GEOG2840 Alps Field Class Fluvial processes in glacial environments Fluvial processes using the glacial environment as a laboratory Introduction Some of the fundamental questions regarding fluvial processes relate to the way in which river flow is impacted upon by the environment through which it flows, and vice versa. It is now recognised that the fundamentally natural pattern of a river is either a braided or a dendritic network: meandering and straight channels only form when the bed sediments or river banks are sufficiently cohesive or resistant to erosion that the erosion rate slows. One of the difficulties of assessing this type of behaviour is the time that it takes for rivers to change. There are commonly two solutions to this problem: (a) study systems that are dynamic and which change quickly (b) use a space for time substitution, where you study channels of different ages by sampling them spatially (i.e. sample in space a set of channels formed at different time periods) Rivers in glacial environments normally have elements of both (a) and (b) – the diurnal cycle of melt creates a discharge cycle that is relatively predictable – spatial variation in slope, as well as a range of sizes of glacier sub catchments, results in a wide variety of river channel patterns and morphologies. Thus, glacier-fed rivers are ideal natural laboratories. The focus of the exercises during this field class will be supraglacial streams: small streams incised into the glacier surface; they are relatively safe to work with as they are small; but extreme care is required when studying them as they can be slippy and fast flowing. Never stand inside one. Follow the usual principle: safety comes before getting good research results; safety is the priority. Classical fluvial geomorphology has a number of basic theories that we will explore in the context of supraglacial streams: (1) the principle of hydraulic geometry; (2) controls on channel pattern in general and meandering in particular; (3) meander morphology; and (4) the riffle-pool sequence. Hydraulic geometry There are two types of hydraulic geometry relationships: at-a-station; and downstream. At-a-station hydraulic geometry states that if the discharge is Q there are predictable relationships between Q and width (w), depth (d) and velocity (v): w aQ b d cQ f v kQ m [1] As Q = wdv, it can be shown that b + f + m = 1 and that ack = 1. These relationships basically describe the sensitivity of width, depth and velocity to changes in discharge. Research suggests that: If b = f = m = 0.33, the cross-section will be broadly parabolic If b = 0.05, f = 0.45 and m = 0.50, the section will be trapezoidal – changes in discharge are weakly related to width changes. If m is very high, then the bed is likely to be very rough, as small changes in Q cause major reductions in flow resistance and hence increases in velocity. These relationships are important as they are the basis of GEOG2840 Alps Field Class Fluvial processes in glacial environments The Manning equation which assumes that Q is a function of d to the power of 5/3 – or d is a function of Q to the power of 0.6 – if not, the Manning equation does not hold. Stage-discharge relations where records of stage are used to estimate changing Q using a stagedischarge regression curve – for this to work, ideally, f>>m and f>>b: i.e. all the changes in Q are achieved through changes in d. The above relationships imply that as discharge increases at-a-station, the width, depth and velocity must increase. Downstream hydraulic geometry takes this one step further by arguing that as Q changes in the downstream direction, w, d and v must also rise to accommodate this increasing flow. It argues that, as with [1], this can also be predicted. Given the above, do hydraulic geometry relationships occur in supraglacial streams? They have special boundaries (ice) which are ‘erodible’ by frictional melt. What sort of cross-sections result? How do they change in the downstream direction? Methods We will divide into 5 pairs along a single supraglacial stream. Four pairs will each have 5 transect sites. This gives 20 sample points. They will need to be systematically chosen – i.e. to always be in riffly sections At each sample point we will measure – flow width, flow depth, section width, section depth and tracer velocity – as well as local slope. These measurements will be done hourly. The 5th pair will position the sections with respect to one another, and note where there must be an increase in discharge due to tributaries entering the stream. They will also do detailed velocity area or salt dilution ngaugings in the reaches between each confluence to get discharge We will then seek to assess the following a. what do supraglacial stream cross-sections look like? b. at each sample point, we will test the at-a-station hydraulic geometry hypothesis – do width, depth and velocity relate to the associated discharge? c. how does cross-section morphology change in the downstream direction? d. do these changes relate to increasing discharge? Channel pattern Predicting the shape of river channel pattern is important as certain river channel patterns are more difficult to manage (e.g. braided rivers) than others. The classic work of Leopold and Wolman (1957) concluded that: points that plot above the line sb 0.013Qb0.44 (sb is valley slope and Qb is bankfull discharge) would be braided and below this would be meandering or straight; and points that plot above the line ** would be braided or meandering and below this would be straight. Subsequent research has shown (Ferguson, 1987) that these relationships are actually thresholds of stream power but also that perimeter sedimentology can make a difference. In the case of supraglacial streams on ice, perimeter sedimentology is unlikely to be important as it is pretty homogeneous, but GEOG2840 Alps Field Class Fluvial processes in glacial environments ice carried debris may disturb this relationship. The question we want to ask, however, is can we use slope and discharge to discriminate between different channel patterns for supraglacial streams? Methods We will identify a set of reaches of supraglacial streams on different slopes and with different discharges. For each reach of each stream, we will record slope For each reach of each stream we will measure the observed discharge using either salt dilution velocity area gauging – this will need to be used to determine bankfull discharge in the evening For each reach of each stream we will measure the flow width and flow depth and the bankfull width and bankfull depth of a number (5) of cross-sections For each reach of each stream, we will work our sinuosity (river length divided by direct line length) The analysis here has a lot of groundwork to assess whether or not discharge and slope can explain the level of sinuosity The measured discharges, slopes, flow widths and flow velocities must be applied to the Manning equation to estimate channel roughness. Q wd wd s 0.5 R 0.67 where R w 2d n The estimated roughness must then be applied to the measured slopes bankfull widths and bankfull velocities to get bankfull velocity estimates. Multiple stepwise regression should then be used to relate sinuosity to (a) measured discharge and slope and (b) estimated bankfull discharge and slope. Is there a relationship? How does it compare with Leopold and Wolman?c Meander morphology One of the things that has puzzled fluvial geomorphologists for some time is what controls meander morphology? The definitions we use for meander morphology are: GEOG2840 Alps Field Class Fluvial processes in glacial environments Meander wavelength, λ Maximum angular deviation, ω Local direction angle, θ A number of controls have been proposed: a. Dury (1956) proposed that meander wavelength could be related to bankfull discharge: 54.3Qb0.5 b. Leopold and Wolman (1957) found a stronger correlation with river width: 12.34w 4w and note that the riffle pool spacing is normally 2πw, which seems to implicate riffle pools in meander development. c. Both (a) and (b) are describing general meander characteristics rather than specific details of meander waveform. Thus, Langbein and Leopold (1966) argued that meander planform could be described by (see Figure above): sin 2 s where s is distance along the bend length. This can be perturbed to capture some of the irregularity in meander pattern by: sin 2 s where ε is some randomly distributed, Gaussian error term. The question emerges as to how well these models do for supraglacial streams. In particular, the above are developed for floodplains with low valley slopes. Supraglacial streams sometimes have steeper slopes. Do they still hold? Methods The main focus of this day will be the measurement of meander bend planform morphology. We will adopt a two-scale of approach: (1) we will do extensive survey of a large number of meander bends to assess the relationship between meander bend wavelength, riffle pool spacing and channel width, and GEOG2840 Alps Field Class Fluvial processes in glacial environments the extent to which it is affected by glacier surface slope; and (2) we will do detailed mapping and analysis of a number of meander bends to explore their planform geometry. 1. The team will split into three: (a) one group doing discharge measurements; (b) one group doing planform geometrical mapping; and (c) one group doing more extensive meander bend measurements. 2. We will identify a meandering reach with no tributary inputs: a. Team 1 will map a part of this using a technique called plane tabling (see Appendix) b. Team 2 will measure the discharge using salt dilution or velocity-area gauging – they also will measure the flow depth and flow width and the bankfull depth and bankfull width c. Team 3 will measure the bankfull width, slope, riffle-pool spacing and wavelength of as many meanders as they can find in the reach. They will also note any factors in each meander measured that might ‘disturb’ the meandering process (e.g. the presence of debris, or a different sort of ice). 3. We will then identify more meandering reaches and repeat 2. 4. The analysis that will be completed upon return can be divided into a. Assessment of the Leopold and Wolman relationship in terms of bankfull width, wavelength, riffle pool spacing relationships and causes of deviation from it b. Determination of bankfull discharge using the technique described in 6a and 6b of river channel pattern. Then assessing if this is related to bankfull width, wavelength, riffle-pool spacing etc c. Digitisation of the meander maps and assessment of the fit to the sine-generated curve (I have a spreadsheet that does this) The velocity reversal hypothesis and riffle pool systems Riffle pool systems are ubiquitous in straight and meandering channels. Indeed, some have argued (e.g. Ferguson, 1993) that braided rivers involve little more than emergent riffles as bar forms, wioth flow diverging around them and converging into confluence scour zones or pools. A key aspect of this system is the transfer of water and sediment downstream. Research has shown that this transfer process is critically linked to the riffle pool system and the way it regulates sediment transfer. In the early 1970s, Keller (1971) observed that as the discharge in a riffle-pool sequence increased, the difference in magnitude between the high velocity flow over the riffle and the slow velocity of the pool is reversed. This is important as sediment will be transported at higher flow events, and increases in velocity through the pool will both help to transfer sediment through the pool system as well as to entrain any material stored in the pool: GEOG2840 Alps Field Class Fluvial processes in glacial environments Pool Riffle Velocity Reversal Discharge Do supraglacial streams have pool-riffle systems? (If so they are like other rivers?). Does veloicityreversal occur. Methods The basic aim of the method is to reconstruct the velocity-discharge curve during a diurnal cycle of meltwater rise. Thus, every 30 minutes for about 6 hours we will measure for two riffle-pool sequences: the discharge (salt dilution gauging or velocity-area gauging) the depth and width-integrated velocity in each riffle and each pool the flow depth in each riffle and each pool Analysis will focus upon constructing the above curve but also thinking about why there might be deviations from it. GEOG2840 Alps Field Class Fluvial processes in glacial environments Techniques 1. Plane Table Mapping We will use a simple technique to create planform maps of the river based upon plane table mapping and a technique called radiation: Line of site to measurement point A levelled plane table Alidade used for sighting Level the plane table Measure the longest dimension of your study area (e.g. the total length of meander bend to study). Measure the length of your paper. Determine the scale of the survey such that the longest dimension will fit on the paper (remember to think about how the table is oriented in plan!) Place a centre of survey mark on the table (the filled black square on the diagram) Identify a measurement point and measure on the ground from beneath the survey mark to the site. Scale this by the scale identified from 4 Site along the alidade to the measurement point Mark a point on the table at the distance identified from 7 along the line of site identified in 8 Move to a new measurement point but remember not to move or to disturb the plane table. 2. Salt Dilution Gauging (a) Principle The basic principle of salt dilution gauging is that the diffusion and dispersion of a tracer depends upon the river’s discharge. In small streams (especially rocky, turbulent ones) discharge can be measured directly by injecting a salt solution and measuring the changing concentration of salt at the GEOG2840 Alps Field Class Fluvial processes in glacial environments gauging section downstream. The commonest approach is the 'gulp' method, in which a large volume of salt solution is thrown into the channel, and the 'wave' passing through the section is monitored. This can be used simply to obtain an estimate of average velocity which is then multiplied by crosssectional area (e.g. Calkins and Dunne, 1970). However, discharge can be estimated directly from the volume and concentration of injected solution and the area under the curve of the plotted salt wave (Ostrem, 1964; Church, 1975; Hongve, 1987). This is the approach that we will adopt. (b) Field methods 1. Take a bucket of water with approximately 100ml of water and measure the conductivity. 3. Add 100g of salt. 4. Mix very well. 5. Measure conductivity, remembering to record the units used. 6. Add 100ml of water and re-measure conductivity. 7. Repeat 6 until 500ml of water has been added to the bucket. 8. Wash the conductivity probe very thoroughly in the river. 9. Person 1 goes to a highly turbulent part of the stream, at the start of a straight section, with the bucket of water. 10. Persons 2 and 3 go to a point some distance downstream (this distance does not need to be known, but should be approximately 400m) 11. Person 1 throws the bucket in and Person 2 starts the stopwatch as this is done. 12. Person 3 records the conductivity every 5 seconds in response to prompting from Person 2, until the saltwave has passed. Remember to record the units used. (c) Computer Analysis Remember to use metric units throughout. 1. First we obtain a regression relationship between conductivity and volume of water in the bucket: Conductivity, S Volume of water V GEOG2840 Alps Field Class Fluvial processes in glacial environments This will be of the following form: S aV b [1] and should have a very good fit to a linear model. The slope of the line is a which is the rate at which conductivity changes with respect to the volume of water added. Thus: a dS dV GEOG2840 Alps Field Class 2. Fluvial processes in glacial environments Our conductivity through time should plot as follows: Conductivity, S Background, Sb Time 3. The area under the curve in Figure 2 is: T A (S Sb)dt t 0 As we have a discrete set of measurements, this can be approximated as a series of polygons. If t=0 is taken as the start of the first polygon The first polygon is: 0.5(S1-Sb)dt The second polygon is: ((0.5(S2-S1))+S1)dt = 0.5(S2+S1)dt The nth polygon is: 0.5(Sn+S(n+1))dt The last polygon is: 0.5(S(n-1)-Sb)dt Adding these up gives the area under the curve. GEOG2840 Alps Field Class Fluvial processes in glacial environments 4. Under the assumption that the background conductivity is negligible, the formula used to determine the discharge is then: V Q 2 . dS dV T Sb V (S Sb)dt 2 . dS dV A Sb t 0 3. Velocity-area gauging (a) Principle This is the standard method of gauging discharge in a natural section. Since velocity both across a section and with height above the bed, several velocity measurements must be taken. In any vertical, a reasonable estimate of mean velocity is obtained from a point measurement at 0.6 of the depth (0.4 of the height above the bed), or from the average of measurements at 0.2 and 0.8 of the depth. Note the support for this that comes from the work that you have done with Chris Keylock on the law-ofthe-wall. Velocity measurements are usually obtained by current meters. These may be mechanical (propellers calibrated in tanks so that the observed number of revolutions per second can be converted to velocity cf. Charlton, 1978), or electromagnetic (based on Faraday's law of electromagnetic induction, and capable of simultaneous and continuous measurement of velocity in two or three orthogonal directions cf. Lane et al., 1993). A completely different method of obtaining mean velocity over a vertical in a deep river involves releasing floats or bubbles at the bed and measuring the time taken to reach the surface; this averages velocity over the depth of flow (see Dyer, 1970). You can always resort to oranges if you done’t have a current meter! Velocity-area gauging requires measurement of mean velocity in the vertical at several verticals across the section. Sufficient verticals are required to ensure that no individual subsection delimited by an adjacent pair passes more than 10% of the total flow. Tests are made in initial gauging exercise for a section to assess the accuracy attained by reducing progressively the number of measured verticals. The mean-section and mid-section methods may be used to calculate discharge from the depths, mean velocities for verticals, and sub-section widths (see Lambie, 1978). Normally, velocityarea methods are employed to 'rate' a stable section so that continuous measurement of water stage can then be used to monitor discharge variation, through use of the stage-discharge rating. Some stagemonitoring systems are sensitive to the rate of change of stage, and control samples so that they sample less frequently during periods of constant low flow. In large rivers, a variety of alternative methods are used such as the moving-boat method, and electromagnetic or ultrasonic methods of measuring velocity (e.g. Hersch, 1976). (b) Field methods We will use the mean section method. 1. The section is divided into segments, each of which is ideally only 5% of total width and passes a maximum of 10% of total discharge. GEOG2840 Alps Field Class Fluvial processes in glacial environments 2. At the division between each segment, measure the position with respect to one bank and the water depth. 3. Work out 0.4 of the water depth above the bed at that segment. 4. Measure velocity with the current meter at 0.4 of the depth above the bed at the dividision between each segment. Velocities are obtained by using a current meter, which consists of a propeller whose revolution speed is a function of the velocity of water moving past. Count the number of revolutions of the meter propeller in 30 seconds, then average the number of revolutions per second. We assume that 30s is long enough to get a stable estimate of the average velocity. (c) Laboratory analysis qi wi di The key to the laboratory analysis is to remember that: Dicharge = width x depth x velocity Q WDV In the above diagram, we have a set of n segments, with each segment having its own discharge (qi). Thus, the total discharge is: n n i 1 i 1 Q qi widivi where wi is the width of each segment, di is the average depth of each segment, and vi is the average velocity in each segment. If we measure the depth and the velocity at the start (i-0.5) and end (i+0.5) of each segment, and the width of each segment is constant across the river, then the Q is: n n i 1 i 1 Q qi W di 0.5 di 0.5 vi 0.5 vi 0.5 2 2 where W is the total width of the river. 1. Estimate velocity from the calibration equation: V=a+bN GEOG2840 Alps Field Class Fluvial processes in glacial environments where V = water velocity (m/sec), N = no. of revs. per sec., and 'a' and 'b' are the calibration constants. 2. Apply the equation in the box. Readings on techniques Calkins, D. and Dunne, T. 1970 A salt-tracing method for measuring channel velocities in small mountain streams. Journal of Hydrology, 11, 379-92. Charlton, F.G. 1978 Current meters. In Hydrometry ed R.W. Herschy, Wiley & Sons, 53-81. Church, M. 1975 Electrochemical and fluorimetric tracer techniques for stream flow measurements. BGRG Technical Bulletin 12, GeoAbstracts. Dyer, A.J. 1970 River discharge measurement by the rising float technique. Journal of Hydrology, 11, 210-12. Herschy, R.W. 1976 New Methods of rover gauging. In Facets of Hydrology ed. J.C. Rodda, Wiley & Sons, 119-61. Hongve, D. 1987 A revised procedure for discharge measurement by means of the salt dilution method. Hydrological Processes 1, 267-70. Lambie, J.C. 1978 Measurement of flow - velocity-area methods. In Hydrometry ed. R.W. Herschy, Wiley & Sons, 1-52. Lane, S.N., Richards, K.S. and Warburton, J., 1993. Comparison between high frequency velocity records obtained with spherical and discoidal electromagnetic current meters. In Turbulence: Perspectives on Flow and Sediment Transport, Clifford, N.J., French, J.R. and Hardisty, J. (eds), pp121-163. Nordin, C.F. and E.V. Richardson 1971 Instrumentation and measuring techniques. Mechanics, Vol. 1 ed. H.W. Shen, 14. 1-14.38. In River Ostrem, G. 1964 A method of measuring water discharge in turbulent streams. Geographical Bulletin 21, 21-43 White, W.R. 1978 Flow measuring structures. In Hydrometry ed R.W. Herschy, Wiley & Sons, 83109. GEOG2840 Alps Field Class Fluvial processes in glacial environments Filth Spreads: The Development of Dirt Cones on Glacial Surfaces Background "Dirt colludes with sunlight to carve snow [and ice] fields into exotic patterns. Dirt is a snowscape architect. Whether the sun sculpts narrow peaks, rounded dimples or a web of ridges in a smooth snow field depends on whether the snow is clean, dirty, or really dirty” Ball (2001) When we think of glaciers, most people will imagine in their mind’s eye a gleaming cascade of ice, tumbling, perhaps romantically, from a jagged mountainscape. Possibly this is the view of glaciers put forward in coffee table books on landscapes, or people have been lucky enough to ski on the upper reaches of an Alpine glacier. However, the facts are much grimier, and more sinister. On walking up to the front of glaciers, most actually greet you with a bizarre and dangerous region of lakes, dead ice, quicksand, potholes and landslides. In short, the fronts of glaciers tend to be pretty filthy things. It is only when you get far up into the accumulation zone (where there isn’t net annual melting) that large areas of white ice are the norm. The sediment in the front (snout) region of glaciers tends to have a variety of origins, including: 1) subglacially and englacially entrained material; 2) sediment moved by streams onto the ice, and by streams englacially; 3) wind blown material from the proglacial areas and mountains; 4) frost shattered materials moved in scree falls, landslides, or avalanches. This material tends to accumulate at the front and surface of the glacier by a variety of processes: 1) ice can move down around sediment and then be moved by internal glacier deformation towards the surface. Melting at the front can reveal this material. 2) subglacially and englacially entrained material can move high into the ice by movement along shear zones, in which initial weaknesses in the glacier fracture and lead to one part of the ice riding up the back of another, often on a smeared mass of sediment; 3) surface material higher up the glacier system is buried by accumulation and then re-exposed by surface melt at the front. Such material can also fall down crevasses and be re-exposed by streams etc. cutting into the ice at the ice front. 4) For a variety of reasons, ice at the front of glaciers tends to move slowly, or be stagnant (because the ice: is thin; often lies on an up-ice deepening basin; is cut free from faster ice by shear zones; is buttressed by moraines). Because of this, material coming down with the faster ice behind the snout tends to accumulate on or around the glacier front. All these sources and processes make the glacier front a ghastly landscape of dead ice interspersed with pits of slurry and sliding boulders. It is these materials that then often form the morainic sequences we see in front of the glaciers and in previously glaciated areas like Snowdonia. By understanding these processes, we can therefore start to try to understand what went on in the recent and quaternary history in terms of ice movements and, ultimately, climate change. However, these sediments are not just passive markers of a glacial front – they have an important role to play. We’ve already said how sediments can move up shear zones in the ice, and such sediments can alter the dynamics of overriding ice masses, speeding them up or slowing them down. In the same GEOG2840 Alps Field Class Fluvial processes in glacial environments way, surface materials also have a profound effect on ice dynamics. Because these materials tend to be darker than ice, but also conduct heat poorly, they can significantly alter the melt rate on the glacier depending on their thickness. This, in turn, controls how faster ice behind the front moves and the stability of the ice mass in general. In this exercise, therefore, we are going to look at the effect sediments have on ice melting, and the origins and dynamics of one of the most important, but overlooked features produced by sediments on glacier surfaces: the Dirt Cone. Dirt cones in Svinafellsjökull, Iceland : Photo by P.M.Colgan What are dirt cones? Dirt cones are masses of ice that stick up above the surrounding ice (see photo above). Usually they have a covering of sediment. Often this sediment will have originally have collected together in a stream pool which will then have developed into the cone, however, it can be the case that the centre of the cone can be built up around a sediment-filled shear zone in the ice, which slowly leaks sediment out onto the enclosing ice. So, how do they form? Research (Drewry, 1972; Betterton, 2001; also see Benn and Evans, 1998, for a summary of others) has shown that the thickness of sediment has an important effect on the melt rate associated with the underlying ice. As a thin layer of dark sediment doesn’t reflect energy as well as the white ice, such a layer will tend to heat up, melting the ice. This causes pits to develop where sediment has collected. However, if the sediment is of a sufficient thickness, the material, which doesn’t transmit heat particularly well, insulates the surface and the underlying ice remains as a pedestal while the surrounding ice carries on melting. The insulated ice should be compared with the case of clean ice, where heating leads to melt and flow of the ice away as water, leaving a fresh surface to melt. At some point, as the dirt cone gets steeper, the sediment will flow off the sides and accumulate elsewhere allowing melting to restart. Overall, this produces a confused topology in which sediment is concentrated and melting reduced by insulation. Fieldwork In the field, we will explore three key questions about the dirt cones and the effect of sediment on melting: Where is the sediment coming from and what kind of material is it? What thicknesses of material lead to different melting conditions? What is the spatial distribution of these sediment bodies? GEOG2840 Alps Field Class Fluvial processes in glacial environments In addition, back at base we’ll try to examine what effect they might have on the development of the glacier snout and the kinds of things we have to take into account when making predictions. Aims: To gain a general understanding of glaciers, the environment and processes, as well as more specific understanding of the glacial snout. To determine the effect of sediment cover on ice evolution. To gain an understanding of the kinds of additional processes needed to predict glacial topology. Objective: To map the spatial patterns in sediment thickness and the resultant topology on Ödenwinkelkees glacier. Methods: Pre-fieldwork planning: Consider the kinds of processes that might be acting on the glacier snout to generate topology. Draw a flow diagram of the kinds of situations and processes that might move water and sediment around the snout / melt-zone system, and how these might interrelate. An example of part of such a diagram is given below. Fieldwork: In the field we will decide on a survey plan. Some people in the group will be measuring sediment thicknesses and ice topology on a wide scale using compass and clinometer surveying, which is a rough and ready mapping technique. Others will be doing detailed mapping and feature measurement on individual dirt cones. Equipment will include: compass clinometers small measuring rulers etc. GEOG2840 Alps Field Class Fluvial processes in glacial environments 3 and 30m tapes laminated maps and aerial photographs digital camera Post-fieldwork analysis: Once back in the classroom we’ll be collating the data, examining what it tells us about the questions formulated above, and what it tells us about how the glacier topology might develop. We’ll also be examining additional data we might need to build up a full picture of glacier melting. References: Ball,P. (2001) Grime every mountain. Nature. http://www.nature.com/nsu/010510/010510-1.html [online] Accessed 1 July 2004 Benn, D.I., and Evans, D.J.A. (1998) Glaciers and glaciation. New York, New York, Wiley, 734 p. Betterton,M.D. (2001) Theory of structure formation in snowfields motivated by penitents, suncups, and dirt cones. Physical Review E, 63. p.63 [online] Accessed 1 July 2004 http://ojps.aip.org/journal_cgi/dbt?KEY=PLEEE8&Volume=63&Issue=5 Drewry,D.J. (1972) A quantitive assessment of dirt-code dynamics. Journal of Glaciology, 11, 63. p.431-446.