Late Pleistocene glacier dynamics of southwestern Montana and adjacent Idaho and paleoclimatic implications by Donald R Murray A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Earth Sciences Montana State University © Copyright by Donald R Murray (1989) Abstract: Application of glacial flow theory to reliable reconstructions of paleoglaciers allows calculation of the dynamics of these glaciers. Effective basal shear stresses calculated along the longitudinal profiles of these glaciers can be used to estimate the component of mass flux due to internal deformation. Assuming basal slip to be zero at the point where deformation mass flux is a maximum, minimum net accumulation and ablation gradients can be calculated. Using the continuity equation, minimum mass flux at the ELA can be estimated. Also, net winter accumulation can be calculated by dividing the mass flux at the ELA by the accumulation area. Because local climate, in part, controls the mass balance and dynamics of a glacier, this model provides information on the climatic setting of paleoglaciers. The model also allows estimation of basal slip as a factor in point estimates of glacial flow. Application of the continuity equation above and below the ELA generates additional estimates of mass flux at discrete points along each glacier. The difference between calculated deformation mass flux and continuity flux at these points yields a first approximation of basal slip, which can be highly variable along the length of a glacier. The . model was developed on the late Pleistocene Big Timber glacier of west-central Montana and tested on five other paleoglaciers in the Northern Rocky Mountains of southwestern Montana and adjacent Idaho. Sensitivity analysis performed on Big Timber glacier shows that the results are accurate within 20%. Low ablation gradients, ranging from 1.9 to 5.4 mm/m for five of the six glaciers, suggest a cold, dry environment in this region during the late Pleistocene. Calculated average annual net accumulation for these glaciers is 20-75% below modern maximum snowpack values, indicating a drier climate during the full glacial period. Basal sliding accounts for most (> 90%) of the glacial flow near the terminus of each glacier, but is variable along the rest of the glacier. While the mass balance values are minima, they are assumed to be reasonable approximations of the actual values, unless very high basal slip rates occurred along the entire length of each glacier.  LATE PLEISTOCENE GLACIER DYNAMICS OF SOUTHWESTERN MONTANA AND ADJACENT IDAHO AND PALEOCLIMATIC IMPLICATIONS by Donald R. Murray A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science ■ in . -. .Ra r t'h S cj. enc.e s MONTANA STATE UNIVERSITY Bozeman,Montana December 1989 /J3V# ii APPROVAL of a thesis submitted by Donald Richard Murray This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. Date Chairperson, Graduate Committee Approved for the Major Department QdC kus /%5 Date Head, Major Department i/ Approved for the College of Graduate Studies Dkbe ^ 7^ Graduate Dean£ iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a master's degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgement of source is made. Permission for extensive quotation from or reproduction of this thesis may be granted by my major professor, or in his absence, by the Dean of Libraries when, in the opinion of either, the proposed use of the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission. VACKNOWLEDGEMENTS The author wishes to thank the members of his committee, Dr. William W. Locke (Committee Chairman), Dr. Katherine Hansen-Bristow, and Dr. Joseph Ashley, for providing support, guidance and helpful criticism throughout the project. Dr. Locke first suggested the topic and was extremely helpful in nurturing the project as it developed. Thanks are also due to Dr. James McCalpin of Utah State University for use of his unpublished work on glacier dynamics of the Sangre de Cristo Range and for discussion of the methods used in this project. I also wish to thank the Department of Earth Sciences for accepting me into the graduate program and for providing a productive and amiable atmosphere in which to work; my parents, William and Barbara Murray, for encouraging me to do whatever makes me happy; my wife, Diana Shellenberger, for field assistance., copyediting and moral support; and finally Bud Ebbett of Lyndon State College for providing the initial spark of my interest in glaciers and geology. This project was partially funded by GSA Research Grant 4051-88 and the D.L. Smith Scholarship through the Department of Earth Sciences. vi TABLE OF CONTENTS Page INTRODUCTION .................................................. I The P r o b l e m .............................................. I Previous W o r k ............................................ 3 Theoretical Considerations ................................ 4 Glacier Dynamics .................................... 4 Equilibrium Line Altitude (ELA)............... . 4 Glacier Flow Velocities ........................ 5 Mass Balance Gradients .......................... 8 The Study A r e a .......................................... 11 Regional Setting .................................... 11 Geology.............................................. 14 Present Climate..................................... 14 Late-Pleistocene Climate ............................ 19 METHODS........................................ 21 Valley Selection .......................................... 23 Glacier Geometry........................-................ 24 Basal Shear Stresses............. 29 Location of the E L A ...................................... 33 Lateral Moraines . .............................. . . 33 Lowest Cirque Elevation .............................. 34 Toe-Headwall Altitude Ratio (THAR) .................. .34 Accumulation-Area Ratios (AAR) . . ■.................. 35 Estimated ELAs . . . ................................ 35 Glacier F l o w .............................................. 36 Ice Deformation...................................... 36 Basal S l i p .......................................... 37 Average Velo c i t y ........................ 39 Ablation/Accumulation Gradients .................... 40 PaleoClimatic Interpretation .............................. 44 RESULTS . .......................................’ .............. 48 Case Study - Big Timber Canyon............................ 48 Valley Selection .................................... 48 Glacier Geometry........................ 48 Basal Shear Stresses................................ 53 Location of the ELA . ..................... 56 Lateral Moraines....................' .......... 57 Lowest Cirque Elevation.................... . . . 57 Toe-Headwall Altitude Ratio (THAR) .............. 57 Page • Accumulation Area Ratio (AAR) .................. 57 Final E L A ...................................... 58 Glacier F l o w ........................................ 59 Ablation/Accumulation Gradients ...................... 59 Sensitivity and Error Analysis .................... . . . . 63 Results from other Valleys .................................. 68 Lemhi Range.......................................... 68 Stroud Greek Glacier ............................ 69 Everson Greek G l a c i e r .......................... 73 Mill Creek Glacier.............................. 76 Meadow Lake G l a c i e r ............................ 80 Beaverhead Range .-.................................. 83 Miner Lakes G l a c i e r ............................ 84 Summary.............................................. 89 DISCUSSION . ............ ' .................................... 90 The M o d e l ..................'............................ 90 Glacier Reconstruction............................ . 90 Basal Shear Stresses................................ 93 Glacier Velocity - Ice Deformation.................. 95 Glacier Velocity - Basal S l i p ........................ 97 Field Observations.............................. 98 Other M e t h o d s .................................. 99 This study........................................ 101 Mass B a l a n c e ................ 102 The R e s u l t s ........; ...................................... 103 Stroud Greek versus Everson Creek .................... 103 Comparison to Mill C r e e k .............................. 104 Comparison of all valleys.............................. 107 Basal S l i p ............................................ 108 Paleoclimatic Interpretations.............................. H O Mass balance gradients . ............................ H O This study....................•.................Ill Other s t u dies.................................... 113 Net Accumulation...................................... 113 Comparison to other studies .................... 115 CONCLUSIONS...................................................... H S Suggestions for Future Study ............................... 120 Dynamics Modeling...................................... 120 Paleoclimatic Modeling : ............................ 121 vii TABLE OF CONTENTS -- continued REFERENCES CITED 123 viii Page APPENDICES....................................■ .............. 129 Appendix A - Topographic maps and air photos............... 130 Appendix B - .Program for calculating theoretical glacier profiles................. 132 Appendix C - Program for calculating shear stress ... 138 Appendix D - Program for calculating ablation gradients . . 144 Appendix E - Program for calculating accumulation gradients.................................... 147 Appendix F - Data for Stroud Creek......................... 150- Appendix G - Data for Everson C r e e k ....................... 152 Appendix H - Data for Mill Creek........................... 154 Appendix I - Data for Meadow L a k e ......................... 156 Appendix J - Data for Miner L a k e s ....................... 158 TABLE OF CONTENTS -- continued ' ix LIST OF TABLES Table Page 1. Analysis of the number of existing lateral moraines in a valley, given differences in climate between two successive glaciations .................. 27 2. Big Timber paleoglacier morphology and rheology ........ 54 3. ELA estimates for Big Timber glacier . . . . .......... 56 4. Average annual net ablation on Big Timber g l a c i e r .................................. 61 5. Mass balance and basal slip results for Big Timber g l a c i e r .................................. 64 6. Error and probability analysis of map-induced errors .................................. 66 7. Mass balance and basal slip results for Stroud Greek glacier ............................. 72 8. Mass balance and basal slip results for Everson Creek glacier ............................. 75 9. Mass balance and basal slip results for Mill Creek g l a c i e r ............... ........ -......... 80 10. Mass balance and basal slip results for Meadow Lake glacier.................................. 84 11. Mass balance and basal slip results for Miner Lakes glacier.................................. 88 12. Values of the flow law constant A ....................... 96 13. Mass balance estimates for Stroud Creek glacier assuming 0, 10, 20 and 50% basal s l i p ................... 106 14. Mass balance summary for all the glaciers.................. 107 15. Net accumulation ........................................114 16. Topographic maps and aerial photographs used for glacier reconstruction .............................. 131 XLIST OF TABLES - - continued Table Page 17. Stroud Greek paleoglacier morphology and rheology . . . . 151 18. Everson Creek paleoglacier morphology and rheology . . . 153 19. Mill Greek paleoglacier morphology and rheology ........ 155 20. Meadow Lake paleoglacier morphology and rheology . . . . 157 21. Miner Lakes paleoglacier morphology and rheology . . . . 159 xi LIST OF FIGURES Figure Page 1. General relationships between climate, glacier response and geologic evidence.............. 2 2. Mass exchange on a steady state glacier ................ 5 tiV 3. Components of glacier f l o w ........................... 6 4. Generalized mass balance gradients for maritime and continental glaciers..........r ................ • 9 5. Location and physiography of the study area . . . . . . . 12 6. General areas of western Montana and adjacent Idaho • covered by ice during the last glacial maximum . . . . 13 7. Streamlines of mean January surface winds over the western United States ............................ 15 8. Climographs and locations of selected cities within the study a r e a ................................ 16 9. Modern snow accumulation gradients.............. . . . 18 10. Flowchart of the methodology.......................... 22 11. Comparison of the shapes- of maritime and continental glaciers ................................ 26 12. Comparison of.the difference in elevation between the ice centerline and the ice margins on modern glaciers ...................................... 28 13. Values of the shape factor (F) for varying parabolic glacier cross-sections .............................. 31 14. Variables used in calculating theoretical glacier prof i l e s ........................... 32 15. Mass balance of the Yellowstone ice c a p ................ 41 16. Continuity theory............' ........................ 43 17. Photo of Big Timber Canyon 49 xii LIST OF FIGURES -- continued Figure Page 18. Map of Big Timber Canyon and the glacier reconstruction ...................................... 50 19. Longitudinal profile of Big Timber glacier . . . . . . . '52 20. Average basal shear stress along Big Timber glacier.............................................. 55 21. Hypsometric curve of Big Timber glacier used to estimate ELA using AAR method........................ 58 22. Mass flux on Big Timber glacier........................ 60 23. Mass balance of Big Timber g l a c i e r ............ 62 24. Map of Stroud and Everson creek valleys and the glacier reconstructions .......................... 70 25. Longitudinal profile of Stroud Creek glacier .......... 71 26. Longitudinal profile of Everson Creek glacier .......... 74 27. Map of Mill Creek valley and the glacier reconstruction ...................................... 77 28. Longitudinal profile of Mill Creek glacier ............ 79 29. Mf.p of Meadow Lake Canyon and the glacier '■reconstruction...................................... 81 30. Longitudinal profile of Meadow Lake glacier ............ 82 31. Map of Miner Lakes valley and the glacier reconstruction ...................................... 86 32. Longitudinal profile of Miner Lakes glacier ............ 87 33. An attempt to model mass balance gradients using mass flux and a r e a ........................’.........100 34. Comparison of calculated net accumulation to modern snowpack accumulation gradients .............. 116 35. FORTRAN program for calculating theoretical glacier profiles................................... 133 xiii Figure Page 36. FORTRAN program for calculating average effective basal shear s t r e s s ..................................139 37. BASIC program for calculating ablation gradients . . . . 145 38. BASIC program for calculating accumulation gradients LIST OF FIGURES -- continued 148 xiv ABSTRACT Application of glacial flow theory to reliable reconstructions of paleoglaciers allows calculation of the dynamics of these glaciers. Effective basal shear stresses calculated along the longitudinal profiles of these glaciers can be used to estimate the component of mass flux due to internal deformation. Assuming basal slip to be zero at the point where deformation mass flux is a maximum, minimum net accumulation and ablation gradients can be calculated. Using the continuity equation, minimum mass flux at the ELA can be estimated. Also, net winter accumulation can be calculated by dividing the mass flux at the ELA by the accumulation area. Because local climate, in part, controls the mass balance and dynamics of a glacier, this model provides information on the climatic setting of paleoglaciers. The model also allows estimation of basal slip as a factor in point estimates of glacial flow. Application of the continuity equation above and below the ELA generates additional estimates of mass flux at discrete points along each glacier. The difference between calculated deformation mass flux and continuity flux at these points yields a first approximation of basal slip, which can be highly variable along the length of a glacier. The . model was developed on the late Pleistocene Big Timber glacier of west-central Montana and tested on five other paleoglaciers in the Northern Rocky Mountains of southwestern Montana and adjacent Idaho. Sensitivity analysis performed on Big Timber glacier shows that the results are accurate within 20%. Low ablation gradients, ranging from 1.9 to 5.4 mm/m for five of the six glaciers, suggest a cold, dry environment in this region during the late Pleistocene. Calculated average annual net accumulation for these glaciers is 20-75% below modern maximum snowpack values, indicating a drier climate during the full glacial period. Basal sliding accounts for most (> 90%) of the glacial flow near the terminus of each glacier, but is variable along the rest of the glacier. While the mass balance values are minima, they are assumed to be reasonable approximations of the actual values, unless very high basal slip rates occurred along the entire length of each glacier. IINTRODUCTION The Problem There is an intimate relationship between climate and glaciation (Fig. I), such that the dynamics (thickness, rate of flow and length) of a glacier are controlled to a large extent by changes in climate (Meier, 1965; Andrews, 1975). Climatic changes during the late Pleistocene (79,000 to 10,000 years ago) have been characterized by several global glacial advances ,and retreats. Evidence that late Pleistocene glaciers existed in some valleys of the Northern Rocky Mountains, which do not have glaciers at present, suggests that the paleoclimate must have been different from the present,. The existence of glaciers does not, however, provide an actual measure of temperature or precipitation, but certain features of glaciers may be used as proxies to these climatic variables. Because the dynamics of a glacier are controlled in part by climate, reconstructed dynamics of paleoglaeiers can be used as proxies to climate. In this study, a model is developed to interpret the ice dynamics of paleoglaeiers. Although this project concentrates on the development of the model, the application of this model to six paleoglaeiers that existed in southwestern Montana and adjacent Idaho during the last glacial maximum (20,000 years ago) suggests the magnitude and direction of late Quaternary changes in precipitation. 2Changes in climate affect the mass balance of a glacier, which is a measure of the balance between mass gain (accumulation) and mass loss (ablation) on a glacier. On alpine glaciers, accumulation occurs mainly as snowfall while ablation takes place mainly through melt (Sugden and John, 1976). Changes in mass balance cause a dynamic response in the glacier. An increase in thickness generally causes an increase in the velocity and mass flux, which cause the glacier to advance (Nye, 1960; Meier, 1965). Thus, mass flux is an indicator of the mass balance of the glacier. For an ideal steady state glacier (one that is neither advancing or retreating), net accumulation is equal to net ablation (gain — loss) . The calculated mass balance (i.e. net accumulation) of a steady state paleoglacier could be used as a crude proxy indicator of the paleoprecipitation on that glacier. CAUSE » E FFECT Loca l g l a c i e r c l ima te G lac ie r geome t ry (g eo log ic r e c o rd ) Gene ra l reg iona l c l ima te Mass b a l a n c e ( a c c umu la t i o n / ab la t i o n ) G lac ie r r e s p o n s e ( v e l o c i t y , mass f l u x ) IN T ER PRE TA T IO N ---------------------------------------------------------------------------------------- E V IDENCE Figure I. General relationships between climate, glacier response and geologic evidence of an individual glacier (left to right) (after Meier, 1965; Andrews, 1975). This study uses the reverse approach by interpreting mass balance and climate from the geologic record. Working backwards through the pattern of study usually applied to climate-glacier relationships (Fig. I), it is feasible to calculate the mass balance of paleoglaciers if the dynamics of these glaciers can be reconstructed (Haeberli and Penz, 1985; McCalpin, 1986). In 3the Northern Rocky Mountains, many alpine paleoglaciers left clear evidence of their maximum areal extent in the form of moraines .and trimlines. This evidence allows reconstruction of the former glacier shape and thickness. Ice velocity and mass flux can be estimated by applying ice flow theory to the reconstructed paleoglacler geometry. Mass flux through the equilibrium-line altitude of the paleoglacler provides an estimate of mass balance (net accumulation and net ablation). As a result, reconstruction of paleoglacler dynamics can be used to provide an approximation of paleoprecipitation (accumulation) on the glacier. Previous Work Only a few studies have used glacial mass balance as inferred from the ice dynamics of paleoglaciers to model paleoclimate. Haeberli and Penz (1985) applied this method to late Pleistocene paleoglaciers in the Alps. Although their methods allowed for only low precision, their estimates showed the Alps to be cold and dry during the last glacial maximum. McCalpin (1986) examined the ice flow dynamics and mass balances of paleoglaciers in the Sangre de Cristo Mountains of south-central Colorado and found that the calculated balances depended greatly on the flow regimes (extending, compressing, or uniform) of the glaciers. Using his method, only balances calculated in uniform flow regimes could be considered valid. Leonard- and others (1986) used a similar method to model the climate of the Colorado Front Range during the late Pleistocene. Their 4results show that area to be much drier during the glacial maximum than was' previously thought. In each of these ice dynamics studies, the method was applied to all the valleys within each study area. In some cases, irregularities in the geometries of the paleoglaciers (icefalls, stepped valleys.) would have produced either overestimates or underestimates of mass balance (McCalpin, 1986). It is suggested that when using ice flow theory to model paleoglaciers, restrictions should be placed on the valley selection (applying the valleys to the method rather than the method to the valleys), thereby ensuring the quality of the results. Theoretical Considerations Glacier Dynamics '' The geometry of a glacier is a response to the flow behavior of ice over the underlying topography. As such, if the geometry of the glacier is known, ice flow laws can be applied to this geometry to calculate values of velocity and mass flux (Nye, 1952; Paterson, 1981) . Equilibrium-Line Altitude (EIA). The equilibrium-line altitude (ELA) is an important descriptor of any glacial system because it is the line where mass balance changes from net accumulation to net ablation. On a steady state glacier (Fig. 2), the net accumulation^ the net ablation, and the mass flux (velocity x cross-sectional area) through the ELA are all equal over a given period of.time (Andrews, 1975).- In this steady state system, net mass balance is zero. The mass flux through the ELA is somewhat less than the total mass 5exchange (where total accumulation = total ablation) on the glacier because there is seasonal accumulation below the EIA and seasonal ablation above the EIA. However, the mass flux at the ELA is equal to the net mass exchange, and balances the net accumulation above the ELA and the net ablation below the ELA. Once the ELA of a paleoglacier is located, the net mass exchange (therefore net accumulation and net ablation) can be determined by calculating the glacier velocity and multiplying it by the cross-sectional area at the ELA. NET MASS FLUX NET ACCUMULATION AT THE ELA ABLATION Figure 2. Mass exchange on a steady state glacier where net accumulation, net ablation and mass flux at the ELA are all equal. Glacier Flow Velocities. Glacier movement can be broken down into components of ice deformation and basal sliding (Fig. 3). When ice is under pressure (stress), deformation takes place along internal planes of ice crystals. From laboratory experiments, Glen (1952) showed that the relationship of the strain rate (rate of deformation) QdCkus/% 5Dkb e^ 7 Total movement Ice deformation Ice surface___________________p 7(£) to shear stress (T) could be expressed as: £ = A T n (I) where A is a temperature dependent constant and n is the slope of the strain rate curve. This is referred to as Glen's flow law. Laboratory and field experiments have shown that a value of n = 3 is appropriate for glaciers (Paterson, 1981). Shear stress is a function of the glacier thickness and surface slope and has been found to vary from about 0.5 to 1.5 bars for modern glaciers (Nye, 1952) . Estimates of the basal shear stresses of paleoglaciers (Mathews, 1967; Pierce, 1979) have indicated that this range is also valid for former glaciers. By integrating the strain rate function; over small increments of glacial thickness (Nye, 1952; Paterson,- 1981) , the velocity at the centerline due to ice deformation can be determined (see Methods). Using this integral, deformation velocity varies as the fourth power of the glacier thickness so accurate estimation of thickness is needed. While Glen's flow law provides a method of determining the deformation velocity, this velocity is only part of the total flow through a cross-section. The other component of glacial flow through a cross-section occurs from the glacier sliding over its bed (Fig., 3). Sliding velocities in modern glaciers have been directly observed in boreholes or have been determined by subtracting the calculated velocity due to deformation from the observed surface velocity (Paterson, 1981). The amount of basal sliding (basal slip) has been measured to account for 3-90% of total" velocity on modern glaciers (Andrews, 1975; Paterson, 1981). Several models have been developed (summarized in Weertman, 81979; Raymond, 1980; Paterson, 1981) to explain basal slip, but there has been little agreement between observations and theory. For paleoglaciers, actual surface velocity cannot be compared with calculated deformation (creep) velocity, therefore, estimation of basal slip on these glaciers presents a problem. Studies using ice dynamics to model mass balance (Haeberli and Penz, 1985; Leonard and others, 1986; McCalpin, 1986, Holmlund, 1988) usually assumed a constant basal slip, yet this did not consider the fact that slip can vary along the length of the glacier. Holmlund (1988) used an average value of 50% slip to calculate the mass balance on the modern Storglaciaren in Sweden, but actual measurements showed that basal slip locally accounted for 80-90% of the motion. Because slip varies along the length of a glacier, local calculations of slip must be made for an accurate assessment of the local flow regime. Mass Balance Gradients. Mass balance provides a key to the activity (velocity, mass flux) of a glacier. Accumulation and ablation gradients show the relationship of net accumulation and net ablation, respectively, to elevation.'. (Fig. ■ 4).., . and as such, provide a measure of a glacier's activity (Andrews, 1975). These gradients are expressed in terms of change of net thickness (mm) of water accumulated or melted with elevation (m) above or below the ELA (thus mm/m) . Both ablation and accumulation tend to , change linearly with elevation, therefore average gradients are reasonable descriptors (Andrews, 1975). High ablation gradients (> 10 mm/m) are typical of glaciers in maritime environments where glacier activity is great due EL EV A TI O N Mass Balance Gradients .......... Con t inen ta l ---------Mar i t ime ELA • i / / . / / LvV0 / / / / / / / / / / / / / Z / / / I I I I. I I I I I___I— -1 0 -5 0 SPECIFIC NET BALANCE (m H2O) i i l - L -15 Figure 4. Generalized mass balance gradients for maritime and continental glaciers. 10 to the increased accumulation. Ablation gradients decrease inland toward more continental environments, to values of 2-7 mm/m (Fig. 4) (Meier and others, 1971; Andrews, 1975). Ablation gradients also decrease in value from temperate to polar climates. An exception to this trend is the occurrence of high gradients on small cirque glaciers in the Rocky Mountains of Wyoming and Montana. This exception can be accounted for by the local microclimates of small niche glaciers like these that produce large amounts of snow accumulation from wind drifting and orographic - precipitation (Meier and others, 1971). On moderate sized glaciers, however, ablation gradients should be more indicative of regional climate (Meier and others, 1971; Andrews, 1975). On modern glaciers, mass balance is directly measured on the glacier surface or can be determined using photogrammetrie, hydrologic, or reconnaissance methods (Paterson, 1981). Balance gradients are determined using the measured mass balance and the areal distribution of the glacier with elevation. On paleoglaciers, these methods cannot be employed because the glacier no longer exists. However, if net accumulation and net ablation can be estimated by calculating net mass exchange at the ELA using reconstructed ice dynamics, average net balance gradients can then be determined. These gradients can be used to determine the annual net accumulation or ablation above or below a point on the glacier. The gradients can also be used as proxies to climate by comparison with modern analogs. 11 The Study Area Regional. Setting The mountains of southwestern Montana and adjacent Idaho (Fig. 5) show extensive evidence of late Pleistocene glaciation (Alden, 1953; Porter and others, 1983). The large number of glaciated valleys provides an opportunity to select only those valleys that comply with the criteria in the model, including valleys with well defined glacial features and constant or slowly varying gradients (see Methods). The study area consists generally of northwest-southeast trending mountain ranges separated by broad valleys. The region is bounded on the east by the flat terrain of the Great Plains and on the west by the Salmon River Mountains of central Idaho. The relatively low, flat area of the Snake River Plain lies to the south. During the late Pleistocene, the region was bounded to the north by the Laurentide and Cordilleran ice sheets (Porter and others, 1983) and to the southeast by the Yellowstone ice cap (Pierce, 1979) (Fig. 6). The presence of these ice sheets significantly altered the regional topography during the last glacial maximum. Six valleys (Fig. 6) were used to model paleoglacier dynamics and the paleoenvironment in this area. Big Timber Canyon (#1) in the Crazy Mountains was used to develop and refine the model. Four valleys (#3-6; Mill Creek, Stroud Creek, Everson Creek, and Meadow Lake) in the Lemhi Range were selected to test the local variability of the model. One other valley (#2; Miner Lakes) in the Beaverhead Range was also• studied to test the regional applicability of the 12 Figure 5. Location and physiography of the study area. 13 44 ° - - IOOkm STUDY VALLEY LOCATIONS Laurentide and I - Big Timber Cordilleran ice sheets 2 - Miner Lakes N r—1 Ice cap, cirque or 3 - Mill Creek valley glaciers 4 - Stroud Creek 5 - Everson Creek 6 - Meadow LakePro-glacial lakes 112° 110° Figure 6. General areas of western Montana and adjacent Idaho covered by ice during the last glacial maximum (Taylor and Ashley, 1986; Waitt and Thorson, 1983; Fullerton and Colton, 1986) and location of the study valleys. 14 model. The model could be applied to many other valleys in this area, but that was not the purpose of this study. Geology The bedrock geology of the mountains in the area ranges from Tertiary volcanic to Precambrian metamorphic rocks (Ross and others, 1955) . Because the model used in this study is only dependent on the valley cross - sectional shape, and the U-shape of a glacial valley is independent of bedrock type (Graf, 1970), differences in lithology from valley to valley should not have affected the results of this study. Varying lithologies along the axis of a glacier could produce irregularities in the profile of the glacier (Flint, 1971), however, stepped valleys were not used in this study because of the problems' in calculating mass flux created by extensional and compressional flow. Present Climate The varied topography of the . study area makes use of the traditional climatic classifications (e.g. Koppen) using mean temperature and precipitation almost impossible. Temperature and precipitation vary over short distances because of the terrain (Harding, 1982) and create a patchwork of climate types which are largely a function of elevation. However, general trends in air mass domination and moisture sources can be determined. Mitchell (1969) classifies the climate of this region using equivalent potential temperature (the temperature that a parcel of air would attain if it were allowed to rise pseudo-adiabatically until it has lost all its moisture and then allowed to descend back to its 15 original pressure (Blair and Fite, 1965)) to distinguish changes in air mass domination over the western United States. Mitchell's (1969) findings show the present study area to be dominated by moist Pacific air masses in the winter (Fig. 7) and drier interior continental air masses in the summer. 1000km - Streamlines of mean | ~ Major mountain January surface winds N masses Figure 7. Streamlines of mean January surface winds over the western United States (after Mitchell, 1969). Harding (1982) notes that the southeastern part of this study area is affected by strong upslope winds from the east during May and June accounting for precipitation maximums at this time (see Bozeman; Fig. 8). An analysis of winter precipitation patterns over western 16 Average monthly temperature Average monthly precipitation - - 44 ° Figure 8. Climographs and locations of selected cities within the study area (data from NOAA, 1985a, 1985b). 17 Montana by Locke (1989) showed a dominance of Pacific moisture to the northwest with decreasing precipitation to the southeast. Locke (1989) also found a secondary source of moisture from the Gulf of Mexico into the southeastern corner of the study area, which agrees with Harding's (1982) observations. Mountain ranges have an effect on the paths that storms follow (Price, 1981; Barry, 1981)'. In the study area, winter storms are steered around the central Idaho uplands allowing an influx of moisture along the Snake River Plain to the south (Fig. 7). Mackay Ranger Station, Idaho, shows a secondary maximum of precipitation in December-January (Fig. 8) which might be accounted for by its position on the northern boundary of the Snake River Plain (Fig, 5). Mountains also affect the local, as well as regional, precipitation patterns. Precipitation usually increases with increasing elevation due to the orographic effects of mountains, thus precipitation differences occur between valleys and mountains. Bozeman,. Montana (Fig. 8) has a 12 cm higher annual average precipitation than Belgrade 13 km to the west because of the orographic effects of the Bridger Range (Harding, 1982). The effect of elevation on precipitation is also noticeable in snowpack records. Plots of maximum winter snowpack versus elevation for the Lemhi, Beaverhead and Crazy Mountains are shown in Figure 9. In addition to the increase of precipitation with elevation,, these gradients also show that precipitation amounts at a given elevation are lower in the western portion of the study area (Lemhi Range) and increase to the east. 18 3000 T 2800 - ■ 2600 - ■ 2400 - ■ A - Crazy • B - Beaverhead * C - Lemhi north ■ D - Lemhi south □ E - Lemhi combined 2200 - • 2000 - - 1800 SNOWPACK (m) Figure 9. Modern snow accumulation gradients for the mountain ranges of the study valleys (data from Soil Conservation Service, 1986a, 1986b). Present climates supporting alpine glaciers within the studied areas exist only in the Crazy Mountains of Montana. Glacial climates do exist outside the study area: in the Beartooth Mountains to the southeast, the Salmon River Mountains of Idaho to the west, and in several of the mountain ranges (Mission, Swan, Flathead) to the north (Field, 1975; Graf, 1976). 19 Late-Pleistocene Climate During the late Pleistocene, the Cordilleran and Laurentide ice sheets that lay to the north of the study area (Fig. 6) were up to 1500 m thick in Montana (Waitt and Thorson, 1983). The presence of this ice not only altered the topography, but probably the temperature gradients as well, which in turn, modified regional windflow and precipitation patterns (Manabe and Broccoli, 1985; Kutzbach and Wright, 1985; Locke and Kemph, 1987). General Circulation Models (GCMs) (Kutzbach and Wright, 1985; Manabe and Broccoli, 1985) show a split jet stream over the Laurentide ice sheet over North America during the last glacial maximum. This split left most of the study area in a zone of weak westerlies with a slight easterly flow on the eastern edge". Such a wind flow pattern would suggest a Pacific moisture source for most of the area, with the possibility of a low level Gulf of Mexico source for the eastern edge. The paleoprecipitation patterns determined by Locke and Kemph (1987) show a trend of increasing moisture to the north and west over western Montana, agreeing faith a dominant Pacific moisture source. Their data also suggest that the Snake River Plain and Gulf of Mexico were locally important sources of moisture. Summer- temperatures in the Northern Rocky Mountains during the last glacial maximum have been estimated to be at least IO0 -C lower than present (Porter and others, 1983; Barry, 1983; Kutzbach and Wright, 1985; Manabe and Broccoli, 1985). ' A strong temperature gradient is interpreted by the GCMs (Kutzbach and Wright, 1985; Manabe and Broccoli, 1985) along the edge of the ice sheets in the Northern 20 Rocky Mountains. While the paleoclimatic analysis in this study did not address climate, temperature is important in modelling the ice dynamics. The present study provides estimates of average annual net accumulation at each glacier locality in addition to an analysis of the dynamics of each paleoglacier. While the sample size in this study is too small to provide accurate regional estimates of paleoprecipitation patterns, it does provide independent spot checks on other paleoclimatic studies in this area (Kutzbach and Wright, 1985; Manabe and Broccoli, 1985; Locke and Kemph, 1987). The results of this study are also compared with modern precipitation data for each locality. 21 METHODS This chapter details the methods, and underlying assumptions, employed in this study. These methods were tested and refined using the Big Timber Canyon paleoglacier on the east flank of the Crazy Mountains, Montana and the results from that test case are discussed in the next chapter. A model involving glacier flow theory can be used to determine mass balance parameters using the geologic evidence left by glaciers (Fig. I) . Underlying each of the steps in this model, are certain assumptions, detailed in this chapter, that affect the accuracy of the results. Figure 10 shows the. flow of the methodology used in this study to reconstruct the glacier dynamics and paleoclimate. Basically, the following steps were taken during the reconstructions: I) selection of a study valley according to specific criteria, 2) reconstruction of the glacier profile and areal extent, 3) calculation of average effective basal shear stress along the longitudinal profile, 4) calculation of mass flux from ice flow equations, 5) calculation of mass balance gradients from the calculated mass flux at the point of highest ice deformation and through the equilibrium-line altitude (ELA), and 6) climatic interpretation of the results by comparison to modern analogs. To ensure the accuracy of the results, a sensitivity analysis was performed during' steps 3, 4, and 5 by varying parameters such as glacier thickness and slope (see Results). 22 Theoretical Basal Shear Stress Lateral moraines Terminal moraines Trimlines SELECT VALLEY LOCATE APPLY CONTINUITY FLOW THEORY Published documentation Field survey Topographic maps A ir photos Highest lateral moraine (minimum) □west cirqi (maximum) AAR THAR CALCULATE NET BALANCE GRADIENTS Well-defined glacial features? Little post-glacial modification? Simple plan? Simple profile? RECONSTRUCT GLACIER GEOMETRY LOCATE MAXIMUM DEFORMATION FLUX BASAL SLIP ESTIMATES MASS FLUX AT THE ELA CLIMATIC INTERPRETATION Figure 10. Flowchart of the methodology used in this study. 23 . Valiev Selection In previous studies (Pierce, 1979; Haeberli and Penz, 1985; Leonard and others, 1986; McCalpin, 1986), all valleys in one area were used because the ice dynamics modelling was secondary to other studies of the glacial geology in those areas. The accuracy of their results was diminished because the flow law of ice is sensitive to changes in glacier thickness and slope, which are in part controlled by the underlying topography. The topography of most valleys in any area is conducive to creating changes in ice slope and thickness over short distances and creates problems when using glacial flow theory in ice dynamics modelling. In an effort to minimize the effects of topographically induced changes in ice thickness and surface slope, criteria were established for selecting valleys that would be suitable to the method. Valleys with stepped longitudinal profiles and many tributary glaciers were not used in order to minimize landform-induced extensionai and compressive flow. Areas where extensionai and compressive flow occur may have had actual velocities differing from those calculated by Glen's flow law (Pierce, 1979; Paterson, 1981).. Mountain range's with small ice caps were not used because of the difficulties in determining the boundaries of the source regioh of ice flowing into the valley glaciers. Some methods used to determine the location of the EIA are dependent on the areal extent, of th.e glacier, and valleys with well defined depositional (moraines) and erosional (trimlines, hanging valleys) glacial features were selected to enhance the accuracy of the . reconstructions of the geometries of the 24 paleoglaciers. Valleys that had minimal mass movement and alluvial valley fill since the glaciers retreated were best suited to this study because the accuracy of thickness and valley shape values was increased if the glacially eroded surface was well exposed. By using these selection criteria, the accuracy of the reconstructions was enhanced, and the final output from the model was as accurate and i- , precise as possible. Glacier Geometry The areal extent of each paleoglacier was determined from USGS topographic maps (scales 1:24000 and 1:62500), aerial photographs (Appendix A), and published documentation (Alden, 1932; Aten, 1974; Knoll, 1977; Ruppel, 1980; Ruppel and Lopez; 1981; Richmond; 19B"6") . Field checking of the geologic evidence of glaciation (terminal and lateral moraines, truncated spurs, hanging tributary valleys and glacial trimlines) was performed on all but two of the valleys during the summer of 1988. There was good correlation between the field checked features and the features observed on the topographic maps and air photographs. In the valleys that were not field checked (Stroud and Everson Creeks, Lemhi Range), glacial geologic evidence was well- preserved; therefore it was assumed that field checking would not significantly alter the results. Because multiple terminal moraines were present in most of the valleys, the crests of the largest, well-defined, morphologically fresh moraines were used as the outer limit of each paleoglacier. Published documentation (when available for a particular site) usually 25 correlated such moraines with the Pinedale-equivalent maxima. While the ages of the early and late Pinedale moraines vary by as much as a factor of three (Porter and others, 1983), differences in the areal extent of the glaciations are minimal. Terminal moraines from the late Pinedale maximum (20,000 years ago) are always present (Porter and others, 1983; Richmond, 1986) in the glaciated valleys of the Northern Rockies. Because the early and late Piriedale terminal moraines are usually indistinguishable, the use of either moraine should produce similar results in determining the glacier.length. Although the glacier length may have been the same, climatic differences between the two periods of glaciation- may have b,e;en substantial. For a given glacier length and valley. shap,e, a glacier in a warm/wet (e.g. maritime) climate should be thicker than one in a cold/dry (e.g. continental) climate, because more mass is exchanged on the glacier (Fig. 11). Any significant difference in climate (i.e. warm/wet versus cold/dry) using the terminal moraines in this study should be reflected in the number of lateral moraines,that grade into these terminal moraines (Table I) . If only one. set of lateral moraines graded into the terminal moraine, then that .set was assumed to represent the late Pinedale ice margin. If two sets of lateral moraines graded into the terminal moraine, the lower (in elevation) of the two sets was used to represent the late Pinedale ice'margin. The longitudinal profiles of both the bedrock arid the centerline, ice elevation along the axis of the each glacier were: drawn based on the reconstruction of the areal extent. In the upper :reaches of the valleys, depositional and erosional features were not as- prevalent as Continenta l ISi ________ KEY: m Net accumulat ion □ Net ablat ion m ELA c ro ss -sec t io na l area [%] Terminal moraine ^ — Flow l ines - Figure 11. Comparison of the shapes of maritime and continental glaciers for a given valley length. 27 Table I. Analysis of the number of lateral moraines that should exist in a valley, given differences in climate between two successive glaciations. Early Pinedale Late Pinedale Number of ice marginal features Warm and wet Warm and wet I Warm and wet Cold and dry 2 Cold and dry Warm and wet I Cold and dry. Cold and dry I ' in the lower reaches and reconstruction of the ice surface contours presented a problem. Theoretical glacier profiles were determined using a model (Schilling and Hollin, 1981) which, calculates ice surface elevations using theoretical average effective basal shear stresses (see next section). Values for the ice surface elevations from the theoretical profile that best fit the actual profile were used to reconstruct the ice surface contours in the upper reaches of ( the valley where geologic evidence of glaciation was lacking. In the lower reaches of- the valley, centerline ice surface elevations w^ ere assumed to be equivalent to the elevations of the ice margins. Comparison of the ■ surface contours of several modern glaciers in Alaska which are similar in plan form to the paleoglaciers used in this study (American Geographical Society, 1960) showed a mean difference of only + 3 . 9 m between the ice centerline and ice marginal elevations below the equilibrium line (Fig. 12). Because of this minimal variance, no correction factor was added to the ice surface elevations below the equilibrium line. Ice centerline elevations are lower than the ice margin above the equilibrium line of modern glaciers (Fig. 12). Because the theoretical ice elevations were lower E le va ti o n d if fe re n ce b e tw e e n ic e ce nt er li ne an d ic e m ar gi n (m ) 10 - ■ O » □ ■ - 1 0 - -20 - -30 - -40 - - 5 0 - ■ McCall-60 - * W e a l G u l k a n a D B e a r L a k e e A v e r a g e-70 - D * -600 -400 -200 E l e v a t i o n b e l o w / a b o v e ELA (m) Figure 12. Comparison of the difference in elevation between the ice centerline and the ice margins against elevation above or below the ELA on modern glaciers (data from American Geographical Society, 1960). 29 than the ice marginal features on the upper portions of many of the paleoglaciers in this study (see Appendices F-J), the theoretical ice elevations were assumed to be appropriate values of ice centerline elevation and were used for calculations of ice thickness, velocity and mass flux. Ice thicknesses were calculated from the difference of ice surface elevation and bedrock elevation. One of the underlying assumptions was that the valley floor elevation has changed little since late Pleistocene, time. Because there have been subsequent glaciations (as evidenced by recessional and re-advance moraines up- valley from the end moraines in most valleys), additional scouring of the bedrock has occurred in the upper reaches since maximum late Pinedale ice extent. The calculated ice thicknesses in this part of each valley were thus maximum values. In the lower reaches of the valley, re-advance and recessional till and post-glacial fill made the estimated ice thickness values a minimum. Where the present stream runs along bedrock (usually in the upper portions of the valleys), fill did not present a problem in estimating ice thicknesses. Basal Shear Stresses Calculation of basal shear stress along the ice centerline provides a check on the accuracy of the reconstructed longitudinal profile. Ice thickness and surface slope, which are determined from the shape of a reconstructed glacier, are used to calculate basal shear stress (Pierce, 1979). On most modern glaciers (Paterson, 1981) and well-studied paleoglaciers (Mathews, 1967; Pierce, 1979), average 30 effective basal shear stresses range from 0.5 to 1.5 bars. Excessively high (> 1.5 bars) or low (< 0.5 -bar) shear stresses could imply an error in reconstruction. While these values are not impossible to attain, Mathews (1967) explained that the dynamics of a glacier provide adjustments of slope and/or thickness over time which will yield values in the normal range. The profiles used in reconstructions are time-averaged (steady state) so the values of average effective basal shear stress should also lie in this narrow range. For a valley glacier, the average basal shear stress (T^ 3) can'be calculated from the relation: Tb = PgH Fsin « , (2) (Paterson, 1981) where p = specific gravity of ice (910 kg m"^) ; g = acceleration due to gravity (9.81 m s*^); H = centerline ice thickness (m) ; F = shape factor to account for drag on the valley sides (dimensionless) (Nye, 1965b) ; and CC = ice surface slope (dimensionless). The shape factor (F) is a function of the glacier half-width divided by the glacier thickness (W) (Fig, 13). Ice thickness, surface slope and valley shape were determined from the reconstructed geometry of the paleoglacier. In strongly extending or compressing flow, longitudinal or transverse normal gradients can add to the total, or effective, shear stress (Pierce, 1979; Paterson, 1981). In this study, basal shear stress calculated from equation 2 was used as the average effective basal shear stress. With uniform flow, effective basal shear stress and total basal shear stress are the same, and most compressing and 31 extending flow regimes should have been eliminated in this study by the valley selection criteria. .7 .8 .9 1.0.3 .4 F Figure 13. Values of the shape factor (F) for varying parabolic glacier cross-sectional shapes, represented by W values (W = glacier half width/glacier centerline thickness) (after Nye, 1965b). A theoretical profile of each glacier was calculated using the relation: ei+l = ei + X b * (3) Fpg Hi which is derived from equation 2 (Schilling and Hollin, 1981). Here Bi and ei+i are the ice surface elevations at steps Xi and xi+i respectively (Fig. 14), and the other variables are the same as in equation 2. This iterative approach is used to calculate glacier thickness (H) at 1000 ft (305 m) intervals (x) along each glacier 32 profile. This step length was used for simplicity (1000 ft = 1/2" on a 1:24000 map) and because no appreciable difference in the ice thickness was determined using a shorter step length. When = 0 (at the terminus), had to be chosen arbitrarily. In this study, the elevation of the lateral moraine at an arbitrary point up valley from the terminus was used as the ice elevation for the first step. The first step was chosen at a point where post-glacial fill did not create errors in thickness calculations. Profiles using various average effective basal shear stresses between 0.5 to 1.5 bars were computed for each valley and compared with the geomorphic evidence. The theoretical (using a constant T^) profile, which had the least difference in ice thickness from the morphologic profile, was used to estimate ice surface elevations in those regions where geomorphic evidence was absent. IaH ROCK Figure 14. Variables used in the iterative scheme for calculating theoretical glacier longitudinal profiles (after Schilling and Hollin, 1981). 33 \ Location of the EIA Once the glacial extent and longitudinal profile were determined, the location of the equilibrium-line altitude (ELA) was determined. Several methods for determining the EIAs of paleoglaciers have been developed and employed (Andrews, 1975; Porter, 1975; Meierding, 1982;. Leonard, 1985; Locke and Kemph, 1987). The paleo-ELAs for the glaciers in this study were estimated from I) highest lateral moraines, 2) lowest cirque floor elevation, 3) toe-headwall altitude ratio (THAR) and 4) accumulation area ratio (AAR). Meierding (1982) tested the accuracy of each of these methods in the Colorado Front Range. He found that the root mean square error of the methods ranged from 80 m for accumulation area ratio (AAR) and toe-headwall altitude ratio (THAR) to 109 m for cirque floor altitudes to 148 m for highest lateral moraines. Accuracy of the location of the paleo-ELA was., important because calculations of net mass exchange could only be made at that point. Lateral Moraines Lateral moraines develop in the ablation zone where ice flows outward towards the margins of the glacier and deposits debris. Because ice flow lines are descending in the accumulation area (Figs. 2,11), the highest point on the lateral moraine would indicate a change from descending to ascending flow and would therefore be an approximation of the location of the ELA (Andrews, 1975; Meierding, 1982) . Although lateral moraines were well developed in the lower reaches of many of the valleys, they have been subject to erosion and 34 mass wasting since the ice retreated. If the EIA was on a-portion of the glacier that was confined within the walls of a canyon, the lateral moraines that developed on this part of the glacier would have been supported by the ice. When the glacier retreated, the lateral moraines would have slumped to the bottom of the valley and the moraines would not have been preserved. The portion of the moraine that was preserved thus represents a minimum estimate of EIA. Lowest Ciraue Elevation Cirque floor elevations are widely used as a measure of the EIA of former cirque glaciers (Andrews, 1975; Meierdingl, 1982) . Meierding (1982) showed that although this method is rapid, it is also highly subjective because cirque floors are not always easily identifiable. Andrews (1975) pointed out that this method is better for a regional approximation of EIA if a trend surface connecting the.lowest north facing cirques is constructed. Also, valley glaciers -extend well, outside the cirques, thus EIAs using this method on valley glaciers should provide a maximum estimate. Most of the valleys used in this study have well-defined cirques, and the elevation of the lowest of these was used as a maximum estimate of the EIA. Toe-Headwall Altitude Ratio (THAR) An empirical relation between the highest and lowest ice limits is used extensively for the rapid determination of the EIA. Highest ice limits were determined from the highest elevation of ice on the reconstructed glaciers. Because the bedrock surface at the termini of the paleoglaciers was covered with till and outwash deposits, the 35 terminal elevation was determined by extrapolating elevations from surrounding pediments to the glacier, front wherever possible. If this was not possible, the elevation of the valley floor on the upstream side of the terminal moraine was used. Meierding (1982) found toe- headwall altitude ratios of .35 and .40 produced the best results. A THAR of .40 was used in this study because ELAs using a value of .35 were lower than the highest lateral moraine in all valleys. Accumulation-Area Ratios (AAR) Studies of modern glaciers have shown that the accumulation area of a glacier is about .65 of the total glacial area, however, this percentage may vary between .60 and .70 (Andrews, 1975). Several studies of paleoglaciers have used the .65 value (Porter, 1975; Meierding, 1982; Leonard, 1984) to estimate the ELA. Meierding (1982) calculated ELAs using values ranging from .50 to .75 and found .65 to have the least error. In this study, the area between successive 200 ft (61 m) and/or 400 ft (122 m) contours (depending on map contour interval) was measured and a cumulative total was plotted against elevation for each glacier. The ELA using an AAR value of .65 was determined for each paleoglacier using this graph. Estimated ELAs The final • ELA for each paleoglacier was estimated using a combination of those methods that agreed most closely in each valley. The highest lateral moraine elevation provided a minimum value for the ELA and was used if the THAR and AAR provided' ' lower values. Similarly, lowest cirque floor elevation is a maximum of ELA, so the 36 final ELA must b.e less than or equal to that value. By nature, THAR and AAR are the least -,subject to modification after deglaciation, and they are considered to produce the best estimates of ELA (Meierding, 1982) . If the ELAs calculated using THAR and AAR lie between the highest lateral moraine and the lowest cirque floor, an average of THAR and AAR was used as the final ELA. Glacier Flow Mass flux through a cross-section of a glacier was calculated by multiplying the average velocity by the cross-sectional area. The cross-sectional area was measured from the topographic maps, but the average velocity had to be estimated using glacial flow theory. As stated previously, glacier movement can be broken down into components of ice deformation and basal sliding, and the contribution pf each to total flow had to be determined. Ice Deformation By integrating Glen's flow law of ice over small increments of glacial thickness, Nye (1952) was able to calculate a vertical profile of the velocity due to deformation within a glacier cross-section. Assuming that the basal velocity is zero, the centerline surface velocity due to ice deformation (uc) can be determined by: uc = 2A( T b)n(H)/(n+l) (4) where: A = temperature^dependent constant of the flow law (0.167 bar-3 a-l at Q0C; Paterson, 1981) H = ice thickness at the centerline (m) 37 Tb = average effective basal shear stress (bar; calculated from equation 2) n = exponential constant of the flow law (n ~ 3 for valley glaciers; Paterson, 1981) By substituting equation 2 for equation 4 becomes: uc = 2A(i0g)n(sinnoc ) (H)n+1/(n+l) (5) and on valley glaciers (with n = 3), uc is proportional to the fourth power of thickness and the third power of surface slope. Therefore, estimation of thickness and surface slope must be as precise as possible. Longitudinal variations in velocity were minimized by averaging ice surface slope over 8-20(H) ■ (Raymond, , 1980). Local values of H are used in this study because they are the same as the average values in most cases. The constant A is a function of the ice temperature and thus should vary through the glacier thickness and length (i.e., with elevation). In a temperate or subpolar glacier, the basal temperature should be at the pressure melting point, approximately O0C. Because most of the motion due to ice deformation occurs near the base of the glacier (Nye, 1952; also Fig. 3), and because the temperature in this region should be near the pressure melting point (0 to -1°C) , A was assumed constant in this study. Basal Slip The other component of glacier flow • is from basal sliding. This has been measured to account for 3-90% of total velocity on modern glaciers (Andrews, 1975; Paterson, 1981). As discussed in the previous chapter, the component of basal sliding (referred to in this 38 study as basal slip) on paleoglaciers cannot be determined by subtracting calculated deformation velocity from measured surface velocity because the surface velocity cannot be measured on a vanished glacier. As a result, estimation of basal slip on paleoglaciers presented a problem. Previous studies (Haeberli and Penz, 1985, Leonard and others, 1986, McGalpin, 1986) assumed a constant percentage of slip along the entire length of the glacier. Basal slip may occur to varying degrees along a glacier depending on the thermal and moisture regime. Even subfreezing conditions do not prevent basal sliding (Echelmeyer and Zhongxiang, 1987), although sliding velocities account for very small portions of the total velocity under those conditions. Therefore, the assumption of a constant sliding velocity at all points on the glacier (Haeberli and Penz; 1985; Leonard and others; 1986; Holmlund, 1988) is not likely to provide an accurate estimation of slip. If the total flux on a glacier changes slowly with distance (Pierce, 1979; Raymond, 1980), highest deformation flux should be offset by lowest slip flux. Highest deformation flux occurs at the point of highest average effective basal shear stress, therefore this should also be the point of least slip. Although basal sliding should be at a minimum here, its actual percentage of total velocity cannot be determined for a paleoglacier. The assumption of zero basal slip at this point introduces the smallest possible bias to all velocity and discharge estimates, which are calculated relative to the point of highest average effective basal shear stress. "Variations in basal 39 slip along the length of the paleoglacier can also be calculated, although these estimates will also be minima. Because basal shear stresses may vary due to extensive and compressive flow arising from irregularities in the bedrock slope, the above approach will not work in a multi-stepped glacial valley. In this study, valleys with gentle, constantly sloping floors were used to avoid compression and extension problems. Therefore, the place where the highest average effective basal shear stress occurred was a reasonable approximation of the place where slip was at a minimum. Average Velocity Assuming no slip at the point of maximum averagb effective basal shear stress, ice surface centerline velocity (uc) was determined using equation 4. Frictional drag on the valley floor and walls causes the average velocity of a glacier through a cross-section to be less than the centerline velocity. Nye (1965b) calculated that the ratios of average velocity through a cross-section (u) to the surface centerline velocity (uc) for glacial channels of varying parabolic shapes average u/uc = 0.63. This ratio assumes no basal sliding and therefore produces a minimum estimate of average velocity. With very high values of slip, this ratio can increase as much as 15 percent (Raymond, 1980) . The calculation of uc from equation 4 also assumes no slip; therefore a value of .63uc should be representative of the average deformation velocity (u) through the cross-section. The deformation mass flux at the point where average effective basal shear stress was a maximum was calculated by multiplying u by the cross - sectional area at that point. If the highest effective 40 basal shear stress occurred at the EIA, then the value derived by this method was also a minimum approximation of net mass exchange (mass balance). If the point of highest average effective basal shear stress occurred elsewhere on the glacier, net mass balance gradients were calculated for the glacier, and the mass flux at - the ELA was extrapolated from this point. Ablation/Accumulation Gradients Net balance (accumulation and ablation) gradients define the specific net balance (average annual net gain or loss of mass) at elevations above and below the ELA, respectively. As 'such, the net accumulation or ablation above or below any point on a glacier can be determined by multiplying the gradients by the surface area (Fig. 15) . Conversely, if the net accumulation or ablation and the surface area are known, then accumulation and ablation gradients can be calculated. In a steady state glacier system, the mass flux at any point on the glacier is equal to the net mass gain (net accumulation - net ablation) upstream from that point. Ideally, mass flux on a glacier increases from zero at the headwall to a maximum at the ELA and then decreases in the ablation zone to zero at the terminus (Raymond, 1980) . In this study, the calculated mass flux at the point of maximum deformation flux (minimum slip flux) provided a minimum estimate of the net accumulation above that point on the glacier. The surface area was determined in the reconstruction; thus net balance gradients were calculated to match the net accumulation above and the mass flux at the point of maximum deformation flux. In this study, cT z o H < > UJ X W 5 E AREA ( IO2 km2/ 152 m (500 ft)) SPECIFIC NET BALANCE (meters of H2O) - 0.6 - 0.4 - 0.2 0 + 0.2 + 0.4 + 0 .6 + 0.8 + 1.0 + 1.2 + 1.4 NET BALANCE (km3/ 152 m (500 ft)) Figure 15. Mass balance of the late Pleistocene Yellowstone ice cap (Pierce, 1979). The reconstructed surface area (left) and the estimated mass flux at the ELA allow the calculation of average mass balance gradients (center), thus specific (center) and net (right) mass balance. 42 the point of maximum deformation flux occurred at of above the ELA, so accumulation gradients were used to produce. the match. If the point of maximum deformation flux occurred below the EIA, ablation gradients would be used. When the point of maximum deformation flux occurred at the EIA, the calculated mass flux provided a minimum estimate of the net mass exchange. Accumulation and ablation gradients that produced net accumulation and net ablation equal to the net mass exchange were then calculated (Fig. 15). When the point of maximum deformation flux occurred above the ELA, the calculated accumulation gradients produced an estimate of net accumulation above the EIA (net mass- exchange) as well as the net accumulation above the point of minimum, slip, because the gradients were calculated relative to the ELA. Ablation gradients that produced net ablation equal to the net mass exchange were then calculated. The gradients also allowed an estimation of basal slip at discrete points along the glacier. From the continuity equation (Fig. 16) , the difference between the mass flux at one cross-section (Q]_) and a down-ice cross-section (Q2) is equal to the amount of mass (A) that is ablated (-) or accumulated (+) over the glacier surface area between these two cross-sections such that: Ql + A = Q2 (6) or Di + Si + A = D2 + Sg (7) Average ice velocities from deformation (u) were calculated at discrete points along the glacier, and when multiplied .by the cross- CONTINUITY EQUATION: Q1 + A = Q. MASS FLUX (Q) = DEFORMATION FLUX (D) + SLIP FLUX (S) A = net accumulation (+) or ablation (-) over the surface area between Q1 and Q2 Figure 16. Continuity theory. 44 sectional area normal to the glacier surface at those points, provided estimates of the mass flux caused by deformation (D) . Starting with the point of maximum deformation flux as D^, deformation mass flux was calculated at the next point downstream (Dg) , either the ELA or the next 200 ft (61 m) or 400 ft (122 m) contour, depending on the topographic map contour interval. The mass balance gradient between these two points determined the amount of mass (A) gained (+) or lost (-) on the glacier surface between them. At the point of maximum deformation flux (D%), was assumed to be equal to zero, and equation 7 reduced to: Sg = + A - D g (8) Equation 8 provided a first approximation of the slip flux component (Sg) of Qg. This method was continued up and down the glacier using the continuity flux as to provide basal slip estimates along the glacier's length. Paleoclimatic Interpretation The mass balance gradients provided an estimation of climate because they are indicative of the precipitation and temperature regimes in which the glacier exists. High ablation gradients (> 10 mm/m) are typical of maritime climates where large amounts of precipitation are offset by the moderate temperatures that induce melting (Meier and others, 1971). As the climate becomes more continental, there is less precipitation. Haeberli and Penz (1985) compared their calculated balance gradients with modern analogs and determined that the climate of the European Alps was cold and dry 45 during the last glacial maximum. Leonard and others (1986) concluded that the Colorado Front Range was also cold and dry by comparing their calculated gradient with modern analogs. The calculated ablation gradients in this study were also compared to modern gradients. Another paleoclimatic variable that was determined from this study was net accumulation, which was equal to the net mass exchange. This value was divided by the accumulation area to provide an estimate of the average annual net accumulation over the accumulation area. These estimates were then compared with modern accumulation data. Average annual precipitation data were not available for all the mountain locations in this study. Climatological reporting stations are usually situated in the valleys (NCAA, 1985a, 1985b) and precipitation data from these stations do not reflect the precipitation received at higher elevations. Modern snowpack records are available from snow courses in the mountain ranges in the study area (Soil Conservation Survey, 1986a, 1986b), providing an estimate of winter precipitation at each site. Total water content of the snowpack is measured on or about the first of each month at these sites. The data are averaged over 25 years (1961-1985) and the maximum average snowpack was used in this study to represent the net winter accumulation at each site. Because' these values represent accumulated snowpack, they may underestimate total winter precipitation. Early season snow may melt rather than accumulate and would not be included in later measurements. The maximum snowfall does not necessarily fall on the first of the month either, thus measured maximum snowpack may be less than the total 46 average annual snowfall in any given year. However, although the values may be minima, they are still the best available precipitation data for the mountain areas in this study. Linear approximations of increasing snowfall with elevation have been used by others (Porter and others, 1983; Leonard, 1984; Locke, 1989) to approximate snowfall at high elevations where snow course data are not available. In reality, the relationship is non-linear at high elevations. The exact relationship can only be defined with adequate measurements and is only valid for a restricted geographic area (Locke, 1989) . Because there are not enough data to define the exact relationship, linear gradients were used to approximate modern net winter precipitation. Plots of net winter accumulation versus elevation were made for the Lemhi, Beaverhead and Crazy mountains (see Introduction, Fig. 8). The closest stations to each of the study valleys were chosen to represent the actual altitudinal distribution , of total winter accumulation. Wherever possible, only stations that were on the same side of the range crest were used. An exception to this was the Crazy Mountains because no snow course data were available for the east side of the range where Big Timber Canyon is located. Data from the northwest end of the range was used instead. The Lemhi Range was divided into a northern and southern gradient to represent the difference between Mill, Stroud and Everson Creeks (northern) and Meadow Lake (southern). An average gradient using all the Lemhi stations was also determined. Calculated average annual net accumulation for each glacier was then compared with extrapolated 47 precipitation values from the gradients. Thus, differences between late Pleistocene and modern precipitation at all of the sites were assessed in this study. 48 RESULTS Case Study - Big Timber Canvon Valley Selection Big Timber Canyon in the Crazy Mountains of Montana (Fig. 4) was used to develop and refine the methods presented in the previous chapter. This glacially sculpted U-shaped valley (Fig. 17,18) was chosen because it has a straight channel, few tributary cirques, well- marked trimlines and a nearly constant, gentle gradient. Glacially eroded bedrock is prevalent in the upper reaches of the valley while in the lower reaches of the valley, some valley fill is present (Fig. 18b) . Big Timber Canyon meets most of the valley selection criteria and provides a good place to test the reconstruction of paleoglacier dynamics using glacial flow theory. Glacier Geometry The geologic evidence left by the Big Timber glacier (Fig. 18b) was determined from USGS topographic maps (scale 1:24000), aerial photographs (Appendix A) , published documentation (Alden, 1932.; Aten, 1974) and field survey. Trimline and moraine crest positions (Fig. 18b) were located using the air photos and plotted on the topographic maps. These features were used as the boundaries of the ice mass. The areal extent of till deposits (Fig. 18b) was determined from Aten's (1974) map. Some modifications of the extent were made using the air photos. Till blanket (Fig. 18b) refers to areas of 49 thick (> 3 m) till deposits (mostly moraines), whereas areas assumed to be overlain by a thin covering of till are labeled as till veneer. Areas covered by post-glacial fill (alluvium), landslide and talus deposits were also identified because those areas indicate regions where the valley shape has been modified since the glacier retreated. Figure 17. Photo looking down Big Timber Canyon showing the transition from a V-shaped to a U-shaped valley. Only deposits from the Pinedale I advance identified by Aten (1974) were used in this reconstruction because that advance was the last major glaciation of the valley and those features are best preserved. Because only one maximum Finedale moraine was identified (Aten, 1974) , it was thought to correlate with the last glacial maximum, approximately 20,000 years ago (Porter and others, 1983; Richmond, 1986). Prior glaciation limits have been obscured by this 50 Sur f ic ia l Geo logy ' V Morainec res t Trim line E 3 Till blanket Talus CD Landslide □ Alluvium 8000' 6000' Figure 18. Map of Big Timber Canyon. Topography (a) and surficial geology (b) were used to reconstruct the glacier surface (c) . < 51' advance and subsequent glaciations were minor. Ice from both Bull Lake and Pinedale I advances filled the valley to approximately the same elevation (Aten, 1974); thus distinction between the trimlines of each advance was not possible from the.air photos. Only one set of trimlines was distinguishable from the the field survey as well. Because these trimlines graded into the Pinedale I ■ moraine crests along the valley sides (Fig. 18b) , the trimlines were assumed to be the same age as the moraines. The areal extent of. the paleoglacier during the last glacial maximum was determined (Fig. .18c) using these geomorphic features. Based on the reconstruction of the areal extent (Fig. 18c), the longitudinal profile of the glacier surface was drawn along the axis of the valley (Fig. 19). Ice surface elevations were determined -from the reconstructed topography (Fig. 18c), and bedrock elevations were taken from the topographic maps. Ice thicknesses near the terminus are minimum values because the bedrock in this part of the valley is covered by post-glacial fill (Fig. 18b). In the upper reaches of the valley, the stream runs along bedrock and calculated values of ice thickness on this part of the glacier are accurate. The reconstructed geometry (Figs. 18c,19) shows that the glacier was approximately 18 km long. Approximately 14 km up the valley from the terminus, the main glacier split into two branches, both of which headed in mainly northeast-facing cirques. Three small cirques fed the main glacier between 10 and 13 km from the terminus. Maximum thicknesses were reached between 9 and 10 km up the .valley from the va tio n .-12 BIG TIMBER GLACIER Theory Trimlines , Moraines Ii -i iu I ■ r w 20- . -6 H-------------------- ----------------------------------------- eTe---------------------------------- Vertical Ex.: 2.5 0 2 4 6 8 10 12 14 16 18 (km) Figure 19. Longitudinal profile of Big Timber glacier 53 terminus (Fig. . 19, Table 2), where the glacier was 3.25-350 meters thick. Basal Shear Stresses Average effective basal shear stresses (T-J3) at the ice centerline (Table 2, Fig. 20) were calculated at 1000 ft (305 m) increments from the terminus to approximately 14.3 km up the valley, where ice marginal features became obscured. Local ice surface slopes averaged over 2000 ft (610 m) were originally used in calculating T-J3, but wide fluctuations in shear stress from step to step occurred (Fig. 20). These fluctuations in shear stress would cause ..fluctuations in deformation velocity from step to step also (equation 4) , yet total flux on a glacier changes slowly with distance. Raymond (1980) showed that using surface slopes averaged over 8-20(H) to calculate Tj3 provided better agreement with longitudinal variations in velocity than using local slopes. The shape factor (F) is also averaged over the same distance. On Big Timber glacier, ice surface slopes and shape factors averaged over 8-20(H) (8000 ft in most cases) were used to calculate the average effective basal shear stresses (Fig. 20). Values for TJ3 under the Big Timber glacier where ice marginal features are well defined (4.0 - 12.0 km from the terminus) range from 0.88 to 1.15 bars (Table 2), well within the range of the expected values. Highest average effective basal shear stresses occur 9.0- 12.0 km up the valley (highest = 1.15 bars, 10.1 km), and coincide with the thickest part of the glacier (Fig. 19), illustrating the relationship of average effective basal shear stress to thickness. In the lowest reaches of the glacier (0 - 4.0 km from the terminus), 54 Table 2. Big Timber paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Step No. Distance Bedrock from Elev- Terminus ation (km) (m) Ice Ice Elev- Thickness ation (m) (m) Shape Factor Calc. ^b1 (bar) Theoretical Ice Elevation^ (m) Diff. (Theo.- Calc.) (m) I 0.3 1628 1664 37 0.84 3 0.9 1646 1725 79 0.85 0.51 5 1.5 1658 1768 H O 0.83 0.49 7 2.1 1670 1780 H O 0.78 0.31 9 2.7 1682 1804 122 0.79 0.34 11 3.4 1695 1823 128 0.77 0.51 13 4.0 1707 1865 158 0.82 0.71 15 4.6 1722 1920 198 0.69 0.88 1927 + 7 17 5.2 1737 1963 226 0.68 0.98 1974 + 11 19 5.8 1752 1999 247 0.69 1.01 2015 f 16 21 6.4 1767 2042 274 0.68 1.00 2052 + 10 23 7.0 1786 2082 296 0.68 1.00 2087 + 5 25 7.6 1804 2109 305 0.69 1.00 2120 + 11 27 8.2 1825 2134 308 0.69 1.01 2151 + 17 29 8.8 1847 2170 323 0.69 1.02 2181 + 11 31 9.4 1877 2207 329 0.67 1.12 2211 + 5 33 10.1 1889 2240 351 0.67 1.15 2241 + I 35 10.7 1926 2268 341 0.66 1.10 2270 + 2 37 11.3 1965 2301 39 11.9 2060 2341 280 0.63 1.08 2335 - 6 41 12.5 2133 2390 256 0.63 1.04 2376 - 14 43 13.1 2200 2422 45 13.7 2258 2487 229 0.67 1.08 2470 - 17 47 14.3 2316 2536 219 0.68 1.07 2521 - 15 49 14.9 2359 2585 226 2570 - 15 51 15.5 2389 2616 53 16.2 2426 2660 55 16.8 2447 2701 57 17.4 2462 2739 59 18.0 2731 2780 ^Calculated from Equation (2) ^Calculated from Equation (3) assuming Ty = 1.0 bar. 55 unusually low values of shear stress (0.31 - 0.51 bar) occurred. The addition of melt water to the glacier bed in this region from Amelong Creek (Fig. 18a) could have significantly increased basal sliding. In such a situation, the average effective basal shear stress would be very low and the glacier surface gradient would have been lowered. Also, ice contact features in this part of the valley have been modified by stream erosion and mass wasting since they were deposited. The reconstruction in this part of the valley is tentative and may lead to the low values of average effective basal shear stress. LU V) <0 1.0 - D 2000 ft slope average • 8000 ft slope average Distance from Term inus (km ) Figure 20. Average basal shear stress along Big Timber glacier with ice surface slope averaged over 2000 and 8000 ft. Gaps indicate places where the geologic evidence of ice thickness was missing. 56 A computer program (Appendix B) was developed (after Schilling and Hollin, 1981) to calculate the theoretical profile of a glacier using equation 3. Several theoretical longitudinal profiles were calculated for Big Timber glacier using this program. The profile using an average effective basal shear stress of 1.0 bar provided the best agreement with the ice marginal features (Table 2). In- the upper reaches of the valley, where geomorphic evidence of glaciation was absent, ice surface elevations and ice thicknesses from the theoretical profile were used to calculate velocity and mass flux. Location of the ELA The paleo-ELA of the late Pleistocene Big Timber glacier was estimated using highest lateral moraine' elevations, lowest cirque floor elevation, THAR and AAR (Meierding, 1982). Several different ratios were used for THAR (.35, .40, .45) and AAR (.60, .65, .75) to test the correlation between methods. Each method was analyzed for its reliability as related to the Big Timber glacier and the rest of the study. For Big Timber glacier, the ELAs estimated using all the methods- ranged from 2163 m to 2452 m with a mean of 2252 m (Table 3) ., Table 3. ELAs for Big Timber glacier using each of the methods listed in the text. Method Lowest Cirque Elevation .35 THAR .40 .45 .60 AAR .65 .70 Highest Lateral Moraine ELA (m) 2432 2163 2241 2318 2170 2231 2286 2170 57 Lateral Moraines. Lateral moraines in Big Timber Canyon (Fig. 18b) are well developed in the lower reaches of the valley and extend some 8.8 km up the valley. The elevation of the highest of these moraines is 2170 m on the southern edge of the glacier. Above this elevation, the glacier was confined within the walls of the canyon. The lateral moraines that may have developed on this part of the glacier would have slumped after retreat of the ice. Therefore, the portion of the moraine that is preserved in Big Timber Canyon represents a minimum estimate of ELA. Lowest Cirque Elevation. In the Big Timber valley, there are many well defined cirques. The lowest cirque occurs/ in the north branch of the main valley at an elevation of 2432 m. Toe-Headwall Altitude Ratio (THAR). THAR estimates were made using the highest ice limits on the north face of Crazy: Peak at an altitude of 3170 m. Because the bedrock surface at the terminus was covered with till (Fig. 18b), the elevation of the toe of the glacier was estimated by extrapolating■the elevation of the surrounding sub- alluvial terrace upon which the till was deposited (Aten, 1974). ELAs using THARs of .35, .40, .45 ranged from 2163 to 2318 m (Table 3). Accumulation Area Ratio (AAR). The area between successive 400 ft (122 m) contours on the glacier surface (Fig. 18c) was measured using a planimeter. A cumulative total of the area from the terminus to headwall was plotted against elevation (Fig. 21).' From this graph, ELAs using values of .60, .65 and .70 were determined, ranging from 2170 to 2286 m (Table 3). 58 3000 - 2 5 0 0 - iii 2000- AAR = .6 5 1500 C um u la tiv e P e rc e n t A re a Figure 21. Hypsometric curve of Big Timber glacier used to estimate the ELA using AAR method. ELA using AAR = .65 is shown. Final ELA■ ELAs using a THAR of .35, the highest lateral moraine and an AAR of .70 were within 7 m of each other (Table 3). There was also good agreement between the THAR of .40 and the AAR of .65, both of which Meierding (1982) found to give the best results. Because mass wasting of the lateral moraines would produce a low estimate for the ELA and because Meierding found the latter two methods to produce the best results, an ELA of 2240 m was used. Mass flux through the cross-section perpendicular to the glacier surface at this elevation was used to estimate mass balance. 59 Glacier Flow Glacier velocity due to deformation was calculated at discrete points along the paleoglacier using equation 2. On the Big Timber glacier, the highest average effective basal shear stress occurs by coincidence at the ELA (Table 2) . Assuming no slip at this location, a calculated average effective basal shear stress of 1.15 bars, and an ice thickness of 351 m, the centerline velocity at the ELA is 44.5 m a'-*-. This value is well within the range of velocities observed on modern glaciers (Paterson, 1981). Total average annual mass flux through the ELA (u x cross - sectional area) is 8.8 x IO^ m^ a" Because basal slip is assumed to be zero at this location, the calculated, velocity and mass flux here are minima. Any value of slip can be assumed at this location, however all other values of slip flux elsewhere on the glacier will also increase. If 50% slip is assumed, uc increases to 89.0 m a-^ and the mass flux at the ELA increases to 22.3 x IO^ m^ a'^ - (Continuity Flux 2, Fig. 22). In that case however, slip would have to account for almost all the velocity in the rest of the glacier. This seems unlikely because the glacier was probably frozen to its bed in the cirque regions. Ablation/Accumulation Gradients On a steady state glacier such as the Big Timber glacier, the mass that moves through the ELA must be lost in the ablation zone. A specific linear ablation gradient will provide the balance between the net mass exchange and the mass ablated over the glacier surface area below the ELA. The area between successive 400 ft (122 m) contours on the glacier was already determined for locating the ELA using AAR. M as s Fl ux ( IO 6 25.0-r Deformation Flux Slip Flux I 20.0 - Slip Flux 2 10.0 _ Flux 5.0 Distance from Terminus (km) Figure 22. Mass flux on Big Timber glacier. Continuity Flux I assumes no slip at the EIA, continuity flux 2 assumes 50% slip at the ELA. Slip flux is the difference between continuity flux and deformation flux. C irque C onfluence 61 The midpoint elevation of each of these areas is used as the average elevation of the entire area. A simple program was developed (Appendix D) to calculate the amount of ablation that would occur on the glacier in each contour interval, using the average altitude of the area below the ELA, a specific linear gradient and the surface area (Table 4). Only one gradient (within the limits of resolution) will produce a balance between the net mass flux at the ELA and the amount ablated on the glacier surface (Fig. 23). The accumulation gradient is calculated in a similar manner, using the surface areas above the ELA (Appendix E). Table 4. Calculation of average annual net ablation on Big Timber glacier using area and elevation. Mass Flux at ELA - 8.,8 X IO6 m3 a-1 Ablation Gradient = 3.0 mm/m (I) (2) (3) (4) (5) Elevation Surface Average Altitude Specific Balance Volume Lost Interval Area below the ELA (3) x gradient (2) x (4) (m) (IO6 m2) (m) (m) (106 m3) 2195-2240 .98 23 0.069 0.067 2073-2195 3.76 106 0.320 1.203 1951-2073 2.3 228 0.686 1.577 1829-1951 1.6 350 1.052 1.682 1707-1829 2.4 472 1.417 3.402 1615-1707 .47 579 1.737 0.817 Net Ablation = 8.749 For the Big Timber glacier, an ablation gradient of 3.0 + 0.6 mm/m produced the balance between net mass exchange and ablation. An accumulation gradient of 1.0 + 0.2 mm/m above the ELA produced a balance between net accumulation and net mass exchange (Fig. 23). O 6 Area ( I O6 m2) 2 4 3 0 0 0 £ 2500 ro > w 2000Kk 1500 Spec i f i c Net Balance i i r i A c c u m u l a t i o n Q A b l a t i o n ELA N e t A c c u m u l a t i o n 8 . 8 x I O6 m3 -I O I (m H2O) - 4 -2 Net Ba lance ( 1 0 m ) ChN> Figure 23. Mass balance of the late Pleistocene Big Timber glacier. 63 Such a low gradient is typical of cold, low-moisture environments (e.g., McCall Glacier, Brooks Range, Alaska: 2 mm/m; Meier and others, 1971) showing that the Big Timber glacier at its peak was sustained more by low temperatures than by high precipitation. The accumulation and ablation gradients also allow an estimation of basal slip around the point of assumed no-slip as outlined in the previous chapter. Table 5 shows the estimated slip up- and down-ice from the ELA on the Big Timber glacier. Slip becomes the dominant part (>90%) of the flow in the lower 4.0 km of the glacier. Because Amelong Creek was supplying meltwater from Amelong Glacier (Fig. 18c) to the bed of Big Timber glacier at this point, a substantial increase in basal slip is logical. Even with the accepted error in the deformation mass flux on this glacier (see next section) , continuity theory shows that basal slip was the dominant part of flow in this part of the glacier. Slip is also required above the ELA, and was responsible for at least 60% of the total flow (Fig. 22, Table 5). Sensitivity and Error Analysis The calculations of T ^ , uc and mass balance are only as good as the data and assumptions on which they were based. All values of surface slope, ice thickness and cross - sectional area were analyzed from topographic maps, using air photos and published documentation as aids in the analysis. Limited field work was done to check whether the valley floor was bedrock or fill. IJsing Big Timber as the test case, sensitivity of the model to changes in surface slope and thickness was performed to determine the error limits of the model. 64 Table 5. Mass balance and basal slip results for Big Timber glacier. Ice surface elevation (m) Centerline ice velocity (uc) (m a" Deformation mass flux (nr a'^) Point of Maximum Shear Stress CELA') 2240 44.5 8.8 x IO6 Mass Balance Accumulation Gradient = 1.0 ± .20 mm/m Ablation Gradient = 3.0 ± .60 mm/m Net Winter Precipitation = .42 ± .08 m Basal Slip Calculations (I) (2) (3) (4) (5) (6) (7) (8) Ice Mass Continuity Cross Average Creep Slip % Elevation Gain(+) Mass Sectional Velocity Velocity Velocity Slip /Loss(- ) Flux Area1 (3)/(4) (5) (5)-(6) (m) (106 m3) (106 m3 a'1) (105 m2) r-f(ti_e (m a*•*■) (m a' )^ 2560-3170 +7.88 7.9 2.86 27.6 11.0 16.6 60 2438-2560 + .518 8.4 1.90 44.2 14.6 29.6 67 2316-2438 + .314 8.7 1.55 56.1 18.5 37.6 67 2240-2316 + .104 8.8 3.17 28.1 28.1 0.0 0 2195-2240 - .067 8.7 3.11 28.0 24.3 3.7 13 2073-2195 -1.20 7.5 2.37 31.8 16.7 15.1 48 1951-2073 -1.58 5.9 1.41 42.2 13.4 28.8 68 1829-1951 -1.68 4.2 0.86 49.8 7.7 42.1 85 1707-1829 1615-1707 -3.40 -0.82 0.80 0 0.47 18.4 1.2 17.2 93 3Cross sections are taken at the lowest elevation in each area. The accuracy of the topographic maps may well be the most important question when determining error. Thompson (1979) shows that for statistical purposes, the allowable root mean square error (RMSE) for a topographic map of scale 1:24,000 is: allowable RMSE - 0.3CI + 24(tan a) (9) where Cl = contour interval and a = slope angle. In the Big Timber 65 study, the maps that were used had contour intervals of 20, 40 and 80 'ft (6.1, 12.2, 24.4 m) . Most of the glacier surface that was calculated from ice marginal features lay within the 40 ft (12.2 m) contour map. Slope angles along the moraine crests were well below 20°; thus tan a is almost negligible. For this map, the allowable RMSE (one standard deviation) is ± 20 ft (6.1 m) assuming a slope of 18°. However, this is the allowable two-tailed vertical error for any single point on a map; thus the probability of the point elevation being greater than or less than this limit is 0.16. .Calculations, of slope and thickness are determined from multiple points, thus -the probabilities of errors being at the extremes are multiplicative. Table 6 shows the calculations of slope, thickness, average effective basal shear stress and centerline velocity based on the maximum errors for 0.5, 0.83 and 1.0 times the RMSE ( Cf ) along with the probabilities of these events occurring. These results show' that even for . 5 cr (± 3.0 m) with a maximum accumulated calculation error of ± 20%, the probability that compounding errors of thickness and slope estimation will occur ■ is only 0.0092. Based on these probabilities, it is likely (p > 99%) that the calculated values of T^ 3, uc and mass flux using the topographic map data are valid ± 20%. Ice surface slope produced the most variance in calculations of average effective basal shear stress (Table 6). Because glacier velocity is not affected by local changes in surface slope (R.aymond, 1980), ice surface slope was averaged over 8-20(H) (2.4 km for Big Timber glacier). Inherent errors from the topographic maps are also 66 Table 6. Error and probability analysis of map-induced errors on basal shear stress and velocity calculations at the ELA for Big Timber glacier. Error none .5cr X ± 3. High Low^ 0 m) P3 .83*(± 4. High Low 6 m) P a (± 6.1 m) High Low p Slope (sina ) Tb (bar) .055 1.15 .057 1.19 .052 1.09 .096 .059 1.23 .051 1.07 .044 .060 1.25 .050 1.05 .026 Thickness (m) Tb (bar) 351 1.15 357 1.17 344 1.13 .096 360 1.18 341 1.12 .044 363 1.19 338 1.11 .026 Tb using combined worst error 1.15 1.21 1.07 .0092 1.26 1.04 .0019 1.29 1.01 .0006 Centerline 44.5 52.8 35.2 .0092 60.1 32.1 .0019 65.0 29.1 .0006 Velocity (m a'1) •*- LU - J LU 2200 - - 2000 - - B Big Timber - A Miner Lakes - B Mill Creek - C Stroud Creek - C Everson Creek - C Meadow Lake - D 1800 0.4 0.6 0.8 SNOWPACK (m) 1.0 Figure 34. Comparison of calculated net accumulation (solid triangles) to modern snowpack accumulation gradients for the mountain ranges in this study (same labels as Figure 9: A = Crazy; B = Beaverhead; C = Lemhi north; D = Lemhi south; E = Lemhi combined). 117 annual precipitation and modern net winter accumulation, the actual decrease was probably greater and therefore closer to the values of Locke and Kemph (1987). In general, there was a greater difference in the Crazy Mountains than in the Beaverhead Range which is consistent with the findings of Locke and Kemph (1987). Expansion of the work of Locke and Kemph (1987) to Idaho (Locke and others, 1989) suggests that paleoprecipitation there was also lower than present, consistent with the results of this study in the Lemhi Range. General Circulation Models (Kutzbach and Wright, 1985; Manabe and Broccoli, 1985) have interpreted a weaker westerly airflow over the study area during the last glacial maximum. A weaker jet stream over this area would reduce the frequency of storms through the area and therefore reduce the total annual precipitation. The results of this study are consistent with that interpretation. Kutzbach and Wright (1985) also interpreted a 25-30% decrease in total annual precipitation over the region. While this is an average value for the northwestern United States, results from this study and others (Locke and Kemph, 1987; Locke and others, 1989) indicate that the average decrease may have been up to 50% less than present. 118 CONCLUSIONS A reliable reconstruction of the areal extent of late Pleistocene glaciers can be made using topographic maps and aerial photographs if evidence such as moraines and trimlines are preserved. From this reconstruction, calculations of average effective basal shear stress, ice velocity and mass balance can be made. Allowable error in -the topographic maps affects the precision of the calculations, but the probability of a 20% error in calculated velocity (thus mass flux) is only 1%. Field survey is useful in determining the extent and -amount of post-glacial modification of the glaciated valleys. Calculations of average effective basal shear stress using the data derived from the maps can be used as a check on the validity of the reconstruction. Normal effective basal shear stresses should range from 0.5 to 1.5 bar if the reconstruction is valid. Unusually low values of basal shear stress in the terminal areas of paleoglaciers can be induced by error in reconstruction. These values may also reflect increased basal sliding in the terminal area from the addition of meltwater to the bed or from basal till deformation. Sensitivity of the flow law equations to.changes in ice slope can be minimized by selecting valleys with constant or slowly varying slope and by averaging slope and valley shape factors over a distance of 8-20 times the ice thickness. Local variations in surface slope do not affect the glacier velocity over short distances. This averaging also reduces the effect of error in the topographic maps by making the 119 vertical error a smaller percentage of the distance over which slope is averaged. For a valley with a floor of constant or slowly varying slope, the mass balance of a paleoglacier can be calculated assuming no basal sliding occurs at the point where the highest deformation mass flux occurs. This point should occur at or near the ELA where lower temperatures should reduce the amount of basal meltwater and where thicknesses and valley cross -sectional areas are generally greatest. If this- point, occurs at a point other than the ElA, net mass exchange at the EIA can be calculated from the continuity equation and an appropriate mass balance gradient. Net mass - exchange is equal to the net accumulation and net ablation on a steady state glacier. Use of the continuity equation also provides an estimation of basal slip along the length of the glacier. Previous studies (Haeberli and Penz, 1985; Leonard and others, 1986; McCalpin1 1986) assumed a constant slip percentage at the EIA but could not model the variation in slip along the rest of the glacier. This study provides the first estimation of the variation of slip along the length of paleoglaciers. The actual values are minima. However, they do show that basal slip varies from glacier to glacier and that the assumption of a constant slip percentage at the EIA for all glaciers is inaccurate. Comparison of calculated mass balance gradients to modern analogs provides an estimation of the climate that occurred when the glaciers were at their peak. Calculated mass balance gradients in this study were similar to modern gradients found in the Brooks Range of Alaska 120 (McCall Glacier: 2 mm/m; Meier and others, 1971). This indicates that the climate of the last glacial maximum in southwestern Montana and northwestern Idaho was much drier and probably colder than present. Mass balance estimates also provide paleoprecipitation values for the glacier accumulation areas. The average annual net accumulation at a point above the ELA can be calculated and compared to modern average precipitation values. Results of this study show that the local paleoprecipitation on these glaciers was less than modern values for the same elevation. Deflection of the westerly Jetstream around the massifs of the Laurentide and Cordilleran ice sheets (Kutzbach and Wright, 1985; Manabe and Brocolli, 1985) potentially caused a decreased frequency of storms in the area, which might be reflected in decreased precipitation. The results of this study also confirm the independent findings of Locke and Kemph (1987) and Locke and others (1989) which yielded paleoprecipitation values roughly 50% of modern values. Suggestions for Future Study Dynamics Modelling The main assumption in the model is that basal sliding is a minimum at the point of maximum deformation flux. The result of this assumption is. that all values of velocity, mass flux and mass balance are minima. A method of determining a reasonable maximum to these values must be found. It is reasonable to assume some slip at the point of maximum deformation. The actual amount must still be determined, within limits. If the amount of slip at this point can be 121 determined, then more precise calculations of mass balance and precipitation can be made. The temperature dependent constant A can vary with elevation and thickness in the glacier. The constant A can change by an order of magnitude with a IO0C change in temperature. Using paleotemperatufe estimates from other models (e.g. GCMs - Kutzbach and Wright, 1985), variations in temperature through the glacier can be modelled. This information could be used in the calculation of deformation velocity which would lead to more accurate calculations of mass flux, mass balance and precipitation. Paleoclimatic Modelling The extensive glaciation of the mountains of southwestern Montana and adjacent Idaho provide numerous valleys in which this model can be applied. If more valleys are studied in this area, regional paleoprecipitation patterns could be determined. The effects of the continental ice sheets on windflow patterns could be better defined by the results from further study. The mass balance of the glacier is a combination of the input through accumulation and the output through ablation. These are usually investigated as functions, of winter precipitation and summer temperature respectively (Sugden and John, 1975). For paleoglaciers, this represents one equation with two unknowns. This model provides values of paleoprecipitation, eliminating one of the unknowns. If paleoprecipitation values are calculated for enough valleys, an estimation of regional paleotemperature may also be possible. 122 The model can be used in other regions as well. Care must be taken to select valleys with constant or slowly varying slope to minimize the modelling errors induced by compressive or extensive flow. Also valleys with well defined glacial geologic features are desirable to facilitate and enhance the accuracy of the glacier reconstruction. 123 REFERENCES CITED 124 Alden, W.C., 1953, Physiography and glacial geology of western Montana and adjacent areas: U.S. Geological Survey Professional Paper ' 231, 200 p. Alford, D., and Clark, R., 1968, The 1967 mass balance of the Grasshopper Glacier, Montana: Northwest Science, v. 42, p. 115- 123. American Geographical Society, 1960, Nine glacier maps: AGS Special Publication No. 34. Andrews, J.T., 1975, Glacial systems: North Scituate, Massachusetts, Duxbury Press, 191 p. Aten, R.E., 1974, Geomorphology of the east flank of the Crazy Mountains, Montana [Ph.D . thesis]: Lafayette, Indiana, Purdue. University, 136 p. Barry, R.G., 1981, Mountain weather and climate: New York, Meuthen, 313 p. .... Barry, R.G. , 1983, Late Pleistocene climatology, in Porter, S. C.., ed. , Late Quaternary environments of the United States: Volume I, The Late Pleistocene: Minneapolis, University of Minnesota Press, p . 390-407. Blair, T.A., and Fite, R.C., 1965, Weather elements, a text in elementary meteorology: Englewood Cliffs, New Jersey, Prentice- Hall, 364 p . Boulton, G.S., and Jones, A.S., 1979, Stability of temperate ice caps and ice sheets resting on beds of deformable sediment: Journal of Glaciology, v. 24, p. 29-43. CLIMAP Project Members, 1981, Seasonal reconstructions of the earth's surface at the last glacial maximum: Geological Society of America Map Chart Series, MG-36. Critchfield, H.J., 1974, General climatology: Englewood Cliffs, New Jersey, Prentice-Hall, 446 p. Davis, P.T., and Osborn, G., 1988, Preface, Holocene glacier fluctuations: Quaternary Science Reviews, v. 7, p. 113-114. Echelmeyer, K., and Zhongxiang, W., 1987, Direct observation of basal sliding and deformation of basal drift at subfreezing temperatures: Journal of Glaciology, v. 33, p. 83-98. Field, W.O., ed., 1975, Mountain glaciation in the northern hemisphere: Hanover, New Hampshire, U.S. Army Corps of Engineers, Cold Regions Research and Engineering Laboratories, 698 p. 125 Flint, R.F., 1971, Glacial and Quaternary geology: New York, John Wiley and Sons, 892 p . Fullerton, D.S., and Colton, R.B., 1986, Stratigraphy and correlation of the glacial deposits on the Montana plains, in Sibrava, V., Bowen, D.Q., and Richmond, G.M., eds., Quaternary Glaciations in the Northern Hemisphere: Quaternary Science Reviews, v. 5, p. 69- 82. Glen, J.W., 1952, Experiments on the deformation of ice: Journal of Glaciology, v. 2, p. 111-114. Graf, W.L., 1970, The geomorphology of the glacial valley cross section: Arctic and Alpine Research, v. 2, p. 303-312. Graf, W.L., 1976, Cirques as glacier locations: Arctic and Alpine Research, v. 8, p. 79-90. Haeberli, W., and Penz, U., 1985, An attempt to reconstruct glaciological and climatological characteristics of 18 ka ice age glaciers in and around the Swiss Alps: Zeitschrift fur Gletscherkunde und Glazialgeologie, v. 21, p. 351-361. Harding, W.G., 1982, Foreword, in Cunningham, C., ed., Montana Weather: Helena, Montana, Montana Magazine, p. 1-10. Holmlund, P., 1988, Is the longitudinal profile of Storglaciaren, northern Sweden, in balance with the present climate?: Journal of Glaciology, v. 34, p. 269-273. Kutzbach, J.E., and Wright, H.E., Jr, 1985, Simulation of the climate of 18,000 years BP; Results for the North American/North Atlantic/European Sector and comparison with the geologic record of North America: Quaternary Science Reviews, v. 4, p. 147-187. Leonard, E.M., 1984, Late Pleistocene equilibrium-line altitudes and modern accumulation patterns, San Juan Mountains, Colorado, U.S.A: Arctic and Alpine Research, v. 16, p. 65-76. Leonard, E.M., Huston, M.M., and Manley, A.E., 1986, Ice dynamics modelling of Rocky Mountain paleoglaciers - methods, results, and paleoclimatic inferences: Geological Society of America Abstracts with Programs, v. 18, p. 671. Locke, W.W., and Kemph, T.S., 1987, Paleo-equilibrium line altitudes , and paleoclimate of Montana during peak glacial times: Geological ,Society of America, Rocky Mountain Section, Abstracts with Programs, v. 19, p. 315. Locke, W.W., 1989, Present climate and glaciation of western Montana, U.S.A.: Arctic and Alpine Research, v. 21, p. 234-244. 126 Locke, W.W., Murray, D.R., and Slaughter, K.W., 1989, Peak-glacial environments of the Rocky Mountains of Montana and Idaho: Abstracts, CANQUA 1989, Edmonton, Alberta. Mathews, W.H., 1967, Profiles of late Pleistocene glaciers in New Zealand: New Zealand Journal of Geology and Geophysics, v. 10, p . 146-163. Manabe S., and Broccoli, A.J., 1985, The influence of continental ice. sheets on the climate of an ice age: Journal of Geophysical Research, v. 90, no. BI, p. 2167-190. McCalpin, J., 1986, written communication, Utah State University, Logan, Utah. Meier, M.F., and Post, A.S., 1962, Recent variations in mass net budgets of glaciers in North America: IASH Commission of Snow and Ice, Publication 58, p. 63-77. Meier, M.F., 1966, Some glaciological interpretations of remapping programs on South Cascade, Nisqually, and Klawatti Glaciers, Washington: Canadian Journal of Earth Sciences, v. 3, p. 811-818. Meier, M.F., 1968, Calculations of slip of Nisqually glacier on its bed: no simple relation of sliding velocity to shear stress: IASH General Assembly of Bern - Commission of Snow and Ice, Publication 79, p. 49-57. Meier, M.F., Tangborn, W.V., Mayo, L.R., and Post, A., 1971, Combined ice and water balances of Gulkana and Wolverine Glaciers, Alaska, and South Cascade Glacier, Washington, 1965 and 1966 hydrologic years: U.S. Geological Survey Professional Paper 715-A, 23 p. Meierding, T.C., 1982, Late Pleistocene glacial equilibrium-line altitudes in the Colorado Front Range: A comparison of methods: Quaternary Research, v. 18, p. 289-310. Mitchell, V.L., 1969, The regionalization of climate in the western United States [Ph.D . thesis]: Madison, Wisconsin, University of Wisconsin, 147 p. NOAA, 1985a, Climatological data, annual summary, Idaho, v. 88, no. 13: Asheville, North Carolina, National Climatic Data Center. NOAA, 1985b, Climatological data, annual summary, Montana, v. 88, no. 13: Asheville, North Carolina, National Climatic Data Center. Nye, J .F ., 1952, The mechanics of glacier flow: Journal of Glaciology, v. 2, p. 82-93. 127 Nye, J .F., 1965a, A numerical method of inferring the budget history of a glacier from its advance and retreat: Journal of Glaciology, v. 5, p. 589-607. Nye, J.F., 1965b, The flow of a glacier in a channel of rectangular, elliptic or parabolic cross-section: Journal of Glaciology, v. 5, p. 661-690. Paterson, W.S.B., 1981, The physics of glaciers. [2nd edition]: Oxford, Pergamon Press, 380 p. Pierce, K.L., 1979, History and dynamics of glaciation in the northern Yellowstone National Park area: U.S. Geological Survey, Professional Paper 729-F, 90 p. Porter, S.C., 1975, Equilibrium-line altitudes of late Quaternary glaciers in the southern Alps, New Zealand: Quaternary Research, v. 5, p . 27-47. Porter, S.C., ed. , 1983, Late Quaternary environments of the United States: Volume I, The Late Pleistocene: Minneapolis, University of Minnesota. Press, 407 p. Porter, S.C., Pierce, K.L., and Hamilton, T.D., 1983, Late Wisconsin mountain glaciation in the western United States, in Porter, S.C., ed., Late Quaternary environments of the United States: Volume I, The Late Pleistocene: Minneapolis, University of Minnesota Press, p."71-111. Price, L.W., 1981, Mountains & man: Berkeley, University of California Press, 506 p . Raymond, C.F., 1980, Temperate valley glaciers, in Colbeck, S.C., ed., Dynamics of snow and ice masses: New York, Academic Press, p . 80- 141. Richmond, G.M., 1965, Glaciation of the Rocky Mountains, in Wright, H.E., and Frey, D.G., eds., The Quaternary of the United States: Princeton, New Jersey, Princeton University Press, p. 217-230. Ross, C.P, Andrews, D.A., and Witkind, I.J., 1955, Geologic map of Montana: U.S. Geological Survey. Ruppel, E .T ;, 1980, Geologic map of the Patterson quadrangle, Lemhi County, Idaho: U.S. Geological Survey Map GQ-1529,. . Ruppel, E.T. , and Lopez, D.A.-, 1981, Geoldgic map of the Gilmore quadrangle, Lemhi and Custer Counties, Idaho: U.S. Geological Survey Map GQ-1543. 128 Schilling, D.H., and Hollin, J .T., 1981, Numerical reconstruction of valley glaciers and small ice caps, in Denton. G.H., and Hughes, T.J., eds., The last great ice sheets: New York, John Wiley and Sons, p. 207-221. Soil Conservation Service, 1986a, Idaho annual data summary, water year 1986 - Idaho Cooperative Snow Survey Data of Federal-State- Private Cooperative Snow Surveys: Boise, Idaho, Soil Conservation Service. Soil Conservation Service, 1986b, Montana annual data summary, water year 1986 - Montana Cooperative Snow Survey Data of Federal- State-Private Cooperative Snow Surveys: Bozeman, Montana, Soil Conservation Service. Sugden, D.E., and John, B.S., 1976, Glaciers and Landscape: London, Halsted Press, 376 p. Taylor, R.L.', and Ashley, J.M., 1986, Geologic map of Montana: Bozeman,- Montana, Department of Earth Sciences, Montana State University. Thompson, M.M., 1979, Maps for America: Washington, D.C., U.S. Government Printing Office, 265 p. Waitt, R.B., Jr, and Thorson, R.M., 1983, The Cordilleran ice sheet in Washington, Idaho, and Montana, in Porter, S.C., ed. , Late Quaternary Environments of the United States,■Volume I: The Late Pleistocene: Minneapolis, University of Minnesota Press, p. 53- 70. Weertman, J . 1973. Creep of ice, in Whalley, E., Jones, S.J., and Gold, L.W., eds., Physics and Chemistry of Ice: Ottawa, Royal Society of Canada, p. 320-337. Weertman, J . 1979. The unsolved general glacier sliding problem: Journal of Glaciology, v. 23, p. 97-115. 129 APPENDICES J APPENDIX A TOPOGRAPHIC MAPS AND AIR PHOTOS 131 Table 16. Topographic maps and aerial photographs used for glacier reconstruction. Aerial Photographs Glacier Topographic Maps (Year) Line Roll/Exposures Date Big Timber Crazy Peak, MT (1972) 49 680/122-125 8-11-81 Amelong Creek, MT (1972) 50 680/142-146 8-11-81 Battleship Butte, MT (1972) 51 480/19-21 7-18-81 680/152-154 8-11-81 52 480/5-9 7-18-81 Miner Lakes Homer Youngs Peak, MT (1966) 9 287/49-53 8-3-87 Miner Lakes, MT (1966) 10 387/115-121 8-3-87 H S 387/99-102 8-3-87 12S 387/141-1435 8-3-87 487/2-45 8-3-87 13S 487/31-355 8-3-87 14S 487/70-735 8-3-87 15C 487/79-S25 8-3-87 Meadow Lake Gilmore, ID„(1987)2 31 680/83-87 7-23-81 Gilmore, ID5 Gilmore SE, ID^ 32 680/50-54 7-23-81 Mill Creek Paterson NW, ID^ 25 480/119-122 7-15-81 Paterson NE, IDz* Paterson, ID5 (1956) 26 480/148-151 7-15-81 Stroud Creek/ Paterson NE, ID^ 25 480/121-123 7-15-81 Everson Creek Paterson SE, IDz* 26 480/151-154 7-15-81 Paterson, ID2 (1956) 27 480/174-177 7-15-81 ^Scale 1:24000 unless noted otherwise ^Provisional Edition ^Scale 1:62500 (15' map) ^Orthophoto quadrangle (scale 1:24000) produced from 15' map 5Scale 1:15840 132 APPENDIX B PROGRAM1' FOR CALCULATING THEORETICAL GLACIER PROFILES O O 133 Figure 35. FORTRAN program for calculating theoretical glacier profiles. C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C ******************************************************************* PROGRAM: ICETOPO.FOR PURPOSE: This program calculates the ice surface elevations for a valley glacier using a perfectly plastic flow law. WRITTEN: 09-OCT-88 (modified from BASIC) LANGUAGE: MS-FORTRAN VER. 3.10 AUTHOR: Donald R. Murray (after Schilling and Hollin, 1981) MODIFIED: 21-OCT-88 DRM: Modified the input so that basal shear stress is input from terminal ratherthan from the file. INPUT FILES: I - File with values of distance from terminus, bedrock elevation, and shape factor. Values must be separated by a space and all values for one step located on the same line in the order listed above. INPUT VARIABLES: DIST GROUND SHAPE TAU ICE distance (in feet) from ice terminus, bedrock elevation at each step (feet). shape factor (F) as defined by Nye. basal shear stress (bars), ice surface elevation at initial point (feet). OUTPUT: Printout or file with the input variables and the calculated ice surface elevation at each step. If the output is to a file, the file must exist or an error will occur. ******************************************************************* DECLARATION OF VARIABLES INTEGER DIST(IOO),GROUND(100),ICE(IOO) INTEGER THICK,YFILE,METRIC REAL*4 SHAPE(100) REAL*4 MTAU,MGRND,MTHICK,CNST,INCR,RHO,G ,TAU CHARACTER*40 CASE CHARACTER*14 FNAME,0UTFILE CHARACTER*! YORN,PORF,NEWRUN,SI LET'S INITIALIZE SOME VARIABLES O O O O O O O O O H O O O RHO - 910. G - 9.810001 134 PRINT OUT THE INSTRUCTIONS FOR RUNNING THE PROGRAM WRITE (*,10) FORMAT (5X,A) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) 'This program calculates ice surface elevations' 'iteratively for valley glaciers using a perfectly' 'plastic approximation' 'The following information must be supplied from a' 'data file:' ' - Distance (in feet) from the ice terminus' ' - Bedrock elevation (in feet)' ' - Shape factor (dimensionless)' ' with each variable separated by a space.' CHECK TO SEE IF WE SHOULD CONTINUE OR NOT WRITE (*,'(1X,A\)') 'Do you wish to continue? ' READ (*, ' (A) ' ) YORN IF ((YORN .EQ. 'n') .OR. (YORN .EQ. 'N')) STOP CONTINUE ON WITH THE PROGRAM AND GET INITIAL INFORMATION 12 WRITE(*,'(/,/,IX,A,/)')'First we need some initial information.' WRITE (*,'(IX,A\)') 'What is the name of this run? ' READ (*,'(A)') CASE GET THE NAME OF THE INPUT FILE WRITE (*,15) 'What is the name of the input file? ' READ (*,'(A)') FNAME OPEN (I,FILE=FNAME) 15 FORMAT (/,IX,A\) 16 FORMAT (/,IX,A,A\) C C CHECK TO SEE IF OUPUT GOES TO THE PRINTER OR A FILE AND IF THE C OUTPUT IS IN ENGLISH OR METRIC UNITS. C YFILE - O WRITE (*,16) 'Do you wish the output to go to a file or to ', + 'the printer (P=»printer/F=file)? ' READ (*,'(A)') PORF IF ((PORF .EQ. 'F') .OR. (PORF .EQ. 'f')) THEN WRITE (*,'(/,IX,A\)') 'What is the name of the ouput file? ' READ (*,'(A)') OUTFIL OPEN (2,FILE=OUTFIL) YFILE = I n o n o o n n n n n o o n 135 ELSE OPEN (2,FILE-'LPTl') ENDIF METRIC - 0 WRITE (*,16) 'Do you wish the output to be in English (E) or + 'SI (S) units? ' READ (*,'(A)') SI IF ((SI .EQ. 'S') .OR. (SI .EQ. 's')) METRIC = I GET THE INITIAL ICE SURFACE ELEVATION WRITE(*,16) 'What is the ice surface elevation at the starting', + ' point (in feet)? ' READ (*,'(15)') ICE(I) GET THE BASAL SHEAR STRESS VALUE WRITE (*,16) 'What is the basal shear stress for this run (in ', + 'bars)? ' READ (*,'(F4.2)') TAU READ THE DATA FROM THE INPUT FILE DO 40, 1-1,100 READ (I,30,END-50) DIST(I), GROUND(I), SHAPE(I) 30 FORMAT(BN,7X,15,7X,15,5X,F4.2) 40 CONTINUE 50 K - I-I THICK - ICE(I) - GROUND(I) Print out the header and the data for step I. WRITE (2,'(IX,A,//)') CASE C IF (METRIC .NE. I) THEN WRITE (2,60) 60 FORMAT (IX,'Distance from',2X,'Bedrock',3X,'Ice Surface',4X, + 'Ice',5X,'Shape',6X,'Basal',/,3X,'Terminus',4X,'Elevation' , + 3X,'Elevation',2X,'Thickness',2X,'Factor',2X,'Shear Stress',/, + 4X,'(ft)',9X,'(ft)',8X,'(ft)',8X,'(ft)' ,15X,'(bar)',/,IX, + '......... ' , 4X,'.........' , 2X, '.......... ',IX,'........ ' , 2X, + '..... ' , 2X, '........... ',/) WRITE (2,65) DIST(I), GROUND(I), ICE(I), THICK, SHAPE(I), TAU 65 FORMAT (3X,16,6X,16,7X,16,5X,16,6X,F3.2,6X,F4.2) ELSE WRITE (2,66) FORMAT (IX,'Distance from',2X,'Bedrock', 3X, ' Ice Surface',4X, + 'Ice',5X,'Shape',6X,'Basal',/,3X,'Terminus',4X,'Elevation', + 3X,'Elevation',2X,'Thickness' ,2X, ' Factor' , 2X,'Shear Stress',/, + 4X,'(m)',10X,'(m)',9X,'(m)',9X,'(m)',16X,'(bar)',/,IX, + '......... \4X,'.........' , 2X, '.......... ',IX,'-....... ' ,2X, 66 n o n n o n n o n n o n n o n n o n 136 + '..... ' . 2X, '............',/) WRITE (2,65) INT(DIST(I)*.3048), INT(GR0UND(1)*.3048), + INT(ICE(1)*.3048), INT(THICK*.3048), SHAPE(I), TAU ENDIF C DO 75, 1-2,K Compute the distance from this step to the next/convert to metric. HLEN = (DIST(I) - DIST(I-I))*.3048 Convert the other values to metric for the calculations. MGRND - GROUND(I) * .3048 MTAU = TAU * 10**5 MTHICK = THICK * .3048 Start calculations CNST = MTAU/(SHAPE(I) * RHO * G) INCR - MLEN/MTHICK ICE(I) - ICE(I-I) + (CNST * INCR * 3.2808) Check out the new ice thickness THICK - ICE(I) - GROUND(I) IF (THICK .LE. 30) THEN WRITE (2,'(1X,A,/)') 'Nunatak or cirque headwall.' GOTO 100 ENDIF Print out the data, check to see if it should be English or Metric IF (METRIC .NE. I) THEN WRITE (2,70) DIST(I), GROUND(I), ICE(I), THICK, SHAPE(I) 70 FORMAT (3X,16,6X,16,7X,16,5X,16,6X,F3.2) ELSE WRITE (2,70) INT(DIST(I)*.3048), INT(MGRND), + INT(ICEd)*. 3048) , INT(THICK*. 3048) , SHAPE(I) ENDIF 75 CONTINUE If the data was written to a printer, send a form feed at the end. 100 IF (YFILE .NE. I) WRITE (2/ (/,A)') 'I' C C If the output was to a file, send a message to the console C saying that the printing has been completed. C IF (YFILE .EQ. I) WRITE (*,125) 'The data has been written to ', + OUTFIL 125 FORMAT (IX,/,IX,A30,A14) O O O O O O 137 See if another run is wanted WRITE (*,15) 'Do you wish to make another run READ (*,'(A)') NEWRUN IF ((NEWRUN .EQ. 'Y') .OR. (NEWRUN .EQ. 'y')) End the program (Y/N)? ' GOTO 12 END APPENDIX C PROGRAM FOR CALCULATING SHEAR STRESS o n 139 Figure 36. FORTRAN program for calculating average effective basal shear stress. C ******************************************************************* C C PROGRAM: TAU.FOR C C PURPOSE: This program calculates the theoretical basal shear C stress at a point on the glacier using the relation: C C TAU - RHO*G*THICK*SIN ALPHA C C where: TAU = average basal shear stress C RHO = ice density (910 kg/m**3) C G = acceleration due to gravity (9.81m/sec**2) C THICK = ice thickness C SIN ALPHA = ice surface slope C C Average basal shear stress (TRUTAU) is calculated by C multiplying TAU by SHAPE. SHAPE is a valley shape factor C that accounts for the frictional effects of the valley C walls. C C WRITTEN: 14-0CT-88 (modified from BASIC) C C LANGUAGE: MS-FORTRAN VER. 3.10 C C AUTHOR: Donald R. Murray C C C C C C C C C C C C C C C C MODIFIED: 21-OCT-88 DRM: Added a feature whereby the user can determine the step length over which to average the ice surface slope. INPUT FILES: I - File with values of distance from terminus, bedrock elevation, ice surface elevation and shape factor at each iterative step. Values must be separated by a space and all values for one located on the same line in the order listed. INPUT VARIABLES: DIST - distance (in feet) from the iceterminus. GROUND - bedrock elevation at each step (feet). ICE - ice surface elevation (feet). SHAPE - shape factor (F) as defined by Nye. OUTPUT: Printout or file with the input variables and the calculated shear stresses at each step. If the C output is to a file, the file must exist or an C error will occur.C ****r*******r********r***** **************************** **************** C C DECLARATION OF VARIABLES ci o o H n o n o o o 140 INTEGER DIST(IOO),GROUND(100),ICE(IOO) INTEGER THICK,YFILE,METRIC,AVSTEP,STEP REAL*4 SHAPE(IOO) REAL*4 CNST,RATIO,ANGLE REAL*4 MTHICK,SLOPE,TAU,TRUTAU CHARACTER*40 CASE CHARACTER*14 FNAME,OUTFILE CHARACTER*! YORN,PORF,NEWRUN,SI,AVE LET'S INITIALIZE SOME VARIABLES RHO = 910. G - 9.810001 PRINT OUT THE INSTRUCTIONS FOR RUNNING THE PROGRAM WRITE (*,10) FORMAT (5X,A) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) WRITE (*,10) 'This program calculates basal shear stresses' 'along the centerline of a glacial valley.' 'The following information must be supplied from a' 'data file:' ' - Distance (in feet) from the ice terminus' ' - Bedrock elevation (in feet)' ' - Ice surface elevation (in feet)' ' - Shape factor (dimensionless)' ' with each variable separated by a space.' ' In calculating the shear stress, ice surface ' ' slope can be averaged over several steps.' CHECK TO SEE IF WE SHOULD CONTINUE OR NOT WRITE (*,'(IX,A\)') 'Do you wish to continue? ' READ (*,'(A)') YORN IF ((YORN .EQ. 'n') .OR. (YORN .EQ. 'N')) STOP C C CONTINUE ON WITH THE PROGRAM AND GET INITIAL INFORMATION C 12 WRITE(*,'(/,/,IX,A,/)')'First we need some initial information.' WRITE (*,'(IX,A\)') 'What is the name of this run? ' READ (*,'(A)') CASE C C GET THE NAME OF THE INPUT FILE C WRITE (*,15) 'What is the name of the input file? ' READ (*,'(A)') FNAME OPEN (I,FILE-FNAME) O O O O O O O O O O 141 15 FORMAT (/,IX,A\) 16 FORMAT (/,IX,A ,A\) C C CHECK TO SEE IF OUPUT GOES TO THE PRINTER OR A FILE AND IF THE C OUTPUT IS IN ENGLISH OR METRIC UNITS. C YFILE - O WRITE (*,16) 'Do you wish the output to go to a file or to ', + 'the printer (P=printer/F=file)? ' READ (*,'(A)') PORF IF ((PORF .EQ. 'F') .OR. (PORF .EQ. 'f')) THEN WRITE (*,'(/,IX,A\)') 'What is the name of the ouput file? ' READ (*,'(A)') OUTFIL OPEN (2,FILE=OUTFIL) YFILE = I ELSE OPEN (2,FILE='LPTl') ENDIF METRIC = O WRITE (*,16) 'Do you wish the output to be in English (E) or ', + 'SI (S) units? ' READ (*,'(A)') SI IF ((SI .EQ. 'S') .OR. (SI .EQ. 's')) METRIC = I See if the ice surface slope should be averaged over several steps. STEP = I WRITE (*,15) 'Do you wish to use an average surface slope? ' READ (*,'(A)') AVE IF ((AVE .EQ. 'y') .OR. (AVE .EQ. 'Y')) THEN WRITE (*,16) 'How many steps do you wish it to be averaged ', + 'over? ' READ (*,'(15)') AVSTEP STEP = AVSTEP/2 ENDIF READ THE DATA FROM THE INPUT FILE DO 40, 1=1,100 READ (1,30,END=SO) DIST(I), GROUND(I), ICE(I), SHAPE(I) 30 FORMAT(BN,7X,15,7X,15,7X,15,5X,F4.2) 40 CONTINUE 50 K = I-I Print out the header. WRITE (2,'(IX1A,//)') CASE C IF (METRIC .NE. I) THEN WRITE (2,60) FORMAT (IX,'Distance from',2X,'Bedrock',3X,'Ice Surface',4X,60 o a o n o n n o n 142 65 C + 'Ice',5X,'Surface',2X,'Theo',2X,'Shape',3X,'True',/,3X, + 'Terminus',4X,'Elevation',3X,'Elevation',2X,'Thickness' + 'Slope',3X,'Tau',3X,'Factor',2X,'Tau',/, + 4X,'(ft)',9X,'(ft)',8X,'(ft)',8X,'(ft)',13X,'(bar)',9X,' + /,IX, '......... ' ,4X,'.........' , 2X,'...........',IX, + '........ ' , 2X, '.......' , 2X, '----' , 2X, '......',ZX,'---- ,3X, (bar)', ,/) ELSE WRITE (2,65) FORMAT (IX,'Distance from',2X,'Bedrock',3X,'Ice Surface',4X, + 'Ice',5X,'Surface',2X,'Theo',2X,'Shape',3X,'True',/,3X, + 'Terminus',4X,'Elevation',3X,'Elevation',2X,'Thickness',3X, + 'Slope',3X,'Tau',3X,'Factor',2X,'Tau',/, + 5X,'(m)',10X,'(m)',9X,'(m)',8X,'(m)',14X,'(bar)',9X,'(bar)', + /,IX,'......... ' , 4X, '.........',ZX,'...........',IX, + '........ ' , 2X, '.......' , 2X, '----',ZX,'......',ZX,'----',/) ENDIF DO 75, I-I1K Compute the thickness of the ice at each point. THICK - ICE(I) - GROUND(I) Compute the step factor. J - I - STEP IF (J .LE. 0) J - I L - I + STEP IF (L .GT. K) L - K Compute the ice surface slope. CNST - FLOAT((ICE(L)-ICE(J))**2 + (DIST(L)-DIST(J))**2) SLOPE - (ICE(L)-ICE(J))/SQRT(CNST) C RATIO - FLOAT((ICE(L)-ICE(J))/(DIST(L)-DIST(J))) C ANGLE - ATAN(RATIO) C SLOPE - SIN(ANGLE) C C Compute the ice shear stresses C TAU - RHO*G*THICK*.3048*SLOPE/10**5 TRUTAU - TAU * SHAPE(I) C C Print out the data, check to see if it should be English or Metric C IF (METRIC .NE. I) THEN WRITE (2,70) DIST(I), GROUND(I), ICE(I), THICK, SLOPE, + TAU, SHAPE(I), TRUTAU 70 FORMAT (3X,16,6X,16,7X,16,4X,16,5X,F5.3,4X,F4.2,4X,F3.2, + 3X.F4.2) ELSE WRITE (2,70) INT(DIST(I)*.3048), INT(GROUND(I)*.3048), n o n n o n n n n n n o n 143 + INT(ICE(I)*.3048), INT(THICK*.3048), SLOPE, TAU, + SHAPE(I), TRUTAU ENDIF 75 CONTINUE If the data was written to a printer, send a form feed at the end 100 IF (YFILE .NE. I) WRITE (2,'(/,A)') 'I' If the output was to a file, send a message to the console saying that the printing has been completed. IF (YFILE .EQ. I) WRITE (*,125) 'The data has been written to ' + OUTFIL 125 FORMAT (IX,/,IX,A30,A14) See if another run is wanted WRITE (*,15) 'Do you wish to make another run (Y/N)? ' READ (*,'(A)') NEWRUN IF ((NEWRUN .EQ. 'Y') .OR. (NEWRUN .EQ. 'y')) GOTO 12 End the program END 144 APPENDIX D PROGRAM FOR CALCULATING ABLATION GRADIENTS 145 Figure 37. BASIC program for calculating ablation gradients. 10 REM*************************************************************** 20 REM Program: ABLATE.BAS 30 REM 40 REM Purpose: This program calculates the corresponding volumes of 50 REM ablation for altitudes below the ELA. 60 REM 70 REM Author: Donald R. Murray 80 REM 90 REM Written: May 23, 1988 100 REM H O REM Modified: 120 REM 130 REM Input: File containing data (in ft) of of elevation below the 140 REM ELA and the surface area (km**2) that this elevation 150 REM represents. 160 REM 170 REM Output: Printout of the elevation, the specific net balance 180 REM there and the volume of ice that is ablated. 190 REM 200 REM************************************************************** 210 DIM ELEVATION(IS),AREA(IS) 220 I - I 230 INPUT "What is the name of the file"; F$ 240 FILE$ = "B:"+F$ 250 INPUT "What is the ELA elevation"; ELA 260 INPUT "What is the initial ablation gradient (mm/m)"; AG 270 INPUT "What is the incremental change"; AGSTEP 280 INPUT "How many iterations do you want"; ITER 290 OPEN FILES FOR INPUT AS I 300 INPUT #1, ELEVATION (I),AREA(I) 310 IF EOF(I) THEN CLOSE:GOTO 340 320 I - I + I 330 GOTO 300 340 X “ I 350 FOR K = I TO ITER 360 TOTAL - 0 370 LPRINT " ELA =",ELA 380 LPRINT " ABLATION GRADIENT = ",AG 390 LPRINT 400 LPRINT "ELEVATION SPECIFIC BALANCE VOLUME LOST" 410 LPRINT " (ft) (m) ( x 10**6 m**3)" 420 LPRINT "............................................ " 430 FOR I - I TO X 440 LOWER - (ELA - ELEVATION(I) )*. 3048 450 NETBAL = LOWER*AG/1000 460 VOLUME - NETBAL * AREA(I) 470 TOTAL = TOTAL + VOLUME 480 LPRINT USING " #.### ";ELEVATION(I),NETBAL,VOLUME 490 NEXT I 146 500 LPRINT " ...... " 510 LPRINT USING " ##.### "; TOTAL 520 LPRINT 530 AG - AG - AGSTEP 540 NEXT K 550 END 147 APPENDIX E PROGRAM FOR CALCULATING ACCUMULATION GRADIENTS 148 Figure 38. BASIC program for calculating accumulation gradients. 10 RElM************************************************************** 20 RElM Program: ACCUM.BAS 30 REM 40 RElM Purpose: This program calculates the corresponding volumes of 50 REM accumulation for altitudes above the ELA. 60 REM 70 REM Author: Donald R. Murray 80 REM 90 REM Written: June 14, 1988 100 REM H O REM Modified: 120 REM 130 REM Input: File containing data (in ft) of elevation above the 140 REM ELA and the surface area (km**2) that this elevation 150 REM represents. 160 REM 170 REM Output: Printout of the elevation, the specific net balance 180 REM there and the volume of ice that is accumulated. 190 REM 200 REM************************************************************** 210 DIM ELEVATI0N(15),AREA(IS) 220 I - I 230 INPUT "What is the name of the file"; F$ 240 FILE$ - "B:"+F$ 250 INPUT "What is the ELA elevation"; ELA 260 INPUT "What is the initial accumulation gradient (mm/m)"; AG 270 INPUT "What is the incremental change"; AGSTEP 280 INPUT "How many iterations do you want"; ITER 290 OPEN FILE$ FOR INPUT AS I 300 INPUT #1, ELEVATION (I),AREA(I) 310 IF EOF(I) THEN CLOSE:GOTO 340 320 I - I + I 330 GOTO 300 340 X - I 350 FOR K = I TO ITER 360 TOTAL = 0 370 LPRINT " ELA =",ELA 380 LPRINT " ACCUMULATION GRADIENT - ",AG 390 LPRINT 400 LPRINT "ELEVATION SPECIFIC BALANCE VOLUME GAINED" 410 LPRINT " (ft) (m) ( x 10**6 m**3)" 420 LPRINT "............................................ " 430 FOR I = I TO X 440 HIGHER = (ELEVATION (I) - ELA)*.3048 450 NETBAL = HIGHER*AG/1000 460 VOLUME = NETBAL * AREA(I) 470 TOTAL = TOTAL + VOLUME 480 LPRINT USING " #.#//# ";ELEVATION(I),NETBAL,VOLUME 490 NEXT I 149 500 LPRINT " ...... " 510 LPRINT USING " ##.### TOTAL 520 LPRINT 530 AG = AG - AGSTEP 540 NEXT K 550 END 150 APPENDIX, F DATA FOR STROUD CREEK 151 Table 17. Stroud Creek paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Distance from Terminus (ft) Bedrock Ice Ice Shape Elev- Elev- Thickness Factor ation ation (ft) (ft) (ft) Calc. Theoretical Diff. Tyl Ice (Theo.- Elevation^ Calc.) (bar) (ft) (ft) 0 7280 7280 0 1000 7400 7580 180 0.84 2000 7440 7640 200 0.84 0.39 7640 0 3000 7500 7760 260 0.80 0.54 7823 63 4000 7540 7840 300 0.77 0.69 7940 100 5000 7590 7960 370 0.76 0.76 8036 76 6000 7660 8080 420 0.75 0.85 8123 43 7000 7720 8160 440 0.74 0.79 8208 48 8000 7760 8240 480 0.71 0.74 8292 52 9000 7840 8320 480 0.71 0.74 8369 49 10000 7900 8400 500 0.72 0.78 8445 45 11000 7960 8480 520 0.73 0.82 8518 38 12000 8060 8560 500 0.75 1.02 8588 28 13000 8160 8640 480 0.77 1.00 8660 20 14000 8360 8800 440 0.79 0.94 8734 - 66 15000 8440 8880 440 0.79 1.00 8833 -47 16000 8520 8960 440 0.80 8926 -34 17000 8680 0.81 9015 18000 8740 0.81 9123 19000 8840 0.81 9217 20000 9000 0.81 9313 21000 9040 0.81 9428 22000 9200 0.81 9521 23000 9280 0.81 9633 24000 10080 0.81 -'-Calculated from Equation (2). ^Calculated from Equation (3) assuming , = .80 bar. 152 APPENDIX G DATA FOR EVERSON CREEK 153 Table 18. Everson Creek paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Distance Bedrock from Elev- Terminus ation (ft) (ft) Ice Ice Elev- Thickness ation (ft) (ft) Shape Factor Calc. Tb1 (bar) Theoretical Ice Elevation^ (ft) Diff. (Theo.- Calc.) (ft) 0 7280 7280 0 1000 7400 7440 40 0.87 2000 7480 7640 160 0.87 0.60 3000 7600 7760 160 0.87 0.60 4000 7760 7920 160 0.86 0.48 5000 7800 8080 280 0.73 0.72 8080 0 6000 7880 8160 280 0.73 0.66 8223 63 7000 7960 8280 320 0.73 0.69 8340 60 8000 8000 8400 400 0.72 0.93 8447 47 9000 8080 8520 440 0.72 0.90 8538 18 10000 8160 8640 480 0.72 0.94 8626 -14 11000 8230 8700 470 0.73 0.84 8712 12 12000 8320 8800 480 0.74 0.77 8794 -6 13000 8390 8880 490 0.76 0.87 8875 -5 14000 8490 8960 470 0.78 0.80 8952 -8 15000 8560 0.79 9032 16000 8620 0.80 9109 17000 8720 0.80 9183 18000 8880 0.80 9262 19000 8920 0.80 9357 20000 8960 0.80 9440 21000 9200 0.80 9516 22000 9520 0.80 9631 22500 9840 0.80 ^Calculated from Equation (2). ^Calculated from Equation (3) assuming - . 80 bar. 154 APPENDIX H DATA FOR MILL CREEK 155 Table 19. Mill Creek paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Distance Bedrock Ice Ice Shape Calc. Theoretical Diff. from Elev- Elev- Thickness Factor T^-*- Ice (Theo. - Terminus ation ation Elevation^ Calc.) (ft) (ft) (ft) (ft) (bar) (ft) (ft) 0 7100 7100 0 1000 7160 7280 120 2000 7200 7460 260 3000 7240 7600 360 4000 7280 7680 400 5000 7330 7760 430 6000 7400 7800 400 7000 7460 7900 440 8000 7500 7920 420 9000 7560 8000 440 10000 7580 8080 500 11000 7600 8160 560 12000 7640 8240 600 13000 7680 8320 640 14000 7720 8400 680 15000 7780 8480 700 16000 7820 8560 740 17000 7880 8640 760 18000 7960 8720 760 19000 8020 8760 740 20000 8080 8840 760 21000 8160 8900 740 22000 8240 8960 720 23000 8320 9040 720 24000 8400 9120 720 25000 8540 9200 660 26000 8680 9280 600 27000 8840 9360 520 28000 8920 9440 520 29000 9080 30000 9280 31000 9360 32000 9520 33000 9900 34000 10400 0.80 0.74 0.54 7460 0 0.74 0.75 7631 31 0.74 0.73 7745 65 0.74 0.78 7840 80 0.76 0.64 7925 125 0.77 0.64 8006 106 0.78 0.62 8083 163 0.79 0.66 8154 154 0.78 0.79 8225 145 0.76 0.84 8292 132 0.74 0.96 8356 116 0.73 1.01 8419 99 0.72 1.06 8481 31 0.71 1.01 8542 62 0.70 1.05 8603 43 0.70 1.05 8663 23 0.70 1.01 8723 3 0.71 1.00 8783 23 0.72 1.04 8843 3 0.73 1.03 8902 2 0.73 1.00 8962 2 0.73 1.07 9024 -16 0.73 1.07 9088 -32 0.74 1.02 9152 -48 0.76 0.99 9222 -58 0.79 0.89 9299 -61 0.79 0.89 9389 -51 0.80 9476 0.80 9580 0.80 9717 0.80 9832 0.80 9964 0.80 ^Calculated from Equation (2). ^Calculated from Equation (3) assuming = .90 bar. 156 APPENDIX I DATA FOR MEADOW LAKE 157 Table 20. Meadow Lake paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Distance Bedrock Ice Ice Shape Calc. Theoretical Diff. from Elev- Elev- Thickness Factor Ice (Theo. - Terminus ation ation Elevation^ Calc.) (ft) (ft) (ft) (ft) (bar) (ft) (ft) 0 7509 1000 7520 2000 7580 3000 7720 4000 7840 5000 7920 6000 7960 7000 8040 8000 8120 9000 8200 10000 8320 11000 8380 12000 8480 13000 8600 14000 8720 15000 8880 16000 8980 17000 9100 18000 9160 19000 9200 20000 9560 21000 9720 22000 10400 7509 7600 80 7800 220 7960 240 8120 280 8260 340 8320 360 8440 400 8600 480 8680 480 8760 440 8840 460 8960 480 9080 480 9160 440 9300 420 9400 420 9520 420 0.90 0.74 0.70 0.74 0.73 0.72 0.71 0.72 0.79 0.69 0.81 0.68 0.77 0.69 0.99 0.69 0.90 0.70 0.75 0.70 0.87 0.71 0.92 0.73 1.09 0.76 0.99 0.78 0.97 0.79 1.08 0.80 1.00 0.80 0.80 0.80 0.80 0.80 7800 0 8002 42 8164 44 8305 45 8429 109 8532 92 8629 29 8722 42 8812 52 8907 67 8995 35 9082 2 9172 12 9265 -35 9373 -27 9477 9586 9682 9767 9966 -43 ■'•Calculated from Equation (2). ^Calculated from Equation (3) assuming Ty - .90 bar. 158 APPENDIX J DATA FOR MINER LAKES 159 Table 21. Miner Lakes paleoglacier morphology and rheology interpreted from topographic maps and comparison with theoretical values. Distance Bedrock Ice Ice Shape Calc. Theoretical Diff. from Elev- Elev- Thickness Factor Ice (Theo.- Terminus ation ation Elevation^ Calc.) (ft) (ft) (ft) (ft) (bar) (ft) (ft) 0 6800 6800 0 1000 6900 7100 200 2000 6920 7120 200 3000 6930 7130 200 4000 6940 7160 220 5000 6955 7170 215 6000 6965 7200 235 7000 6955 7240 285 8000 6955 7260 305 9000 6955 7300 345 10000 6955 7320 365 11000 6960 7410 450 12000 6965 7440 475 13000 6965 7480 515 14000 6970 7560 590 15000 6970 7600 630 16000 6970 7660 690 17000 6990 7700 710 18000 6990 7740 750 19000 6990 7780 790 20000 6990 7820 830 21000 7000 7840 840 22000 7040 7880 840 23000 7040 7920 880 24000 7060 7940 880 25000 7080 7960 880 26000 7080 8000 920 27000 7100 8040 940 28000 7120 8080 960 29000 7130 8120 990 30000 7140 8200 1060 31000 7150 8260 1110 32000 7200 8300 1100 33000 7260 8360 1100 34000 7380 8420 1040 35000 7520 8480 960 36000 7560 8520 960 37000 7600 8560 960 38000 7620 39000 7640 40000 7670 41000 7700 8760 1060 0.95 0.95 0.10 0.99 0.13 0.99 0.14 0.99 0.16 0.99 0.26 0.95 0.27 0.94 0.34 0.94 0.42 0.93 0.50 0.91 0.58 0.90 0.62 0.88 0.73 0.87 0.67 7560 0 0.85 0.76 7618 18 0.85 0.73 7671 11 0.84 0.69 7720 20 0.84 0.72 7767 27 0.84 0.66 7811 31 0.83 0.62 7854 34 0.83 0.62 7894 54 0.83 0.64 7933 53 0.82 0.64 7973 53 0.82 0.69 8011 71 0.82 0.81 8048 88 0.81 0.87 8085 85 0.80 0.93 8121 81 0.79 1.05 8157 77 0.78 1.15 8193 73 0.76 1.26 8229 29 0.76 1.25 8264 4 0.76 1.25 8298 -2 0.76 1.09 8333 -27 0.76 1.05 8368 -52 0.76 1.05 8407 -73 0.76 1.01 8450 -70 0.76 0.99 8493 -67 0.75 8536 0.73 8579 0.73 8621 0.70 8665 -95 160 Table 21. (continued). Distance from Terminus (ft) Bedrock Elev­ ation (ft) Ice Ice Elev- Thickness ation (ft) (ft) Shape Factor Calc. Tb1 (bar) Theoretical Ice Elevation^ (ft) Diff. (Theo.- Calc.) (ft) 42000 7800 8840 1040 0.68 8709 -131 43000 7870 8880 1010 0.68 0.77 8756 -124 44000 7900 8920 1020 0.68 0.78 8804 -116 45000 7910 8960 1050 0.67 0.72 8852 -108 46000 7950 9020 1070 0.67 8898 -122 47000 7970 0.67 8944 48000 8010 0.67 8988 49000 8030 0.68 9032 50000 8030 9100 1070 0.69 9074 51000 8360 0.71 9113 52000 9000 0.73 9166 53000 9300 0.75 9401 54000 10000 0.78 ^Calculated from Equation (2). ^Calculated from Equation (3) assuming Ty = .80 bar. MONTANA STATE UNIVERSITY LIBRARIES 762 101 8883 7