Propagation of Flexural Gravity Waves in Sea Ice

V E R N O N A. SQUIRE and ALASTAIR J. A L L A N

ABSTRACT

When ocean waves and swell are incident on a fast ice boundary, much of the wave energy is reflected but some is propagated into the ice sheet as a flexural gravity wave with altered dispersive properties. A treatment is given of the theoretical boundary value problem at the ice edge in which a number of evanes­cent waves appear. The subsequent propagation of the flexural gravity wave is then considered for ice sheets of varying viscoelastic properties. A decay rate is found which is dependent on the viscous parameters used, and a dispersion relation which depends on the elastic modulus. Experimental data were obtained in January-March 1977 from Notre Dame Bay, Newfoundland, using an array of novel strainmeters based on a wire strainmeter developed for earth tide measure­ment (Goodman et al., 1975; Allan and Winsor, 1978). A comparison is made between theory and experiment for the wavelength of the flexural gravity wave. An approximate open water amplitude spectrum is derived by a semi-quantitative method.

I N T R O D U C T I O N

The cracking of sea ice due to flexure by incident sea swell is the primary cause of ice breakup. Parallel cracks form normally to the direction of wave propagation and are regularly spaced some 10-40 m apart. The resultant strips of ice are often hundreds of meters in length, but after rotation across the wave path they are soon broken into characteristic angular cakes. To obtain a relationship between crack spacing and incident wavelength it is necessary to know the wavelength in ice as a function of period, which will depend on the physical properties of the sea ice at the time of breakup.

The process applies to pack ice as well as fast ice. In regions exposed to constant wave action, such as the Labrador Sea, where floes are a short-lived

327

328 V E R N O N A. SQUIRE and ALASTAIR J . ALLAN

phenomenon, any sizable floe will be reduced to 10 m ice cakes after a few hours of wave activity. During periods of quiescence and low temperatures, the debris is refrozen into another brecciated floe, only to be broken again by the onset of further wave action.

Although the breaking of ice by flexure is a complex process, a simple vis-coelastic model of a gravity wave propagating through ice is presented here and the theory compared with experimental data obtained on fast ice at Twillingate, Newfoundland.

VISCOELASTIC MODEL

Biot (1955) derives by a variational approach a general equation for the flexural bending of an isotropic linear viscoelastic thin plate of thickness h. The stress-strain relation is

a-a = 2Qeu'+ R8uekk (1)

where Q and R are material operators and rational functions of d/dt, crl} is the stress, e„ is strain, and 8U is the Kronecker delta.

A parameter B, is derived which is equivalent to the flexural rigidity for an elastic plate:

Bx = M2(Q + R)hs (2)

12(2Q + R)

Suppose that we consider ice to be a material which is elastic under hydrostatic stresses. The material operators Q andi? are then simply related:

R=K-^Q, (3)

where K is the bulk elastic modulus. This leads to

B, = 4g(g + 3K)h3 ( 4 )

12(4Q + 3K)

so that as K -* oc (Nevel, 1966), Bl -> hsQ/3. For a uniaxial tension test on a material which is elastically incompressible to

hydrostatic stresses (Fliigge, 1975),

ex,, = 3Qe.rx. (5)

Then with the usual definitions of deviatoric stress and strain, we have

(T'XJ. = 2Qe'xx, (6)

where the prime implies deviatoric.

Propagation of Flexural Gravity Waves in Sea Ice 329

-vAAA/V-

vww

>

Figure 1. Maxwell-Voigt spring-dashpot model.

We assume that a Maxwell-Voigt spring dashpot system (Fig. 1) most closely approximates the behavior of sea ice in such a tension test. This was found to be true for freshwater ice in the laboratory (Jellinek and Brill, 1956) and for sea ice during in situ bending beam tests (Tabata, 1955, 1958).

Our sea ice model may therefore be expressed as

S-dt2 dt

+ a2) cr/,r = — - + bx— )€,„. dt2 dt

(7)

where the a, and b-{ are material constants which may be related to the elastic moduli Ex and E2 or the viscosities 17, and 17, of individual components in the model. Explicitly,

ax = ^L + A + £?_, Vi V2 V2

a, = EXE,

V1V2

b0 = £ , , (8)

* , = Mi. V2

Returning to Biot (1955), we may write the equation for flexural bending of a viscoelastic plate on a fluid foundation as

Bt dx4 Pi

d2w dt2 P(x, t) (9)

where w is the vertical displacement, p, is the density of the plate, and P(x, t) is the pressure exerted on the plate by the fluid. Hence, using the limit form of B,

330 VERNON A. SQUIRE and ALASTAIR J. ALLAN

with equations (6) and (7) leads to an equation for the flexure of a thin sea ice sheet of constant thickness:

h'A ,, d2 , , <3, d4w , ,.d2 , d , ,d2w — (bn 1- b, —) h p,h( 1- ax 1- a ,) 6 dt2 dt dxA dt2 dt ' dt2

d2 d = ( — + a , — + a,)P(x, t).

dt2 dt

(10)

We seek a solution to the problem of an infinitesimal monochromatic deep water wave propagating in the positive *-direction towards a linear, semi-infinite, viscoelastic ice sheet.

HYDRODYNAMICS

If the hydrodynamics are assumed to be linear and irrotational, we may define a velocity potential $ within the water such that at every point Laplace's equa­tion

V24> = 0, (11)

is satisfied (Stoker, 1957). The water is assumed to be deep so that

^-1 = 0 , Qy \y-*oc

and at the surface the kinematic condition is

(12)

dw

dt d\ v=0. (13)

The problem may be divided naturally into two distinct regions (Fig. 2) with different surface boundary conditions (Hendrickson and Webb, 1963).

Incident

Region 1

Figure 2. Schematic representation of wave/ice interaction.

y r

t

R e g i o n 2

ill x "

Propagation of Flexural Gravity Waves in Sea Ice 331

Region 1

In region 1 (— oc < x < 0), we assume a free surface, so that Bernoulli's

theorem provides a second surface boundary condition

gWl 9 *

dt (14)

v=0.

where the subscript refers to the region and g is the acceleration due to gravity. A feasible solution is therefore

<I>, = [exp(ikx) + Rc exp(-ikx)] exp(-fcy) exp( - /W) , (15)

where Rc is the reflection coefficient for amplitude, k is the wave number, and u> is the circular frequency. This leads to the well-known deep-water dispersion equation

k =^L. (16) 8

Region 2 In region 2 (0 < x < oc), we assume that the sea ice has no submergence, a

valid assumption so long as the waves are long compared with the ice thickness. Bernoulli's theorem therefore gives the boundary condition along the interface between the viscoelastic ice and water. From equation (10)

h tu °>2 . u d ^*w2 , . .d2 , d ^ »2w., — {Pa H 0i —) + p;h ( h a, h a.,) -6 dt2 dt dx4 dt2 dt - dt2

-p( h a , h a,) ( = dt2 dt ' dt

+ gw2), (17) =0

where p is the density of sea water. There will be no frequency change in an ocean swell impinging on the ice, so that we may assume a solution of the form

<ï>, = [A„ e\p(iknx) + B„ exp(—ik„x)] exp(—k„y) exp(—tof) (18)

where summation with respect to n is implicit. In this expression, A„ and/?,, are amplitude coefficients for the nth mode and k„ is the wave number (possibly complex) associated with the nth mode.

Substitution into the boundary condition (17) yields the dispersion equation, a quintic polynomial in kn :

h et-. — (bn(o + ib^k,,5 + (to + iax) ( pg — p//îw2) k„ 6 to

— p(co 'L + ia^w2 = 0. (19)

332 VERNON A. SQUIRE and ALASTAIR J. ALLAN

Clearly, we must have boundedness as y -» =c so that only roots of this equation with positive real parts are permitted. Also, since the sea ice is assumed to occupy the region 0 ^x < ^ and we require waves propagating in the positive x-direction, we may restrict solution (18) and write

<ï>2 = [Ax exp(*,.r) exp(-fciy) + A, exp(ik2x) expOfc2y) (20) + Bs exp(—ik3x) exp(—fc3y)] exp(—mt)

where

*1 = f l + ' C l . * 2 =Ï2 +%2,k3 = £ 3 %*, (21)

andf„, £„ are real positive constants providing phase and attenuation information, respectively.

Values of the material constants £ , , ~qx, E2, -q.2 were chosen from Tabata (1958), whose measurements were made over the same range of temperatures encountered, so that

a, = 1.4 x 10-2 s"1, a, = 0.07 x 10~4 s~2, i , = 1.2 x 109 Nm'2 , b\ = 1.02 x 107 Nm~2 s"1.

Using these values the dispersion equation was solved numerically for k„. It is found that all three modes have £„ > 0, so that attenuation is always

present. However, £, is several orders of magnitude smaller than the correspond­ing £2 and £3, and so it most closely corresponds to a propagating wave. A graph illustrating the dependence of the attenuation coefficient £, on period for various

0 1.0 2.0 3 0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 Period , s.

Figure 3. The attenuation coefficient f, versus period for a variety of ice thicknesses.

Propagation of Flexural Gravity Waves in Sea Ice 333

ice thicknesses is shown in Figure 3. Each curve shows a clear frequency de­pendent peak reflecting the interaction between Maxwell and Voigt units in the viscoelastic model. For periods greater than that corresponding to the maximum in each curve, the attenuation coefficient is seen to decrease as the period of the incident swell increases, so that long waves will propagate much farther than short waves.

Matching

In order to evaluate^,, and/vc we match potential functions and velocities at x = 0. Matching may be carried out only atv = 0, because & and/:, (k2,k3) are not, in general, equivalent. This implies that the potentials <P1 and 4>2 are approxi­mate and should contain an additional unknown potential to facilitate matching along the entire x = 0 line. Wadhams (1973) discusses this approximation in more detail and shows that it is reasonable since most of the incident wave's energy is near the surface. Hence,

<I>, = <t>2 x=0 y=0,

8<Pj_ = Ô*2_

dx dx x=0 >>=0.

At the free end of the plate there is no bending moment or shear so that

d2w2 „i dsw.z n i

dx2 U=o, dx3 \x=0.

(22)

(23)

These four boundary conditions lead to a system of equations which may be solved numerically.

The amplitude ratio (Fig. 4), a measure of the distribution of transmitted

0 1.0 2.0 3.0 4 0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 Period , s.

Figure 4. The amplitude ratio versus period for the three flexural-gravity waves in 0.5 m of sea ice. Curve l most corresponds to a propagating wave.

334 VERNON A. SQUIRE and ALASTAIR J. ALLAN

energy between the three possible flexural gravity modes relative to the incident deep water wave, may then be computed from

(amplitude ratio)„ = -^-—^,—Lp- > (24) I k„ J \A„ I

where « = 1, 2, 3. For short periods, the energy transmitted across the ice edge is divided, to the

same order of magnitude, among the three possible wave solutions. As the period of the incoming wave increases, however, the wave most closely corresponding to a propagating wave (i.e., smallest attenuation) begins to dominate and the two strongly evanescent waves become less significant.

EXPERIMENT

Phase From (17), the velocity potential of the monochromatic flexural wave prop­

agating in the x-direction which shows least attenuation is

A{ exp(/£,x) exp(—/:,>') exp(—«W)>

so that the phase change between points x, andx, at an instant in time is given by

f i f e ~ xx).

Hence given the ice thickness, we may find theoretically the wavelength of the flexural wave for a variety of periods (Fig, 6).

A linear array of three wire strainmeters (Goodman et al., 1975; Goodman, 1977) was deployed on fast ice at Main Tickle near Twillingate, Newfoundland, with the aim of using the measured phase difference between strainmeters to find experimentally the wavelength for waves of known period. A separation of 10 m was chosen so that the phase differences between strainmeters 1 and 2 and between strainmeters 1 and 3 were restricted to the same cycle for all the wavelengths likely to be encountered (Fig. 5). For such an experiment the forc­ing is not a single sinusoidal oscillation but a whole spectrum of incident wave energies. It was therefore hoped to obtain values of wavelength for a variety of measured periods.

Three possible methods of analysis were considered (Allan and Squire, in

Impinging 1 2 3

f l exu ra l 10m. w a v e -« ».

^ 20m. Figure 5. Linear array of strainmeters used in phase experiment.

Propagation of Flexural Gravity Waves in Sea Ice 335

press): the cross correlation; the phase spectrum; and the transfer function. The cross correlation proved to be difficult to interpret for all but a monochromatic wave. The phase spectrum method was slow and also carried with it large statisti­cal errors. The method finally adopted was that of the transfer function (J. R. Rossiter, personal communication, 1977) and the associated coherence function. This approach is often used in engineering to compare the input and output of a black-box system. Given signals from strainmeters 1 and 2, say, at a known separation, we define the transfer function as

transfer function = "12 . (25)

where G12 is the average cross power spectrum between strainmeters 1 and 2, and G n is the average auto power spectrum for strainmeter 1. Averaging was carried out over approximately one hour, a reasonable time given the océanographie conditions over the duration of the experiment.

A measure of the validity of the transfer function is provided by the coherence function (Bendat and Piersol, 1971) defined as

17̂ 12 coherence function = J_ 121 __ , (26)

Gn • G22

where G22 is the average auto power spectrum for strainmeter 2. The transfer function therefore provides a comparison between the magnitude and phase of two strainmeter signals, and the coherence function gives an indication of their similarity.

Using this approach, the wavelength for a variety of periods, corresponding to a minimum coherence of 0.96, was plotted for both 10 m and 20 m separation (Fig. 6). Within the standard error, the experimental data do not differ sig­nificantly from the theoretical curves and do not show a bias to either side. A chi-squared test

X2 = T (A-o - K)2 ^ (27) K

where \ 0 is the observed wavelength, k, is the theoretical wavelength, and n is the number of samples, gave values for %2 In of 7.99, 12.15, and 16.97 for 0.5 m of ice, 1.0 m of ice, and the deep-water wave respectively. A thickness of approximately 0.5 m was measured over the experimental site, agreeing with the lowest chi-squared value.

Strain

The theoretical strain at the surface of the sea ice is given approximately by

e = _A fe) (28) 2 ax2

336 VERNON A. SQUIRE and ALASTAIR J. ALLAN

500

200

100

?50

> 5

20

10

Standard error

è-*^^'^^

1 0 m / ^ ^ / *

s^KSm. /

/ D e e p water

A ^ - ^

Ay^ .s' A

*S^^ A

A 10m. separation • 20m. separation

A

-

•

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 Period, s.

Figure 6. A comparison of theory and experiment for wavelength as a function of period.

(Allan and Squire, in press). Given a value for the incident wave amplitude, therefore, it is possible to predict the infinitesimal surface strain as a function of period for a particular distance into the ice sheet.

An experiment was set up to monitor continuously the surface strain at dis­tances of 5 m, 10 m, and 15 m from the ice edge and to compare the observed strains with the theoretical strains. The infinitesimal surface strain calculated from (28) was corrected to a finite strain measured over 2 m, the physical length of the wire strainmeter used. The observed strain was obtained as a function of period from the magnitude spectrum averaged over approximately one hour. Since no wave rider buoy data were available to provide information about the incident spectrum, it is not possible to compare theory and experiment abso­lutely. However, the ratio of observed strain to theoretical strain may be found at the three strainmeters.

If an amplitude of unity is used in the theoretical model, then, since we have assumed linearity, a plot of this ratio against period may be regarded as a semi-quantitative approach to evaluating the incident wave amplitude spectrum. A typical example is shown in Figure 7.

All three spectra show the same characteristic shape and reflect the observed océanographie conditions at the time of the experiment, i.e., small-amplitude, locally generated waves with no long period component. The 5 m line appears to indicate a larger amplitude for the incident waves than the 10 m and 15 m lines. This would be expected close to the ice edge due to the possible existence of evanescent waves.

Propagation of Flexural Gravity Waves in Sea Ice 337

5.0 6.0 Period , s.

Figure 7. Semiquantitative amplitude spectrum for the incident deep water wave.

A better understanding of the propagation of gravity waves in ice is required both in a study of the natural break-up process and in a number of applied problems such as the wave produced by rapid aircraft movement on ice runways or artificially induced fracture by air cushion vehicles. This paper has attempted to examine some of the more fundamental characteristics of the flexural gravity wave propagating through a sheet of fast ice.

ACKNOWLEDGMENTS

This work was carried out while Vernon Squire was in receipt of a research studentship from the Natural Environmental Research Council of Great Britain. We are indebted to Dr. G. de Q. Robin, Harold Snyder, and Dr. Peter Wadhams for their continuous support. We also thank members of the Twillingate field party, whose good humour and encouragement contributed to a successful exper­iment.

REFERENCES

Allan, A. J., and V. A. Squire. Naturally induced surface strain in fast ice. C-CORE Publication, in press.

Allan, A. J., and W. D. Winsor. 1978. Industrial applications of ice strain measurements. In Proceedings of the Fourth International Conference on Port and Ocean Engineering under Arctic Conditions (ed. D. B. Muggeridge), vol. 2, pp. 629-37, Memorial Uni­versity of Newfoundland, St. John's.

Bendat, J. S., and A. G. Piersol. 1971. Random Data: Analysis and Measurement Procedures, 407 pp., Wiley-Interscience, New York.

338 VERNON A. SQUIRE and ALASTAIR J. ALLAN

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Fliigge, W. 1975. Viscoelasticity, 2nd éd., 194 pp., Springer-Verlag, Berlin. Goodman, D. J. 1977. Creep and fracture of ice, and surface strain measurements on

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Jellinek, H. H. G., and R. Brill. 1956. Viscoelastic properties of ice. Journal of Applied Physics, 27(10), 1198-1209.

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Wadhams, P. 1973. The effect of sea ice cover on ocean surface waves, Ph.D. disserta­tion, 223 pp., Cambridge University, England.