Laboratory tests on ridging and rafting of ice sheets

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. C9, 3125, doi: /2001jc000848, 2002 Laboratory tests on ridging and rafting of ice sheets Jukka Tuhkuri Laboratory for Mechanics of Materials, Helsinki University of Technology, Helsinki, Finland Mikko Lensu Ship Laboratory, Helsinki University of Technology, Helsinki, Finland Received 13 February 2001; revised 5 November 2001; accepted 28 November 2001; published 19 September [1] The mechanical deformation of a sea ice cover takes place through ridging and rafting. These processes have been studied in an ice basin by pushing two identical ice sheets together. Nonuniform ice sheets consisting of floes of thickness t 1 and thin ice of thickness t 2 connecting the floes were used. The major thickness t 1 and the thickness ratio t 2 /t 1 were varied. Ice sheets of uniform thickness (t 2 /t 1 = 1) never formed ridges; they only rafted. However, when ice sheets of nonuniform thickness were used, initial rafting transformed into ridging. In general, high values of t 1 and low values for t 2 /t 1 favored ridging, while low values of t 1 and high values for t 2 /t 1 favored rafting. The forces during the tests were measured. During the initial rafting stage the force increased linearly with displacement. The experiments also suggest that the ridging force has a maximum value. This limit can be related to horizontal growth of the ridge or onset of ridging in another site. The relation between force and ice sheet thickness has also been analyzed. Further, from the force and the measured ridge profiles it was possible to estimate the ratio of work to change in potential energy. This ratio was about 15 for ridging and about 35 for rafting. INDEX TERMS: 4540 Oceanography: Physical: Ice mechanics and air/sea/ice exchange processes; 4594 Oceanography: Physical: Instruments and techniques; KEYWORDS: pressure ridges, rafting, model ice, ice tank, ice cover deformation Citation: Tuhkuri, J., and M. Lensu, Laboratory tests on ridging and rafting of ice sheets, J. Geophys. Res., 107(C9), 3125, doi: /2001jc000848, Introduction [2] The main deformation modes of sea ice cover under compression are ridging and rafting. Ridges are elongated piles of ice blocks and rafting is overriding of one ice sheet by another. The study of these processes is important because the energy expended in deformation determines the large scale strength of the ice pack, and because the horizontal stresses in the ice cover and the strength and thickness of ridges are important in the design and operation of Arctic vessels and offshore structures. [3] Several ridging models have been proposed. Parmerter and Coon [1972] considered a process where two ice sheets move toward each other closing a lead filled with broken ice pieces. Kovacs and Sodhi [1980] suggested that when a moving ice sheet hits a preexisting ridge or a thicker sheet, the ice sheet will buckle and form ice blocks, which then pile up to form a ridge. Sayed and Frederking [1988] modelled ridges as two-dimensional wedges and assumed that a critical state of the wedge simulates an actively forming ridge. Hopkins [1994, 1998] used a discrete element method to study ridge formation when thin lead ice was driven against a thick floe. Common to the above approaches is the assumption of a preexisting obstacle, while Copyright 2002 by the American Geophysical Union /02/2001JC less attention has been given to ridge formation from two ice sheets of equal thickness, which is a common mechanism in the Baltic Sea. Parmerter [1975] suggested that, when pushed together, two similar ice sheets raft if they are thin and form a ridge if they are thick. Ridging has also been studied in ice model basins. In these tests a scaled model ice sheet was either forced to break and accumulate against a plate or ice rubble was compressed between two ice sheets [Abdelnour and Croasdale, 1986; Timco and Sayed, 1986]. [4] This paper will describe and analyze a set of scaled ridging and rafting experiments. The experiments consisted of two parts. First an attempt was made to produce ridges from an ice sheet of uniform thickness. As this was not successful, the attention shifted to study ice sheets of nonuniform thickness. Parts of the test program and some results have already been described by Tuhkuri and Lensu [1998] and Tuhkuri et al. [1998, 1999]. Parallel computer simulations were also performed and are reported by Hopkins et al. [1999]. 2. Ice Tank and Model Ice [5] The ice tank of the Ship Laboratory at the Helsinki University of Technology was used in the tests. The tank is a40m 40 m water basin equipped with a cooling system and an xy carriage. Water depth in the basin is 2.8 m. The x carriage has a span of 40 m and is mounted on rails along 8-1

2 8-2 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Table 1. Thickness t, Bending Strength s f, and Young s Modulus E of the Uniform Ice Sheets Test t, mm s f, kpa E, MPa B C D E F the two sides of the basin. A smaller y carriage hangs under the x carriage and is mounted on round steel rails. [6] In order to obtain model ice with scaled mechanical properties (strength, Young s modulus), the basin is filled with a mixture of ethanol (0.4%) and water. Temperature of the solution is maintained at about 0.2 C. The model ice used was granular fine grained ice which is produced by spraying the basin water into air from about 300 nozzles attached into the x carriage while it is moving at a constant velocity back and forth across the basin. During spraying the air temperature is maintained at about 10 C and thus the water droplets partly freeze in the air. This procedure results in a layer of fine grained granular model ice with uniform structure and grain size. When a desired thickness is reached, the air temperature is lowered to about 15 C and the ice is left to harden. After that, a target model ice strength is obtained by tempering the ice at a higher temperature [Jalonen and Ilves, 1990; Li and Riska, 1996]. [7] In ice modeling different methods may be used to scale the results, depending on the physical processes addressed. As the ridging and rafting processes are not fully understood, the required scaling cannot be fully specified either. Therefore, in the experiments reported here Froude scaling, which is the commonly used scaling method in ice modeling, was used. In this method similitude between the ratios of inertial and gravitational forces in the model scale and full scale is required [Newman, 1986]. Then, denoting the geometric pffiffiffiscale factor as l, the forces scale as l 3, velocities as and bending strength and Young s modulus as l. The downward bending strength s f for each ice sheet used in the test was determined from cantilever beam tests. The Young s modulus E was determined from load deflection curves with plate deflection theory [Sodhi et al., 1982]. This method, however, has a broad error margin as the deflections are very small. The measured model ice properties are given in Tables 1 and 2. Using Froude scaling, the current experiments can be interpreted as model tests where l Tests With Uniform Ice Sheets 3.1. Experimental Setup [8] Following earlier experimental work on ridging, an attempt was made to produce ridges from ice sheets of uniform thickness. Ice sheets with different thicknesses and mechanical properties were produced. In each test the model ice sheet was cut into two 20 m 40 m ice sheets and one of the sheets was pushed against the other at the velocity of about 1 mm/s. The initial 40 m long lead was either irregular and about mm wide with loose floes floating in the lead, or straight and narrow, or zigzag shaped. In one test the angle between the lead and the pushing direction was 69, in the other tests the angle was 90. Only visual observations were made during these tests. Table 1 gives the test variables Test Results [9] In all the tests with uniform ice thickness only finger rafting and no ridging was observed. The ice sheets did fracture when lifted/submerged on top/below the other ice sheet, but the fractured floes kept their arrangement when pushed and did not start ridging. [10] Parmerter [1975] assumed that rafting of two ice sheets can occur only if the maximum stress caused by the deformation does not exceed fracture levels. If fracture occurs, ice blocks will break off the sheet and provide the rubble to form a ridge. However, if the ice sheet is thin enough, the deformation needed to raft ice sheets will not cause fracture. According to this model, the maximum thickness for a rafting ice sheet is t max ¼ 14:2 1 n2 P r w g where n P is the Poisson s ratio for ice, r w is the density of water, s is the (bending) strength of ice, and E is Young s modulus. If n P, r w, and g are assumed to be constant, the s 2 E ð1þ Table 2. Thickness t 1 and t 2, Bending Strength s f, Young s Modulus E, Thickness Ratio, and Deformation Type in the Tests With Nonuniform Ice Sheets Ice Sheet Tests t 1, mm t 2, mm s f, kpa E, MPa t 2 /t 1 Types 1 1, B2,A 2 3,4, A,A,A 3 6,7, B1,B1,A 4 9,10, C,B2,B1 5 12,13, B1,B2,B1 6 15,16, B1,B1,B1 7 18,19, B2,A,A 8 21,22, A,A,A 9 24,25, A,A,A 10 27,28, A,A,A 11 30,31, B1,B2,B ,34, B2,C,C 13 36,37, A,A,A Figure 1. Crossover thickness between ridging and rafting suggested by Parmerter [1975] (solid line) and test results (asterisks). In all the tests the ice sheets rafted.

3 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-3 steel frame was attached into the y carriage of the ice basin. In most of the tests the pushing velocity was 10 mm/s. The ridge initiated at the transverse cut. The load on the pusher plate and its displacement were measured. Each test was also recorded with a video camera. [13] In some tests the front sheet, which was pushed with the pusher plate, started to move sideways (in the y direction, Figure 2). This displacement might have been caused by the forces at the ridging site. Unfortunately, this displacement created a contact with the surrounding ice sheet and a frictional force at one side of the front sheet. As in this test setup this frictional force cannot be separated from the ridging or rafting force, these cases are not reported here. Figure 2. The experimental setup used in the ridging and rafting tests with ice sheets of nonuniform thickness. crossover thickness between ridging and rafting is a function of s and E only. Figure 1 shows the crossover thickness between ridging and rafting calculated from Parmerter s equation and also from the test results given in Table 1. Even though the model ice sheets were thicker than the crossover thickness, no ridging occurred. This result suggests that the Parmerter s concept of crossover thickness should be used with caution. 4. Tests With Nonuniform Ice Sheets 4.1. Experimental Setup [11] As model ice sheets with uniform thickness did not ridge, the attention moved to using ice sheets with nonuniform thickness. These sheets consisted of floes of thickness t 1 and thin ice of thickness t 2 connecting the floes. These ice sheets were produced in two stages. First an initial ice sheet of thickness t 1 t 2 was produced and broken into square shaped floes of size 500 mm 500 mm which were randomly distributed across the tank. No ice was removed, but as the edges of the floes were crushed and some of the floes were pushed on top of the others, areas of open water were created. Then another ice sheet of thickness t 2 was produced on top of the floes creating the ice sheet with thicker floes (t 1 ) in a thinner matrix (t 2 ). The brash remnants created by the cutting of the initial sheet and rearranging the floes, produced roughness to the floe rims but did not otherwise affect the mechanical properties of the floes. [12] Figure 2 illustrates the test setup. For each test a 6 m wide area of nonuniform ice was cut loose from the surrounding ice sheet by cutting parallel slots, leaving the far end uncut. Then the strip, which length was about 26 m, was divided into two parts by making a transverse cut. During an experiment the strip was compressed with a vertical pusher plate. The pusher plate was constructed from a 25 mm thick plywood attached into a steel frame. The 4.2. Test Program and Analysis Procedures [14] Table 2 shows the test program and lists the key test variables. 38 tests were performed in altogether 13 ice sheets. As illustrated in Figure 2, it was possible to run three tests in each ice sheet. The values for the bending strength s f and Young s modulus E were measured for Tests 1 17 from the initial sheet of thickness t 1 t 2. However, these values are uncertain as they may have changed during the freezing-tempering cycles involved in the creation of the secondary layer t 2. Therefore, for Tests s f and E were measured from the sheet with thickness t 1. For a few ice sheets the floe size ( mm) and pushing velocity (10 60 mm/s) were varied, but no effect on the results was found [Tuhkuri and Lensu, 1998]. [15] After completing the tests with one ice sheet, the ridge cross sections were studied. This was done by cutting channels into the rubble and recording the cross sections with an underwater video camera. Before cutting, the formations were left to consolidate. Images were then captured from the video recordings so that cross-sectional profiles could be constructed. [16] Figure 3 shows cross sections of an ice sheet before a test and of a ridge formation after a test. During a test the pusher plate moved a distance of l 0 and an amount of ice pushed into a ridge was A 0 = tl 0, where t is the average thickness of the parent ice sheet. The cross-sectional area of the ridge A TOT is defined to be A TOT ¼ A R þ tl 1 where l 1 is the section length and A R = A K + A S is the crosssectional area of the rubble. A K and A S are the keel and sail areas, respectively. The cross-sectional area of a profile measured after a test was not A TOT but only the underwater part of A TOT denoted as A 1. A TOT was calculated by assuming isostatic balance: A TOT ¼ 1 þ r W r ICE A 1 r W where r W (1000 kg/m 3 ) and r ICE (930 kg/m 3 ) are densities of water and model ice, respectively [Li and Riska, 1996]. The porosity n of a ridge is defined as the porosity of the keel and sail mass only and n can be obtained from the above two equations and the relationship ð2þ ð3þ ð1 nþa R ¼ tl 0 ð4þ

4 8-4 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Figure 3. formation. Schematic presentation of cross sections of the ice sheet before and after a test of ridge [17] When the ridge profile and porosity are known, the increase in potential energy during the test can be calculated, and as the work done on the system can be calculated from the measured force and displacement, the energy losses can be estimated. Possible energy sinks in the system include frictional dissipation and inelastic deformation of the model ice. The energy balance during a test is W ðlþ ¼E P ðlþþe D ðlþ where W is the work done, E P is the change in stored potential energy, E D is the dissipated energy, and l is the displacement of the pusher plate. The stored potential energy can be estimated by neglecting porosity variations within the ridge. The potential energy of a section of width x (Figure 3) is then Z 0 Z h1 E P ðxþ ¼gð1 nþ ðr ICE r W Þ zdz þ r ICE zdz h 2 0 By assuming further that a section x is in isostatic balance, h 1 can be given as a function of h 2, r W, and r ICE and equation 6 can be put into the form E P ðþ¼ x 1 2 gð1 nþ ð r W r ICE Þ r W h 2 2 r ICE [18] The potential energy for a cross section was calculated as the integral of equation (7). The cross-sectional area and shape may vary within a ridge, and therefore at least two cross sections were made for each ridge and the average potential energy was determined Test Results Description of the Observed Deformation Process [19] Characteristic to the laboratory experiments was that at first the two ice sheets rafted at the initial transverse cut. This rafting was either finger rafting or simple rafting. The ð5þ ð6þ ð7þ rafting continued until one sheet started to fail and form a ridge. Length of this initial rafting depended on the major thickness t 1 and the thickness ratio t 2 /t 1 (1). Small t 1 and high t 2 /t 1 favored long lengths of rafted ice. In tests where t 2 /t 1 = 1, only rafting was observed. However, if t 1 was large enough and t 2 /t 1 was low enough, the initial rafting transformed into ridging. Figure 4 illustrates the deformation types observed and Figure 5 shows how they depend on t 1 and t 2 /t 1. The observed deformation types types for each test are listed in Table 2. The deformation types can be characterized as follows: Type A: Initially finger rafting but then ridging which formed both a sail and a keel. Also shear zones formed. Type B1: Initially finger rafting but then rubble piles formed under each finger. A sail did not form; The rubble piles formed under intact ice sheets. Type B2: Similar to B1, except that the deformation started as simple rafting. Type C: Only simple rafting, no rubble formation. [20] Types A and C are obvious ridging and rafting processes, respectively, but Types B1 and B2 are processes between these modes. Only Type A had a clear sail formed from discrete ice blocks. In Type B ridges, buoyancy of the keel lifted up the ice sheet on top, but slope of the above water part was so gradual that no clear sail formed. Interestingly, from above, as in Figure 4, Type B1 formations look just like finger rafting even though there is a large keel. During a single test it was often possible to observe two deformation types. A Type B2 process is always preceded by a Type C process, and a Type B1 process can transform into in a Type A process. [21] As an example, a Type A ridging observed during Test 2 is described in Figures 6 and 7. At first the two sheets rafted at the 6 m wide initial cut and formed three fingers (Figures 6a and 7a). This initial rafting transformed into ridging when the underriding ice sheet started to fail into ice blocks. With continued displacement, the ridge was growing in depth but not much in the pushing direction. In the later stages of the test the widths of the fingers were changing as one of the original three fingers merged into

5 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-5 the middle finger (Figures 6c and 7b) and also new fingers formed (Figure 7c). The ridge in Figure 6e is basically similar with that in Figure 6d, although about 4 m of ice has been added to the ridge. There are some changes in the order of overriding and in the horizontal shape of the ridge and, naturally, the keel has grown. Interestingly, even though the ice sheets were under uniaxial compression, a shear zone formed in the ridge (Figures 6f and 7d). At the shear zone, the ice sheets did not overlap Ridging Force [22] The ridging and rafting processes can be described in more detail by analyzing the measured force displacement records. An F(x) record of Type C (simple rafting) is shown in Figure 8. When this Test 34 started, the front sheet went under the back sheet at the initial cut. This rafting continued until x 6.8 m. Then the front sheet failed under the back sheet and started to raft a third layer under the second layer. The F(x) record shows that a threshold force value was required to initiate the rafting. There was also an initial force peak in all the tests. After this initial phase, the rafting force increased linearly with displacement until the ice sheet failed and the load dropped abruptly. With continued rafting, the force again increased linearly. The force drop at x 6.8 m is linked with the ice sheet failure over the whole width of the sheet. In other tests where the ice sheet failure was more localized, such extreme force fluctuations were not observed. In other words, this test was kind of two dimensional. [23] Figure 9 shows the force displacement records of Type C (Test 9) and Type B2 (Tests 1, 10, 13) tests. During Test 9 the ice sheets rafted into two layers, and thus the F(x) record is linear after the initial force peak. Similarly, the initial parts of the F(x) records of Type B2 tests are also linear and the of these two types are about equal, but after a displacement l R the forces of Type B2 processes deviate from the linearly increasing trend. For Test 10, l R 10 m which is also the measured length of the ridge profile. For Tests 1 and 13, l R 5 m which is again about the length of the ridge profiles. It is thus reasonable to hypothesize, that during these tests the pushing force was increasing linearly with displacement as long as rafting was taking place and that the rafting continued until the horizontal force was high enough to break the ice sheet and initiate rubble formation. Figure 4. Sketches of the deformation types observed in tests on compression of ice sheets. The dashed lines show extent of keel or overlap. Pluses refer to an ice sheet on top, and arrows refer to movement. Figure 5. The deformation types in compression of ice sheets as a function of t 2 /t 1 and t 1.

6 8-6 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Figure 6. Sketches of ridging observed during Test 2. Time of the snapshot is given in seconds. The keel extent is shown with dashed lines. The lines in Figure 6F show locations of the cross-sectional profiles (see Figure 13). [24] During rafting into two layers, the F(x) record was increasing linearly (Figure 9). Similarly, the contact area between the two sheets was increasing linearly with x. This suggests that the constitutive relationship at the ice-ice interface can be described using the relationship F = mf n, where F is the tangential force and F n is the normal force. m is the coefficient of kinetic friction which in this case can be calculated from the the difference between densities of model ice and basin water, and the ice sheet dimensions. For Test 9, m = 0.36 is obtained. [25] An example of the F(x) record for Type A ridging is shown in Figure 10. This record includes all the features typical to Type B processes described above, albeit the geometry of the ridge types are different: There is an initial force peak, a threshold force value is needed to initiate the process, and the force increases first linearly but then deviates from the linear trend. Figure 10 suggests also that the ridging force reaches a maximum level toward the end of the test and then maintains that level. There is visual evidence, that this constant force level is related to the horizontal growth of a ridge. The force increases with the amount of ice pushed into the ridge until a limit height/ depth is reached and the ice sheet starts to pile up/down against the ridge side. [26] Figure 11 shows the effect of ice thickness on the ridging and rafting force. Both the force needed to initiate a ridging process and the ridging force are increasing with ice thickness. The same data is also shown in Figure 12. For each thickness, the lower and upper groups of points are average forces for 100 mm displacements in ranges 1000 mm < x < 2000 mm and 6000 < x < 7500 mm, respectively. Also shown are a first order polynomial fitted to the lower group and a second order polynomial fitted to the upper group. The data suggests thus, that during the initial stages of a ridging process, the force-thickness relationship is linear. This is in line with the observation that the ridging processes started as rafting. The data gives also support of an assumption that during the latter stages of ridging the force-thickness relationship is nonlinear Cross-Sectional Profiles and Porosities of the Ridges [27] Ridge and raft cross sections were examined after selected tests. Table 3 lists the measured dimensions and shows that mostly Type A ridges were studied. As an example, Figure 13 shows the cross sections of the ridge formed during Test 2. The porosities were calculated by equation (4). [28] The average porosity of all Type A ridges in Table 3 is 0.21 and the minimum and maximum porosities are 0.03 and 0.40, respectively. In addition to this variation from ridge to ridge, the rubble cross section areas A 1 and consequently the calculated porosities within one ridge varied also. This can be caused by the flow of ice pieces in across channel direction during ridge formation. In addition, the ridge geometry can cause deviation to the data. [29] As the porosity of Type C deformation, where only rafting occurs, is by definition small, also porosity of Type B1 and B2 ridges should be lower than the porosity of Type A ridges. The results from Tests 11 and 12 support this kind of reasoning. In Test 11 the final ridge had a large section length, it included large areas of rafted ice, and hence the porosity was low (n = 0.11). Test 12, in turn, resulted into a more compact ridge, and the porosity was higher (n = 0.26). [30] Table 3 gives also the maximum keel depth h K, sail height h S and the ratios h K /h S and h K /t 1. The average values of the ratios for Type A ridges were: h K /h S = 5.4 and h K /t 1 = 9.0. The keel depths for Type B ridges were smaller than those of Type A. No clear h K to t 1 relationship can be formed from the test results. Rather, the data suggests that the Type A ridges reached a limit height of about 0.8 m irrespective of the ice sheet thickness Energy Balance [31] Table 3 gives the calculated values for the ratio of work to stored potential energy for Type A ridges. The average value of W/E P was The lowest measured value was 13.1 and the highest W/E P for Type A ridges showed no dependency on thickness of ice sheet t 1. [32] The value of the ratio W/E P for Type C rafting can be estimated from Test 9. By assuming porosity to be zero, the value W/E P = 34.6 is obtained. The ridge which formed during Test 11 was of type B2 but included large

7 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-7 a) b) Figure 7. Photographs taken during Test 2. Squares of thicker ice (light shade) are frozen into a matrix of thinner ice (dark shade): (a) after displacement of about 1.9 m, (b) after displacement of about 3.4 m, (c) after displacement of about 4.8 m, and (d) after displacement of about 6.5 m.

8 8-8 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS c) d) Figure 7. (continued)

9 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-9 Figure 8. The force displacement record of Test 34. The ice sheets rafted into three layers; t 1 = 30 mm, and t 2 = 12 mm. areas of rafted ice, and thus the measured W/E P = 26.4 is between the values for Type A and Type C, as expected. 5. Discussion 5.1. Ridging Versus Rafting [33] The key result of this study was that, for the first time, ridges were formed in an ice tank from two ice sheets of equal thickness. This was achieved through using ice sheets with nonuniform thickness. Even though such ice sheets can be found in nature, it is not suggested that only they form ridges. Instead, the nonuniform model ice should be seen as a test method to model inhomogeneity. In addition to thickness inhomogeneity, ice in nature contains cracks, brine channels, and the snow cover can have a variable thickness. As shown above and also through systematic numerical simulations by Hopkins et al. [1999], the inhomogeneity is an important parameter governing the likelihood of ridging and rafting along with the ice thickness, strength, and elastic modulus identified by Parmerter [1975]. [34] It must be recognized however, that the classical concept, in which ice sheets raft if they are thin and ridge if they are thick, is backed by several field observations. Both rafting and ridging occur in nature but thin ice sheets seem Figure 10. The force displacement record of Test 29 during which a Type A ridge formed; t 1 = 95 mm, and t 2 = 31 mm. to be rafting more often than thick ice sheets. What is important, though, is that even if it is rare that thick ice sheets raft, it has been observed as reported already by Parmerter [1975]. Maybe rafting of thick ice sheets is rare, not only because they are thick, but because thick ice sheets are rarely uniform enough to raft. Thin smooth ice sheets are usually recently formed during a cold calm period. Thicker ice sheets are more often created from ice that has already experienced some deformation, for example from consolidated floes. Thicker ice has also experienced more cycles of temperature variation which creates thermal cracks in the surface layer of the ice. [35] If ridging really initiates as rafting, a part of a ridge cross section should have rafted layers of the parent ice sheet near the waterline. The ridge sail and keel which are formed from ice blocks, would then grow above and below these rafted layers. There is some evidence that in some ridges this in fact is the case. Kankaanpää [1997] studied thin sections made from ice samples taken from Baltic ridges. She studied two ridges and showed that in one ridge two ice sheets had rafted and in the another there were 4 to 5 rafted layers. In order to further study this hypothesis, Figure 9. The force displacement records of Tests 1, 9, 10, and 13. The signals are low pass filtered at f N, where f N is the Nyquist frequency. Figure 11. The force displacement record from tests with different ice thicknesses. The data shown are average forces from two tests (95 mm: 28 and 29; 78 mm: 36 and 38; 71 mm: 18 and 20; 48 mm: 31 and 32; 30 mm: 34), except for the thinnest ice. The signals are also low pass filtered at 0.02 f N, where f N is the Nyquist frequency.

10 8-10 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Figure 12. The ridging force shown in Figure 11 as a function of ice thickness. The data points are average forces during a displacement of 100 mm. Tuhkuri et al. [1999] studied three ridges in the Northern Baltic. The ridges were formed from 80 mm thick ice and had keel depths of almost 4 m. Also these ridges included large amounts of rafting. At least five rafted ice layers were found. [36] The observation that rafting was an initial stage of ridging suggests that ridging and rafting are not two different and separate processes but rather different stages in a single process. The terms rafting and ridging are defined by WMO and used to describe an ice cover, but they are not necessarily very good terms to describe the deformation processes of an ice cover. [37] Recently, Timco and Cornett [1997] investigated the influence of nonuniform ice thickness on ice loads on a sloping structure. They defined an effective thickness as the thickness of a hypothetical uniform ice sheet which would give the same force as the nonuniform sheet. This approach cannot be applied to the present ridging problem, because an ice sheet with nonuniform thickness had a different failure process than an ice sheet with uniform thickness Deformation Modes A, B, and C [38] The major difference between Type A and Type B is in the shape of the above water parts (Figure 4). Type A ridges had both keels and sails which were formed from ice blocks, while in Type B only the keels were formed from ice blocks which accumulated under intact ice sheets. Due to buoyancy, the keel of a Type B feature naturally lifted the intact ice sheet upwards, but only a small block sail, if any, formed. The classical geometrical ridge model based on a triangular keel and a triangular sail in isostatic balance, is not well suited to describe these Type B ridges. [39] Another distinct feature was that Type A ridges were curvilinear and not straight. Further, the ridge cross sections were nonsymmetric. The block sails were not above the deepest points of the keels, but rather closer to one keel edge. [40] Although most of the ridge studies in the field have concentrated on measuring the cross sections of welldefined linear ridge segments close to their highest points, also other types of ridges have been observed and reported. Type A and B ridges were identified already by Fukutomi and Kusunoki [1951], ridges with large separation between sail and keel were found in the Arctic by Weeks et al. [1971], and curvilinear ridges were found and measured in the Baltic by Lensu et al. [1998]. The latter ridges resemble the ridges formed in the laboratory. In fact, undulating Type A ridges are common in many sea areas. In fresh undulating ridges water is often seen to rise to the outer curve side of the ridge while the inner curve side is elevated. This pattern alternates from side to side along the ridge. [41] The curvilinearity of Type A ridges was a remnant from the initial finger rafting stage. Maybe this curvilinearity is a property of a ridge in its initial stage of forming. With more and more ice pushed into a ridge, width of the fingers change, some fingers can merge together, and the continuing movement can alter the keel and sail shapes. In this way the linear ridge sections observed in the field can develop in the later stages of ridge formation. [42] The observation that rubble piles can exist under intact ice sheets may have practical implications. An error may result if the volume of ice in a ridge is estimated from the dimensions of a triangular sail only. Bowen and Topham [1996] reported that a ridge they surveyed contained about 3 times as much ice as would be estimated by applying the classical triangular geometry. It is also possible, that an Table 3. Ice Thickness t 1, Push Length l 0, Number of Cross Sections Studied n, Section Length l 1, Keel Depth h K, Sail Height h S, Cross- Sectional Area A 1, Porosity n, Ratio of Work to Change in Potential Energy W/E P, and Deformation Type Test t 1,mm l 0,m n l 1,m h K,m h S,m h K /t 1 h K /h S A 1,m 2 n W/ E P Type B A A A A B B A A A A A A A A A A

11 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-11 Figure 13. Cross-sectional profiles of the ridge formed during Test 2 (see Figure 6).

12 8-12 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Figure 14. Different stages in a ridging process. observer in field may not consider a Type B ridge as a ridge at all, but assume it to be finger rafting. [43] This result may also have implications to remote sensing of ice where it is a common practice to identify an independent ridge as a feature having at least twice the elevation as the shallowest troughs on either side [Tucker et al., 1979; Wadhams, 1981]. In addition, a cutoff value have been assumed in order to distinguish between ridges and other surface irregularities. Typical cutoff values range from 0.5 m to 1 m. Such ridge identification methods do not identify Type B ridges. From this point of view, a bottom topography data can better be used to describe an ice sheet Geometry of Keel and Sail [44] The ratio of keel depth h K to sail height h S for the Type A model scale ridges was 5.4. This is close to the values h K /h S 6.3 suggested for Baltic ridges by Kankaanpää [1997] and h K /h S 4.4 given by Timco and Burden [1997] for first year ridges. [45] The ratio of keel depth h K to ice thickness t 1 for Type A laboratory ridges was 9.0. Kankaanpää [1997] has suggested that for Baltic ridges h K /t The ratio h K / t 1 is a somewhat problematic measure, as it depends on the push length. It can be argued that if enough ice is pushed into a ridge, the ridge becomes fully grown and h K /t 1 reaches a limit. In the experiments this probably was the case, but it is not known whether the field results are obtained from fully grown ridges Characterization of the Ridge Formation Process [46] The visual observations and the measured forces can be used to characterize pressure ridging. Figure 14 illustrates the observed failure sequence in two dimensions and Figure 15 shows the related forces. The observed ridging process has four parts. At first a threshold force value F 0 must be reached before the initial rafting can start. In the second phase the two sheets raft up to a displacement l R and the force increases linearly with = a. This initial rafting ceases when the lower ice sheet starts to fail and pile down. This keel formation is then followed by sail formation near the edge of the upper sheet. During this third phase, the ridging force increases at a lower rate than during rafting. In the fourth phase the ridge is growing laterally as the ice sheets fail against the ridge. This sets the maximum ridging force F MAX and also the maximum ridge depth and height. F MAX is a function of ice sheet thickness and mechanical properties. Also the rafting length l R is a function of ice thickness. The thicker the ice, the shorter is l R. [47] The above characterization is two dimensional and cannot capture all the aspects of Type A and B1 ridges. It is believed, however, that while the third dimension adds new features to the ridging process, it does not affect the phases described above in a major way. [48] In the experiments, the three dimensionality of ridging grew from the initial finger rafting. In finger rafting one sheet overrides the other and the order of overriding varies along the front. Thus the sheets split into strips and, from a fracture mechanics point of view, both the sheets contain cracks under Mode III loading (tearing mode; Irwin [1958]). Both in finger rafting and in Type B1 ridging, the cracks propagated in the direction of the differential movement of the ice sheets. Thus in Type B1 processes the ridge keel sections can be clearly separated from each other (Figure 4). In Type A ridging in turn, after the initial finger rafting phase, the cracks did not propagate in the direction of the differential movement and thus the ice sheets did not divide into separate zones. Rather, in Type A processes the moving ice sheets started to fail into blocks and pile up and down at the area where the order of overriding changed during the initial finger rafting. This resulted in a curvilinear ridge with one continuous keel. [49] This discussion can be summarized by two hypotheses: (i) Ridging starts as rafting, and (ii) the three-dimensional shape of a ridge is affected by crack propagation. It is understood that inhomogeneity is an important parameter. Inhomogenous ice may have weak zones and a rough surface, but inhomogeneity also affects the crack propagation. In homogenous ice cracks tend to propagate in the direction of the push, resulting into finger rafting and Type B ridges, while Type A ridges are created from more inhomogenous ice where the cracks easily follow the preexisting flaws Relation to Geophysical Scale [50] The applicability of the results of this work to the large-scale geophysics of ice deformation is discussed next. Figure 15. Schematic presentation of the force displacement relation during the different phases of a ridging process shown in Figure 14.

13 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS 8-13 With geophysical scale it is here referred to the scale of continuum approximation which is made by most dynamic ice models. This scale varies between 10 and 100 km depending on the considered sea area. The amount of ridging is quantified by ridge density or the number of ridges per unit linear distance. If the arrangement of ridges is independent of direction, the length of ridges per unit area is p/2 times ridge density [Mock et al., 1972]. If ridge density is 6/km, which is a typical value, the total length of ridges in a km 2 area is about 1000 km. The ridges arrange into a crisscrossing connected network where the identification of individual ridges is rather a convention of description. However, the local length scale in ridge formation can be taken to be such that the direction and magnitude of local forces during ridge buildup do not vary much spatially. It is assumed that this scale is 100 m which about corresponds to the 6 m model scale of the ice tank experiments. The creation of the 1000 km of ridges corresponds then to 10 4 ridge formation events comparable to the ice tank experiments. [51] In other words, in relation to the individual ridges the ice cover on a geophysical scale is a macroscopic system composed of a large number of microsystems. In order to understand ice deformation on a geophysical scale, the macroscopic properties must be linked to the ridge formation events on a smaller scale. The macroscopic properties include ridge density, distributions of ridge spacing and ridge size, aggregate strength and energy dissipation. [52] One example of a link from the current ice tank experiments to the geophysical scale is the energy dissipation, which can be estimated using the ratio W/E P from the experiments. Another example is the aggregate strength of ice cover. The aggregate strength may be estimated by a dissipation model [Thorndike et al., 1975] or it may be considered as a model tuning parameter [Hibler, 1979]. In both cases it is found that the aggregate strength is about two magnitudes smaller than the observed local strength. Here it is assumed that the aggregate strength is a real macroscopic parameter that can be observed if the ice cover with all relevant details is scaled down to allow experiments. The objective is then to relate the aggregate strength to local ridge building forces. This includes not only the magnitude of forces but also the onset and cessation of the local ridge buildup processes. There are at least two ways of approaching this: through a statistical physical theory and through discrete element (DE) simulations. A proper statistical theory does not yet exist whereas recent developments of DE methods are promising. In a DE simulation the strategy can be to model each floe in the sea area studied [Hopkins, 1996]. Deformation of the domain produces areas of localized failure and areas of open water. Pressure ridging is an important part of the localized failures, and the present model scale experiments can be used to model these ridging events. As the DE model can be used to derive yield curves in the geophysical scale, the DE method provides a link between small scale ridging and geophysical scale response. 6. Summary and Conclusions [53] This paper has described laboratory experiments on ridging and rafting of an ice cover. During each test, a 6 m wide sheet of floating model ice was pushed against another similar ice sheet and either rafting or ridging was observed. The main results can be summarized as follows: Ice sheets of uniform thickness did not form ridges, they only rafted. When ice sheets of nonuniform thickness were used, an initial rafting transformed into ridging. Type A ridges were curvilinear. This feature was a remnant of initial finger rafting. Type B ridges were piecewise linear and resembled finger rafting when observed from above. In Type B ridges the keel was under an intact ice sheet. During the initial rafting stage, the measured force increased linearly. The experiments also showed that there is a threshold force value which must be reached before the rafting starts. The ridging force continued to grow also after the initial linear trend, albeit at a lower rate. In some tests the force reached a plateau value connected with onset of failure of the surrounding ice sheet against the ridge. The ratio of work done to change in potential energy was about 15 for ridging and about 35 for rafting. [54] Acknowledgments. The experimental work described here was a part of the project Local Ice Cover Deformation and Mesoscale Ice Dynamics supported by the European Commission, DG XII, through the MAST III programme. The partners in the project were Ship Laboratory, Helsinki University of Technology, Finland; Nansen Environmental and Remote Sensing Center, Norway; Scott Polar Research Institute, University of Cambridge, UK; Department of Geophysics, University of Helsinki, Finland; and Engineering Research Institute, University of Iceland, Iceland. During the data analysis, JT was funded by the Academy of Finland through a Research Felowship. References Abdelnour, R., and K. Croasdale, Ice forces associated with ridge building: Model tests and results compared with theoretical models, in Proceedings of International Association for Hydraulic Research Symposium on Ice, vol. 3, pp , Inst. of Hydraul. Res., Univ. of Iowa, Iowa City, Bowen, R. G., and D. R. Topham, A study of the morphology of a discontinuous section of a first year arctic pressure ridge, Cold Reg. Sci. Technol., 24, , Fukutomi, T., and K. Kusunoki, On the form and the formation of the hummocky ice ranges (in Japanese with English summary), Low Temp. Sci., 8, 59 88, Hibler, W. D., III, A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, , Hopkins, M. A., On the ridging of intact lead ice, J. Geophys. Res., 99, 16,351 16,360, Hopkins, M. A., On the mesoscale interaction of lead ice and floes, J. Geophys. Res., 101, 18,315 18,326, Hopkins, M. A., Four stages of pressure ridging, J. Geophys. Res., 103, 21,883 21,891, Hopkins, M. A., J. Tuhkuri, and M. Lensu, Rafting and ridging of thin ice sheets, J. Geophys. Res., 104, 13,605 13,613, Irwin, G. R., Fracture, in Handbuch der Physik, Band VI, edited by S. Flügge, pp , Springer-Verlag, New York, Jalonen, R., and L. Ilves, Experience with a chemically-doped fine-grained model ice, in Proceedings of International Association for Hydraulic Research Symposium on Ice, vol. 2, pp , Helsinki Univ. of Technol., Dep. of Mech. Eng., Espoo, Finland, Kankaanpää, P., Distribution, morphology and structure of sea ice pressure ridges in the Baltic Sea, Fennia, 175, , Kovacs, A., and D. S. Sodhi, Shore ice pile-up and ride-up: Field observations, models, theoretical analyses, Cold Reg. Sci. Technol., 2, , Lensu, M., J. Tuhkuri, and M. Hopkins, Measurements of curvilinear ridges in the Bay of Bothnia during the ZIP-97 experiment, Rep. M-231, 64 pp., Helsinki Univ. of Technol., Ship Lab., Espoo, Finland, Li, Z., and K. Riska, Preliminary study of physical and mechanical properties of model ice, Rep. M-212, 100 pp. + append., Helsinki Univ. of Technol., Ship Lab., Espoo, Finland, 1996.

14 8-14 TUHKURI AND LENSU: LABORATORY TESTS ON RIDGING AND RAFTING OF ICE SHEETS Mock, S. J., A. D. Hartwell, and W. D. Hibler, Spatial aspects of pressure ridge statistics, J. Geophys. Res., 77, , Newman, J. N., Marine Hydrodynamics, 402 pp., MIT Press, Cambridge, Mass., Parmerter, R. R., A model of simple rafting in sea ice, J. Geophys. Res., 80, , Parmerter, R. R., and M. D. Coon, Model of pressure ridge formation in sea ice, J. Geophys. Res., 77, , Sayed, M., and R. M. W. Frederking, Model of ice rubble pileup, J. Eng. Mech., 114, , Sodhi, D. S., K. Kato, F. D. Haynes, and K. Hirayama, Determining the characteristic length of model ice sheets, Cold Reg. Sci. Technol., 6, , Thorndike, A. S., D. A. Rothrock, G. A. Maykut, and R. Colony, The thickness distribution of sea ice, J. Geophys. Res., 80, , Timco, G. W., and R. P. Burden, An analysis of the shapes of sea ice ridges, Cold Reg. Sci. Technol., 25, 65 77, Timco, G. W., and A. M. Cornett, The influence of variable-thickness ice on the loads excerted on sloping structures, Cold Reg. Sci. Technol., 26,39 53, Timco, G. W., and M. Sayed, Model tests of the ridge-building process of ice, in Proceedings of International Association for Hydraulic Research Symposium on Ice, vol. 1, pp , Inst. of Hydraul. Res., Univ. of Iowa, Iowa City, Tucker, W. B., W. F. Weeks, and M. Frank, Sea ice ridging over the Alaskan continental shelf, J. Geophys. Res., 84, , Tuhkuri, J., and M. Lensu, Ice tank tests on ridging of non-uniform ice sheets, Rep. M-236, 130 pp., Helsinki Univ. of Technol., Ship Lab., Espoo, Finland, Tuhkuri, J., M. Lensu, and M. Hopkins, Laboratory and field studies on ridging of an ice sheet, in Ice in Surface Waters, edited by H. T. Shen, vol. 1, pp , A. A. Balkema, Brookfield, Vt., Tuhkuri, J., M. Lensu, and S. Saarinen, Laboratory and field studies on mechanics of ice ridge formations, in Proceedings of the 15th International Conference on Port and Ocean Engineering Under Arctic Conditions, edited by J. Tuhkuri and K. Riska, Rep. M-241, vol. 3, pp , Ship Lab., Helsinki Univ. of Technol., Espoo, Finland, Wadhams P., Sea-ice topography of the Arctic Ocean in the region 70 W to 25 E, Philos. Trans. R. Soc. London, Ser. A., 302, 45 85, Weeks, W. F., A. Kovacs, and W. D. Hibler, Pressure ridge characteristics in the Arctic coastal environment, in Proceedings of the 1st International Conference on Port and Ocean Engineering Under Arctic Conditions, vol. I, pp , Tech. Univ. of Norway, Trondheim, M. Lensu, Ship Laboratory, Helsinki University of Technology, P. O. Box 5300, FIN HUT Espoo, Finland. (mikko.lensu@hut.fi) J. Tuhkuri, Laboratory for Mechanics of Materials, Helsinki University of Technology, P. O. Box 4100, FIN HUT Espoo, Finland. ( jukka. tuhkuri@hut.fi)