meter

power

• rate of doing work, one watt = one joule/ one sec
• a small force over a long period of time can produce the same result as a large force over a short amount of time, but the latter requires more power (work per unit time)

energy

• the ability or capacity to do work in units of joules
• all processes in the physical environment require energy and involve transformations of energy
• potential energy = mgh
  • the acceleration of mass from a height represents work, where the force is gravity (and thus acceleration is g)
• kinetic energy = 1/2mv²
  • work is performed in bringing a moving body to rest (deceleration)
• e.g. energy along the length of a stream
  • the total energy of a unit volume of water is the sum of potential and kinetic
  • the proportion changes from all potential to all kinetic with decreasing elevation and increasing velocity
  • the longitudinal, cross-sectional and planimetric geometries of a stream channel respond to these changes in energy

Distribution of energy: the laws of thermodynamics

• all processes require energy and involve transformations of energy
• all systems tend towards a state of equilibrium with specific distributions of energy and conditions (e.g. temperature, pressure) called state variables and forms (e.g. soil and hillslope profiles, climax vegetation)
• these characterize the equilibrium state and do not reflect the previous history of the system
• the theoretical basis for the transfer and distribution of energy are the laws of thermodynamics:

First law of thermodynamics

• energy is neither created nor destroyed by conserved
• input = output - changes in storage or, in terms of thermodynamics:
• energy absorbed = work performed + change in internal energy
• this law is the basis for continuity equations that express the balances of energy or mass
• e.g. hydrological balance, glacier mass balance, soil budget on a hillslope to express changes in surface elevation and soil thickness in terms of in inputs and outputs
• when processes (creep, slope wash) are expressed in terms of slope angle and downslope distance, the continuity equation can be used to simulate slope evolution

Second law of thermodynamics

• energy cannot be completely converted into work
• systems move towards equilibrium, a state of lower free energy (energy that can be used to do work)
• no process occurs spontaneously unless there is a degradation of energy from a concentrated from to a dispersed form (i.e. an increase in entropy)

entropy

• a measure of the degree of disorder or randomness within a system
• the entropy of a perfectly ordered system is zero
• energy can be created but not destroyed, thus as the result of natural processes the amount of entropy in the world is continuously increasing
• conversely the amount of energy available for work is continuously decreasing
• entropy is a logarithmic function of the possible configurations of a system (each configuration has a different amount of total potential energy)
• energy is required to order a system (e.g. cultural landscapes)
• e.g. (from Strahler & Strahler, 1974)
  
  
  • oranges in pyramid, slight disturbance causes a create change from a highly ordered state to maximum disorder, there is maximum total mechanical energy equal to the sum of the potential energies of all the oranges, with the oranges scattered across the floor the total mechanical energy drops to zero and entropy is maximized
  • energy has been lost overcoming friction between the oranges
  • the direction of change is dictated by the tendency of the system to achieve a lower energy state and a more random configuration
  • the process cannot be reversed by disturbing one of the oranges lying on the floor
  
  
• this is the statistical nature of entropy, from the perspective of thermodynamics, entropy is the ratio of heat to temperature, thus the units of entropy are joules per Kelvin degree, e.g. when one kg of ice melts ar 0^o C, the increase in entropy is
• 333.7 kJ/ 273.15 K = 1.22 kJ K^-1

Applications of thermodynamics to geomorphology

• the laws of thermodynamics are the basis for the idea that geomorphic systems evolve to an equilibrium where there is minimum free energy and thus stable conditions
• e.g., hydraulic geometry
• the longitudinal profile is predicted by maximum entropy, as mechanical energy decrease with decreasing elevation and increasing distance downstream
• velocity and thus kinetic energy increase relatively slowly and are transformed as work is performed on the stream bed and in transporting sediment
• entropy is maximized when a graded profile is achieved
• the most probable configuration of energy loss is a uniform increase in entropy (and thus uniform expenditure of energy), this will occur when a stream has an exponential longitudinal form and discontinuities in energy loss are minimized
• thus waterfalls represent disequilibrium and meanders represent adjustment towards a smooth energy gradient

law of average stream fall

under dynamic equilibrium, the ratio of the average fall between any two different orders of streams in the same basin is unity

law of least rate energy expenditure

during the evolution towards its equilibrium condition, a natural stream chooses its course of flow so that potential energy expenditure per unit mass of water along this course is a minimum

The Mathematical Basis of Geomorphology

• mathematics enables unambiguous expression of relationships; then predictions can be made with known degrees of accuracy and statements of relationships can be translated into scientific laws
• mathematics is a logical and consistent language.
• the building blocks of math
  • the numbers used in arithmetic
  • the English and Greek letters used in algebra
  • the spatial measures (points, lines, planes and solids) used in geometry

Applications to Geomorphology

1. Arithmetic
   
   
   • measurement and reduction of data
   
   
2. Geo-Trig
   
   
   • trigonometry: ratios between sides of triangles (triginon (Gr.): triangle).
     • the gravitational stresses on earth materials
     • location by distances and angles
     • reducing slope distance to rise and run
   
   
3. Algebra
   
   
   • using symbols to derive and express functions

   linear functions

   • y = ax + b or mx + k
   • lapse rates
     • a = change in temperature with altitude
     • b = T_o (initial temperature)
     • T = az + T_o
     • DALR = 9.8^o/100 m , thus T = -9.8z + T_o
     
     
   • Coulomb equation
   • continuity equation (budgets or balance)
     • inputs = outputs +/- changes in storage

   non-linear functions

   • most relationships in physical geography are non-linear, some of these can be approximated by linear functions
   
   
   1. power functions: y = xⁿ
      • y = x² , a parabolic function describes the cross-sectional form of a glaciated valley (Graf, '76)
      • allometric growth (versus isometric or constant change)
        • change as a function of form and size (scale dependence)
        • e.g. stream bed slope is a power function (increasingly decreases) of drainage area, whereas meander wavelength is a linear function of channel width irrespective of drainage area (i.e., it is isometric)
      
      
   2. logarithmic functions: y = log_ax
      • fall sorting on talus slopes: partcile size = log (distance from top of slope)
      • the phi-scale of particle size: PHI = -3.32 log D
      • negative log makes phi positive for small (i.e., common) particle sizes
        • the range of diameters (D) in each particle size class gets increasingly larger with increasing particle size, but the more important particle sizes are the smaller ones
        • logarithmic scales suppress the range of large values (e.g., high stream discharges), enabling the display of smaller values (particle sizes or low flows)
        • the following table gives the range of diameters (mm) and phi values for four particle sizes
        
        
        

        size class	D	phi
        boulder	> 256 mm	< - 8.0
        pebble	64-4 mm	- 6.0 to -2.0
        medium sand	0.5 - 0.25 mm	1.0 to 2.0
        medium clay	0.00195-0.00098 mm	9.0 to 10.0

      
      
   3. exponential functions: y = e^x, where e is the natural logarithm (2.71828...)
      • describes rates of decay of radioactive isotopes, growth of stream networks, decline of hillslopes, longitudinal profiles of graded streams
      
      
   4. trigonometic: y = sinx, y = cosx, y = tanx or y = sin^-1x (inverse; y is the angle)
      • gravitational forces
      • rise = d sin(slope angle), run = d cos(slope angle), where d is the surface distance
      
      
   5. polynomial equations: e.g, y = ax² + bx + c is a quadratic polynomial
      • fitting curves to landscapes; the number of inflections in the curve is the largest exponent minus one (e.g. a cubic polynomial is the equation of a curve with two changes in direction)
   
   
4. Calculus: solution of non-linear functions

   integration (integral calculus)

   • the cumulative effect of one parameter upon another with the integration of a continuous function, e.g. the function y = 1/x2 has a finite value or limit as x approaches infinity
   • area under curves: topographic cross-sections or theoretical frequency distributions, e.g., the hypsometric integral, the area under an elevation frequency distributions is characteristic for a given landscape
   • volumes under earth surfaces
   • trend surface analysis: the summation or integration of a series of periodic functions to characterize the trends in a landscape; based on the concept that as dx approaches zero the sum of the curves converges on a straight line
   • cumulative denudation: integration of a function that expresses erosion or mass wasting processes (integration over time)
   
   

   differentiation (differential calculus)

   • deriving, from a non-linear continuous function, the rate of change of one parameter with another
   • the first derivative of elevation (dz/dx) is slope, the second derivative is slope curvature; the second derivative of horizontal distance (dy/dx) is contour or plan curvature.
   • the first derivative of velocity is acceleration; thus there are differential equations for the acceleration (falling or transport) of earth materials.
   • solving for dy/dx = 0, i.e., maxima or minima
     • stationary points: changes in the rate or direction of a function, e.g., rates of erosion
     • the highest and lowest points (elevations) in a curve representing a landscape
   
   
5. Probability and statistical relationships
   
   
   • a true mathematical function is exact and unambiguous; it can be arrived at theoretically with absolute precision
   • however, most relationships in environmental sciences (geography being a good example) are arrived at by inductive (observation and measurement) rather than deductive reasoning
   • thus there are geography courses in statistical methods but not in other branches of mathematics
   • rarely is it possible to establish a unique relationship between two geomorphic variables
   • there is usually too much noise: variance in the dependent variable caused by other parameters and by inaccuracy and inadequacy of measurement
   • also it is not possible to measure all occurrences and conditions of two variables (stream discharge, soil texture, slope gradients, etc.) and, therefore, empirical relationships are established from samples of observations or statistics
     
     
   • frequency distribution of elevation: classes of equal size plotted on a graph of frequency versus elevation
     • the shape can be described in terms of deviations from a normal distribution: skewness, kurtosis, unimodal versus multimodal
     • a normal distribution would correspond to a landscape where intermediate elevations are most common and the higher or lower an elevation, the less common (probable) it is, so there are no broad plateaus or valley bottoms
     • the normal distribution is an example of a theoretical frequency distribution; its characteristics are defined for an infinite populations of items or events (mode = mean = median, 2/3 of population within one standard deviation, 95% of population within two standard deviations, and so on)
     
     
   • most natural phenomena in physical geography (e.g. rates of soil movement, particle size of weathered debris, flood magnitudes) conform to a log-normal distribution: right-skewed such that logarithms of the original data are normally distributed (thus often the transformed data can be subjected to parametric statistical methods, also then we are faced with explaining a relationship between logarithms)
   • right-skewness is common because there is an absolute lower limit of zero for the magnitude or frequency of natural phenomena, but the upper limit (rare catastrophic events) is governed only by the physics and chemistry of the earth and its atmosphere
   • when a phenomenon is known to conform to a theoretical frequency distribution, the characteristics of the distribution represent a theoretical framework for the study of that phenomenon

   empirically-derived relationships

   • correlation
     • variables co-vary either directly (+ve r) or inversely (-ve r), but cause and effect are not implied
     • statistical significance of the correlation depends on the degrees of freedom (n - 2), because as the sample size (n) increases, obtaining a high correlation by chance (randomly) becomes less probable
     
     
   • regression
     • statistically derived functions:the equation of a curve (line) that best fits a scatter of points
     • the terms are given by the basic shape of the curve, while the coefficients, constants and exponents are derived from the empirical data
     • e.g.,stream channel (hydraulic) geometry
       • regression of hydrometric data revealed that stream slope is an inverse function of discharge, slope = Q^-0.5 to ^-1.0, i.e., with increasing volume to surface area, the flow of water requires less gravitational stress to overcome boundary and fluid resistance
       • similarly, Q is empirically related to velocity, width and depth (Leopold and Maddock, 1953, USGS Professional Paper 252)
       
       
     • since frequency distributions and statistical relationships are based on samples of measurements, we are never certain that they truly represent the entire population, they are only probable
     • therefore frequency distribution are termed probability distributions, and the probability that a statistical relationship is due simply to chance must be within acceptable limits, usually < 5% (19 times out of 20, in the case of public opinion poles)
     • the likelihood of statistical significance depends on whether a relationship truly exists, but also on the size and homogeneity of the sample and thus on sampling

References

• Carson, M.A. 1971. The Mechanics of Erosion, Pion, London, 174 pp.
• Ferguson, J. 1988. Mathmatics in Geolog, Allen & Unwin, London, 299 pp.
• Wilson, A.G. and Kirkby, M.J. 1975. Mathematics for Physical Geographers and Planners. Clarendon Press, Oxford, 325 pp.
• Scheidegger, A.E.. 1991. Theoretical Geomorphology. Springer-Verlag, Berlin, 434 pp.
• Sumner, G.N. 1978. Mathematics for Physical Geographers. John Wiley & Sons, New York, 236 pp.