1
Response of a circular ice floe to ocean waves
Michael H. Meylan and Vernon A. Squire
Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
Abstract. A new model is presented to reproduce the behavior of a solitary,
circular, flexible ice floe brought into motion by the action of long-crested sea
waves. The intended application of the work is ultimately a fully three-dimensional
analogue of a marginal ice zone (MIZ) through which ocean waves propagate,
allowing the attenuation and directional advance to be forecast and validated
against observations. (Existing theory does not treat directional changes correctly.)
To enable a check to be made on the model, two independent methods are
developed: an expansion in the eigenfunctions of a thin circular plate, and the more
general method of eigenfunctions used to construct a Green’s function for the floe.
Displacement and three-dimensional scattering patterns in the water surrounding
the floe are given for several floe geometries. The model is also used to investigate
the strain field generated in the floe, its surge response, and the energy initiated
in the water encircling it. Finally, with the aim of understanding how floes herd
together to form cohesive structures in the MIZ, the force induced on floes of various
thicknesses and diameters is plotted.
Introduction
Given that the principal determinant of the distribution of floe sizes within the marginal
ice zone (MIZ) is ocean wave activity, that wave-induced radiation stress at the ice margin
is an important ingredient in the balance of forces [Liu et al., 1993], and that the attributes
of the ice cover are significant to weather and climate, observational and theoretical research
on wave-ice processes requires little justification. Unfortunately, although several data sets
exist describing features of the process [e.g., Squire and Moore, 1980; Wadhams et al., 1986,
1988], they are not underpinned by an altogether satisfactory theoretical foundation. No
current theories deal entirely adequately, for example, with directional changes to seas and
swell moving through the ice; indeed, most predict the opposite of what is actually observed,
a collimation in contrast to a spectral broadening to isotropy. The exception is the work
of Masson and LeBlond [1989] discussed below, which finds that the wave spectrum rapidly
becomes isotropic in form in an open ice field for a range of values of ka, where k denotes
the wave number and a the floe radius.
The two theoretical strategies in common use both have serious deficiencies. The pragmatic approach, whereby the properties of the MIZ in relation to waves are supposed to
be due to the integrated effect of all its component ice floes, suffers from the serious defect
of lacking a fully three-dimensional, flexible ice floe model, the rigid floating body probably being a poor analogue to a real ice floe. Consequently, apart from the Masson and
LeBlond [1989] study, the validity of any modelling exercises is limited to normally incident,
long-crested seas at all frequencies; an unlikely physical scenario. Attempts to use twodimensional theories to study directional features of penetrating wave spectra have yielded
results in conflict with observations [Wadhams et al., 1986; Squire and Meylan, 1994]. The
alternative approach, which has greater mathematical appeal and more elegance, represents
the MIZ as a single surface boundary condition with altered dispersive and attenuative character to that for the open sea. Here the problem is even more fundamental: we are not yet
able to characterize the behavior of the ice pack mathematically, the thin, homogeneous
elastic plate in common use falling well short of a good parameterization [Squire, 1995a, b].
Recent unpublished results that allow the elastic properties, density, and thickness of the
2
plate to change spatially offer an improvement (M. Barrett, personal communication, 1995),
but this work has not been thoroughly validated to date.
The paper by Masson and LeBlond [1989] justifies more discussion, as it signals the first
serious attempt to characterize wave generation and propagation in an MIZ. Therein, the
energy balance equation of Hasselmann [1960], namely,
∂F
+ cg · ∇F = Sin + Sds + Snl + Sice
∂t
(1)
is used, where F is the directional frequency spectrum which evolves in time and space,
cg is the group velocity, Sin is the rate of energy input, Sds is the rate of dissipation,
Snl represents energy transfer by nonlinear interactions among spectral components, and
Sice denotes redistribution and dissipation of energy due to the ice. Terms Sin and Sds
are parameterized relatively straightforwardly; Snl is more problematical, as the commonly
used parameterizations will not hold in an MIZ, so a complete Boltzmann energy transfer
integral is required; and Sice is new. The final term assimilates the effect of the many ice
floes forming the MIZ and thus depends on the ability of single floes to scatter a fraction
of the incoming wave energy. Dissipative processes specific to the ice field due to, e.g.,
ridge formation, rafting, or waves breaking over floes, must also be included. Masson and
LeBlond [1989] represent each ice floe as a rigid disk, invoking the model of Isaacson [1982]
to compute the various rigid body motions and thereby the scattering cross section for each
floe. The present paper, while falling short of integrating contributions from the many floes
present and ultimately employing equation (1), treats the ice floes themselves differently,
namely, as compliant floating bodies rather than as rigid ones, as observations suggest that
significant bending does occur [Squire and Martin, 1980]. Since it is unlikely that rigid ice
floes will scatter waves in the same manner as flexible ones unless their aspect ratios are
such that they behave relatively stiffly, the predicted results of Masson and LeBlond [1989]
are limited to situations where ka < 1. Likewise, the drift forces calculated by Masson
[1991] are subject to the same constraint. We believe that the flexure of individual floes
generally is influential in regulating their scattering behavior and, concomitantly, that the
parameterization of Sice should be amended from that used by Masson and LeBlond [1989].
Our aim then in this paper is to set the scene for a new three-dimensional model for wave
propagation within MIZs. To do this, we must focus initially on the dynamics of solitary
ice floes, as it is their individual actions when coalesced that determine the qualities of the
MIZ as a whole. Later papers will apply these results to MIZs, either in the sophisticated
manner of Masson and LeBlond [1989] or by alternative means.
To leave the security of two-dimensional theory requires us to choose an archetypal shape
for ice floes; we have selected a circular disk, as we feel that its motions and flexure, as well
as its capacity to diffract and scatter waves, will embody features of real ice floes while being
geometrically and algebraically realizable. Even so, the computation of the behavior of the
disk in a planar ocean wave field is nontrivial because of the different coordinate systems
implicit in the formulation.
An extension of Meylan and Squire [1993a, b, 1994] to a fully three-dimensional geometry
will be presented, considering a single ice floe as a thin, flexible, circular disk of constant
thickness acted upon by long-crested ocean waves. Linear diffraction theory will be used to
derive a linear equation relating the velocity potential φ and its normal derivative φn beneath
the floe. Following Meylan and Squire [1993a, b, 1994] and Stoker [1957], we invoke the
shallow draft approximation, wherein the disk submergence is neglected. This introduces a
slight modification of the basic equations, as described by Buchner [1993] in connection with
the motion of rigid bodies of shallow draft moving under the action of waves. The vibration
of a circular thin plate, governed by the Bernoulli-Euler equations of motion, with free edge
conditions is well documented [Itao and Crandall, 1979], and the eigenfunctions (modes)
of the plate are known. These eigenfunctions allow us to derive the necessary relationship
3
between φ and φn to solve the problem, much as the rigid body motion is solved by using
the six rigid body modes corresponding to the body’s six degrees of freedom.
We report here predictions from the new flexible ice floe model of circular geometry acted
upon by long-crested ocean surface waves.
Boundary Value Problem
Consider the motion of a flexible, circular ice floe floating on the surface of the water and
subjected to an incoming train of long-crested ocean waves. This is shown in Figure 20,
which also shows the coordinate systems used. Since we have a plane wave input andFig.
a 20
circular disk, it is natural to use both Cartesian (x, y, z) and cylindrical (r, θ, z) coordinate
systems. The linearized boundary value problem for the velocity potential Φ, assuming
irrotational and inviscid flow, is the following:
−∞<z <0
∇2 Φ = 0
∂Φ
=0
z → −∞
∂z
∂w
∂Φ
=
z = 0
∂z
∂t
¶
µ
∂p
∂Φ ∂ 2 Φ
+ 2 =−
ρ g
∂z
∂t
∂t
(2a)
(2b)
(2c)
z = 0
(2d)
In (2), w is the surface displacement, p is the pressure on the water surface, which is assumed
constant except beneath the floe, ρ is the density of the water, and g is the acceleration due
to gravity. We also require appropriate conditions to be met as r → ∞ and at t = 0.
The floe is modelled as a thin, circular disk of radius a satisfying the Bernoulli-Euler
equation of motion:
∂2w
(3)
D∇4 w + ρ0 h 2 = p
∂t
where h is the disk’s thickness, ρ0 is its density, and its flexural rigidity is given by D =
Eh3 /12(1 − ν 2 ), where E denotes Young’s modulus and ν is Poisson’s ratio. The free edge
boundary conditions at r = a are [Itao and Crandall, 1979]
µ
¶
1 ∂w
∂2w
1 ∂2w
+ν
(4a)
+ 2 2 =0
∂r2
a ∂r
a ∂θ
µ 2 ¶
∂ ¡ 2 ¢ 1−ν ∂ 1∂ w
=0
(4b)
∇ w +
∂r
a ∂r r ∂θ2
From (2) and (3) the boundary condition beneath the floe is
¶
µ
∂ ∂Φ
∂Φ
∂Φ ∂ 2 Φ
+ 2 = ρ0 h 2
+ D∇4
−ρ g
∂z
∂t
∂t ∂z
∂z
z = 0
0<r<a
(5a)
whereas that in the open sea is
µ
¶
∂Φ ∂ 2 Φ
−ρ g
+ 2 =0
∂z
∂t
z = 0
(5b)
a<r<∞
Equations (2) together with boundary conditions (4) and (5) are now nondimensionalized
using
r
r
z
Φ
g
r̄ =
t̄ = t
z̄ =
Φ̄ = √
a
a
a
a ag
4
and defining
γ=
ρ0 h
ρa
β=
D
gρa4
√
√
With separable and periodic Φ̄(x̄, z̄, t̄) = φ̄(x̄, z̄)e−it̄ α , where α is the nondimensional
radian frequency, the boundary value problem to be solved becomes
∇2 φ = 0
−∞<z <0
∂φ
=0
z → −∞
∂z
∂φ
− αφ = 0
z = 0
1<r<∞
∂z
¢ ∂φ
¡ 4
β∇ + 1 − αγ
− αφ = 0
∂z
z = 0
0≤r≤1
µ
¶
∂ ∂φ
∂ 2 ∂φ
∂ 2 ∂φ
+ν
+ 2
=0
∂r2 ∂z
∂r ∂z
∂θ ∂z
r¶= 1
µ
¶
µ
2
∂
1
∂
∂φ
∂φ
∂
∇2
+ (1 − ν)
=0
∂r
∂z
∂r r ∂θ2 ∂z
r=1
(6a)
(6b)
(6c)
(6d)
(6e)
(6f)
where overbars have been omitted for clarity. With the standard Sommerfield radiation
condition [Sarpkaya and Isaccson, 1981] as r → ∞, denoting the incident potential φi , we
may write formally
µ
¶
√
∂
r
as
r→∞
(7)
− iα (φ − φi ) = 0
∂r
unit amplitude travelling in the
Henceforth we shall confine φi to be a long-crested wave of√
positive x direction on deep water. Then φi (x, y, z) = −(i/ α)eiαx eαz .
Solution
Two methods of solution have been conceived, thereby providing a check of numerical
results. Each will be reported, although in practice method 1 will be used predominantly,
as it is computationally more economical.
Method 1
From John [1950] and Sarpkaya and Isaccson [1981], and using Buchner [1993], the potential for a circular, elastic ice floe satisfies
Z
³
´
φ(P) = φi (P) + G(P; Q) αφ(Q) − φz (Q) dSQ
(8)
∆
where ∆ is the unit circle on the surface of the water, dSQ is the surface element with respect
to the Q coordinates, P = (x, y, z) and Q = (ξ, ς, η) in Cartesians for example, and φn , the
normal derivative of φ, becomes simply ∂φ/∂z |z=0 which we denote by φz . In deriving (8)
the relationship, Gn = αG has been used.
G(P; Q), the infinite water depth Green’s function, is given by [Buchner, 1993]
G(P; Q) =
¶
µ
´
³
1
2
− πα H0 (αR) + Y0 (αR) + i2παJ0 (αR)
4π R
(9)
5
p
where R = (x − ξ)2 + (y − η)2 , J0 and Y0 are respectively Bessel functions of the first
and second kind of order zero, and H0 is the Struve function of order zero [Abramowitz and
Stegun, 1970, p. 496].
As well as (8) the potential must satisfy the equation of motion of the floe and its edge
conditions:
¢
¡ 4
(10a)
β∇ + 1 − αγ φz = αφ
µ
¶
2
2
∂ φz
∂φz
∂ φz
+ν
=0
(10b)
+
∂r2
∂r
∂θ2
r=1
0 ≤ θ < 2π
µ 2 ¶
¢
¡
∂
∂ 1 ∂ φz
=0
(10c)
∇2 φz + (1 − ν)
∂r
∂r r ∂θ2
r=1
0 ≤ θ < 2π
To solve the boundary value problem specified by (8) and (10) we expand in terms of the
eigenfunctions of a circular thin plate of unit radius, given by Itao and Crandall [1979], as
ψj,p (r, θ) = Aj,p sin pθ
¶
µ
p<0
× Jp (λj,p r) + Cj,p Ip (λj,p r)
ψj,p (r, θ) = Aj,0
µ
¶
× J0 (λj,0 r) + Cj,0 I0 (λj,0 r)
p=0
ψj,p (r, θ) = Aj,p cos pθ
µ
¶
× Jp (λj,p r) + Cj,p Ip (λj,p r)
p>0
(11a)
(11b)
(11c)
where j (j ≥ 0) and p (−∞ < p < ∞) are integers and Jp (λ) and Ip (λ) are respectively
Bessel functions and modified Bessel functions of the first kind of order p [Abramowitz and
Stegun, 1970, p. 358].
of j and p are the rigid body modes
√ The three√exceptions in the ranges √
ψ0,−1 = 2r sin θ/ π, ψ0,0 = 1/ π, and ψ0,1 = 2r cos θ/ π. The frequency parameter λj,p
and the mode shape parameter Cj,p are fixed by the eigenvalue problem, and the amplitude
parameter Aj,p is fixed by the normalization requirement. They are each given by Itao and
Crandall [1979], whose normalization expression is wrong, although generally results in the
paper are correct. The correct expression is given by Meylan [1995].
Since the problem we are considering is symmetric about the x axis, we need only consider
modes for which p ≥ 0. This gives us the following expressions for φz and φ
XX
aj,p ψj,p (P)
(12a)
φz (P) =
p
j
XX
bj,p ψj,p (P)
(12b)
which may be substituted into (10a) to obtain
¢
¡ 4
βλj,p + 1 − αγ aj,p = αbj,p
(13)
φ(P) =
p
j
Finally, we substitute (13) into (8) and take an inner product. Then
³
XXDZ
bj,p = hφi , ψj,p i +
αG(P; Q) bk,q
k
q
∆
E
´
bk,q
−
(Q)dQ,
ψ
(P)
ψ
k,q
j,p
βλ4k,q + 1 − αγ
(14)
6
where the inner product is defined
hf, gi =
Z
f (P)g(P)dSP
∆
The solution of (14) is straightforward by truncation of the expansion at the point where
errors become negligibly small.
Method 2
An alternative method of solution is to use the eigenfunctions to construct a Green’s
function for the circular ice floe. We seek a function satisfying the following equation
∇4 g(P; Q) +
1 − αγ
g(P; Q) = δ(P − Q)
β
(15)
subject to boundary conditions (4). This Green’s function follows from the eigenfunctions
as described by Courant and Hilbert [1953] and is given by
g(P; Q) =
X X ψj,p (P)ψj,p (Q)
λ4j,p + (1 − αγ)/β
p
j
(16)
so that (8) becomes
φ(P) = φi (P) +
α
−
β
Z
∆
Z
∆
³
G(P; Q) αφ(Q)
´
g(Q; R)φ(R)dSR dSQ
(17)
This is a two-dimensional Fredholm integral equation that can be solved straightforwardly
by integration.
Computation
In nearly all the results to follow we shall use the first 76 modes, which correspond to
all the modes whose eigenvalue λj,p < 22. (For 400-m-diameter floes we use 135 modes.)
Solution by the eigenfunction expansion method, i.e., by method 1 (equation (14)), or by
using the Green’s function for the disk, i.e., method 2 (equation (17)), gives essentially
the same results, and this acts as a check on our theory. For floating objects of arbitrary
geometry, method 2 is the only one possible, but for the circular disk the eigenfunction
expansion method is computationally preferable, and for this reason it is used here.
Throughout this work, the elastic moduli for sea ice, i.e., Young’s modulus and Poisson’s ratio herein, are set respectively at 6 GPa and 0.3, the density of seawater is set at
1025 kg m−3 , and the density of sea ice is set at 922.5 kg m−3 . Results do not vary greatly
within the range of physically plausible, partially relaxed elastic moduli [Squire, 1993], although an unrealistically small E that makes the ice too compliant will diminish its effect
on the wave train and too large an E will cause the floe to behave more like a rigid floating
raft in the manner of Masson and LeBlond [1989] and Masson [1991]. Because the flexural
rigidity D varies linearly with E but as the cube of thickness h, changes in thickness are far
more influential than changes in E.
The solution is a function of three nondimensionalized coordinates, α, β, and γ, representative of the wavenumber, the floe stiffness, and the floe mass, respectively.
7
Results
Floe and Water Displacement Patterns
In the sequence of Figures 21—24, ice floe displacement is shown for different diameters and
Figs. 21—24
thicknesses of floe at a moment in time. Each subplot in Figure 21, for example, shows the
displacement and hence the deformation experienced by a 50-m floe at thicknesses of 0.5 m,
1 m, 2 m, and 5 m. For the 50-m floe, bending is minimal at each thickness plotted, although
it is apparent that the degree of flexure increases as thickness is decreased. Bending for
floes of 100-m diameter (Figure 22) is more pronounced; the 0.5-m floe conforms especially
well to the profile of the passing waves. Because the floes shown in Figures 23 and 24 have
diameters greater than the length of the incoming ocean wave train beneath the ice, the
floe is subjected to a full wave cycle or more. This is most conspicuous for the thinnest
ice, where large deformations are evident and the geometric shape of the waves, i.e., their
long-crestedness, appears to be relatively unaffected by the presence of the thin ice. As the
ice becomes thicker, however, the wave crests cease to be parallel and the cylindrical shape
of the floe influences the three-dimensional structure of the waves to a larger extent. There
is also a noticeable decrease in the deformation induced in the ice floe for thicker ice as was
seen for small-diameter ice floes.
The large deflections seen in Figures 22a, 23a, and 24a occur because the floe diameter
is roughly an integer multiple of the wavelength beneath the ice. Accordingly, a kind of
resonance occurs that for real ice floes would be limited by the ability of the ice to withstand
the curvature imparted by the underlying wave. There are many instances of sea ice floes
being fractured by incoming waves (see for example Squire and Martin [1980]); indeed V.A.S.
has been aboard a floe when this actually happened. Since current wisdom suggests that
floes can withstand strains up to about 10−5 —10−4 before they break up [Squire, 1995c], it
is unlikely that the deflections of Figures 22a to 24a could be sustained.
It is beneficial to look at Figures 23b and 24b in the context of in situ field experiments
to determine the seakeeping and flexural behavior of solitary ice floes in ocean waves. While
few such experiments have taken place, those that have have furnished data which have
proved to be difficult to analyze. The several field experiments of the Scott Polar Research
Institute, University of Cambridge, to study wave propagation in MIZs generally, for example, invariably utilized strain gauges, accelerometers, and tiltmeters deployed on individual
ice floes. Interpretation of these data has been hindered by their complexity, and limited
success has been had in their elucidation [Squire, 1983]. Indeed, the distortions to simple
long-crestedness seen in Figures 23b and 24b suggest that no easy interpretation may be
possible when ice thickness is sufficient to influence the attributes of the flexure.
In Figure 25 the floe deformation and motion have been suppressed in the plotting Fig.
to 25
allow features of the water wave field surrounding the ice floe to be shown. This is done
respectively for 1-m-thick ice floes of 50-, 100-, 200-, and 400-m diameter, all in waves of
length 100 m. In each case the ocean wave train which is forcing the motion is subtracted
out, so that the consequence of the floe’s presence is seen clearly. What is plotted therefore
is the sum of the scattered and diffracted wave fields, which will be referred to hereinafter as
the initiated field. Because incoming plane waves proceed in the x direction, i.e., their crests
are parallel to the y axis, all subplots are symmetric about the x axis. The initiated wave
fields in each case are complicated, although at the smallest diameter (Figure 25a, 50 m) the
pattern surrounding the ice floe is approximately concentric. Larger-diameter floes create
distorted wave patterns because several cycles of flexural oscillation can be induced in the
floe, initiating a confused mix of cylindrical and planar wave fronts in the water.
8
Principal Strains
The strain field generated in a circular ice floe by a long-crested ocean wave train can be
rather complicated and is highly dependent on the relative dimensions of the circular floe
in comparison to the length of the incoming waves. Because the surface strain is tensorial,
being composed of three linearly independent components, it is convenient to diagonalize
the strain tensor to eliminate shear strain. Then the three components will comprise two
principal strains, together with their direction relative to, say, the x axis.
In terms of the local Cartesian coordinate frame defined by the polar coordinate system,
the values of the principal strains are given by the eigenvalues of the following matrix:


1 ∂2w
∂2w
1 ∂w
−
h
∂r2
r ∂r∂θ r2 ∂r 

= 
(18)

2
2 1 ∂ w
1 ∂w
1 ∂ 2 w 1 ∂w 
+
−
r ∂r∂θ r2 ∂r
r2 ∂θ2
r ∂r
and the orientations of the principal strains are the corresponding eigenvectors.
In Figures 26—29 we have plotted for two wave cycles the principal surface strains and their
Figs. 26—29
orientation at various locations on ice floes of different diameters and thicknesses. Figures
are composed of four pairs of time series, each illustrating the maximum and minimum
principal strains (a1, b1, c1, and d1) and the direction of the axes of principal strain (a2,
b2, c2, and d2). The pair a1, a2 refers to a point at the center of the ice floe, the pair b1, b2
to a site at the nondimensionalized point x̄ = 0.5, ȳ = 0, the pair c1, c2 to a point x̄ = 0,
ȳ = 0.5, and the pair d1, d2 to x̄ = −0.5, ȳ = 0. Thus points a, b, and d lie along the x axis
on the centerline of the ice floe, while point c lies on the y-axis halfway between the center
and the edge. Because the wave train is travelling parallel to the x axis, the orientation
of the axes of principal strain is consistently along the x axis for stations a, b, and d, i.e.,
θ = 0, whereas at point c in each case the strain field is more interesting.
Figure 26 illustrates the strains generated in a 50-m-diameter floe of 1-m thickness at the
locations described above. The displacement induced in such a floe is shown in Figure 21b,
where it is seen that the bending is relatively straightforward. Maximum deflection occurs
near the floe’s center, becoming smaller as the edge is approached. This is seen clearly in
Figure 26, noting that time series a1 corresponds to the center of the floe while series b1 and
d1 are halfway out to the edge: significantly greater flexure is experienced at site a than at
sites b and d. Strains at c are similar in magnitude to those at a but are affected by the
circular geometry of the floe in relation to the planar nature of the incoming wave field. This
results in a (varying) nonzero shear stress at some periodic times during the wave cycle and,
concomitantly, a rotation of the axes of principal strain which is evident in all Figures 26c
to 29c but which is particularly clear in Figure 27c.
In accord with our comments of the previous section it is unlikely that surface strain fields
observed during in situ experiments on sea ice floes can be interpreted straightforwardly; the
deflections set up within an arbitrarily shaped ice floe of significant thickness in long-crested
seas are just too disorganized. The only hope is to synthesize theoretical results with data.
Although in the 50- and 100-m-diameter cases of Figures 26 and 27, the maximum strain
occurs at the center of the ice floe, it is important to appreciate that this is not always
so and that when the floe experiences more than one cycle of flexure across its length, it
is an unlikely scenario. See Figure 28, corresponding to the displacement field shown in
Figure 23b, for a counter example where the strains at point b exceed those at sites a, c,
and d.
Notwithstanding this and acknowledging that slightly greater strains may be set up in
large floes, Figure 30 illustrates the strain amplitude at the center of the ice floe for different
Fig. 30
floe diameters as thickness is varied. Note that each curve peaks at a well-defined thickness
9
but that multiple maxima are possible when the floe diameter is longer than the wavelength
and multiple cycles of strain occur within the floe. At great thicknesses the strain amplitude
becomes small as the floe behaves in a quasi-rigid manner. The thickness at which the strain
amplitude peaks in each case is rather insensitive to diameter, recalling of course that all
curves are for 100-m waves only and the strain magnitudes are referenced to 1-m amplitude.
Because of this it appears that floes with thicknesses in the approximate range 0.5—1.5 m are
most likely to be fractured and consequently destroyed by these waves, and that ice floes of
smaller or greater thickness may survive. Longer waves would displace the “range of likely
fracture” toward greater thickness. In a real sea, composed of many wavelengths present at
different energies, the range would depend on the ocean’s spectral form, but typically the
picture would not be so different from that shown in Figure 30.
Surge Response
An estimate of the surge motion may be calculated by considering the potential around
the floe and integrating the force from Bernoulli’s equation. The normalized surge response,
i.e., the surge divided by the wave amplitude, is then
Z 2π
1
S=
φ(1, θ) cos θdθ
(19)
iαπ 0
Fig. 31
and is plotted in Figure 31 for 100-m-length waves. When the thickness of the floe is large
so that bending is minimal, the surge response is uncomplicated; an increase in diameter
leading to a reduced response. The 50-m curve is also particularly simple because its aspect
ratio is such that it behaves relatively stiffly at all wave periods. A more intricate response
is seen when the floe can bend significantly, either with just a single cycle in the case of the
100-m curve (see also Figure 22a), or with the multiple cycles seen in Figures 23a and 24a
and appearing here as the 200- and 400-m curves. Detail in the surge response is a direct
result of bending and suggests again that field experiments to study the seakeeping motions
of ice floes must be interpreted with care.
Energy and Time-Averaged Force
The Russian worker N. E. Kochin [Buchner, 1993; Wehausen and Laitone, 1960] defined
the following function (in dimensional coordinates):
Z
H(θ) = (kφ − φz ) eik(x cos θ+y sin θ) dSP
(20)
∆
which is now known as the Kochin function. H(θ) allows compact formulae to be written
down for many physical quantities of interest. For example, the energy radiated by the
initiated potential per unit angle per unit time is given by [Newman, 1967]
E(θ) =
ρω 3
|H(π + θ)|2
8πg
and the force component on the body is given by
Z
ρk 2 2π
ρωA
|H(θ)|2 cos θdθ +
Xav =
ImH(π)
8π 0
2
(21)
(22)
Figs. 32—35
The radial distributions of initiated energy shown in Figures 32—35 are normalized with
respect to the energy passing beneath each floe per unit time; i.e., the energy is divided by
ρag 2 A2 /ω, where A is the amplitude of the input wave. For the smallest diameter considered,
namely, 50 m in Figure 32, the radiation patterns are different, as except for the thinnest
10
case (Figure 32a) corresponding to h = 0.5 m, the floe behaves like a rigid body. The energy
reflected back, comprising the area within the petal on the 180◦ radius of each subplot,
increases as the thickness increases. (Note that the radial scale in each subplot is different
to allow the pattern to be seen more clearly.) The initiated energy in the direction of the
forward-going vector also increases with thickness, as does the energy associated with the
sidelobes. The balance between each of these energies changes with thickness, the sidelobes
especially becoming more important as thickness is increased until at 5 m, for example,
the sidelobes obscure the forward-going petal centered on 0◦ entirely. Figure 32b may be
compared directly with the three-dimensional wire frame plot Figure 25a, noting that the
latter is a plot of displacement rather than of energy.
Subsequent plots in the series Figures 33—35 illustrate clearly that the dominant far field
effect of larger ice floes is to generate forward-going energy. Further, the width of the
associated lobe narrows as the diameter of the ice floe is increased.
For the 100-m floe illustrated in Figure 33, particularly, nearly all the energy is accommodated in the forward-going petal centered about 0◦ , presumably because the flexural
response is conspicuously long-crested at this diameter and wavelength (see Figure 22).
While 200- and 400-m floes also diffract and scatter most energy forward (see Figures 34
and 35) because of the wave-making potential of the far side of the ice floe, significant petals
of energy are present for the two thicker floes at other angles. These are especially visible in
the 400-m case, where Figures 35c and 35d, representing 2- and 4-m examples, respectively,
show interesting and complicated radiation patterns due to multicycle bending of the floe.
The scattered energy plots shown in Figure 36 for a 50-m floe of thickness 0.5 m, 1Fig.
m, 36
2 m, and 5 m assume the floe to be infinitely stiff. Comparing these results with those of
Figure 32, it is immediately apparent that a rigid floe scatters significantly more wave energy
than a flexible one, except where the floe is already sufficiently thick that any bending is
negligible, e.g. for the 5-m-thick floe. Note that these results are for 100-m waves. At
longer wavelengths, greater thicknesses would be required for a theory based on rigid ice
floes to be valid. Accordingly, for the archetypal wavelengths and floe sizes of the MIZ,
we conclude that floe bending is important and that a fully flexible theory is needed to
accurately determine the scattering functions.
Perhaps the most far-reaching calculation that can be done with the energy radiation
patterns of Figures 32—35 and the Kochin function H(θ) is the calculation of force, as
provided by equation (22). This is done in Figure 37, where the force due to a 1-m amplitude
Fig. 37
incident wave has been computed for 1-m-thick floes of 50-, 100-, 200-, and 400-m diameter.
The value of these results lies in the interpretation of field and laboratory experiments which
observe that an assembly of ice floes and cakes tends to segregate into accumulations of floes
such as bands or streamers, i.e., herding occurs. The current model allows this herding to
be studied theoretically. The curves appearing in Figure 37 are fairly predictable; again
the example provided is for 100-m-wavelength waves. Larger-diameter ice floes experience
greater horizontal drift force than smaller floes of the same thickness, and the force tends to
increase with thickness, although some fine structure occurs in the curves when appreciable
flexure is present. The 50-m ice floe bends very little because its diameter is much smaller
than the wavelength (see Figure 21). As a result, the force simply increases monotonically
with ice thickness. The most detail is seen in curves when more than one cycle of bending
is occurring across the width of the ice floe.
It is of some value to compare the force induced on a flexible ice floe, as shown in Figure 37,
with its counterpart when the floe is infinitely stiff, i.e. perfectly rigid. The latter is
illustrated in Figure 38. Two results are immediately apparent: first, when the floes are
Fig. 38
sufficiently thick, in this case at a thickness of around 5 m, the stiff and flexible results are
the same; and second, in the limit of no thickness the stiff and flexible results differ quite
markedly. Neither result is surprising. An ice floe that is perceived by waves of a particular
11
wavelength to be thick will be so stiff that any bending is negligible. On the other hand,
an ice floe which is perceived to be so thin that it easily deforms to the curvature of the
incoming wave will lead to zero force. A totally rigid ice floe, however, no matter how
thin, will always be subject to significant wave forcing. Because a thin floe cannot truly be
infinitely stiff, accurate estimates of force can be found only using a theory which permits
the floe to bend to the wave’s profile.
Geophysical Implications
The principal geophysical implication from this study, recognizing that it is an ingredient
of a larger program to study ocean wave propagation in marginal ice zones, is that the
flexure experienced by ice floes can be an important feature of the processes of scattering
and diffraction. Put another way, there is an additional potential due to the compliancy
of the ice over and above the diffracted and forced, i.e., heave, pitch, and surge, potentials
that affects Sice . This potential is most significant for floes of size comparable to, or larger
than, the wavelength of the incoming sea. Thus we would argue that the model of Masson
and LeBlond [1989] is inaccurate for all but small floes insofar as Sice but that its framework
and formulation still offer an outstanding vehicle to study wave growth and propagation in
MIZs into which an amended Sice incorporating floe bending can be embedded. While the
reader might question this as a geophysical conclusion, it is an important and far-reaching
result which has hitherto been suspected but not demonstrated. Moreover, it influences all
future second-generation MIZ models which include wave generation and action. Herein we
demonstrate its legitimacy.
A secondary implication of our study concerns the interpretation of strain measurements
carried out aboard solitary ice floes and icebergs. Although few such data sets exist, those
that do exist have proved to be very difficult to analyze and little has appeared in the
scientific journals. The illustrative examples of the deflections in Figures 21—24, and of
the principal strains and their directions in Figures 26—29, suggest why such data are so
problematical. In short, the flexural contortions of a floe in long-crested sea waves, with
its seldom simple three-dimensional geometry, are unlikely to be mapped to any degree of
understanding by a single strain gauge rosette placed at its approximate center unless the
floe is smaller than all the wavelengths present in the sea during the experiment. Indeed,
our simulations show that the maximum strains achieved during each wave cycle may even
be underestimated.
While the model reported in this paper computes only the wave-induced motions of solitary floes, the forces upon them, and the wave field in the waters around them, some
qualitative statements can be made about what will occur when many such floes coexist.
Here we stress that these ideas are qualitative, although the two-dimensional work of Meylan and Squire [1994] adds weight to our assertions as it suggests that floes may be quite
close before theory breaks down. There are three aspects to the curves of Figure 37 which
have relevance; their fine structure caused by the floe’s compliance, their dependence on
floe diameter, and their dependence on floe thickness. Very loosely speaking, the thickest,
largest floes experience the greatest force, which is quite reasonable physically, though this
is confused by the inherent resonances in the system. Although these resonant peaks would
in reality be damped by hysteresis during flexure, they do signal the possibility that floes
with favored geometries can behave quite differently from their neighbors. This apart, floes
of similar diameter experience similar forces, so one might expect zones of similar ice morphology to form within the MIZ. The absolute magnitude of the forces involved are large in
comparison to those due to the wind (approximately 3 × 103 N for a 50-m floe, 104 N for a
100-m floe, 5 × 104 N for a 200-m floe, and 2 × 105 N for a 400-m floe), because all the curves
in Figure 37 are referenced to a monochromatic 1-m amplitude, i.e., 2-m wave height, 100-m
length wave, which is rather extreme.
12
Conclusions and Summary
The theoretical model introduced in this paper describes the motion and flexure of a
circular disk in long-crested sea waves. It is proposed that this is a good analogy to a
solitary ice floe, perhaps located within the MIZ. The model has been used to investigate
several aspects of the response of ice floes of various diameters and thicknesses and to study
the wave patterns in the waters around the floe. Of significance, the force on single ice floes
has been computed. The following conclusions are evident:
1. A flexible circular ice floe responding to ocean waves may bend in a highly confused
manner, depending on its diameter and thickness relative to the spectral content of the
incoming long-crested sea. This is due to interaction between the two geometries which
underpin the motion, i.e., the linear and parallel nature of the wave fronts as opposed to the
circular shape of the floe.
2. Thin ice floes which deform easily to the sea surface curvature tend not to affect the
long-crested nature of the incoming waves, while thicker floes introduce distortion because
of their stiffness.
3. The wave field initiated in the water is strongly affected by the character of the bending
induced in the ice floe, as flexure of the floe initiates outgoing waves. This is particularly
important for floes which are of similar size or bigger than the wavelength of the incoming
waves. It suggests that the Sice of Masson and LeBlond [1989] should be adapted to take
the compliance of the ice into account when floes are not small.
4. The force on a circular floe or ice cake due to long-crested ocean waves generally
increases as floe diameter or thickness is increased. One application of these results is to
the herding of pancake ice in the Weddell Sea, for example. Ice pancakes are particularly
well suited to the current model because they are often very close to being perfectly circular
disks.
Acknowledgments. We are grateful to the University of Otago; the New Zealand Foundation
for Research, Science and Technology; and the Royal Society of New Zealand for their continued
financial support. Helpful comments and recommendations in the thorough review by Dr Diane
Masson are acknowledged. The paper was completed while V.A.S. was a guest of the Department
of Civil and Environmental Engineering, Clarkson University, Potsdam, New York, supported in
part by a grant from the National Science Foundation.
References
Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, 7th ed., Dover, Mineola,
New York, 1970.
Buchner, B., An evaluation and extension of the shallow draft diffraction theory, in Proceedings
of the 3rd International Offshore and Polar Engineering Conference, vol. 3, p. 230, Int. Soc. of
Offshore and Polar Eng., Golden, Colo., 1993.
Courant, R., and D. Hilbert, Methods of Mathematical Physics, 1, 358 pp., Wiley-Interscience, New
York, 1953.
Hasselmann, K., Grundgleichungen der Seegangsvoraussage, Schiffstechnik, 7, 191, 1960.
Isaacson, M. de St. Q., Fixed and floating axisymmetric structures in waves, J. Waterw. Port
Coastal Ocean Div. Am. Soc. Civ. Eng., 108, 180, 1982.
Itao, K., and S. H. Crandall, Natural modes and natural frequencies of uniform, circular, free-edge
plates, J. Appl. Mech., 46, 448, 1979.
John, F., On the motion of floating bodies, II, Simple harmonic motion, Commun. Pure Appl.
Math., 3, 45, 1950.
Liu, A. K., S. Häkkinen, and C. Y. Peng, Wave effects in ocean-ice interaction in the marginal ice
zone, J. Geophys. Res., , 98, 10,025, 1993.
Masson, D., Wave-induced drift force in the marginal ice zone, J. Phys. Oceanogr., 21, 3, 1991.
Masson, D., and P. H. LeBlond, Spectral evolution of wind-generated surface gravity waves in a
dispersed ice field, J. Fluid Mech., 202, 43, 1989.
Meylan, M., The motion of a floating flexible disk under wave action, in Proceedings of the 5th
International Offshore and Polar Engineering Conference, vol. 3, p. 450, Int. Soc. of Offshore and
13
Polar Eng., Golden, Colo., 1995.
Meylan, M., and V. A. Squire, Finite-floe wave reflection and transmission coefficients from a semiinfinite model, J. Geophys. Res., , 98, 12,537, 1993a.
Meylan, M., and V. A. Squire, A model for the motion and bending of an ice floe in ocean waves,
Int. J. Offshore Polar Eng., 3, 322, 1993b.
Meylan, M., and V. A. Squire, The response of ice floes to ocean waves, J. Geophys. Res., , 99,
891, 1994.
Newman, J. N., The drift force and moment on ships in waves, J. Ship Res., 11, 51, 1967.
Sarpkaya, T., and M. de St. Q. Isaacson, Mechanics of Wave Forces on Offshore Structures, Van
Nostrand Reinhold, New York, 1981.
Squire, V. A., Dynamics of ice floes in sea waves, J. Soc. Underwater Technol., 9, 20, 1983.
Squire, V. A., A comparison of the mass-loading and elastic plate models of an ice field, Cold Reg.
Sci. Technol., 21, 219, 1993.
Squire, V. A., Geophysical and oceanographic information in the marginal ice zone from ocean wave
measurements, J. Geophys. Res., , 100, 997, 1995a.
Squire, V. A., Reply, J. Geophys. Res., , 100, 8851, 1995b.
Squire, V. A., Engineering repercussions of ocean wave propagation in ice-infested seas, in Proceedings of the 5th International Offshore and Polar Engineering Conference, vol. 2, p. 1, Int. Soc. of
Offshore and Polar Eng., Golden, Colo., 1995c.
Squire, V. A., and S. Martin, A field study of the physical properties, response to swell, and
subsequent fracture of a single ice floe in the winter Bering Sea, Sci. Rep. 18, Dept. of Atmos.
Sci. and Oceanogr., Univ. of Wash., Seattle, 1980.
Squire, V. A., and M. Meylan, Changes to ocean wave spectra in a marginal ice zone 2, in Proceedings of the 4th International Offshore and Polar Engineering Conference, vol. 3, p. 142, Int. Soc.
of Offshore and Polar Eng., Golden, Colo., 1994.
Squire, V. A., and S. C. Moore, Direct measurement of the attenuation of ocean waves by pack ice
Nature, 283, 365, 1980.
Squire, V. A., J. P. Dugan, P. Wadhams, P. J. Rottier, and A. K. Liu, Of ocean waves and sea ice,
Annu. Rev. Fluid Mech., 27, 115, 1995.
Stoker, J. J., Water Waves: The Mathematical Theory With Applications, Wiley-Interscience, New
York, 1957.
Wadhams, P., V. A. Squire, J. A. Ewing, and R. W. Pascal, The effect of the marginal ice zone on
the directional wave spectrum of the ocean, J. Phys. Oceanogr., 16, 358, 1986.
Wadhams, P., V. A. Squire, D. J. Goodman, A. M. Cowan, and S. C. Moore, The attenuation of
ocean waves in the marginal ice zone, J. Geophys. Res., , 93, 6799, 1988.
Wehausen, J. V., and E. V. Laitone, Surface waves, in Encyclopedia of Physics, vol. IX, p. 446,
Springer-Verlag, New York, 1960.
M. H. Meylan and V. A. Squire, Department of Mathematics and Statistics, University of Otago, P.O.
Box 56, Dunedin, New Zealand.
(email: [email protected])
(Received December 15, 1994; revised October 24, 1995;
accepted October 30, 1995.)
Copyright 1996 by the American Geophysical Union.
Paper number 95JD030706.
0148-0227/96/95JC-03706$05.00
14
Figure 1. The coordinate frames for the problem.
Figure 1. The coordinate frames for the problem.
Figure 2. Three-dimensional plots of a deforming circular ice floe with diameter of 50 m and thickness of
(a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m. Wavelength is
100 m.
Figure 2. Three-dimensional plots of a deforming circular ice floe with diameter of 50 m and
thickness of (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m. Wavelength is 100 m.
Figure 3. As Figure 2 for a 100-m-diameter ice floe.
Figure 3. As Figure 2 for a 100-m-diameter ice floe.
Figure 4. As Figure 2 for a 200-m-diameter ice floe.
Figure 4. As Figure 2 for a 200-m-diameter ice floe.
Figure 5. As Figure 2 for a 400-m-diameter ice floe.
Figure 5. As Figure 2 for a 400-m-diameter ice floe.
Figure 6. Initiated wave patterns created by a 1m-thick ice floe with diameter of (a) 50 m, (b) 100 m,
(c) 200 m, and (d) 400 m in waves of length 100 m. The
subplots are created by subtracting out the (plane) wave
field which is causing the ice floe to move and flex.
Figure 6. Initiated wave patterns created by a 1-m-thick ice floe with diameter of (a) 50 m,
(b) 100 m, (c) 200 m, and (d) 400 m in waves of length 100 m. The subplots are created by
subtracting out the (plane) wave field which is causing the ice floe to move and flex.
Figure 7. Time series of principal strains and the angle
θ of the principal strain axes to the x-axis for a 50× 1 m
ice floe in 100-m-long ocean waves. Time series a1 and
a2 are located at the center of the floe; b1 and b2 are at
x̄ = 0.5, ȳ = 0; c1 and c2 are at x̄ = 0, ȳ = 0.5; and d1
and d2 are at x̄ = −0.5, ȳ = 0. The symbol T denotes
the period of the forcing.
Figure 7. Time series of principal strains and the angle θ of the principal strain axes to the
x-axis for a 50 × 1 m ice floe in 100-m-long ocean waves. Time series a1 and a2 are located at
the center of the floe; b1 and b2 are at x̄ = 0.5, ȳ = 0; c1 and c2 are at x̄ = 0, ȳ = 0.5; and d1
and d2 are at x̄ = −0.5, ȳ = 0. The symbol T denotes the period of the forcing.
Figure 8. As Figure 7 for a 100 × 1 m ice floe.
Figure 8. As Figure 7 for a 100 × 1 m ice floe.
Figure 9. As Figure 7 for a 200 × 1 m ice floe.
Figure 9. As Figure 7 for a 200 × 1 m ice floe.
15
Figure 10. As Figure 7 for a 400 × 1 m ice floe.
Figure 10. As Figure 7 for a 400 × 1 m ice floe.
Figure 11. The strain at the center of a floe due to 100m-long ocean waves, plotted for different floe diameters
against ice thickness.
Figure 11. The strain at the center of a floe due to 100-m-long ocean waves, plotted for different
floe diameters against ice thickness.
Figure 12. The surge response plotted as a function of
ice thickness for different floe diameters in 100-m-long
ocean waves.
Figure 12. The surge response plotted as a function of ice thickness for different floe diameters
in 100-m-long ocean waves.
Figure 13. The far-field initiated energy plotted as
a function of angle for 100-m-long waves interacting
with a 50-m-diameter ice floe. Ice thickness is (a) 0.5 m,
(b) 1 m, (c) 2 m, and (d) 5 m.
Figure 13. The far-field initiated energy plotted as a function of angle for 100-m-long waves
interacting with a 50-m-diameter ice floe. Ice thickness is (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m.
Figure 14. As Figure 13 for a 100-m-diameter ice floe.
Figure 14. As Figure 13 for a 100-m-diameter ice floe.
Figure 15. As Figure 13 for a 200-m-diameter ice floe.
Figure 15. As Figure 13 for a 200-m-diameter ice floe.
Figure 16. As Figure 13 for a 400-m-diameter ice floe.
Figure 16. As Figure 13 for a 400-m-diameter ice floe.
Figure 17. As Figure 13 for an infinitely stiff, i.e.,
perfectly rigid, ice floe.
Figure 17. As Figure 13 for an infinitely stiff, i.e., perfectly rigid, ice floe.
Figure 18. The time-averaged force exerted on ice floes
of various diameter, plotted as a function of thickness
in waves of amplitude 1 m and length 100 m.
Figure 18. The time-averaged force exerted on ice floes of various diameter, plotted as a function
of thickness in waves of amplitude 1 m and length 100 m.
Figure 19. As Figure 18 for an infinitely stiff, i.e.,
perfectly rigid, ice floe.
Figure 19. As Figure 18 for an infinitely stiff, i.e., perfectly rigid, ice floe.
16
Figure 1. The coordinate frames for the problem.
Figure 1. The coordinate frames for the problem.
Figure 2. Three-dimensional plots of a deforming circular ice floe with diameter of 50 m and
thickness of (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m. Wavelength is 100 m.
Figure 2. Three-dimensional plots of a deforming circular ice floe with diameter of 50 m and
thickness of (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m. Wavelength is 100 m.
Figure 3. As Figure 2 for a 100-m-diameter ice floe.
Figure 3. As Figure 2 for a 100-m-diameter ice floe.
Figure 4. As Figure 2 for a 200-m-diameter ice floe.
Figure 4. As Figure 2 for a 200-m-diameter ice floe.
Figure 5. As Figure 2 for a 400-m-diameter ice floe.
Figure 5. As Figure 2 for a 400-m-diameter ice floe.
Figure 6. Initiated wave patterns created by a 1-m-thick ice floe with diameter of (a) 50 m,
(b) 100 m, (c) 200 m, and (d) 400 m in waves of length 100 m. The subplots are created by
subtracting out the (plane) wave field which is causing the ice floe to move and flex.
Figure 6. Initiated wave patterns created by a 1-m-thick ice floe with diameter of (a) 50 m,
(b) 100 m, (c) 200 m, and (d) 400 m in waves of length 100 m. The subplots are created by
subtracting out the (plane) wave field which is causing the ice floe to move and flex.
Figure 7. Time series of principal strains and the angle θ of the principal strain axes to the
x-axis for a 50 × 1 m ice floe in 100-m-long ocean waves. Time series a1 and a2 are located at
the center of the floe; b1 and b2 are at x̄ = 0.5, ȳ = 0; c1 and c2 are at x̄ = 0, ȳ = 0.5; and d1
and d2 are at x̄ = −0.5, ȳ = 0. The symbol T denotes the period of the forcing.
Figure 7. Time series of principal strains and the angle θ of the principal strain axes to the
x-axis for a 50 × 1 m ice floe in 100-m-long ocean waves. Time series a1 and a2 are located at
the center of the floe; b1 and b2 are at x̄ = 0.5, ȳ = 0; c1 and c2 are at x̄ = 0, ȳ = 0.5; and d1
and d2 are at x̄ = −0.5, ȳ = 0. The symbol T denotes the period of the forcing.
Figure 8. As Figure 7 for a 100 × 1 m ice floe.
Figure 8. As Figure 7 for a 100 × 1 m ice floe.
Figure 9. As Figure 7 for a 200 × 1 m ice floe.
Figure 9. As Figure 7 for a 200 × 1 m ice floe.
17
Figure 10. As Figure 7 for a 400 × 1 m ice floe.
Figure 10. As Figure 7 for a 400 × 1 m ice floe.
Figure 11. The strain at the center of a floe due to 100-m-long ocean waves, plotted for different
floe diameters against ice thickness.
Figure 11. The strain at the center of a floe due to 100-m-long ocean waves, plotted for different
floe diameters against ice thickness.
Figure 12. The surge response plotted as a function of ice thickness for different floe diameters
in 100-m-long ocean waves.
Figure 12. The surge response plotted as a function of ice thickness for different floe diameters
in 100-m-long ocean waves.
Figure 13. The far-field initiated energy plotted as a function of angle for 100-m-long waves
interacting with a 50-m-diameter ice floe. Ice thickness is (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m.
Figure 13. The far-field initiated energy plotted as a function of angle for 100-m-long waves
interacting with a 50-m-diameter ice floe. Ice thickness is (a) 0.5 m, (b) 1 m, (c) 2 m, and (d) 5 m.
Figure 14. As Figure 13 for a 100-m-diameter ice floe.
Figure 14. As Figure 13 for a 100-m-diameter ice floe.
Figure 15. As Figure 13 for a 200-m-diameter ice floe.
Figure 15. As Figure 13 for a 200-m-diameter ice floe.
Figure 16. As Figure 13 for a 400-m-diameter ice floe.
Figure 16. As Figure 13 for a 400-m-diameter ice floe.
Figure 17. As Figure 13 for an infinitely stiff, i.e., perfectly rigid, ice floe.
Figure 17. As Figure 13 for an infinitely stiff, i.e., perfectly rigid, ice floe.
Figure 18. The time-averaged force exerted on ice floes of various diameter, plotted as a function
of thickness in waves of amplitude 1 m and length 100 m.
Figure 18. The time-averaged force exerted on ice floes of various diameter, plotted as a function
of thickness in waves of amplitude 1 m and length 100 m.
Figure 19. As Figure 18 for an infinitely stiff, i.e., perfectly rigid, ice floe.
Figure 19. As Figure 18 for an infinitely stiff, i.e., perfectly rigid, ice floe.
18
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MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
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MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
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MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES
MEYLAN AND SQUIRE: RESPONSE OF A CIRCULAR ICE FLOE TO OCEAN WAVES