Investigation on Characteristics of Pack Ice Motion with Distributed Mass/Discrete Floe Model

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1 Investigation on Characteristics of Pack Ice Motion with Distributed Mass/Discrete Floe Model Comparison to Circulation Water Channel Experiment by Chang Kyu Rheem *, Member Hajime Yamaguchi *, Member Hiroharu Kato *, Member Summary A recently proposed Distributed Mass/Discrete Floe model (DMDF model) is a model for practical computations of pack ice motion. The floes are divided into bunches, in which they are assumed to be distributed uniformly. As a result, larger number of floes can be treated in comparison with discrete element models. The ice interaction force is formulated from the momentum conservation in the collisions of floes in a bunch and between the bunches. The movement of floes is calculated in combination with water flow using a multi-layer model to express the surface flow more accurately. This paper describes an improved disk floe version. In a circulation water channel, drift tests of physical model floes were performed in order to investigate characteristics of their motion and interaction with the structure. The results are summarized as follows : The floe motion near the structure depends on the floe shape. Disk floes show a lateral motion in front of the structure. They flow out from both sides. As a result, the number of the floes is front of the structure decreases with time. Rectangle floes, however, have no motion in front of the sturucture. The number of the floes in front of the structure remains constant with respect to time. These experiments indicate that when the motion of pack ice is simulated around a structure, it is important to choose the floe shape. The disk floe motion and the rectangle floe motion can be regarded as the two extreme cases of real pack ice motions. An actual pack ice motion may be between these two extreme cases. Two computations using DMDF model were made. Simulation results were compared with the circulating water channel experiments and the forecast of the operational model of Environment Canada. The DMDF predicted the circulating water channel drift tests quite closely. The DMDF results also compared quite well with the computations at the Environment Canada. Nomenclatures Total ice area in a bunch Length of a bunch in x-direction Length of a bunch in y-direction Ca : Friction coefficient between the air and ice Ci : Ice concentration of a bunch Friction coefficient between the water and ice dcfx : Distance between center of neighboring disk floes in x-direction dcfy: Distance between center of neighboring disk floe rows in y-direction dlic: Diameter of the disk floe dt : Time step for the computation f : Coriolis parameter Ice interaction force ( internal stress ) g : Acceleration hi : Ice thickness of gravity Unit vector in the depth direction Mass of a bunch Number of floe rows in a bunch in x-direction Number of floe rows in a bunch in y-direction Number of floe rows to collide in a bunch in x -direction Number of floe rows to collide in a bunch in y -direction Wind velocity at a height of 10 m Ice velocity Water velocity Ice/ice friction coefficient Water surface elevation Density Density of air of ice University of Tokyo 1. Introduction Received 10th Jan Read at the Spring meeting 17, 18th May 1995 When the motion of pack ice is simulated, it is very important to evaluate the interaction among the frag-

2 132 Journal of The Society of Naval Architects of Japan. Vol. 177 ments of pack ice and a solid body such as a coastline or an ocean structure. So far, researchers involved with such problems have treated the pack ice as a continuum. Several continuum models have been proposed and applied to long-term and wide-area predictions1),2),3) and short-term and narrow-area predictions4),5),6). However, since the pack ice has discrete features, such models have certain limitations. On the other hand, Serrer et al.7), Frederking & Sayed8) and LOset9) proposed a discrete element model approximating each ice floe by a disk. Although this approach is interesting, practical applications of it might be difficult because it requires much computation time. A Distributed Mass/Discrete Floe model is a model for numerical simulation of mesoscale pack ice rheology. It has the advantages of both a continuum model and a discrete model. It can express the discrete nature of pack ice which is difficult for a continuum model to treat. It can treat larger number of floes in much shorter computation time compared to discrete element model. A rectangular floe version of this model was reported in previous papers10),11), and a disk floe version in another paper12). This paper describes an improved disk floe version. Figure 1 shows the computational procedure of this model. The drift tests of floe models are done in a circulation water channel. Two computations are made in order to investigate characteristics of pack ice motion with Distributed Mass/Discrete Floe model. The first is made for the same conditions as the drift test of the physical model floes around an ocean structure made at the circulating water channel. The others is pack ice movement simulation along an irregular boundary on the east coast of Canada. Simulation results are compared with the forecast of the operational model of Environment Canada 13). 2. Distributed Mass/Discrete Floe Model 2. 1 Ice model In the present model, the pack ice is divided into rectangular bunches in which the floes, all of equal size, are assumed to be distributed uniformly. A bunch is characterized by the values of its center position, area and ice concentration which is the ratio of the total floe area to the bunch area : ( 1 ) ( 2 ) ( 3 ) ( 4 ) where, bl.,bly is the bunch area, N,Ni, is the number of disks in the bunch, and dl ic is the diameter of a disk. Figure 2 shows such a bunch with 9 disk floe elements. The ice floes are denoted by the gray disks in the bunch. One of them is divided into two semi-disks to describe the uniform distribution of floes. A floe such as A, whose right half is inside the bunch, is treated as one floe in the same manner as the others in the bunch. On the other hand, floe B belongs to the neighboring bunch. The triangle abc is a regular triangle and 0 is a vertical angle of a regular triangle Equations of bunch motion Pack ice movement with a relatively short time-scale (of the order of several days at most) can be numerically well simulated with a dynamical model where the Coriolis force, the sea surface inclination force, the interaction forces among the floe bunches and the stress due to the wind and water current are taken into account, neglecting ice growth and ablation. The momentum change of the ice floe bunch in a time interval of dt can be expressed by the following equations : where, ( 5 ) ( 6 ) ( 7 ) Fig. 1 Computational procedure of DMDF model Fig. 2 Disk floe bunch

3 Investigation on Characteristics of Pack Ice Motion with Distributed Mass/Discrete Floe Model 133 ( 8 ) mi is the bunch mass while ra and eu, denote the shear forces acting on the bunch due to the wind and the sea current respectively. -MigV77 is the force due to the inclination of sea surface. -Mifk ~ ƒëi is the Coriolis force which acts in the right-hand direction perpendicular to the bunch movement in the Northern Hemisphere. Pi denotes the ice internal force, i. e. the interaction force of floes in a bunch and between bunches1 1)12) 2. 3 Bunch movement and deformation The bunches move following the above equations and change the shapes due to the interaction. In the present model, the bunch movement and deformation are ex- pressed by the movement of the four edges of bunch. One bunch, the size and velocity at the time t are blirba and ƒëti respectively, is considered in Figure 3. The bunch collides with the neighboring bunch within one time step of dt. The axial interaction force exerted on the edge of E is FL. The y-direction lateral expansion force12) exerted on the edge of N and S are expressed by, ( 9 ) where pi is the friction coefficient between floes. Assuming that the force exerted on the floes except the interaction force is Fb, the velocity of the edges of the bunch at time t + dt are expressed as, (10) (11) (12) (13) Assuming that the velocity of the edges varies linearly, the moving distances of the edges are determined by the following equations, (14) (15) and the size of the bunch changes as follows : (17) (18) (19) 2. 4 Model improvement In the previous paper12), assuming that the floe distribution in a bunch is invariant within one time step dt, the ice interaction forces are formulated using a fully explicit scheme as follows, (20) (21) where the superscript t denotes values at the time t and +1 denotes values of the neighboring bunch. However, the floe distribution in the bunch would be altered within one time step dt due to the lateral expansion of floes. In order to take into consideration the change of the floe distribution, the ice interaction forces are formulated as follow ; (22) (23) where m'r is the mass of the floes taking part in the collision within one time step dt. The interaction forces dt and the mass mt are evaluated 3. Drift test of physical model floes by iteration. The aim of the drift test of model floes was to investigate characteristics of floe motion and interaction between floe and ocean structure for different floe shapes Experimental set-up The experiments were performed in a circulation water channel with a test section of 8 m long ~ 1.8 m wide ~ 0.9 m deep. Figure 4 shows the principal arrange- (16) (a) t = t (b) dt Fig. 3 Bunch movement and deformation Fig. 4 Measurement system arrangement for an ocean structure model

4 134 Journal of The Society of Naval Architects of Japan, Vol. 177 ment for these tests which are more thoroughly described in Rheem14). The disk and rectangle floe models were made out of polypropylene with a specific weight of The size of disk was 50 mm diameter ~ 15 mm deep and that of rectangle was 50 mm ~ 50 mm ~ 15 mm deep. The ocean structure model of size 30 cm ~ 30 cm ~ 5 cm deep was made from stainless steel mesh to pass water. A threecomponent dynamometer was used to support the structure. The wind speed was measured by a Pitot tube anemometer and the water speed was measured by a blade wheel current meter. Figure 5 shows the measuring section and the initial conditions of the tests. The floes were at rest and the wind and the current were steady. The initial concentration of floes was 0.99 for the case of rectangles and 0.9 for the case of disks Experimental results Figures 6 and 7 show the measured wind and current distribution in the absence of floe on the water surface. Both are slightly higher at the inside and outside of the channel than at the center. The downstream water speed is higher than the upstream due to the wind. Figure 8 shows the time variation of drift in both cases. The initial surface water speed at the position of the structure were reduced to 98 mm/sec for the case of disk floes and 103 mm/sec for the case of rectangle floes, because they were at rest. They begin to drift driven by the wind and current, and collide with the structure. The drift speed at the inside and outside of the channel is higher than at the center due to the difference of the surface water speed and the form drag. Then the floe models accumulate in front of the structure. In the case of the disk floes shown in Figure 8 ( a ), the floe models in front of the structure have active motion, because collision between disks is along the diametral line. As a result, they flow out from both sides and the number of floes in front of the structure decreases with time. In the case of the rectangles, however, the floe models have no motion in front of the structure, because they collide facially. The number of the floes in front of the structure is constant with respect to time as shown in Figure 8 ( b ). Figure 9 shows the time variation of the x-direction force of F, exerted on the structure (see Figure 5), each peak corresponds to a collision of a floe model or floe models. The horizontal axis denotes the elapsed time from the start of the force measurement. The negative peak is due to the vibration of the structure. In the present test, all the floes in front of the structure collided with the structure simultaneously, because the floe concentration was very high. Hence higher peaks appear intensively at the beginning of contact. In the case of rectangle floes, the force Fx is stable compared to the case of disk floes. This is due to the behavior of their motion in front of the structure : the rectangle floes had no motion, while the disk floes had active motion. 4. Numerical Results Fig. 5 Measuring section and initial condition of drift test of floe models around an ocean structure of 30 cm ~ 30 cm ~ 5 cm Two computations were made in order to investigate the characteristics of pack ice motion with Distributed Mass/Discrete Floe model. One was for the same conditions as the drift test just described and the other for the Canadian coast Drift of floe models The drift observed in the model floe experiments were numerically simulated for the two shapes with the same Fig. 6 Wind distribution Fig. 7 Current distribution

5 Investivation on Characteristics of Pack Ice Motion with Distributed (a) Disk floe model Fig. 8 Floe Model (b) Rectangle floe model Drift of Hoe models around the structure (b) Rectangle (a) Disk floe model Fig. 9 Mass/Discrete Measured interaction force on the structure floe model 135

6 136 Journal of The Society of Naval Architects of Japan, Vol. 177 wind and current condition as in the circulating water channel. One is for the disk floe model and the other is for the rectangle floe model. The friction coefficient Cw was 0.2 for both disk and rectangle floe models and Ca. was for disk and for rectangle. These values were obtained from drag measurements on a single model floe. Ca was determined in such a way that the computed single floe motion agreed with the experimental one. The number of water layers was 5 and the height of each layer was 30 mm. The measured speed of the current at 150 mm depth was used as the bottom condition. In the present computation, assuming that the friction coefficients were the functions of ice concen- tration, the following equations were used ; (24) (25) (26) where subscript * denotes direction x or y and superscript + denotes upstream neighboring mesh. These equations were formulated in such a way that the computed surface water speed at the position of the structure agree with the experiment. Figure 10 shows the time variation of ice concentration contours. The outer contour line denotes 0.01 and the contour interval is 0.2. The results is similar to the circulating water channel experiment. In the case of the disk floes shown in Figure 10 ( a ), the floes accumulate in front of the structure and flow out from both sides. The number of floes in front of the structure decreases with time. The number of rectangle floes in front of the structure remains constant with time as observed. The drift speed at the inside and outside of the channel is higher than the center for both cases of disk and rectangle floes. Figure 11 shows the time variation of the x-direction force exerted on the structure. The horizontal axis denotes the elapsed time from the start of floe motion. The highest peak value agrees (a) Disk floe (b) Rectangle floe Fig. 10 Computated floes motion around the structure ; the outer contour line denots 0.01 and the contour interval is 0.2

7 Investigation on Characteristics of Pack Ice Motion with Distributed Mass/Discrete Floe Model 137 nearly with the circulating water channel experiment for both shapes. The highest peak for the case of rectangles is higher than that of disks similar to the experiment. There are two reasons. One is the mass of the floe : the rectangle floe model is 1.1 times as heavy as the disk floe model. The other is the floe motion in front of the structure : the disk floe models started the lateral motion simultaneously with their first contact with the structure. Therefore, the momentum directed to the structure is less than the case of rectangle floe model. The highest peak appeared subsequent to the first contact. The predicted floe motions in front of the structure were stable compared to experiments. According to computations, the number of floes flowing out from both sides of the structure were fewer and higher peaks of the interaction force appeared intensively at the beginning of contact. This difference is due to the dynamic response of the ocean structure : the DMDF model regards the structure as a fixed boundary, while the experiment allows the structure to vibrate. As a result, in the experiments, the floe motions in front of the structure became active due to the dynamic response of the structure model and a larger number of peaks appear compared to the results of the computation Pack ice movement along an irregular boundary This simulation was done using constant wind velocity and zero water current. A boundary representing an area along the east coast of Canada is used. Comparison was made with the prediction of the Regional Ice Model (RIM) of Environment Canada. Details of this study are given by Neralla et al.13). Figure 12 shows the boundary geometry and initial ice concentration contour. The mesh size was 10 km, the friction coefficients Ca and C. were and respectively. Simulation was run using wind velocity with an easterly component of 9 m/sec and a northerly component of 4 m/sec. Figure 13 shows the resulting ice concentration contours after 48 hours. Predicted ice concentrations agree closely except near the northwest boundary. The DMDF simulation predicts the development of an open water area near the north-west boundary which is not predicted by RIM. It should be noted that at a fixed boundary, the RIM uses the "noslip" condition, while the DMDF model allows floes to slide. The "no-slip" boundary condition of the RIM probably prevents the development areas near a fixed boundary. of such open water 5. Conclusions (a) Disk floe model An improved disk floe version of the DMDF model for pack ice motion simulation was described. In a circulation water channel, the drift tests of model floes were performed in order to investigate characteristics of their motion and interaction with a structure. The results are summarized as follows : The floe motion near the structure depends on the floe shape. The disk floes have lateral motion in front of the (b) Rectangle floe model Fig. 11 Computated interaction force on the structure Fig. 12 Initial ice concentration contours

8 138 Journal of The Society of Naval Architects of Japan, Vol. 177 Acknowledgments This study is supported by the INSROP (International Northern Sea Route Programme) organized by the FridtjƒÓf Nansen Institute, Norway, the Central Marine Research & Design Institute, Russia, and the Ship and Ocean Foundation, Japan. The experiments were made at the circulating water channel of the Institute of Industrial Science, University of Tokyo. The authors thank Prof. Maeda and Mr. Suzuki for their kind help throughout the experiments. The authors would like to express their sincere gratitude to Dr. Neralla, Environment Canada and Dr. Sayed, National Research Council Canada for their cooperation in the comparative calculation with the Canadian (a) DMDF Model operational model. The authors' gratitude is extended to Mr. Maeda, Mr.Miyanaga and Mr.Usami for their invaluable advice and help in the experiment. References 1) Campbell, W. J., "The Wind-Driven Circulation of Ice and Water in a Polar Ocean", J Geophys Res, Vol. 70, No. 4, pp , ) Hibler III, W. D., "A Dynamic Thermodynamic Sea Ice Model", J Phys Oceanogr, Vol. 9, pp , ) Flato, G. M. and W. D. Hibler III, "Modeling Pack Ice as a Cavitating Fluid", J Phys Oceanography, Vol. 22, pp , ) Thomson, N. R., J. K. Sykes and R. F. McKenna, "Short -Term Ice Motion Modeling with Applica- tion to the Beaufort Sea", J Geophys Res, Vol. 93, (b) Regional Ice Model Fig. 13 Ice concentration contours after 48 hours structure, they flow out from both sides. As a result, the number of the floes in front of the structure decreases with time. The rectangle floes, however, have no motion in front of the structure. The number of the floes in front of the structure is constant with respect to time. These experiments indicate that when the motion of pack ice is simulated around a structure it is important to choose the floe shape. The disk floe motion and the rectangle floe motion can be regarded as the two extreme cases of real pack ice motions. An actual pack ice motion may be between these two extreme cases. Two computations using 'DMDF model were made. Simulation results were compared with the circulating water channel experiments and the forecast of the operational model of Environment Canada. The DMDF predicted the circulating water channel drift tests quite closely. The DMDF results also compared quite well with the computations at the Environment Canada. pp , ) Bruno, M. S. and O. S. Madsen, "Coupled Circulation and Ice Floe Movement Model for Partially Ice-Covered Continental Shelves", J Geophys Res, Vol. 94, pp , ) Rheem, C. K., H. Yamaguchi, H. Kato and H. Horikome, "A Numerical Study on Pack Ice Movement Using a Dynamic Ice Model as a Continuum", J Soc Naval Archi Jpn, Vol. 173, pp , ) Serrer, M., S. B. Savage and M. Sayed, "Visualization of Marginal Ice Zone Dynamics", Proc 1st Int Conf and Exhibution VIDEA 93, South- ampton, UK, ) Frederking, R. and M. Sayed, "Numercal Simulations of Mesoscale Rheology of Broken Ice Fields", Proc 12th Int Conf POAC, Vol. 2, pp , ) LƒÓset, S., "Discrete elememt modelling of a broken ice field", Cold Reg. Sci. Technol., Vol. 22, pp , ) Yamaguchi, H., C. K. Rheem and H. Kato, "Pack Ice Movement Simulation Using a Distributed Mass/Discrete Floe Model", Proc 12th Int Conf POAC, Vol. 2, pp , ) Rheem, C. K., H. Yamaguchi and H. Kato, "Numerical Simulation of Rectangle Ice Floes Movement Using a Distributed Mass/Discrete Floe Model," J Soc Naval Archi Jpn, Vol. 175, pp , ) Rheem, C. K., H. Yamaguchi and H. Kato, "A

9 Investigation on Characteristics of Pack Ice Motion with Distributed Mass/Discrete Floe Model 139 Distributed Mass/Discrete Floe Model For Rheology Computation of Pack Ice Consisting of Disk floes", Proc 4th Int Conf ISOPE, pp , ) Neralla, V. R., M. Sayed, M. Serrer and S. B. Savage, "The Influence of ice Rheology on Ice Forecating", Int Conf Sea Ice, Beijin, China, ) Rheem, C. K., "Study of Numerical Model for Rheology of Pack Ice Taking the Discrete Nature into Consideration", Doctorate thesis, Univ. of Tokyo, 1994.

Keywords: sea ice deformation, ridging, rafting, finger-rafting, ice rubble, pile-up, ridge keel, ridge sail UNESCO-EOLSS

ICE RIDGE FORMATION Jukka Tuhkuri Aalto University, Department of Applied Mechanics, Finland Keywords: sea ice deformation, ridging, rafting, finger-rafting, ice rubble, pile-up, ridge keel, ridge sail