Ice in the Environment: Proceedings of the 16th IAHR International Symposium on Ice Dunedin, New Zealand, 2nd–6th December 2002 International Association of Hydraulic Engineering and Research

ON THE EXCITATION OF SHELF EDGE WAVES DUE TO SELF-INDUCED OSCILLATIONS OF ICE FLOES

Aleksey Marchenko1, Alexander Makshtas2 and Lewis Shapiro3

ABSTRACT The model of the excitation of gravity waves in shallow water by irregular displacements of ridged ice is developed. The amplitudes of shelf edge waves are estimated in sill model of shelf zone near Point Barrow, when the frequency and the amplitude of rough ice displacements are given. Frequencies and amplitudes of wind-induced displacements of rough ice are estimated using the model of ice ridges buildup between circular floes.

1. INTRODUCTION Filed observations of the landfast sea ice near Point Barrow, Alaska, recorded vertical displacements of several centimeters amplitude and of the order of 600 s (Bates and Shapiro, 1980) prior to a large ice push episode. Further, over about 5 years of nearly continuous radar observations of near shore ice motion in that area, similar oscillations were always observed to occur for at least several hours before the start of movement of the landfast ice or adjacent pack ice. This suggests that oscillations of the ice within some range of periods and amplitudes, are a precursor to ice motion and could be useful in providing warning of potential hazards to offshore or coastal installations and operations, and to hunting parties moving over the ice. However, periods of oscillations were only rarely followed by ice movements, so if observations of the oscillations are to be used to warn of potentially hazardous situations, then it is necessary to understand the relationships between the oscillations and the associated ice motions. As a step toward that goal, in this paper, we examine some possible sources of the oscillations. Typical periods of ocean waves (wind waves and swell) are much shorter than those described here. Therefore we assume that the source of observed waves was not related to swells propagating from ice free areas. Analysis of dispersion properties of gravity waves propagating in shelf zones shows that there are a countable number of shelf edge waves propagating in shelf regions of the ocean (Munk et al., 1956). The energy of the

1 Theoretical Department, General Physics Institute of RAS, Vavilova str. 38, 119991 Moscow, Russia

2 International Arctic Research Center, University of Alaska Fairbanks, 930 Koyukuk Dr., P.O.Box 757335, Fairbanks, AK 99775

3 Geophysical Institute, University of Alaska Fairbanks, 930 Koyukuk Dr., P.O.Box 757335, Fairbanks, AK 99775

edge waves is localized on the shelf, and their amplitudes decays away from the shelf-break. The edge waves cannot be excited by Poincare waves coming to the beach from offshore, but their excitation is possible under the influence of localized source of water surface perturbations. It was found that relative motion of ice floes composing continuous ice has oscillations, which periods are varied from several seconds to several tens minutes (Martin and Drucker, 1991). Numerical modeling of continuous ice com-pression accompanied by the buildup of infinitely long ridges has shown oscillating motion of ice fields separated by the ridges (Marchenko and Makshtas, 2001). We assume that such motion can excite gravity waves due to the transfer of floes impulse into the water. These waves will propagate from the region of ridges buildup to the coastline as suggested by Bates and Shapiro (1980), and a part of their energy can be transformed into the energy of the edge waves propagating along the coast. This paper is devoted to the investigation of the excitation of the edge waves due to oscillating motions of ice floes caused by ice ridge buildup. In the next section (Part 2) the dispersion properties of the edge waves are analyzed using parameters typical of the shelf topography near Point Barrow, after which the excitation of edge waves by localized periodical source of external pressure is investigated in Part 3. Then, in Part 4 the results of numerical simulations of wind induced ridge buildup between four circular floes are described (the model of ridge buildup between circular floes is formulated the Appendix) to explain the existence of wind-induced oscillations of floes motion with periods about several tens minutes. Finally, the conclusion section includes a discussion of the main results of the work. 2. DISPERSION PROPERTIES OF SHELF EDGE WAVES Linearized shallow water equations are written in the form Fu +∇−=∂ ηgt , 0)( =⋅∇+∂ uHtη , (1) where ),( vu=u is the vector of water velocity, ),,( yxtη is the elevation of water surface, g is gravity acceleration, ),( yx FF=F is the vector of external forces acting on the water, H is water depth, t is the time, x and y are horizontal coordinates. Differential operator ∇ is defined as ( )yx ∂∂=∇ , . Excluding vector u from equations (1) one finds

0)()( =⋅∇+∇⋅∇−∂ FHgHtt ηη . (2)

It is assumed that the coastline is coinciding with line lx −= , and the shelf-break is located near line 0=x (Fig. 1a). Water depth is equal to 0H when 0>x and 1H when

)0( l,x −∈ . From the continuity of mass flux and water pressure at the shelf-break and coastline it follows 0=u , lx −= ; uHuH

xx 0001 limlim+→−→

= ; ηη00

limlim+→−→

=xx

. (4)

Periodical waves propagating along the coastline are described as

(a) (b) Figure 1: (a) Schematic diagram of shelf region under the consideration. (b) The buildup of three ridges between four circular floes in ridged ice zone. c.c.)( )( += + tyki yex ωUu , c.c.)( )( += + tyki yex ωςη , (5) where c.c. denotes complex conjugation. Substituting formulas (5) into equation (2) and assuming 0=F one finds that function )(xς is written as xikxik eCeCx 11

21)( −+=ς , 0<<− xl ; xikxik eCeCx 0043)( −+=ς , 0>x , (6)

where 21

12

1 )( ykgHk −= −ω and 210

20 )( ykgHk −= −ω . The function )(xU is found

from (1) taking into account formulas (5) and (6). The rays 21

2ykgH=ω and

20

2ykgH=ω separate half-plane 0>ω into three regions I, II and III (Fig. 2). In the

region I both wave numbers 0k and 1k are real, in the region II 1k is real and 0k is imaginary, in the region III 0k and 1k are imaginary. Without loss of generality we set

0Im 0 ≥k and 0Im 1 ≥k . From the boundedness of the solution it follows that constant 04 =C when yk and ω are in the region II or III. In this case there are three equations

(4) for the finding of three constants 1C , 2C and 3C . Using formulas (5) and (6) from contact conditions (4) one finds three linear ordinary algebraic equations for the finding of 1C , 2C and 3C . Nonzero solution of the system exists when the determinant is equal to zero ( ) ( ) 0sincos),( 111100 =−≡∆ lHlHky λλλλω , (8) where )/()( 0

2220 gHky ωλ −= and 2

122

1 )()/( ykgH −= ωλ . Condition (8) is dispersion equation for shelf edge waves. This equation has solution in the region II at the plane

),( ωyk , which is performed by denumerable set of dispersion curves )( yj kωω = . The

beginnings of theses curves are located at the line ykgH0=ω . Each dispersion curve

tends to the line ykgH1=ω when | |yk Æ• .

Dispersion curves of the four edge wave modes are shown in Figure 2 when m 301 =H , m 3000 =H and km 10=l . These parameters approximately describe sea bottom

topography near Point Barrow. One can see that there are only two edge wave modes, which frequency is 0.01 s–1 (period is 600 s). Wave number of the first wave mode is

-11 m 00056.0≈yk (wavelength is ~11.2 km)., and wave number of the second mode is -12 m 00037.0≈yk (wavelength is ~16 km). There is only one edge wave mode, which

period is 0.0025 s–1 (period is 42 min). Wave number of this mode is -10 m 00007.0≈yk (wavelength is ~89.7 km).

Figure 2: Dispersion curves of four edge wave modes are denoted as 1, 2, 3 and 4.

3. EXCITATION OF SHELF EDGE WAVES BY ICE FLOES OSCILLATIONS In the case when the sea surface is covered by ice, external forces F in equation (1) are related to the friction between the water and the seabed ( wbF ) and the water and the ice ( wiF ). Assuming the square-law for shearing drag forces we set wiwb FFF += ,

1|| −= HCwbwb uuF and 1)(|| −−−= HC iiwiwi uuuuF , where iu is ice cover velocity,

wbC and wiC are drag coefficients. Typically drag coefficients are equal to 0025.0=wbC and 005.0=wiC (Lepparanta, 1994). Assuming 1| | 0.1 ms-ªu ,

1| | 0.1 ms ,i-- ªu u 10.01 sω -ª and m 10>H one finds that 1 2| | 10 mst

- -∂ ªu , 6 2| | 2.5 10 mswb- -< ◊F and 6 2| | 5 10 mswi

- -< ◊F , i.e. the influence of drag forces on gravity waves with periods of the order 600 s is very small. When the ice bottom is rough, such as in a ridge keel, the form resistance

1)(|| −−−= HC iiwrwr uuuuF with drag coefficient 1≈wrC becomes important (Kawai et al., 1999). In part 4 of the paper we will show that ice floes can oscillate because of wind induced ridge buildup between them. In this case ice floe velocity is written as

0i i δ= +u u u , where 0

iu is slow varied quantity, and | | i tue ωδ δªu . Period ω of these oscillations can reach several tens minutes. The form resistance of the keel is parameterized as the second-degree dependence of the pressure due to the current velocity near the keel. In problems of wave excitation by floating bodies it is possible to model the forcing by an equivalent external pressure field ),,( tyxp applied to the water surface p∇=F (Whitham, 1974). The pressure

gradient is estimated as 2 1| | wrp C u Hδ -— ª . Integrating this formula over a ridge gives an estimate of the water pressure at the ice bottom in the vicinity of the ridge

2 1wr rp C u L Hδ -ª , where m 50≈rL is a typical ridge width (Fig. 4). We let the ridged

ice be located in the region S of the ocean surface near the shelf-break ),0( rxx −∈ (Fig. 1b). Therefore 1=wrC when ),0( rxx −∈ , and 0=wrC away from this region. Assuming time periodicity of ice floes displacements in the region S and the irregularity of their distribution along y -direction we study gravity wave excitation by ridged ice located in the wide region near the line 0=y : ),0( rxx −∈ , (0, )y yδŒ . We set

c.c.)()(2 0 += tieyxpp ωδ , 2 ( ) yik yyy e dkπ δ

•

-•= Ú , (9)

where const=ω and 2 10 0 1rp P u L yHδ δ -= ∫ when ),0( rxx −∈ , and 00 =p away this

region. Substituting formula p∇=F , where the pressure p is defined in (9), into equation (2) and matching conditions (4) and using Fourier transform along the y -axis one finds the solution in the shelf region as

1 1 110 0

c.c.+

1 cos[ ( ' )] ( ') ' sin[ ( ')] ( ') '4

r yx ik yxi t

yex l p x dx x x p x dx dk e

gω

η

λ λπ λ

∆ ∆∆

-•

-•

=

Ê ˆ+ + -Á ˜

Ë ¯Ú Ú Ú (10)

where )cos()cos(),,( 1001111 xHxHxky λλλλω −=∆ , and 20( ) ( ( ) ) ( )xx yp x k p x= ∂ - .

The existence of edge waves in the solution (10) follows from the existence of the poles of the integrand expression coinciding with the roots of the dispersion equation (8). Near coastal amplitude of the edge wave related to the root ewy kk = is equal to

( )2

1 01 12

1 1

( , , )( ) sin[ ( )] sin[ ]

( ) 'y ew

yew r

ew k k

k l PA k l x l

g Hω ω

λ λλ

∆∆

=

-= - - ,

( , )' ,

y ew

yew

y k k

kk

ω∆∆

=

∂=

∂ (11)

Formula (11) estimates the amplitude of edge waves excited by the oscillating motion of ridged ice in the wide region ),0( rxx −∈ , (0, )y yδŒ . The amplitude of the water surface elevation induced by edge waves excited by ridged ice from the region S is estimated as 1( ) ( )t ew ew rA k A k yδΛ -ª , where rΛ is the extension of the region S in the y -direction.

Estimations in SI show that 1 6 2( ) 7 10t y rA k uδ Λ-ª ◊ and 2 4 2( ) 1.7 10t y rA k uδ Λ-ª ◊ when

-1s 01.0=ω and km 1=rx , and 0 5 2( ) 2 10t y rA k uδ Λ-ª ◊ when -1s 0025.0=ω and

km 5=rx . One can see that cm 17)( 2 ≈yt kA and cm 2)( 0 ≈yt kA when km 100=Λ r and -10.1 msuδ = . Thus it is shown that the amplitude of the second edge wave mode is

much more than the amplitude of the first edge wave mode when they are excited by rough ice oscillations of 10 minutes period. The amplitude of the second mode reaches

several centimeters when the rough ice occupies sufficiently narrow band region on shallow sea near shelf break of the area about one hundred square kilometers. 4. WIND INDUCED RIDGE BUILDUP BETWEEN CIRCULAR FLOES Figure 1b shows the problem under consideration. Four circular floes are being compressed against the edge of the landfast ice by pressure from a zone of compacted floes of width L. The offshore boundary of this zone is designated as Line B in Figure 1b, and it is assumed that there are additional dispersed floes further offshore. It is also assumed that pressure due to wind drag on the ice is accumulated between the coastline and Line B, causing ridge buildup between the circular floes. As a result, the free edges of the floes are shortened, and the floe volumes decrease as ice is added to the ridges. The floes are enumerated from 1 to 4 from offshore to the fast ice edge, and the ridges between them are enumerated from 1 to 3 in the same sense. It is also assumed that the velocity of floe number 1 is the same as that of the compacted floes, and that the velocity of floe number 4 is zero because it is in contact with the landfast ice. Finally we assume that the top and bottom surfaces of floes 2 and 3 are rough. Equations describing the motion of the n floe ( 41 <≤ n ) are written as

extnnn

nni FFF

dtdU ,1 +−= −vρ , 1

nn n

d xδ+= -v v

dt, (12)

where -3mkg 930 ⋅≈iρ is sea ice density, nv is the velocity of the n floe, nxδ is relative displacement of the n and the n+1 floes, nF is the force acting on the n floe from the n + 1 floe, extnF , is external drag force acting on the n floe from the wind and

the water. Floe volumes are defined as 21 ,1 / 2 2t cU h R U RLhπ= - + and 2

nU h Rπ= - , 1 ,( ) / 2t n t nU U- + ( 42 −=n ), where 2h Rπ is the initial volume of circular floe, 2/1,tU

is a half of the volume of the first ridge, ch and cRLh2 are the thickness and volume of the ice in the zone of compacted floes. The volume of the n ridge ntU , is defined by formula (A2). Using formulas (A1), the volume ntU , can be represented as a function of nxδ . The force nF on the n ridgeline is directed offshore and defined by formula (A5). The external force extnF , is equal to a sum of wind drag and water drag forces applied to the surfaces of the circular floes and to the floes in the compacted zone:

21, ,1( )( /(2 ) 2 )ext a w tF F F R U h RLπ= - - + , 2

, ,( )( /(2 ))n ext ar wr t nF F F R U hπ= - - , 1>n . (13) The surface densities of wind and water drag forces are defined as

2

aaaa VCF ρ= , 2vwww CF ρ= , 2aaraar VCF ρ= , 2vwrwwr CF ρ= , (14)

where drag coefficients are 003.0=aC , 005.0=wC , 03.0=arC , 1=wrC , air density is

-31.2 kg maρ = , and sea water density is ρw = 1200 kg m–3.

The width L of the compacted zone increases with time. The simplest description of the time-variations of L is that of a plane kinematic wave propagating offshore. Line B on Fig.1b shows the location of the kinematic wave. The velocity of this wave ( D ) and the variation of L are described by

1

11 )1)((),( −−−−= AAD dd vvvv , ),(/ 11 dDdtdL vvv += , (15)

where A and dv are the compactness and the velocity of the ice offshore from L, and

1v is the velocity of the first circular floe.

Figure 3: Dependencies of floe velocities 1v , 2v and 3v , and maximal values of ridge drafts 1

max,kh , 2max,kh and 3

max,kh from the time during the ridge buildup process. For numerical simulations it is assumed that m 7.0=h , м 2=ch , m 100=R , 7.0=A ,

10.1tanh 0.1 ms0.5 3600d

t -Ê ˆ= +Á ˜Ë ¯

v , 125 tanh ms0.5 3600a

tV -Ê ˆ= Á ˜Ë ¯◊, 0

03,2 ==t

v ,

00

==tixδ and km 5

0=

=tL .

Examples of the numerical simulations are shown in Figure 3. One can see that velocities 1v , 2v and 3v increase up to 6 and 3 cm s–1 and then decrease to 2 cm s–1 during the first 12 minutes of the motion, while the velocity 1v falls from 10 cm s–1 to 2 cm s–1 during this period. Further, the floes oscillate with a period of about 42 minutes and the amplitudes of the oscillations decrease with time. The ridge drafts approach a limit of about 8 m during 3 hour of ridges buildup, with ridge 3, closest to shore the highest, and ridge 1 the lowest. 5. CONCLUSIONS The problem of the excitation of shelf edge waves caused by the irregular displacements of rough ice is considered. The shelf length and the water depth in the shallow and deep areas are based on the shelf topography near Point Barrow. It is shown that given oscillating motion of a rough ice can excite shelf edge waves of the first or the second modes with amplitudes about several centimeters. This can occur when the amplitude of ice cover velocity is of the order 10 cm s–1, and the ridges occupy an area of the order 100 km2 in the shallow water region near the shelf break. The amplitude of the second edge wave mode (~17 cm) is much greater then the amplitude of the first mode, when the frequency of the rough ice oscillations is equal to 0.01 s–1 (period ~10 min). The

amplitude of the first edge wave mode is about 2 cm, when the frequency of the rough ice oscillations is equal to 0.0025 s–1 (period ~42 min). Numerical simulations of wind-induced compression of four similar floes have shown that typical period of their oscillating motion is about several tens minutes, and only the duration of initial floes pulsations is closed to 10 minutes. This time interval can be considered as typical time of ice ridge buildup, which is calculated as a time, which is necessary for the motion of ice cover over space distance equal to typical ridge width. Although constructed model is idealistic, it seems to give good demonstration of the rise of oscillating component of ice cover motion caused by constant wind drag. We found that periods of the floes oscillations are closed to each other, while the amplitudes of oscillating component of the floes velocities are decreasing in the time. Note that ridge buildup process is accompanied by the breakup and crushing of the edges of the level ice floes, which cause high frequency elastic waves in the ice cover (Smirnov, 1996) and a noise in the water (Pritchard, 1990). Above considered oscillations of the ice floes and water can be responsible for low frequency modulations of these high frequency signals. ACKNOWLEDGMENT The Frontier Research System for Global Change supported this work. We thank Roger Colony for help and permanent attention and support of our work in International Arctic Research Center, University of Alaska Fairbanks. REFERENCES Bates, H.F. and Shapiro, L.H. Long-period gravity waves in ice-covered sea. J.

Geophys. Res. 85(2): 1095–1100 (1980). Kawai, T., Makita, S., Saeki, H., Hara, F. and Ohtsuka, N. Experiments on the fluid

resistance of rubble field of ice. In Ice in Surface Waters, H.T. Shen ed., Balkema (1999) 883–888.

Lepparanta, M. The dynamics of sea ice. In Physics of Ice-Covered Seas, M. Lepparanta, ed., Helsinki University Printing House (1994) 305–342.

Marchenko, A. and Makshtas, A. Ice ridging over various space scales. J.P. Dempsey and H.H. Shen, eds., In IUTAM Symposium on Scaling Laws in Ice mechanics and Ice Dynamics, Kluwer Academic Publishers (2001) 103–114.

Munk, W.H., Snodgrass, F.E. and Carrier, G. Edge waves on the continental shelf. Science 123: 127–132 (1956).

Pritchard, R.S. Sea ice noise-generating processes. J. Acoust. Soc. Am. 88(6): 2830–2842 (1990).

Smirnov, V.N. Dynamic Processes in Sea Ice (in Russian). Gidromeyteoizdat, St.-Petersburg (1996) 162 p.

Whitham, G.B. Linear and Nonlinear Waves. Wiley-Interscience Publishers, New-York (1974).

APPENDIX. THE MODEL OF ICE RIDGE BUILDUP BETWEEN TWO CIRCULAR FLOES Compression of two identical circular floes (discs) of thickness h and radiuses R is shown at Fig. 4. The ridge is formed along the contact line of the floes. The ridgeline is rectilinear due to the symmetry of the problem. The velocities of the disks are equal to

nv and 1+nv . Relative displacement of the floes is denoted as ( ,0)n nxδ δ=x . Using

evident from Fig. 4 expressions for ridge length αsin2, Rnr =Λ and displacement 2 (1 cos )nx Rδ α= - we can write

( )2 2 1

,2 1 1 (4 )n r nx R Rδ Λ -= - - , ( )2 20 2 1 1x R y Rδ -= - - , (A1)

where 0 0 ( )x x yδ δ= is equal to the value of nxδ in the moment when ridge buildup starts up in the points with coordinates y± . Total ridge volume is calculated as

( )( ),

,

/ 2 1 2 2 1, , , , ,/ 2

2 arcsin /(2 ) (2 ) 1 (4 )r n

r nt n r n r n r n r nU U dy hR R R R

Λ

ΛΛ Λ Λ- -

-= = - -Ú . (A2)

where , 0( )r n nU h x xδ δª - is area of vertical cross-section of the ridge. Simple algeb-raic calculations show that quantity hU nt /, is equal to the area of gray region on Fig. 4. Using the triangular approximation of vertical cross-section of the ridge (Fig. 4), where

nsnk hh ,, 4.4= , 9.32=kϕ and 6.26=sϕ (Timco and Burden, 1997), the variation of ridge draft along the ridgeline is written as

2 2 2 2 1, ,0.77 2 ( 1 1 (4 ) )k n r nh hR y R RΛ- -ª - - - . (A3)

Figure 4: Scheme of ridge buildup between two circular floes. Total compressive force between the disks can be written as (Marchenko and Makshtas, 2001)

,

,

/ 2

,/ 22 r n

r nn i k nF g h h dyρ λ

Λ

Λ-= - Ú , kiiw ϕµρρρλ 1tan2/)( −+−= , (A4)

where 3.0≈µ is drag coefficient between ice blocks composing the ridge. Estimations show that force nF can be written as ,max , ,max , 0

0.4 , .n nn i k r n k k n y

F g hh h hρ λ Λ=

ª - = (A5)