EMPIRICAL FORMULA OF DISPERSION RELATION OF WAVES IN SEA ICE

Ice in the Environment: Proceedings of the th IAHR International Symposium on Ice Dunedin, New Zealand, nd th December International Association of Hydraulic Engineering and Research EMPIRICAL FORMULA OF DISPERSION RELATION OF WAVES IN SEA ICE Shigeki Sakai and Kohta Hanai INTRODUCTION The dispersion relation of waves propagating in ice-covered sea differs from that in open water. Liu et al. (989) showed that the relation between the wave length observed by SAR and the wave period measured by an accelerometer installed on a buoy in the Labrador Sea very closely approxates the theoretical dispersion relation of waves under an infinite-length elastic floating plate. In their model experent, Sakai et al. (99) demonstrated that applicability of the theoretical dispersion relation depends on the ice plate size. However, discussion has not yet addressed quantitative evaluation of the effect of ice on the dispersion relation. It is essential to consider the effects of ice when investigating the dispersion relation in ice-covered seas when dealing with various phenomena in such icy sea areas: the estation of external force against coastal structures induced by the pact of ice floes moved by wave forces, and the calculation of behavior of spilt oil in icy sea regions. Thus, the present study investigates celerity (phase velocity) characteristics of waves that propagate in a model sea area covered with ice. It also examines ice plate length and thickness effects and elastic coefficient effects on the dispersion relation. On the basis of results of these examinations, we propose an empirical equation for the dispersion relation that can express these characteristics and effects comprehensively. EXPERIMENTAL EQUIPMENT AND PROCEDURE The present experent employed a wave flume that was m long,.8 m wide and. m deep, which is illustrated in Figure. A model ice sheet was placed on the water surface in the center of the flume. Profiles of waves that propagated in this sulated sea ice area were investigated. A wave maker, which could generate waves to sulate any condition, was installed at the front of the flume. A wave sensor was set on a surface of paddle of the wave maker. According to the water elevation measured by the wave sensor, a computer sent a synthesized signal to elinate the reflected wave and to Associate Professor, Iwate University, --5 Ueda, Morioka -855 Japan. Fax: +8-9-5-8, email: sakai@iwate-u.ac.jp Consulting Engineer, Nikken Consultants Ltd., -7-9 Shinbashi, Tokyo 5- Japan. Fax: +8--55-8, email: hanai@nikken-con.co.jp

make an incident wave. This system could make the area between the wave maker and the front edge of the model ice free from multiple reflection condition. Furthermore, another instrument of the same function was also installed at the other end of the flume; it elinated the reflection waves from the back end. This allowed the characteristics of waves propagating in the model sea ice area to be investigated without any influence of reflection waves..8 m model ice floe wave maker wave gauge ultrasonic sensor wave absorber wave gauge m 8 m m Figure : Experental equipment In the present experents, polyethylene sheets were used as model ice, because a specific gravity of polyethylene is silar to that of the actual ice. Two kinds of polyethylene sheets of differing thickness were used; one was 5 mm thick; the other was mm thick. Although the elastic modulus of a polyethylene plate generally changes depending on temperature, it was found that the elastic modulus was approxately 85 MPa (8 88 MPa) for a 5 mm plate and 5 MPa ( 7 MPa) for a mm plate. The specific gravity of polyethylene was.9 for both plates. The polyethylene plates had six different lengths: 8,,,.5, and.5 m. The numbers of plates used were:,,, 8,, and, respectively. These plates were placed to cover 8 m of the water surface in the center of the flume. Because polyethylene plates available in the market were m long, the 8 m plate was created by connecting four m-long plates; silarly, m plates were created by combining two plates. Plates less than m long were created by cutting a m plate. For example, to prepare a.5 m plate, a m plate was cut into fourths by length. We measured vertical displacement of the polyethylene plates in the sulated icy sea area using nine ultra-sonic sensors. The first sensor was installed at a distance of.5 m from one edge of the polyethylene plates; the other eight sensors were placed at m intervals. After measuring, we moved the eight sensors by.5 m and measured vertical displacement under the same incident wave condition. Furthermore, vertical displacement was measured with the sensors moved by.5 m. We gained te histories of vertical displacements at 5 positions in all. Wave profiles in the front and end open water were measured with wave gauges. The incident wave period had six types that ranged from. to. sec. Wave steepness was of three kinds that ranged from.5 to.. Table shows combinations of wave period and wave steepness that were employed as experental conditions. DISPERSION RELATION OF WAVES IN SEA ICE Figures and demonstrate the distribution of wave celerity in the sulated sea ice area. The figures show cases where lengths of the plates were 8,,,,.5, and.5 m (from the left) and periods were.,.8,.,.,., and. sec (from the top). The

horizontal axes indicate distance from the edge of the sulated sea ice area. In each case, the distance up to 8 m was covered by the plates. It must be noted that measurements for the wave period of. sec were significantly smaller than for other periods when the lengths of the polyethylene plates were.,.5, and.5 m. This occurred because the.-sec period waves were greatly attenuated in the sulated ice sea region; therefore, the ultrasonic sensor with. mm resolution that was used in the experent could not measure wave profiles accurately. The chain lines in Figures and indicate the wave celerity calculated Table : Wave conditions thickness : mm sheet length wave period (s) 8m m m m.5m.5m.,.5..5..5.,.5.8..5.5,..,..5.8.5,..,..5.8 thickness : 5mm sheet length wave period (s) 8m m m m.5m.5m.,.5..5..5.8.,.5,..,.5...5..5.,.,..5,.,.5 Model ice length li (m) Model ice thickness : mm Elastic plate (5MPa) Open water Mass loading Wave period T (s) T=. li=8. li=. li=. li=. li=.5 li=.5 T=.8 T=. T=. Wave celerity (m/s) T=. T=. 8 8 8 8 8 8 Distance (m) Figure : Distribution of wave celerity in the sulated sea ice area (Model ice thickness: mm)

by the following equation of dispersion relation on the basis of the linear theory of waves under an infinitely-length elastic floating plate (hereinafter referred to as the elastic plate). Broken lines refer to the wave celerity obtained under the condition where the mass was attached to the surface of water (i.e., mass loading), which was calculated with the elastic modulus in the dispersion relation equation being treated as zero. Solid lines indicate the wave celerity in open water. 5 Mk + ρ gk ω = () ρcoth kh + ρikhi where ω : angular frequency, k : wave number, h : water depth, h i : thickness of model ice, ρ : density of water, ρ i : density of model ice, g : gravitational acceleration, ratio ( =. ). M = Eh ( - ν ) : bending stiffness, E : Young s ratio, ν : Poisson s i The leftmost values in the rows in Figure, which demonstrate results of the experent with the mm-thick plates, show the case in which there was one 8 m plate. It is evident that wave celerity fluctuated concomitant with wave celerity below the elastic plate; the period and amplitude are somewhat constant. This fluctuation may have corresponded to the mode in bending vibration of elastic Model ice length li (m) Model ice thickness : 5mm Elastic plate (85MPa) Open water Mass loading Wave period T (s) T=. li=8. li=. li=. li=. li=.5 li=.5 T=.8 T=. T=. Wave celerity (m/s) T=. T=. 8 8 8 8 8 8 Distance (m) Figure : Distribution of wave celerity in the sulated sea ice area (Model ice thickness: 5 mm)

body. Values in the second column, which show the case where two m plates were placed to cover 8 m of the water surface, demonstrate silar patterns. The wave celerity was found to be slow in the area of m at the ice sheet edge. This is because an open-water area was momentarily created after the waves passed the first plate. Wave celerity propagating in this open-water area was always slower than that of waves passing under the plates. However, the observed wave celerity in this area did not correspond to the theoretical value in the open-water region. Values in the third column demonstrate cases where four m plates were placed to cover 8 m of the water surface. Areas of,, and m on the horizontal axis are points of discontinuity. As in the previous two cases, wave celerity was found to be slower in the open water area than below the plates. Mode-like fluctuations observed between the m plates were obscured, compared to those observed between 8 m plates and between m plates. Values in the fourth column show experental conditions with eight m plates. In these cases, neither mode-like fluctuations nor discontinuity of wave celerity in the open-water region were observed. Wave celerity was almost constant when the period was short. It was also found that average wave celerity was significantly lower than that under the elastic plate and was about the same as celerity in the open water area in the case of the longer period. Values of the fifth and the sixth columns, which show results of experents using even shorter plates, the tendency observed in the m plate experent became more salient. It should be noted here that, in the case with the shortest plates, wave celerity with the short period was barely lower than that in the open-water region. This wave celerity was equal to that in mass loading. To sum up, when plates were longer, mode-like fluctuations were observed in distribution of wave celerity. At the same te, discontinuity of wave celerity appeared when waves passed through the open-water region. In contrast, when the plates were shorter, these fluctuations became obscured. When plates were even shorter, both mode-like fluctuations and discontinuity became unclear and wave celerity was held constant. In cases where both the length of the plates and the period were short, the wave celerity was silar to that in mass loading. This tendency was qualitatively true for cases with the plates that were 5 mm thick; these are shown in Figure. 5 Elastic plate(5mpa) Open water Mass loading Model ice thickness : (mm) 5 Elastic plate(85mpa) Open water Mass loading Model ice thickness : 5(mm) Wave number Kex (/m) 5 8mmm mmm* mmm* mmm*8.5mmm*.5mmm* Wave number Kex (/m) 5 8m5mm m5mm* m5mm* m5mm*8.5m5mm*.5m5mm*.5.5 Frequency f (Hz).5.5 Frequency f (Hz) Figure : Dispersion relation (Model ice thickness : mm) Figure 5: Dispersion relation (Model ice thickness : 5 mm)

Next, we examine the average wave celerity over the whole sulated icy sea area. Figures and 5 show the relationship between wave celerity and wave period with plate length treated as the parameter. The celerity in these figures is the sple average of the corresponding celerity in Figures and. Figure demonstrates results of experents using mm-thick plates. When plates were longer, wave celerity was equal to that below the elastic plate. As plate length became shorter, celerity approached the wave speed in the open-water region. When the plates were even shorter, wave celerity became close to the speed in the mass loading. Results of experents with the 5 mm-thick plates are shown in Figure 5. It was found that variations caused by thickness of plates were smaller compared to experents using mm-thick plates. However, basic characteristics of variations were qualitatively identical for both cases. EMPIRICAL FORMULA FOR THE DISPERSION RELATION As shown in Figures and, when the plates were short, discontinuity of wave celerity passing the open water area became obscured. The area containing a number of plates to create an 8 m elastic plate could be considered to be covered by one plate with a certain elastic modulus. Thus, the relation between wave celerity and thickness of plates was examined. We calculated backward to obtain the equivalent elastic modulus E eq that agreed with the observed wave celerity; we used the wave celerity calculated by the linear solution of Equation () below the elastic plate. After calculating the modulus, we computed the ratio E( = Eeq / E) of this value to the observed elastic modulus E ; we then illustrated the relation between celerity and plate thickness. This relation is shown in Figure. In the case where plates were 5 mm thick, the ratio was close to zero when the plates were long. However, when it exceeded a certain value, the ratio was in proportion to log( l i ); it could be approxated as the chain line in the figure. In the case in which plates were mm thick, the same tendency was generally observed, but E had variations when plates were long. This tendency could be approxated as the broken line in the figure. As the two approxated lines clearly demonstrate, the range in which E is zero and the increasing rate depend on plate thickness. Thus, in order to express the effect of plates on the range and the rate, we will use the plate length li l c and plate thickness hi / l c, which was made densionless by the characteristic length l c regarding bending fluctuation of the elastic foundation. The characteristic Model ice thickness : mm, : 5mm Model ice thickness : mm, : 5mm E E - li Figure : Effects of ice length on equivalent elasticity -.5.5 IF Figure 7: Combined effects of ice length and thickness on equivalent elasticity

length can be defined as the distance from the loading point to the position at which the negative bending moment is maxum when a load acts at the center of the elastic foundation. It is calculated as / lc = ( Ehi / ρg( - ν )). () If the equation, / IF = ( hi / lc) log( li / lc), () is defined as the parameter that comprehensively expresses the effects of both plate length and plate thickness upon E, then the relation between E and IF will be as that shown in Figure 7. The range in which E is approxately zero is identical for both the 5 mm condition and the mm condition. In the range where E increases linearly, values for E are more scattered in the 5 mm condition than in the mm condition. However, the values can be generally approxated as the line depicted in the figure; they can be expressed as follows: E Ï IF.5 Ô IF -.5 = Ì.5 < IF. Ô.55 ÔÓ.< IF () Using the empirical equation above, the equivalent elastic modulus E eq can be obtained from the plate length, plate thickness, and elastic coefficient. Figure 8 shows the comparison between the wave celerity C es calculated by this equivalent elastic modulus and Equation (), and the observed wave celerity C ex. As the figure clearly indicates, both the estated wave celerity and the observed celerity agree with each other extremely well. Therefore, it can be concluded that Equation () expresses combined effects of length and thickness of plates upon wave Ces (m/s) Model ice thickness : mm, : 5mm Cex (m/s) Figure 8: Validity of empirical formula Model ice thickness : mm Wave celerity (m/s) 5 Wave period T=. (s) Wave period T=.8 (s) Wave period T=. (s) 5 Wave period T=. (s) Wave period T=. (s) Figure 9: Relationship between li and E E Wave period T=. (s)

celerity. Equation () is based on results shown in Figure 7. Nevertheless, it can express the wave celerity in Figure 8 better with high accuracy. Unlike the high predictive accuracy of wave celerity in Figure 8, celerity varies somewhat widely in Figure 7. To resolve this dilemma, we investigated the degree to which variation in the elastic modulus affects wave celerity. Figures 9 and show results of calculation of the relation between wave celerity for mm plates and for the 5 mm plates and the elastic modulus, using Equation (). Increase or decrease in wave celerity caused by variation in the elastic modulus was moderate, except for the case where the elastic modulus was extremely small. Therefore, it can be concluded that a slight difference in the wave celerity in Figure 8 is expressed as a great variation in the elastic modulus in Figure 7. Model ice thickness : 5mm Wave period T=. (s) Wave period T=.8 (s) Wave period T=. (s) Wave celerity (m/s) Wave period T=. (s) Wave period T=. (s) Wave period T=. (s) 8 8 E 8 Figure : Relationship between IF and E CONCLUSIONS In the present study, we calculated the equivalent elastic modulus with the elastic modulus of ice and with an empirical equation for the dispersion relation. Using this modulus and an equation of the dispersion relation on the basis of the linear theory of wave motion under an infinite-length elastic floating plate, we successfully estated wave celerity propagation in the sea ice region. Previous studies examined the effects of ice upon the dispersion relation only qualitatively. Thus, the present investigation, which evaluated the issue quantitatively, is of great significance. However, this study leaves a number of questions unresolved. One such question is related to the fact that the waves in an actual sea region are random waves. It is necessary to investigate whether the empirical equation proposed in this study is valid for each frequency component of such random waves. Furthermore, our study employed a condition wherein plates were placed in a sulated sea ice region with no intervening space; in other words, we studied a condition in which ice coverage of the water surface with is %. However, distribution of drifting sea ice on the coast of Sea of Okhotsk indicates that ice coverage varies in terms of space and te. Therefore, prior to application to an actual sea region, effects of ice coverage upon the dispersion relation must also be examined. Finally, this study used model ice with uniform length and thickness. The diverse size of ice plates in actual icy sea areas must also be considered prior to application to an actual icy sea area.

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