- Email: [email protected]
Onshore ice pile-up: a comparison between experiments and simulations
Cold Regions Science and Technology 26 Ž1997. 205–214
Onshore ice pile-up: a comparison between experiments and simulations Mark A. Hopkins
)
US Army Corps of Engineers, Cold Regions Research and Engineering Laboratory, 72 Lyme Road, HanoÕer, NH 03755-1290, USA Received 10 February 1997; accepted 3 July 1997
Abstract Recently computer models have been used to simulate the Arctic pressure ridging process. The results of these simulations have led to revised estimates of the energy dissipated in pressure ridging. This is important in large-scale ice ocean modeling, where the internal strength of the ice pack depends on the energy expended in pressure ridging. However, there has been no experimental data available to establish the accuracy of the simulations. This lack of data is due to the difficulty of modeling the pressure ridging process in the laboratory and of measuring ridge formation in the field. In this work the results of computer simulations of the closely related process of ice pile-up on an inclined ramp are directly compared with the results of a similar series of physical experiments conducted in an ice basin. In the experiments and simulations an inclined ramp is pushed against a long, stationary strip of intact, floating ice. The forces exerted on the ramp, the total energy expended, and the increase in the potential energy of the ice piled on the ramp are measured. q 1997 Elsevier Science B.V. Keywords: sea ice mechanics; discrete element modeling ŽDEM.; ice ridging; ice pile-up; pressure ridging
1. Introduction The strength of the Arctic ice pack depends on the energy expended in pressure ridging. In large-scale ice ocean models ŽHibler, 1980; Flato and Hibler, 1995., based on the ice thickness distribution theory of Thorndike et al. Ž1975., a parameter p ) defines the compressive strength of the ice pack. The parameter p ) is determined from an equation developed by Rothrock Ž1975.: p ) div v s cHh2c Ž h . d h )
Ž 1.
Tel.: q1 Ž603. 646 4249; fax: q1 Ž603. 646 4477; e-mail: [email protected]
It equates the rate of work by compressive deformation to the rate of increase of potential energy due to ridging. In Eq. Ž1., h is ice thickness, c is a constant that depends on the density of ice and sea water and c Ž h. defines the change in the ice thickness distribution from ridging. However, this equation neglects the energy dissipated during ridge creation. Hibler Ž1980. and Flato and Hibler Ž1995. account for dissipation by multiplying the right-hand side of Eq. Ž1. by an assumed ratio of the rate of work to the rate of increase of potential energy. Most estimates have assumed that work is approximately twice the change in potential energy ŽRothrock, 1975., a conclusion largely based on a kinematic ridge model developed by Parmerter and Coon Ž1972..
0165-232Xr97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 2 3 2 X Ž 9 7 . 0 0 0 1 5 - 3
206
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Recently, Hopkins Ž1994. developed a discrete element computer model of pressure ridge formation, in which an intact ice sheet covering a refrozen lead was pushed at constant speed against a thick multiyear floe. The thin sheet, breaking repeatedly in flexure, created the rubble blocks which form the ridge sail and keel. In the simulations the ratio of work to potential energy varied from 10 to 17 depending on ice thickness and ridge size. This represents a great increase in the estimated strength of the ice pack compared with the estimate of 2 given above. Unfortunately, there is no field or laboratory data to directly compare the simulation results. The ridging process has not been successfully modeled in the laboratory because it depends on buckling, which is difficult to scale correctly using model ice. The closely related process of onshore ice pile-up is more easily modeled in the laboratory, because it is relatively insensitive to the type of ice used, since any ice will form a rubble pile on a slope. It shares two important features with pressure ridging. First, rubble blocks broken from an intact sheet are piled on a surface, in one case a ramp and in the other a floe. Second, in both processes the dominant dissipative mechanism is frictional sliding. Onshore ice pile-up is an important process in its own right. It occurs when an intact ice sheet is pushed up an
inclined surface such as a beach, river bank, or the berm of an offshore structure. A large collection of observations of ice pile-up and an analysis of the process were made by Kovacs and Sodhi Ž1980.. In the experiments and simulations described in this work, an inclined ramp moving at constant speed was pushed against a stationary, intact sheet of ice. The experiments were performed in the CRREL ice basin using urea-doped ice. The forces exerted by the ice on the ramp were measured at short intervals. The total energy expenditure was calculated from the force record. The increase in the potential energy of the ice was calculated from measurements of the final rubble pile. In the parallel set of simulations, a computer model of ice pile-up was adapted from the pressure ridging model of Hopkins Ž1994. by substituting a ramp for the multi-year floe. The apparatus dimensions, ice dimensions, and ice properties used in the simulations were taken from the physical experiments.
2. Experimental apparatus The physical experiments were performed in the 36 m long by 9 m wide refrigerated basin in the CRREL Ice Engineering Facility. An inclined ramp,
Fig. 1. Diagram of the experimental apparatus, consisting of a moving, inclined ramp and a stationary ice sheet.
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214 Table 1 Dimensions and properties of the ice sheet and ramp Parameter
Value
ice sheet width h Žnominal ice sheet thickness. u Žice sheet speed. characteristic length ice tensile strength Žtop. ice tensile strength Žbottom. r i Žice density. r w Žwater density. ramp slope
1.2 m 40 mm 0.1 mrs 0.5"0.05 m 100"10 kPa 50"5 kPa 920 kgrm3 1000 kgrm3 258
shown in Fig. 1, was suspended from a movable carriage by load cells to measure the horizontal and vertical loads exerted on the ramp by the ice. The surface of the ramp was roughened by attaching 10 mm tall transverse strips at 0.2-m intervals. Walls, 0.3 m in height, were attached to the sides and upper end of the ramp to confine the ice pile. Slots were cut in the ice sheet ahead of the ramp, leaving a 30 m long strip of ice slightly narrower than the width of the ramp. The end of the ice sheet opposite the ramp was left uncut. The ramp was pushed toward the long strip of ice at constant speed. Following each experiment, a grid was placed above the pile to measure the three-dimensional relief of the top surface of the pile. The experiments, which used 1% urea-doped model ice, were performed at the slightly-above-freezing air temperatures used to reduce the strength of the ice sheet. There was no evidence of frozen contacts in the rubble pile following the experiments. The dimensions and material properties of the model ice sheet and the dimensions of the ramp are listed in Tables 1 and 2. Characteristic lengths were calculated from deflection measurements on floating ice sheets, and Table 2 Parameters used in the computer model in addition to those in Table 1 Parameter
Value
ice sheet width h Žice sheet thickness. elastic modulus ms Žstatic friction. md Ždynamic friction. m w Žunderwater friction.
1m 40 mm 100 MPa 1.0 0.33–0.50 0.33
207
tensile strengths were calculated from bending tests on floating cantilever beams.
3. Description of the computer model The computer model of ice ride-up is based on an existing two-dimensional, discrete element model of the pressure ridging process ŽHopkins, 1994.. The important features of this model are: a dynamic linear viscous–elastic model of a floating ice sheet; flexural failure Žincluding buckling. of the ice sheets; realistic block lengths broken from the parent sheet at points where tensile stress exceeds strength; secondary flexural breakage of rubble blocks; inelastic contacts between rubble blocks; frictional sliding contacts between blocks; separate friction coefficients for submerged and above-water contacts; buoyancy of the ice sheet and rubble; and water drag. The brief description of the key features that follows is taken from Hopkins Ž1994.. In the simulations, an intact ice sheet is driven at a constant speed into an inclined ramp. As the leading edge of the ice sheet rides up the ramp, it breaks repeatedly in flexure. The accumulation of ice rubble forms a pile on the sloping ramp. The ice sheet, and rubble blocks broken from the sheet, are each composed of single rows of uniform, rectangular blocks that are attached to neighboring blocks by viscous–elastic joints. The discretization of the floating ice sheet and rubble blocks is shown in Fig. 2. Relative displacements between adjacent component blocks create forces and moments, internal to the sheet and rubble blocks. The internal forces on the component blocks are added to external forces exerted by the surrounding ice rubble, gravity, and buoyancy. When the tensile stress in a joint at either surface of the sheet or a rubble block exceeds the specified strength, the joint is broken. The block created by the fracture becomes part of the rubble and is added to the pile. While the cracks must occur at joints, the length of a rubble block is variable since it may contain any number of component blocks. The dashpots attached to each block in the sheet dissipate elastic waves caused by buckling and flexural failure. The dashpot damping constants decrease exponentially, approaching zero at the free Žright-hand. end.
208
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Fig. 2. Discretization of the floating ice sheet and rubble blocks into uniform rectangular blocks, showing the boundary conditions on the sheet. The tip of the sheet is beveled Žpoint A. to facilitate the sheet riding up the ramp or over rubble blocks.
Contact forces between rubble blocks, between rubble and sheet, or between sheet and ramp use a different force model that supports no tensile force. Two blocks are defined to be in contact if the polygons defining their shapes intersect. The force between two intersecting blocks is calculated in a local coordinate frame with axes normal and tangential to a contact surface connecting the intersection points. The force acts at the centroid of the area of intersection. A viscous–elastic normal force model is used with a Coulomb friction, tangential force model. The internal forces and moments at each joint in the ice sheet and rubble blocks and the external, contact forces between blocks are calculated at each time step. Equations of motion, derived from a Taylor series expansion about the current time, are used to find the updated positions and velocities. The elastic modulus was computed from the nominal ice thickness and average characteristic length listed in Table 1. The various coefficients of friction used in the computer model were chosen in the following manner. The slope of the ramp in the basin experiments was 258. For an ice block to sit on the slope without moving, the coefficient of friction must be at least equal to the tangent of the slope angle, which is 0.47. In a simple test performed with the urea-doped model ice, a section of the ramp was covered with a sheet of smooth ice. Small blocks from the same sheet were set on the smooth ice and given an initial downward velocity. The blocks did
not seem to accelerate and occasionally stopped on the inclined sheet. Based on these observations, dynamic friction coefficients md in the range 0.33 to 0.5 were used in the simulations. Variation of md in this range had no discernible effect on the results. In the physical experiments, the ice piles formed on the ramp were very stable. Ice, once piled, seldom slid back into the water. In an attempt to obtain similarly stable behavior in the simulations, the static friction coefficient was set to a rather high value of 1. To prevent rubble blocks from sliding down the smooth ramp, the square blocks which made up the ramp were offset 10 mm from their neighbors as shown in Fig. 2. This had an effect similar to that of the transverse strips attached to the ramp in the basin experiments. The less critical underwater friction coefficient was arbitrarily set at 0.33. 4. Results of ice basin experiments Twelve experiments were performed in the ice basin. The experiments, of varied duration, were terminated by the failure of the fixed end of the ice sheet. The final profile from the first experiment listed in Table 3 is shown in Fig. 3. The profile is an average over the width of the ramp. The forces per unit width measured in the same experiment are shown in Fig. 4. The vertical force shows the weight of the accumulating ice rubble supported by the ramp. The graph of the horizontal force exhibits
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
209
Table 3 The results of the physical experiments Exp.
h Žmm.
DT Žs.
Weight ŽNrm.
Work ŽJrm.
PE ŽJrm.
W r PE
V ≠ Žm3rm.
V x Žm3 rm.
1yn
1 2 3 4 5 6 7 8 9 10 11 12 Avg
41 44 43 43 40 41 39 35 35 42 44 41
220 200 210 170 265 110 145 75 100 137 130 275
3991 4580 4741 3099 5430 2002 2811 1820 2455 3018 2788 5497
10847 11015 8795 7356 17088 4087 7274 2519 4533 7725 6581 11735
1261 1490 1415 701 1787 487 670 506 631 721 756 1595
8.6 7.4 6.2 10.4 9.6 8.4 10.8 5.0 7.2 10.7 8.7 7.3 8.3
0.66 0.78 0.69 0.48 1.00 0.33 0.50 0.41 0.34 0.50 0.50 0.90
0.42 0.26 0.18 0.23 0.49 0.15 0.22 0.08 0.03 0.21 0.16 0.40
0.71 0.67 0.78 0.75 0.63 0.70 0.65 0.50 0.81 0.70 0.64 0.70 0.69
several periods of gradual increase and precipitous decrease. The periods of gradual increase correspond to periods when the sheet is pushed smoothly onto the growing pile. The precipitous drops, which mark the ends of these periods, are caused by buckling of the sheet. The increasing magnitude of the forces prior to buckling reflects the growing size of the rubble pile. Plots of the forces from the other experiments show similar gradual increases followed by abrupt drops, but differ regarding peak forces and
V ≠ Ž1 y n . huDT 0.52 0.59 0.60 0.49 0.59 0.51 0.57 0.79 0.79 0.60 0.56 0.56 0.59
frequency of failure. The frequent occurrence of buckling is a limitation of small-scale model tests, because thin ice fails easily in that mode. In full-scale tests buckling failure might not be as prevalent. In the simulations described below, in which a simulation of experimental conditions is attempted, frequent buckling of the thin ice also occurs. At the end of each experiment the ramp was moved back to its original position. In the process, any floating rubble which was not supported by the ramp was left behind. The potential energy of the rubble piled on the ramp was estimated by measuring the depth of the rubble pile at many points. A three-dimensional profile of the rubble pile was constructed using a grid laid atop the box used to confine the rubble on the ramp. The height of the rubble was measured at the intersection points of the grid. The volumes of ice rubble above, V ≠, and below, V x, the waterline were calculated. A rubble porosity n was determined for each experiment using the equation: mass s r iV ≠ Ž 1 y n . y Ž r w y r i . V x Ž 1 y n .
Fig. 3. Profile of the final ice pile from experiment 1 in Table 3.
Ž 2.
where r i is the ice density and r w is the water density. The mass was found from the weight of the ice on the ramp at the end of an experiment. The porosity from Eq. Ž2. was used to calculate the potential energy of the rubble in each 0.1 = 0.1 m cell defined by the grid. The energy expended in
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
210
tity in the last column is the volume of ice piled above the waterline, V ≠Ž1 y n., divided by the total volume of ice broken in the experiment, huDT, where u is the carriage speed. The average volume of ice piled below the waterline, V xŽ1 y n., divided by the total ice volume huDT was 0.22. As mentioned above, some ice was left behind when the ramp was returned to its initial position. The average amount of ice left behind, expressed as a fraction of the total ice volume huDT, was 0.19. This lost ice had some potential energy. Since it was nearly the same fraction of the total ice volume as the submerged ice remaining on the ramp V x, it would be conservative to assume that it also had the same potential energy. The average potential energy of the submerged ice on the ramp was 2% of the total. Fig. 4. Force versus time from experiment 1 in Table 3.
5. Results of computer simulations creating the pile was calculated by integrating the product of the horizontal force and the constant carriage velocity over the duration of the experiment. The results of the physical experiments are given in Table 3. In Table 3, h is the sheet thickness, DT is the duration of the experiment, PE denotes the potential energy, and W denotes work. The weight, work, potential energy, and ice volumes are expressed as quantities per unit width of the ice sheet. The quan-
Eleven simulations were performed with the computer model. A random variation Ž"5%. of the elastic modulus at each joint in the ice sheet was used to create unique outcomes in simulations using the same initial configuration of ice and the same parameters. A snapshot showing the final state of the first simulation listed in Table 4 is shown in Fig. 5. The forces measured in the same simulation are shown in Fig. 6. The vertical force shows the weight of the accumulating ice rubble supported by the
Table 4 The results of the simulations Simulation
DT Žs.
Weight ŽNrm.
Work ŽJrm.
PE ŽJrm.
W r PE
V ≠ Žm3 rm.
V x Žm3rm.
1yn
1 2 3 4 5 6 7 8 9 10 11 Avg
200 200 200 200 180 160 200 200 200 200 200
4451 4134 4918 3960 4089 2903 3367 3051 3049 2661 3350
10137 7496 6918 9407 6469 4766 6274 5658 5490 5995 6959
1365 1365 1591 1153 1477 644 967 780 916 635 998
7.4 5.5 4.3 8.2 4.4 7.4 6.5 7.3 6.0 9.4 7.0 6.7
0.63 0.57 0.66 0.60 0.61 0.44 0.53 0.49 0.48 0.44 0.45
0.43 0.47 0.38 0.47 0.38 0.44 0.58 0.65 0.62 0.68 0.63
0.75 0.77 0.77 0.75 0.73 0.73 0.72 0.70 0.73 0.71 0.74 0.74
V ≠ Ž1 y n . huDT 0.59 0.55 0.63 0.56 0.62 0.50 0.48 0.43 0.44 0.39 0.42 0.51
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
211
Fig. 5. Snapshot of ice pile-up on an inclined ramp from simulation 1 in Table 4. The blocks which form the ramp are 0.2 m wide.
ramp. The horizontal force is a bit more ragged than the force measured in the physical experiment that is shown in Fig. 4. As in the experiments the forces from the other simulations show similar gradual increases followed by abrupt drops, but differ regarding peak forces and frequency of failure.
The energy expended in creating the rubble pile was calculated by integrating the product of the horizontal force and the ice velocity over the duration of the simulation. The change in potential energy was explicitly calculated ŽHopkins, 1994. at each time step during the simulation. The energy
Fig. 6. Force versus time from simulation 1 in Table 4.
212
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
dissipated by frictional and inelastic contact forces was calculated ŽHopkins, 1994. and used to determine the accuracy of the overall energy balance in each simulation. The difference between the calculated work and the sum of the various energy sinks was less than 1% of the work in all simulations. The results of the simulations are listed in Table 4.
6. Discussion In the process of piling ice rubble, energy is dissipated primarily by friction in sliding contacts between ice blocks. Although it is impossible Žin a physical experiment. to measure frictional dissipation directly, the total amount of dissipated energy can be calculated by comparing the work required to create a rubble pile with the potential energy of the pile. The average ratio of work to potential energy in the physical experiments, given in Table 3, was 8.3. The accuracy of this calculation depends on the accuracy of the separately measured values of work and potential energy. The work calculation depends on the accuracy of the force measurement. The load cells used to measure forces have a claimed accuracy of "2% and were calibrated to the known weight of the test rig prior to each experiment. The potential energy calculation depends on the accuracy of the measurements of the profile of the rubble piles. Each profile required approximately 200 depth measurements over an irregular surface. Since the variations of the rubble surface surrounding each measurement point were random, "10% accuracy seems reasonable. With these assumptions, the expected value of the ratio of work to potential energy falls in the range 8.3 " 1. The average value of the ratio of work to potential energy in the simulations, given in Table 4, was 6.7, significantly below the ratio measured in the physical experiments. The average porosity was also lower in the simulations. This is not unexpected since two-dimensional systems typically have lower porosities than similar three-dimensional systems. All else being equal, lower porosity results in a more compact rubble pile with less potential energy, which tends to increase the ratio of work to potential energy. The effect of porosity on work is unclear. The other average measure in Tables 3 and 4 which
is comparable is the ratio of the ice rubble piled above water to the total ice volume V ≠Ž 1 y n.rhuDT. The value from the simulations, 0.51, is lower than the value from the basin tests, 0.59. However, if the two anomalous values from basin tests 8 and 9 are dropped, the average from the basin tests decreases to 0.56. The direct comparison is hampered by the irregular durations of the basin experiments. Consider separately the first three experiments, whose duration is similar to the simulations. The average potential energy in these three experiments is 1389 Jrm, and the average above-water volume ratio V ≠Ž1 y n.rhuDT is 0.57. The first five simulations have an average potential energy of 1390 Jrm and an average volume ratio of 0.59. Although these measures are nearly identical, the work tells a different story. The average work in the basin experiments is 10219 Jrm, while the average in the simulations is just 8085 Jrm. Thus it would appear that, all else being equal, the two-dimensional computer model requires less energy, or equivalently, less force, to create a similar rubble pile. This is shown in Fig. 7, where the average work expended is plotted versus the length of ice pushed into the pile. The results of all twelve experiments and eleven simulations were av-
Fig. 7. A comparison of the average ice piling work versus the length of ice pushed into the pile from the experiments and the simulations.
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
eraged. Following the initial divergence of the two graphs their slopes are nearly equal. The last six simulations in Table 4 have a distinctly different character from the first five. As a group they have a smaller average fraction of rubble piled above the water, 0.44, and a lower average potential energy of 823 Jrm. In the second group of simulations, more of the early rubble piled on the slope was observed to slide back into the water. The large amount of floating rubble in front of the ramp made it more difficult for the intact sheet to push blocks onto the pile on the ramp resulting in more frequent premature failure, with the result that less ice was piled on the ramp.
7. Conclusions The evidence suggests two conclusions. First, the simulations tend to pile a smaller fraction of the rubble above the water than the physical experiments. Second, the rubble piling forces tend to be lower in the simulations, even in simulations where the amount of rubble piled above the water is similar. These differences appear to be a consequence of the two-dimensionality of the computer model. A two-dimensional model is, by definition, uniform in the third dimension. This means that every block surface is a potential slip plane. In the three-dimensional physical experiments, the transverse roughness of the rubble pile makes slippage less likely. This difference has two probable effects. First, the rubble pile forming on the slope will slide back into the water more frequently in the simulations. This was observed. Second, just as the rubble slides down the slip planes more easily, it may also be pushed up the slip planes more easily, requiring less force. The forces and energy expended in the simulations were some 20–30% lower than in the experiments. Ice sliding back into the water reduces the potential energy already paid for in work, increasing the ratio of work to potential energy, while lower forces reduce the work without affecting potential energy. The two effects tend to cancel leaving the ratio of ice piling work to the increase in potential energy of the piled ice unchanged. As mentioned in the introduction, the ratio of ridge-building work to the increase in potential en-
213
ergy of ridged ice is a critical parameter in large-scale sea ice models. In the ice piling experiments presented here the ratio was approximately 8 : 1. In the simulations the ratio was approximately 7 : 1. This range probably represents an approximate lower bound on the ratio in ice rubble piling events. If the ramp slope were increased, more ice would fall back into the water increasing the ratio of work to potential energy. If the ramp slope were decreased, it would probably have little effect on the ratio because it would not change the ultimate slope of the rubble pile. In simulations of pressure ridging using a similar computer model ŽHopkins, 1994., the ratio varied from 10 : 1 to 17 : 1 depending on ice thickness and ridge size. The higher ratio in pressure ridging is probably due to the fact that a significant amount of work goes into building the keel which contains little potential energy compared with the amount in the sail. Based on this comparison of experiments and simulations of ice piling, the computer model produces a reasonably accurate simulation of the physical experiments. The broad differences between the results of the experiments and simulations are probably due to the two-dimensionality of the computer model. Based on this comparison it is likely that the pressure ridging simulations, described in Hopkins Ž1994., underestimate ridge-building forces, but predict the ratio of work to potential energy fairly accurately.
Acknowledgements This work was supported by grant N00014-92MP-2402 from the Office of Naval Research and by Civil Works Work Unit CWIS-32548, ‘Ice Effects on Riprap’, of the U.S. Army Corps of Engineers Ice Engineering Research Program. The apparatus used in the experiments was designed by Dr. Devinder S. Sodhi and constructed by Peter Stark. The experiments were performed by Jessie M. Stanley.
References Flato, G.M., Hibler III, W.D., 1995. Ridging and strength in modeling the thickness distribution of Arctic sea ice. J. Geophys. Res. 100, 18611–18626.
214
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Hibler III, W.D., 1980. Modeling a variable thickness sea ice cover. Mon. Weather Rev. 108, 1943–1973. Hopkins, M.A., 1994. On the ridging of intact lead ice. J. Geophys. Res. 99, 16351–16360. Kovacs, A., Sodhi, D.S., 1980. Shore ice pile-up and ride-up: field observations, models, theoretical analyses. Cold Regions Sci. Technol. 2, 209–288.
Parmerter, R.R., Coon, M.D., 1972. Model of pressure ridge formation in sea ice. J. Geophys. Res. 77, 6565–6575. Rothrock, D.A., 1975. The energetics of the plastic deformation of pack ice by ridging. J. Geophys. Res. 80, 4514–4519. Thorndike, A.S., Rothrock, D.A., Maykut, G.A., Colony, R., 1975. The thickness distribution of sea ice. J. Geophys. Res. 80, 4501–4513.
Onshore ice pile-up: a comparison between experiments and simulations Mark A. Hopkins
)
US Army Corps of Engineers, Cold Regions Research and Engineering Laboratory, 72 Lyme Road, HanoÕer, NH 03755-1290, USA Received 10 February 1997; accepted 3 July 1997
Abstract Recently computer models have been used to simulate the Arctic pressure ridging process. The results of these simulations have led to revised estimates of the energy dissipated in pressure ridging. This is important in large-scale ice ocean modeling, where the internal strength of the ice pack depends on the energy expended in pressure ridging. However, there has been no experimental data available to establish the accuracy of the simulations. This lack of data is due to the difficulty of modeling the pressure ridging process in the laboratory and of measuring ridge formation in the field. In this work the results of computer simulations of the closely related process of ice pile-up on an inclined ramp are directly compared with the results of a similar series of physical experiments conducted in an ice basin. In the experiments and simulations an inclined ramp is pushed against a long, stationary strip of intact, floating ice. The forces exerted on the ramp, the total energy expended, and the increase in the potential energy of the ice piled on the ramp are measured. q 1997 Elsevier Science B.V. Keywords: sea ice mechanics; discrete element modeling ŽDEM.; ice ridging; ice pile-up; pressure ridging
1. Introduction The strength of the Arctic ice pack depends on the energy expended in pressure ridging. In large-scale ice ocean models ŽHibler, 1980; Flato and Hibler, 1995., based on the ice thickness distribution theory of Thorndike et al. Ž1975., a parameter p ) defines the compressive strength of the ice pack. The parameter p ) is determined from an equation developed by Rothrock Ž1975.: p ) div v s cHh2c Ž h . d h )
Ž 1.
Tel.: q1 Ž603. 646 4249; fax: q1 Ž603. 646 4477; e-mail: [email protected]
It equates the rate of work by compressive deformation to the rate of increase of potential energy due to ridging. In Eq. Ž1., h is ice thickness, c is a constant that depends on the density of ice and sea water and c Ž h. defines the change in the ice thickness distribution from ridging. However, this equation neglects the energy dissipated during ridge creation. Hibler Ž1980. and Flato and Hibler Ž1995. account for dissipation by multiplying the right-hand side of Eq. Ž1. by an assumed ratio of the rate of work to the rate of increase of potential energy. Most estimates have assumed that work is approximately twice the change in potential energy ŽRothrock, 1975., a conclusion largely based on a kinematic ridge model developed by Parmerter and Coon Ž1972..
0165-232Xr97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 6 5 - 2 3 2 X Ž 9 7 . 0 0 0 1 5 - 3
206
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Recently, Hopkins Ž1994. developed a discrete element computer model of pressure ridge formation, in which an intact ice sheet covering a refrozen lead was pushed at constant speed against a thick multiyear floe. The thin sheet, breaking repeatedly in flexure, created the rubble blocks which form the ridge sail and keel. In the simulations the ratio of work to potential energy varied from 10 to 17 depending on ice thickness and ridge size. This represents a great increase in the estimated strength of the ice pack compared with the estimate of 2 given above. Unfortunately, there is no field or laboratory data to directly compare the simulation results. The ridging process has not been successfully modeled in the laboratory because it depends on buckling, which is difficult to scale correctly using model ice. The closely related process of onshore ice pile-up is more easily modeled in the laboratory, because it is relatively insensitive to the type of ice used, since any ice will form a rubble pile on a slope. It shares two important features with pressure ridging. First, rubble blocks broken from an intact sheet are piled on a surface, in one case a ramp and in the other a floe. Second, in both processes the dominant dissipative mechanism is frictional sliding. Onshore ice pile-up is an important process in its own right. It occurs when an intact ice sheet is pushed up an
inclined surface such as a beach, river bank, or the berm of an offshore structure. A large collection of observations of ice pile-up and an analysis of the process were made by Kovacs and Sodhi Ž1980.. In the experiments and simulations described in this work, an inclined ramp moving at constant speed was pushed against a stationary, intact sheet of ice. The experiments were performed in the CRREL ice basin using urea-doped ice. The forces exerted by the ice on the ramp were measured at short intervals. The total energy expenditure was calculated from the force record. The increase in the potential energy of the ice was calculated from measurements of the final rubble pile. In the parallel set of simulations, a computer model of ice pile-up was adapted from the pressure ridging model of Hopkins Ž1994. by substituting a ramp for the multi-year floe. The apparatus dimensions, ice dimensions, and ice properties used in the simulations were taken from the physical experiments.
2. Experimental apparatus The physical experiments were performed in the 36 m long by 9 m wide refrigerated basin in the CRREL Ice Engineering Facility. An inclined ramp,
Fig. 1. Diagram of the experimental apparatus, consisting of a moving, inclined ramp and a stationary ice sheet.
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214 Table 1 Dimensions and properties of the ice sheet and ramp Parameter
Value
ice sheet width h Žnominal ice sheet thickness. u Žice sheet speed. characteristic length ice tensile strength Žtop. ice tensile strength Žbottom. r i Žice density. r w Žwater density. ramp slope
1.2 m 40 mm 0.1 mrs 0.5"0.05 m 100"10 kPa 50"5 kPa 920 kgrm3 1000 kgrm3 258
shown in Fig. 1, was suspended from a movable carriage by load cells to measure the horizontal and vertical loads exerted on the ramp by the ice. The surface of the ramp was roughened by attaching 10 mm tall transverse strips at 0.2-m intervals. Walls, 0.3 m in height, were attached to the sides and upper end of the ramp to confine the ice pile. Slots were cut in the ice sheet ahead of the ramp, leaving a 30 m long strip of ice slightly narrower than the width of the ramp. The end of the ice sheet opposite the ramp was left uncut. The ramp was pushed toward the long strip of ice at constant speed. Following each experiment, a grid was placed above the pile to measure the three-dimensional relief of the top surface of the pile. The experiments, which used 1% urea-doped model ice, were performed at the slightly-above-freezing air temperatures used to reduce the strength of the ice sheet. There was no evidence of frozen contacts in the rubble pile following the experiments. The dimensions and material properties of the model ice sheet and the dimensions of the ramp are listed in Tables 1 and 2. Characteristic lengths were calculated from deflection measurements on floating ice sheets, and Table 2 Parameters used in the computer model in addition to those in Table 1 Parameter
Value
ice sheet width h Žice sheet thickness. elastic modulus ms Žstatic friction. md Ždynamic friction. m w Žunderwater friction.
1m 40 mm 100 MPa 1.0 0.33–0.50 0.33
207
tensile strengths were calculated from bending tests on floating cantilever beams.
3. Description of the computer model The computer model of ice ride-up is based on an existing two-dimensional, discrete element model of the pressure ridging process ŽHopkins, 1994.. The important features of this model are: a dynamic linear viscous–elastic model of a floating ice sheet; flexural failure Žincluding buckling. of the ice sheets; realistic block lengths broken from the parent sheet at points where tensile stress exceeds strength; secondary flexural breakage of rubble blocks; inelastic contacts between rubble blocks; frictional sliding contacts between blocks; separate friction coefficients for submerged and above-water contacts; buoyancy of the ice sheet and rubble; and water drag. The brief description of the key features that follows is taken from Hopkins Ž1994.. In the simulations, an intact ice sheet is driven at a constant speed into an inclined ramp. As the leading edge of the ice sheet rides up the ramp, it breaks repeatedly in flexure. The accumulation of ice rubble forms a pile on the sloping ramp. The ice sheet, and rubble blocks broken from the sheet, are each composed of single rows of uniform, rectangular blocks that are attached to neighboring blocks by viscous–elastic joints. The discretization of the floating ice sheet and rubble blocks is shown in Fig. 2. Relative displacements between adjacent component blocks create forces and moments, internal to the sheet and rubble blocks. The internal forces on the component blocks are added to external forces exerted by the surrounding ice rubble, gravity, and buoyancy. When the tensile stress in a joint at either surface of the sheet or a rubble block exceeds the specified strength, the joint is broken. The block created by the fracture becomes part of the rubble and is added to the pile. While the cracks must occur at joints, the length of a rubble block is variable since it may contain any number of component blocks. The dashpots attached to each block in the sheet dissipate elastic waves caused by buckling and flexural failure. The dashpot damping constants decrease exponentially, approaching zero at the free Žright-hand. end.
208
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Fig. 2. Discretization of the floating ice sheet and rubble blocks into uniform rectangular blocks, showing the boundary conditions on the sheet. The tip of the sheet is beveled Žpoint A. to facilitate the sheet riding up the ramp or over rubble blocks.
Contact forces between rubble blocks, between rubble and sheet, or between sheet and ramp use a different force model that supports no tensile force. Two blocks are defined to be in contact if the polygons defining their shapes intersect. The force between two intersecting blocks is calculated in a local coordinate frame with axes normal and tangential to a contact surface connecting the intersection points. The force acts at the centroid of the area of intersection. A viscous–elastic normal force model is used with a Coulomb friction, tangential force model. The internal forces and moments at each joint in the ice sheet and rubble blocks and the external, contact forces between blocks are calculated at each time step. Equations of motion, derived from a Taylor series expansion about the current time, are used to find the updated positions and velocities. The elastic modulus was computed from the nominal ice thickness and average characteristic length listed in Table 1. The various coefficients of friction used in the computer model were chosen in the following manner. The slope of the ramp in the basin experiments was 258. For an ice block to sit on the slope without moving, the coefficient of friction must be at least equal to the tangent of the slope angle, which is 0.47. In a simple test performed with the urea-doped model ice, a section of the ramp was covered with a sheet of smooth ice. Small blocks from the same sheet were set on the smooth ice and given an initial downward velocity. The blocks did
not seem to accelerate and occasionally stopped on the inclined sheet. Based on these observations, dynamic friction coefficients md in the range 0.33 to 0.5 were used in the simulations. Variation of md in this range had no discernible effect on the results. In the physical experiments, the ice piles formed on the ramp were very stable. Ice, once piled, seldom slid back into the water. In an attempt to obtain similarly stable behavior in the simulations, the static friction coefficient was set to a rather high value of 1. To prevent rubble blocks from sliding down the smooth ramp, the square blocks which made up the ramp were offset 10 mm from their neighbors as shown in Fig. 2. This had an effect similar to that of the transverse strips attached to the ramp in the basin experiments. The less critical underwater friction coefficient was arbitrarily set at 0.33. 4. Results of ice basin experiments Twelve experiments were performed in the ice basin. The experiments, of varied duration, were terminated by the failure of the fixed end of the ice sheet. The final profile from the first experiment listed in Table 3 is shown in Fig. 3. The profile is an average over the width of the ramp. The forces per unit width measured in the same experiment are shown in Fig. 4. The vertical force shows the weight of the accumulating ice rubble supported by the ramp. The graph of the horizontal force exhibits
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
209
Table 3 The results of the physical experiments Exp.
h Žmm.
DT Žs.
Weight ŽNrm.
Work ŽJrm.
PE ŽJrm.
W r PE
V ≠ Žm3rm.
V x Žm3 rm.
1yn
1 2 3 4 5 6 7 8 9 10 11 12 Avg
41 44 43 43 40 41 39 35 35 42 44 41
220 200 210 170 265 110 145 75 100 137 130 275
3991 4580 4741 3099 5430 2002 2811 1820 2455 3018 2788 5497
10847 11015 8795 7356 17088 4087 7274 2519 4533 7725 6581 11735
1261 1490 1415 701 1787 487 670 506 631 721 756 1595
8.6 7.4 6.2 10.4 9.6 8.4 10.8 5.0 7.2 10.7 8.7 7.3 8.3
0.66 0.78 0.69 0.48 1.00 0.33 0.50 0.41 0.34 0.50 0.50 0.90
0.42 0.26 0.18 0.23 0.49 0.15 0.22 0.08 0.03 0.21 0.16 0.40
0.71 0.67 0.78 0.75 0.63 0.70 0.65 0.50 0.81 0.70 0.64 0.70 0.69
several periods of gradual increase and precipitous decrease. The periods of gradual increase correspond to periods when the sheet is pushed smoothly onto the growing pile. The precipitous drops, which mark the ends of these periods, are caused by buckling of the sheet. The increasing magnitude of the forces prior to buckling reflects the growing size of the rubble pile. Plots of the forces from the other experiments show similar gradual increases followed by abrupt drops, but differ regarding peak forces and
V ≠ Ž1 y n . huDT 0.52 0.59 0.60 0.49 0.59 0.51 0.57 0.79 0.79 0.60 0.56 0.56 0.59
frequency of failure. The frequent occurrence of buckling is a limitation of small-scale model tests, because thin ice fails easily in that mode. In full-scale tests buckling failure might not be as prevalent. In the simulations described below, in which a simulation of experimental conditions is attempted, frequent buckling of the thin ice also occurs. At the end of each experiment the ramp was moved back to its original position. In the process, any floating rubble which was not supported by the ramp was left behind. The potential energy of the rubble piled on the ramp was estimated by measuring the depth of the rubble pile at many points. A three-dimensional profile of the rubble pile was constructed using a grid laid atop the box used to confine the rubble on the ramp. The height of the rubble was measured at the intersection points of the grid. The volumes of ice rubble above, V ≠, and below, V x, the waterline were calculated. A rubble porosity n was determined for each experiment using the equation: mass s r iV ≠ Ž 1 y n . y Ž r w y r i . V x Ž 1 y n .
Fig. 3. Profile of the final ice pile from experiment 1 in Table 3.
Ž 2.
where r i is the ice density and r w is the water density. The mass was found from the weight of the ice on the ramp at the end of an experiment. The porosity from Eq. Ž2. was used to calculate the potential energy of the rubble in each 0.1 = 0.1 m cell defined by the grid. The energy expended in
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
210
tity in the last column is the volume of ice piled above the waterline, V ≠Ž1 y n., divided by the total volume of ice broken in the experiment, huDT, where u is the carriage speed. The average volume of ice piled below the waterline, V xŽ1 y n., divided by the total ice volume huDT was 0.22. As mentioned above, some ice was left behind when the ramp was returned to its initial position. The average amount of ice left behind, expressed as a fraction of the total ice volume huDT, was 0.19. This lost ice had some potential energy. Since it was nearly the same fraction of the total ice volume as the submerged ice remaining on the ramp V x, it would be conservative to assume that it also had the same potential energy. The average potential energy of the submerged ice on the ramp was 2% of the total. Fig. 4. Force versus time from experiment 1 in Table 3.
5. Results of computer simulations creating the pile was calculated by integrating the product of the horizontal force and the constant carriage velocity over the duration of the experiment. The results of the physical experiments are given in Table 3. In Table 3, h is the sheet thickness, DT is the duration of the experiment, PE denotes the potential energy, and W denotes work. The weight, work, potential energy, and ice volumes are expressed as quantities per unit width of the ice sheet. The quan-
Eleven simulations were performed with the computer model. A random variation Ž"5%. of the elastic modulus at each joint in the ice sheet was used to create unique outcomes in simulations using the same initial configuration of ice and the same parameters. A snapshot showing the final state of the first simulation listed in Table 4 is shown in Fig. 5. The forces measured in the same simulation are shown in Fig. 6. The vertical force shows the weight of the accumulating ice rubble supported by the
Table 4 The results of the simulations Simulation
DT Žs.
Weight ŽNrm.
Work ŽJrm.
PE ŽJrm.
W r PE
V ≠ Žm3 rm.
V x Žm3rm.
1yn
1 2 3 4 5 6 7 8 9 10 11 Avg
200 200 200 200 180 160 200 200 200 200 200
4451 4134 4918 3960 4089 2903 3367 3051 3049 2661 3350
10137 7496 6918 9407 6469 4766 6274 5658 5490 5995 6959
1365 1365 1591 1153 1477 644 967 780 916 635 998
7.4 5.5 4.3 8.2 4.4 7.4 6.5 7.3 6.0 9.4 7.0 6.7
0.63 0.57 0.66 0.60 0.61 0.44 0.53 0.49 0.48 0.44 0.45
0.43 0.47 0.38 0.47 0.38 0.44 0.58 0.65 0.62 0.68 0.63
0.75 0.77 0.77 0.75 0.73 0.73 0.72 0.70 0.73 0.71 0.74 0.74
V ≠ Ž1 y n . huDT 0.59 0.55 0.63 0.56 0.62 0.50 0.48 0.43 0.44 0.39 0.42 0.51
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
211
Fig. 5. Snapshot of ice pile-up on an inclined ramp from simulation 1 in Table 4. The blocks which form the ramp are 0.2 m wide.
ramp. The horizontal force is a bit more ragged than the force measured in the physical experiment that is shown in Fig. 4. As in the experiments the forces from the other simulations show similar gradual increases followed by abrupt drops, but differ regarding peak forces and frequency of failure.
The energy expended in creating the rubble pile was calculated by integrating the product of the horizontal force and the ice velocity over the duration of the simulation. The change in potential energy was explicitly calculated ŽHopkins, 1994. at each time step during the simulation. The energy
Fig. 6. Force versus time from simulation 1 in Table 4.
212
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
dissipated by frictional and inelastic contact forces was calculated ŽHopkins, 1994. and used to determine the accuracy of the overall energy balance in each simulation. The difference between the calculated work and the sum of the various energy sinks was less than 1% of the work in all simulations. The results of the simulations are listed in Table 4.
6. Discussion In the process of piling ice rubble, energy is dissipated primarily by friction in sliding contacts between ice blocks. Although it is impossible Žin a physical experiment. to measure frictional dissipation directly, the total amount of dissipated energy can be calculated by comparing the work required to create a rubble pile with the potential energy of the pile. The average ratio of work to potential energy in the physical experiments, given in Table 3, was 8.3. The accuracy of this calculation depends on the accuracy of the separately measured values of work and potential energy. The work calculation depends on the accuracy of the force measurement. The load cells used to measure forces have a claimed accuracy of "2% and were calibrated to the known weight of the test rig prior to each experiment. The potential energy calculation depends on the accuracy of the measurements of the profile of the rubble piles. Each profile required approximately 200 depth measurements over an irregular surface. Since the variations of the rubble surface surrounding each measurement point were random, "10% accuracy seems reasonable. With these assumptions, the expected value of the ratio of work to potential energy falls in the range 8.3 " 1. The average value of the ratio of work to potential energy in the simulations, given in Table 4, was 6.7, significantly below the ratio measured in the physical experiments. The average porosity was also lower in the simulations. This is not unexpected since two-dimensional systems typically have lower porosities than similar three-dimensional systems. All else being equal, lower porosity results in a more compact rubble pile with less potential energy, which tends to increase the ratio of work to potential energy. The effect of porosity on work is unclear. The other average measure in Tables 3 and 4 which
is comparable is the ratio of the ice rubble piled above water to the total ice volume V ≠Ž 1 y n.rhuDT. The value from the simulations, 0.51, is lower than the value from the basin tests, 0.59. However, if the two anomalous values from basin tests 8 and 9 are dropped, the average from the basin tests decreases to 0.56. The direct comparison is hampered by the irregular durations of the basin experiments. Consider separately the first three experiments, whose duration is similar to the simulations. The average potential energy in these three experiments is 1389 Jrm, and the average above-water volume ratio V ≠Ž1 y n.rhuDT is 0.57. The first five simulations have an average potential energy of 1390 Jrm and an average volume ratio of 0.59. Although these measures are nearly identical, the work tells a different story. The average work in the basin experiments is 10219 Jrm, while the average in the simulations is just 8085 Jrm. Thus it would appear that, all else being equal, the two-dimensional computer model requires less energy, or equivalently, less force, to create a similar rubble pile. This is shown in Fig. 7, where the average work expended is plotted versus the length of ice pushed into the pile. The results of all twelve experiments and eleven simulations were av-
Fig. 7. A comparison of the average ice piling work versus the length of ice pushed into the pile from the experiments and the simulations.
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
eraged. Following the initial divergence of the two graphs their slopes are nearly equal. The last six simulations in Table 4 have a distinctly different character from the first five. As a group they have a smaller average fraction of rubble piled above the water, 0.44, and a lower average potential energy of 823 Jrm. In the second group of simulations, more of the early rubble piled on the slope was observed to slide back into the water. The large amount of floating rubble in front of the ramp made it more difficult for the intact sheet to push blocks onto the pile on the ramp resulting in more frequent premature failure, with the result that less ice was piled on the ramp.
7. Conclusions The evidence suggests two conclusions. First, the simulations tend to pile a smaller fraction of the rubble above the water than the physical experiments. Second, the rubble piling forces tend to be lower in the simulations, even in simulations where the amount of rubble piled above the water is similar. These differences appear to be a consequence of the two-dimensionality of the computer model. A two-dimensional model is, by definition, uniform in the third dimension. This means that every block surface is a potential slip plane. In the three-dimensional physical experiments, the transverse roughness of the rubble pile makes slippage less likely. This difference has two probable effects. First, the rubble pile forming on the slope will slide back into the water more frequently in the simulations. This was observed. Second, just as the rubble slides down the slip planes more easily, it may also be pushed up the slip planes more easily, requiring less force. The forces and energy expended in the simulations were some 20–30% lower than in the experiments. Ice sliding back into the water reduces the potential energy already paid for in work, increasing the ratio of work to potential energy, while lower forces reduce the work without affecting potential energy. The two effects tend to cancel leaving the ratio of ice piling work to the increase in potential energy of the piled ice unchanged. As mentioned in the introduction, the ratio of ridge-building work to the increase in potential en-
213
ergy of ridged ice is a critical parameter in large-scale sea ice models. In the ice piling experiments presented here the ratio was approximately 8 : 1. In the simulations the ratio was approximately 7 : 1. This range probably represents an approximate lower bound on the ratio in ice rubble piling events. If the ramp slope were increased, more ice would fall back into the water increasing the ratio of work to potential energy. If the ramp slope were decreased, it would probably have little effect on the ratio because it would not change the ultimate slope of the rubble pile. In simulations of pressure ridging using a similar computer model ŽHopkins, 1994., the ratio varied from 10 : 1 to 17 : 1 depending on ice thickness and ridge size. The higher ratio in pressure ridging is probably due to the fact that a significant amount of work goes into building the keel which contains little potential energy compared with the amount in the sail. Based on this comparison of experiments and simulations of ice piling, the computer model produces a reasonably accurate simulation of the physical experiments. The broad differences between the results of the experiments and simulations are probably due to the two-dimensionality of the computer model. Based on this comparison it is likely that the pressure ridging simulations, described in Hopkins Ž1994., underestimate ridge-building forces, but predict the ratio of work to potential energy fairly accurately.
Acknowledgements This work was supported by grant N00014-92MP-2402 from the Office of Naval Research and by Civil Works Work Unit CWIS-32548, ‘Ice Effects on Riprap’, of the U.S. Army Corps of Engineers Ice Engineering Research Program. The apparatus used in the experiments was designed by Dr. Devinder S. Sodhi and constructed by Peter Stark. The experiments were performed by Jessie M. Stanley.
References Flato, G.M., Hibler III, W.D., 1995. Ridging and strength in modeling the thickness distribution of Arctic sea ice. J. Geophys. Res. 100, 18611–18626.
214
M.A. Hopkinsr Cold Regions Science and Technology 26 (1997) 205–214
Hibler III, W.D., 1980. Modeling a variable thickness sea ice cover. Mon. Weather Rev. 108, 1943–1973. Hopkins, M.A., 1994. On the ridging of intact lead ice. J. Geophys. Res. 99, 16351–16360. Kovacs, A., Sodhi, D.S., 1980. Shore ice pile-up and ride-up: field observations, models, theoretical analyses. Cold Regions Sci. Technol. 2, 209–288.
Parmerter, R.R., Coon, M.D., 1972. Model of pressure ridge formation in sea ice. J. Geophys. Res. 77, 6565–6575. Rothrock, D.A., 1975. The energetics of the plastic deformation of pack ice by ridging. J. Geophys. Res. 80, 4514–4519. Thorndike, A.S., Rothrock, D.A., Maykut, G.A., Colony, R., 1975. The thickness distribution of sea ice. J. Geophys. Res. 80, 4501–4513.