Introduction
Interactions between sea ice and waves are a defining characteristic of the
marginal ice zone (MIZ), loosely defined as a region of the sea ice cover
adjacent to the ice edge and directly influenced by the neighboring open
ocean. In recent years, as the sea ice extent in polar and subpolar regions
of the Northern Hemisphere decreases and thick, multiyear ice is replaced
with thinner, weaker seasonal ice, conditions typical for MIZ (ice
concentration lower than 90 %, small floe sizes, patchy distribution of floes
on the sea surface, etc.) tend to occur over larger and larger areas. There
is a growing observational and modeling evidence that wave–ice interactions
play an important role in the observed expansion of MIZ and negative trends
in sea ice extent see,
e.g.,. Thin,
fragmented sea ice is susceptible to further breaking and, depending on
ambient weather and oceanic conditions, melting, which facilitates faster ice
drift, a decrease in ice concentration, and an increase in wind fetch and
thus creates more favorable conditions for wave propagation and generation,
leading to still stronger fragmentation. These – and many other – feedbacks
suggest that it is crucial to include (the effects of) wave–ice interactions
in numerical ocean–sea ice–atmosphere models in order to be able to
reliably reproduce the observed processes and forecast future changes on both
synoptic and climate scales. Parameterizations of wave–ice interactions for
large-scale continuum models (i.e., those in which ice is treated as a
continuous mass rather than as discrete particles) are crucial for further
development of those models. However, although appreciable effort has been
made in that direction in recent years
,
our understanding of many aspects of wave–ice interactions is still too
limited to allow formulating such parameterizations, especially those
suitable for a wide range of conditions. Strong fragmentation of the ice into
many small floes, and highly energetic environment due to the presence of
waves make the MIZ a very difficult, demanding location for field work. Due
to their low temporal resolution in polar regions, satellite data only
provide snapshots of sea ice conditions, making it difficult or impossible to
infer details of processes acting on timescales comparable with a typical
wave period. Therefore, in spite of recent advances in remote sensing
techniques to monitor waves in the MIZ e.g.,, the
amount of observational data necessary for validation of numerical models
remains very limited. Consequently, many crucial processes and their
large-scale effects are only poorly understood. As the overview of the
relevant literature in the following paragraphs clearly shows, one of them is
sea ice breaking by waves and the resulting floe-size distribution (FSD) –
the main subject of this paper.
Review papers by and provide a
good overview of the state-of-the-art research related to wave–ice
interactions. Problems studied in this context include, but are not limited
to, wave propagation, attenuation, and scattering by various ice types, e.g.,
continuous ice sheets, broken compact ice, (groups of) individual ice floes,
and inhomogeneities like pressure ridges, cracks, etc.; motion of ice floes
(and other floating objects, including very large floating structures) on
waves and wave-induced floe collisions; sea ice breaking by waves.
Considering relatively large amounts of literature on wave propagation in sea ice and
wave-induced motion of ice floes and sheets see, e.g.,and references
there, as this list is by far not
complete,
the number of studies on sea ice breaking by waves is remarkably limited and
– as aptly put it – they are to a large degree based on
“anecdotal evidence”. In a series of papers published in the 1980s, V. Squire
analyzed wave propagation in continuous land-fast ice and basic mechanisms
of wave-induced ice breaking, related to the presence of secondary
ice-coupled waves affecting the wave envelope close to the ice edge and
rapidly decaying away from the edge see, e.g.,.
In their review paper, describe qualitatively the process of
breaking of land-fast ice by swell waves, in which elongated, parallel strips
of ice are progressively separated from the initially continuous ice sheet.
They write that “the width of the strips, and hence the diameter of the
floes created by the process, is remarkably consistent and appears in the
sparse evidence available to be rather insensitive to the spectral structure
of the sea, but highly dependent on ice thickness.” Consistently, their
modeling results showed that the location of maximum flexural strain in the
ice relative to the ice edge depends mainly on ice thickness rather than wave
period. Notwithstanding these conclusions, a close relationship between the
incoming wavelength and floe sizes produced by breaking is usually assumed,
as for example in the above-mentioned parameterizations by
, , and others. It is worth
stressing that these models do not directly simulate the sea ice breaking
process. Instead, they simulate the effects of breaking by testing if
the conditions are favorable for breaking (criteria based on the wave height
and thus strain that the ice experiences) and, if these conditions are
fulfilled, by modifying the maximum floe size Lmax according to
certain prescribed rules. The shape of the FSD for floe sizes
Lo<Lmax is prescribed as well, so that Lmax is the
only variable parameter characterizing the FSD. In other words, these models
are suitable for analyzing the consequences of wave-induced breaking of sea
ice (i.e., the influence of the evolving FSD on ice dynamics and/or
thermodynamics) given the assumed relationships between the FSD and the wave
forcing. Thus, as with any parameterization, our understanding of the
processes involved decides upon the validity and accuracy of the modeling
results.
Since the pioneering works described above, few studies have been devoted
specifically to the analysis of sea ice breaking by waves. In a modeling
study of ice motion on waves, analyzed flexural strain
variability in ice floes of different sizes and thicknesses.
analyzed experimentally and numerically the fatigue
behavior of first-year sea ice subject to repeated bending stress and
demonstrated that the time history of strain acting on the ice is crucial for
predicting its breaking. In a subsequent work,
extended their earlier work to estimate lifetime of landfast ice subject to
waves with given characteristics. Based on ship observations of ice breaking
during a strong-wave event in the Barents Sea, analyzed
the role of nonlinear wave processes and the resulting strong modulation of
wave amplitude in ice breaking, in accordance with much earlier observations
and theoretical results of .
estimated ice breaking probabilities in the Arctic
sea ice in function of the distance from the ice edge based on the
probability density functions of the sea surface curvature. This approach,
employed also by , assumes a simple relationship
between strain (estimated directly from the shape of the wave profile) and
stress in the ice. Finally, sea ice breaking is included in the recent model
of wave–sea ice interactions by . In simulations of
wave propagation and multiple scattering by circular ice floes in MIZ, they
used strain-based floe breaking criteria and obtained approximately normal
FSDs without any a priori assumptions regarding their shape.
In this paper, a coupled sea ice–wave model is proposed suitable for
simulating ice–wave interactions in the time domain, including computation
of instantaneous stresses in ice and ice breaking. The model consists of a
bonded-particle discrete-element sea ice model, similar to that of
, and a wave model based on the code of the Non-Hydrostatic
WAVE (NHWAVE) model by . The two parts are
coupled with proper boundary conditions exchanged at every NHWAVE time step.
The type of a discrete-element model (DEM) used here, in which bonds
connecting grains behave as elastic “rods”, is particularly suitable for
studying sea ice–wave interactions due to oscillatory nature of these
processes, prohibiting inelastic effects from becoming significant
see, e.g.,.
Apart from providing a detailed description of the model, the main goal of
this work is, first, to analyze spatiotemporal variability of wave-induced
stress in ice floes with varying thickness and sizes and, second, to analyze
the time evolution of breaking and the final breaking patterns produced by
regular and irregular waves. The paper is structured as follows:
Sect. contains the definitions and assumptions underlying
the model, followed by the description of the model equations and coupling
between the wave and ice modules. The results of simulations are presented in
Sect. . Finally, Sect. provides a
discussion and a summary.
Model description
The model consists of two parts, the sea ice module and the wave module,
exchanging information at every time step. The wave part is based on
version 2.0 of the NHWAVE model developed by
and available at
https://sites.google.com/site/gangfma/nhwave. NHWAVE solves
three-dimensional incompressible Navier–Stokes equations in vertically scaled
σ coordinates (see further Sect. ). For the
purpose of this work, NHWAVE has been extended to allow non-free surface
boundary conditions under the (floating) ice, as described in detail further
in Sect. . The second component is a discrete-element
bonded-particle sea ice model. It is based on similar ideas and assumptions
as the DESIgn model by , with certain modifications crucial
for representing ice motion and bending on the oscillating sea surface (in
DESIgn, which is essentially two-dimensional in the horizontal plane, these
effects are treated in a very rudimentary way, with a number of unrealistic
assumptions).
Recently, and implemented in NHWAVE
equations for floating objects and other solid “obstacles”. Their method is
based on immersed boundary techniques ,
suitable for modeling interactions between fully or partially submerged solid
bodies (fixed or moving) and the surrounding fluid. The algorithms of
are not yet included in the publicly available version
of NHWAVE (although the code does contain basic treatment of fixed
obstacles); the present model, developed independently, shares many features
with their approach but, due to a number of assumptions related to the shape
and the characteristics of motion of the floating objects, it is much less
general, suitable for the specific configuration analyzed in this work. In contrast, the model of assumes that floating
objects are rigid bodies, making it unsuitable for an analysis of ice
deformation and breaking, which is crucial for the present study.
Definitions and assumptions
The model is two-dimensional in the xz plane. The waves are unidirectional
and propagate along the x axis; the z axis is directed vertically upward,
with z=0 at the mean water level. The sea ice is composed of discrete
elements (called grains) of cuboidal shape that are floating on the water
surface and may be bonded to their neighbors with elastic bonds. The grains
are rigid bodies, so that the deformation of the sea ice is accommodated only
by the bonds, which may break during the simulation if stresses acting on
them exceed their strength.
Sketch of the grid organization and spatial arrangement of variables
in the coupled wave–ice model, for the case of three constant-thickness
uppermost layers (Nl,ice=3) accommodating the ice “grains” (dashed
boxes). Crosses denote velocity points, while dots are the pressure points. Locations in
which the immersed-boundary forcing is applied are shown in red, while pressure
points affected by the boundary are in blue (note that, in accordance with the
immersed boundary method, the model equations are solved everywhere inside
the model domain, independently of ice being present in a given grid cell or
not). See text for more details.
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In the present version of the model it is assumed that the horizontal
resolution of the wave model, Δx, and the sizes of the grains are
adjusted; i.e., every one of the i=1,…,Nx grid cells of the wave model
is either ice-free or fully covered with ice (Fig. ). Let
us denote a set of indices of ice-covered cells as Ig.
All grain-related variables and equations referenced further are relevant for
i∈Ig. Similarly, as bonding is possible only between
grains occupying neighboring cells, we may define a set of bond indices
Ib so that i∈Ib if and only if
both i∈Ig and (i+1)∈Ig. (To
avoid renumbering of bonds during a simulation, broken bonds are not removed
from the list, but their strength is set to zero; see
Sect. .)
Top view (a) and side view (b) of two neighboring grains (white
rectangles) connected with a bond (blue rectangle). Relative translational
and angular velocity differences relevant for the present study are shown in
red (rotation in the xz plane and vertical displacement), while the remaining
velocity components are in gray. Red dots with labels “C” and “T” in panel (b) mark
the locations of maximum compressive and tensile stress, respectively, acting
on the bond when the relative rotation is directed as shown in
red.
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The grains have length 2li=Δx, thickness hi, and mass density
ρi (Fig. ). The model equations are formulated
for an ice “strip” with unit width in the y direction. The position of
the center of the ith grain is [xi,zi], and the deviation of its
orientation from the horizontal position due to rotation in the xz plane is
denoted with θi. The motion of the grains is described by the
translational velocity [ui,wi] and the angular velocity ωi. For
each grain, the center of mass and the center of rotation are assumed
identical, so that the off-diagonal elements of the mass and buoyancy
matrices vanish. For rotation within the xz plane, the moment of inertia
per unit grain width is Ig,i=ρilihi6(hi2+4li2). The
mass per unit grain width is mi=2ρilihi. The assumption regarding
the grains' positions relative to the wave model cells implies that
ui≡0 and xi is constant, which makes the model applicable only to
compact sea ice in which the drift and oscillatory surge motion is
insignificant. Obviously, this is true in a continuous, unbroken ice sheet;
in broken ice at high ice concentration, i.e., with densely packed floes,
horizontal motion is suppressed by collisions between neighboring floes.
These limitations will be relaxed in the future versions of the model.
All bonds are cuboid (Fig. ) and their geometric
properties are thickness hb,i and length
lb,i=λ(li+li+1)=λΔx, where λ∈(0,1] is a
coefficient deciding whether the elastic deformation is distributed across
the grains (λ=1) or limited to narrow zones at the grains' boundaries
(λ→0). As in the case of grains, it is assumed that the
bonds have unit widths in the y direction. Additionally, the bonds have the
following material properties: Young's modulus Eb,i, the ratio of the normal
to shear stiffness λns,i; tensile strength σt,br,i;
compressive strength σc,br,i; and shear strength τbr,i.
From this set of properties, the normal and shear stiffness can be
calculated: kn,i=Eb,i/lb,i and kt,i=kn,i/λns,i,
respectively. Finally, the relevant moments of inertia (again, per unit bond
width) are Ib,i=112hb,i3.
Due to the assumption of no motion along the x direction, no contact model
is necessary for neighboring grains that are not bonded to each other.
If surge is taken into account, repulsive contact forces between
touching grains should be implemented, e.g., the Hertzian model, as used
in.
In the vertical direction, the model domain is bounded by z=-H(x) and
z=η(x,t), where H(x) denotes the (time-independent) water depth and
η(x,t) denotes the instantaneous water surface elevation. The total
instantaneous water depth is D(x,t)=H(x)+η(x,t).
Equations and boundary conditions
Wave model
As already mentioned, the wave-related part of the model is based on NHWAVE.
Its full description can be found in ;
therefore, only a summary of the most important model features is given here.
NHWAVE solves incompressible, nonhydrostatic Navier–Stokes equations in a
three-dimensional domain, formulated in Cartesian horizontal coordinates and
boundary-following vertical σ coordinates, defined as
σ=(z+H)/(H+η)=(z+H)/D,
for z∈[-H(x),η(x,t)]. In the xz space, in which the present coupled
ice–wave model is formulated, the governing equations are the mass and
momentum conservation equations:
∂D∂t+∂(Du)∂x+∂ω∂σ=0,∂(Du)∂t+∂(Du2+12gD2)∂x+∂(Duω)∂σ=gD∂H∂x-Dρ∂p∂x+∂p∂σ∂σ∂x+DSτx,∂(Dw)∂t+∂(Duw)∂x+∂(Dwω)∂σ=-1ρ∂p∂σ+DSτz,
where g denotes acceleration due to gravity, p is the dynamic pressure,
u and w are water velocity components in x and z direction,
respectively, ω is the velocity component perpendicular to the
σ surfaces, and Sτx and Sτz are turbulent diffusion
terms, assumed equal to zero in the present work. The free surface is
obtained explicitly from the vertically integrated continuity Eq. (). To close the system of Eqs. ()–() are supplemented by the Poisson
equation for pressure .
At the bottom, z=-H, the kinematic and free-slip boundary conditions for
velocity, and the Neumann boundary condition for pressure are
w=-u∂H∂x,∂u∂σ=0,∂p∂σ=-ρDdwdt.
Boundary conditions at the free surface, z=η, not covered with ice are
w=∂η∂t+u∂η∂x,∂u∂σ=0,p=0.
In the model applications presented in this work, sponge layers are applied
at the left and right boundary, and waves are generated inside the model
domain with a so-called internal-wavemaker technique, in which a source term
is added to the model equations at the wave generation location, and the
waves propagate out of this location in both directions .
Sea ice model
The sea-ice-related part of the model can be formulated as a set of the
following ordinary differential equations:
dθidt=ωi,i∈Ig,dzidt=wi,i∈Ig,Ig,idωidt=Mwv,i+Mb,i-Mb,i-1+li(Ft,i-Ft,i-1),i∈Ig,midwidt=Fwv,i+Fz,i-Fz,i-1,i∈Ig,dMb,idt=-kn,iIb,i(ωi-ωi+1),i∈Ib,
dFt,idt=kt,ihb,ivt,i,i∈Ib,dFz,idt=kn,ihb,ivz,i,i∈Ib.
Equations () and () are definitions of the angular
and translational velocities of the grains, respectively. The
angular-momentum Eq. () describes changes of ωi due
to moments of forces acting on the grains. Analogously, the linear-momentum
Eq. () describes changes of the vertical velocity wi due
to forces acting on the grains. The terms on the right-hand side of
Eqs. () and () can be calculated from the remaining
Eqs. ()–(). As in all DEMs, the bonds
transmit both torques and forces. Relevant in the present configuration are
bending moments Mb,i, resulting from the relative rotation (rolling) of
the bonded grains in the xz plane; torques liFt,i acting on the grain
boundaries due to tangential forces resulting from translational shear
displacement of the grains (with velocity vt,i); and the vertical
component of the sum of normal and tangential forces, Fz,i, resulting
from relative displacement of the grains (with vertical velocity vz,i).
As can be seen, in Eqs. ()–() linear relationships
between displacement and force are assumed, which is typical for DEM models
(see , and, for a detailed algorithm for calculating the
displacements and forces in a fully 3-D case, , and
). Finally, the first terms on the right-hand side of
Eqs. () and () denote the net moment of forces and the
net vertical force, respectively, from the wave motion underneath the ice.
They are calculated by integrating the contribution from waves over the
wetted surface of the grains. Their detailed formulation is given further in
Sect. .
Note that, in a general case, although the value of Ft,i characterizes
the bond connecting two neighboring grains, the torque related to this force
acting on these grains would be different if li≠li+1. Note also
that the horizontal component of the normal and tangential forces would be
relevant only for horizontal displacements of the grains, which are not taken
into account here.
As noted earlier, all forces and moments are formulated for a unit width of grains and bonds.
The stresses acting on bonds are calculated according to the classical beam theory, so that
τi=|Ft,i|hb,i,i∈Ib,σc,i=Fn,ihb,i+|Mb,i|hb,iIb,i,i∈Ib,σt,i=-Fn,ihb,i+|Mb,i|hb,iIb,i,i∈Ib,
where Fn,i denotes the normal force (i.e., along the bond length). The
stresses are evaluated for every bond at every model time step. If at least
one of the three stress components exceeds the bond strength, i.e., if
τi>τbr,i or σc,i>σc,br,i or
σt,i>σt,br,i, the bond breaks. In bonded-particle models
this is typically achieved by instantaneously setting the Young's modulus, as
well as the forces and moments transmitted by this bond, to zero. This
approach, based on an assumption that breaking happens infinitely fast, is
well known to produce too-brittle behavior, unrealistic in many materials.
Some models therefore introduce a softening mechanism, ensuring that stress
in broken bonds drops gradually instead of instantaneously see,
e.g.,. In the present model, breaking is extended in
time by assuming that stresses acting on a bond that undergoes breaking drop
to zero gradually over a certain time tbr. Numerical tests showed that
tbr∼0.1 s is enough to remove spurious effects associated with
instantaneous breaking. The influence of tbr on the model behavior is
demonstrated in Sect. .
Sea ice–wave coupling
In the present model, the discretization of the model domain in the vertical
direction is modified so that a prescribed number Nl,ice out the total
of Nl layers is used to accommodate the ice (Fig. ).
That is, the uppermost Nl,ice layers have a constant thickness equal to
hf/Nl,ice, where hf denotes the draft of the ice. The remaining
Nl-Nl,ice layers are divided uniformly from the bottom, z=-H(x) to
z=η(x,t)-hf. Thus, the thickness of the upper model layers does not
vary in time and at each time step the ice grains' boundaries coincide with
boundaries of the cells of the wave model. This fact significantly simplifies
the formulation of boundary conditions along the horizontal and vertical ice
surfaces. At the lower surface of the ice we have
w=wi,∂u∂σ=0,1D∂p∂σ=-ρdwidt.
Analogously, at the vertical ice surfaces,
u=ui,∂w∂x=0,∂p∂x=-ρ∂ui∂t.
(Note that ui=0 in the present model version.) As can be seen, a free-slip
condition is assumed for velocity components tangential to the ice surface.
In the immersed-boundary method, the influence of the ice on the surrounding
water is taken into account by adding an additional forcing term Fice to
the momentum equations at the second step of the two-step second-order
Runge–Kutta scheme, used in NHWAVE to numerically integrate the governing
equations . By definition, Fice≠0 only
along the boundaries of floating and submerged objects (points marked with red
crosses in Fig. ). Details of the formulation of this
force can be found in and in references cited there. Linear
interpolation of velocities close to ice boundaries is used, as recommended
by and .
To close the wave–ice interaction problem, the forcing from water to the ice
has to be passed to the ice model. This forcing can be obtained by
integrating the dynamic pressure p over the surface area of an submerged
object. Due to the specific geometry and assumptions described in previous
sections, the formulation of this forcing is relatively straightforward. As
the horizontal motion of the grains is not taken into account and the tilt of
the grains is likely to remain small (so that sinθi is close to zero
and cosθi close to one), contribution of pressure force and momentum
acting on the vertical surfaces of end grains can be omitted. Thus, the
moment Mwv,i used in () and the vertical component of the
wave-induced force Fwv,i in () are
Mwv,i=∫xi-lixi+lip(l)ni×ridl,i∈Ig,Fwv,i=cosθi∫xi-lixi+lip(l)dl,i∈Ig,
where l denotes distance along the lower grain surface,
ni=[-sinθi,cosθi] is a unit vector normal to that
surface, and ri is a vector of length l tangential to it.
Assuming linear variability of pressure between pi-1 and pi, as
well as between pi and pi+1, it is straightforward to evaluate the
integrals in Eqs. () and () to obtain
Mwv,i=li33Δx(pi+1-pi-1),i=1,…,Ng,Fwv,i=2lipi+2li8Δx(pi+1-2pi+pi-1)cosθi,i=1,…,Ng.
Simplified flowchart of the coupled wave–ice
model.
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Numerical implementation
The code of the sea ice model is written as an additional module included in
NHWAVE. A simplified flowchart of the coupled model is shown in
Fig. . Due to more strict stability requirements of the
sea ice part of the model, it is solved with a shorter time step Δtice=γtΔtwave, with γt<1. In
simulations presented in this paper, γt=1/150 was used. The time step
of the ice model is limited by the grain size used and by mechanical ice
properties, with more stiff ice (higher Eb) requiring smaller Δtice.
Model parameters used in the simulations in Sect. .
Variable
Value
Constant parameters
Water depth H
10 m
Basin length
1500 m
Horizontal grid size Δx
0.5 m
Number of σ layers Nl
30
Number of “ice layers” Nl,ice
max{3,3hi}
Width of sponge layers
125 m
Internal-wavemaker location
290 m
Bond length parameter λ
0.5
Normal to shear stiffness ratio λns
1.5
Young's modulus Eb
1.0×109 Pa
Time step ratio γt
150
Wave amplitude a
0.025 m
Variable parameters
Floe length Lo
5–500 m
Ice thickness hi
0.3–3.0 m
Open-water wavelength Lw,0
25–84 m
Bond tensile strength σt,br
1500–3000 Pa
(∞ in simulations
without breaking)
Results
Model setup
In this section, the model is applied to a series of simulations in which a
single ice floe with a given thickness hi and length Lo is moving on
waves with a given open-water wavelength Lw,0. A summary of the model
setting is given in Table . The water depth is constant
H=10 m, and the water column is divided into Nl=30 layers. The number of
“ice layers” Nl,ice depends on the ice thickness but is never lower
than 3. The horizontal resolution of the model, i.e., the cell size of the
wave model Δx and the horizontal dimensions of the grains 2li,
equals 0.5 m. Preliminary simulations with the stand-alone NHWAVE model were
performed to verify whether Δx is sufficiently small and Nl
sufficiently large to reproduce the shortest waves considered with
satisfactory accuracy. The results showed that for the whole range of
wavelengths analyzed, no significant loss of energy during propagation was
observed. Thus, the attenuation present in the results described further in
Sect. and originates in the
sea ice module due to damping in the bonds, which are not perfectly elastic.
This undoubtedly is an undesired property of the numerical scheme used in the
sea ice module; however, it has been shown in tests with artificially
modified damping in bonds that it does not influence the results in terms of
the floe sizes obtained; see Sect. for a discussion.
It is also worth stressing that – as in all DEM models – in simulations
that are designed to reproduce the behavior and macro-properties of any
particular specimen of a brittle material (its strength, elastic modulus, and
so on), the microscopic properties of grains and bonds have to be carefully
calibrated see, e.g.,for examples of how the coefficients of the bond
and contact models are calibrated in order to take into account their
dependence on grain size. As the results
presented here are not calibrated to any real-world case, this issue is not
further investigated. For realistic applications of the model, its parameters
(λ, λns, Eb, σt,br, and so on) can be adjusted
to obtain desired macroscopic sea ice properties.
In the simulations described in Sect.
and , a number of combinations of hi, Lo, and
Lw,0 are considered, with the range of values 0.3–3.0, 5–500 m, and
25–84 m, respectively. For H=10 m, the range of Lw,0 corresponds to
wave periods between 4.04 and 9.19 s and to kH values between 2.5 and 0.75
(where k denotes the wave number). The thickness of both grains and bonds
is identical.
The simulations were performed first without ice breaking in order to analyze
the spatiotemporal variability of stress in the ice, as described in
Sect. . Subsequently, the bonds' strength was reduced
to a number of values to study ice breaking pattern, analyzed in
Sect. .
Stress variability in continuous ice
During the motion of the modeled ice floe on waves, the bonds undergo
tensile, compressive, and shear stress related to the relative displacement
and rotation of neighboring grains. In the simulations described here, the
compressive and tensile stresses had comparable amplitudes, whereas the shear
stress was 2–3 orders of magnitude lower. All bond breaking events in
simulations from Sect. happened due to tensile
failure and therefore σt,i is analyzed here as the most relevant
stress component.
Simulated space–time variability of the ice vertical displacement
zi (a) in centimeters and the tensile stress σt (b) in Pa for an ice floe
with length Lo=500 m, ice thickness hi=0.5 m, and open-water wavelength
Lw,0=42 m. Lower diagrams show the amplitude of zi and σt in
function of the distance from the ice edge. Magenta dot in panel (b) marks
σt,max.
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Simulated maximum tensile stress σt,max (a, c) and
location (distance from the up-wave ice edge) at which it occurs (b, d) for
different ice thickness hi and floe length Lo values; open-water
wavelength is Lw,0=42 m. Note that the x axis in panel (c, d) is
logarithmic.
![]()
Figure shows the vertical displacement of the ice
and the tensile stress acting on bonds in function of time and distance from
the ice edge. As can be seen in the diagrams, the amplitude of stress acting
on bonds increases from zero at the ice edge (where the amplitude of zi is
largest) towards a maximum value σt,max at a certain
distance from the ice edge (see the pink dot in the lower plot in
Fig. b). Figure a, c show the
value of σt,max for different combinations of ice
thickness and floe lengths; the location of the stress maximum (measured
relative to the ice edge) is shown in Fig. b, d. For a
given ice thickness, the value of σt,max increases with
increasing floe size, as the floes' response changes from rigid motion (very
small floes) to flexural motion (larger floes). Up from a certain floe size,
equal to between one and two wavelengths, no further increase of
σt,max is observed, i.e., the stress saturates to a value
specific for a given ice thickness. For a given floe length, the influence of
ice thickness on σt,max is less trivial: there is a
certain value of hi for which σt,max reaches the
highest value, and for larger floes this maximum
(Fig. a) shifts towards thicker ice. The reason for the
drop of stress in very thick ice is that a lot of wave energy is reflected at
the ice edge, leading to lower amplitudes within the ice itself. Moreover,
thick floes are more rigid, with reduced strain and thus lower stress levels.
For very small floes, σt,max occurs in the middle of the
floe and thus its location is independent of ice thickness; for larger floes,
location of σt,max moves further from the ice edge with increasing
ice thickness (Fig. d). For a given ice thickness,
the location of σt,max moves away from the ice edge with
increasing floe size (Fig. b).
Simulated maximum tensile stress σt,max (a, c) and
location (distance from the up-wave ice edge) at which it occurs (b, d) for
different floe length Lo and open-water wavelength Lw,0 values; ice
thickness is hi=0.5 m (a, b) and hi=1.0 m (c, d). Note that the x axis is
logarithmic.
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Apart from the ice properties, the value and location of
σt,max are influenced by the characteristics of the
incoming waves, as shown in Fig. for two selected ice
thicknesses and for a range of floe lengths. For a given open-water
wavelength Lw,0, σt,max increases with increasing
floe length up to a certain “saturation” value
(Fig. a, c). In contrast, for large floes there is a
certain open-water wavelength producing maximum tensile stress (assuming the
same incident wave amplitude). Again, this is related to both wave reflection
at the ice edge and the response of the ice itself. For very short waves,
strong reflection leads to lower wave amplitude within the ice; for very long
waves, however, reflection and damping within the ice are weaker,
but the wave steepness is small as well, leading to less intense flexural
motion of the ice see also. Most importantly, the
location of σt,max is almost independent of the incoming
wavelength (Fig. b, d; note that the size of the grains,
and thus the effective resolution of the model, equals 0.5 m, so that the
differences seen in the figures, especially in the case of hi=0.5 m,
amount to just two–three grains).
Amplitude of the tensile stress σt in function of the
distance from the up-wave ice edge for different floe length Lo; open-water
wavelength is Lw,0=42 m, and ice thickness is hi=0.5 m. The plot in panel (b) is a
close-up of the region to the left of the dashed line in panel
(a).
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For large floes, a few stress maxima with decreasing amplitude can be
observed behind the main one, as shown in Fig. .
Sufficiently far from the floe edge, the stress amplitude decreases
gradually, depending on the damping rate (which depends on ice thickness and
wave characteristics; see also Fig. and
Sect. for the discussion on the sources of damping in
the present model version). At the rear side of large floes, small-amplitude
ripples are observed before the stress drops to zero – similar increase of
the amplitude of the vertical motion of elastic plates at their down-wave ends
has been observed and modeled, e.g., by and
. As already mentioned, small floes (Lo<Lw,0/2) have
only one stress maximum, as they undergo bending around their symmetry axis
(Fig. b). As the floe size exceeds Lw,0/2, the
symmetry gradually vanishes and the second maximum appears when Lo is
close to Lw,0.
Simulated space–time variability of the tensile stress σt
(Pa) for an ice floe with length Lo=500 m undergoing progressive breaking;
ice thickness is hi=0.5 m, open-water wavelength is Lw,0=42 m, and bond
strength 2500 Pa. The plot in panel (b) is a subset of that in panel (a); breaking events
are marked with magenta dots and broken bonds with dashed
lines.
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Breaking of uniform ice by regular waves
The spatiotemporal variability of tensile stress in the ice, described above,
is crucial for the evolution of ice breaking and the resulting FSD. Figure illustrates how breaking of a
large floe (Lo=500 m) progresses from the ice edge deeper and deeper into
the ice, producing small floes with lengths comparable to the distance of
σt,max to the ice edge. An individual wave is
“responsible” for a few breaking events (between one and three in the case
shown in Fig. ; up to five in other analyzed cases)
and thus produces a few new ice floes. In thinner ice, the number of new
cracks per wave period tends to be larger, i.e., breaking progresses into the
ice faster than in stronger, thicker ice. Moreover, as can be expected, the
final width of the zone of broken ice is dependent on ice strength as well and,
in the cases analyzed, increases roughly linearly with decreasing bond
strength (not shown). The resulting breaking pattern is not perfectly
regular, but the FSD is very narrow. In the simulation
presented in Fig. , in which the distance of
σt,max from the ice edge equaled 8 m (yellow curve in
Fig. b), only four floe sizes were obtained, 6.5, 7.0,
7.5, and 8.0 m, with the mode of the distribution at 7.0 m. Generally, the
location of σt,max appears to constitute an upper bound on
the size of floes detached from the edge of continuous ice, and breaking
takes place not farther than a few grains in front of that limiting location.
Amplitude of the tensile stress σt (a) and vertical ice
displacement (b) in simulations without and with ice breaking. Floe length
is Lo=500 m, ice thickness is hi=0.5 m, open-water wavelength is Lw,0=42 m,
and bond strength is 2500 Pa.
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Once the small floes break off the receding ice edge, they begin to move as
almost-rigid bodies, changing their vertical position and rotating around
their symmetry axis (Fig. ). In the present
model, in which the horizontal component of ice motion is not included,
neighboring grains do not interact with each other if they are not bonded.
Thus, a very important mechanism of wave-energy attenuation is not taken into
account: floe–floe collisions. Consequently, the model produces lower
attenuation rates in broken ice than in the initial continuous ice sheet
(Fig. b). This behavior is fully consistent
with the model assumptions, but not realistic. As a result, the width of the
zone of broken ice is likely overestimated in the present model version.
However, this drawback hardly influences the overall breaking patterns, as
they are very robust to changes of the model configuration. As an example,
Fig. shows the results of a simulation analogous to that
presented in Fig. , but with incoming waves with a
Jonswap energy spectrum (one of widely used idealized models of wave-energy
spectra, suitable for a wide range of wind and fetch conditions). As can be
seen, even though the waves are irregular and breaking takes places in short
episodes (associated with wave groups) separated by quieter periods without
formation of new cracks, the final FSD is as regular as
that produced by sine waves. Another important mechanism not taken into
account in the present version of the model is multiple wave scattering by
small ice floes detached from the ice edge. As have
recently shown, scattering may lead to both destructive and constructive
interference, thus contributing to local decrease or increase of the wave
amplitude and strain of the ice. The net effects of these processes on the
wave attenuation rates and ice breaking patterns are hard to estimate and
presumably sensitive to the details of any particular configuration. (Note
that the present model is capable of simulating multiple scattering but not
in the configuration used here, in which the grains of the sea ice module
occupy full cells of the wave module, so that no water–ice boundary
conditions are applied at the vertical walls of neighboring grains.)
As in Fig. but for irregular incoming
waves with Jonswap energy spectrum (wave height and peak period corresponding
to those of sine waves used in simulation from
Fig. ).
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Finally, it is worth noticing that the regular floe pattern described above
is obtained only in simulations in which the “delayed” bond breaking
mechanism, described at the end of Sect. , was
activated. Figure compares the results of
two similar simulations, one with instantaneous and one with “delayed” bond
breaking. If breaking is instantaneous, a sudden drop to zero of all stress
components at the broken location produces short-wave disturbance propagating
out of this location in both directions
(Fig. b). The excess stress related to that
disturbance, combined with stress induced by the propagating wave, leads to
rapid bond breaking in neighborhood of the initial breakage, producing very
small ice floes, typically two–three grains in size (compare
Figs. a to b).
If, to the contrary, the drop of stress during bond breaking is extended over
a time period of just less than 0.1 s, it is sufficient to suppress the
amplitude of the breaking-induced disturbance to insignificant levels
(Fig. c). Consequently, no additional
breaking takes place around the initial crack.
Comparison of the model behavior in simulations with instantaneous
and “delayed” bond breaking: space–time variability of the tensile stress
σt (Pa) in a simulation analogous to that shown in
Fig. but with instantaneous bond breaking (a); and
details of σt in the vicinity of a selected breaking event from a
simulation with instantaneous (b) and “delayed” (c) bond breaking. The
curves in panels (b, c) show σt along a selected fragment of the ice floe
before (blue) and shortly after (red and yellow) breaking; dashed black lines
mark the location where breaking took
place.
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Discussion and conclusions
In this paper, a coupled wave–ice model was used to analyze wave-induced
stress in sea ice and the resulting patterns of sea ice breaking. The most
important results can be summarized as follows: (i) breaking of a continuous
ice sheet by waves produces floes of almost equal sizes, dependent on the
thickness and strength of the ice, but not on the characteristics of the incoming
waves. (ii) This breaking pattern results from the fact that maximum tensile
stress experienced by the ice is located at a distance from the ice edge that
does not depend on incoming wavelength. (iii) The incoming wave
characteristics, together with ice properties, control the value of the
maximum stress, thus deciding whether breaking takes place or the ice remains
intact. (iv) For a given floe size, there exist ice thickness and incident
wave length for which the stress reaches maximum and thus breaking is most
likely to occur.
As no attempt at calibrating the model against observational data was made,
the numbers obtained as a result of the simulations might be unrealistic.
Also, as has been already mentioned in the previous section, there are a
number of mechanisms of wave-energy dissipation that are not included in the
present version of the model (floe–floe collisions, ice–water friction,
multiple scattering by the floes already broken off the ice edge, etc.).
However, these facts do not affect the general conclusions formulated above.
The present results agree with the findings of , described in
the introduction, and provide another evidence – obtained with a very
different model than that of Squire and colleagues – in favor of the
hypothesis that it is the ice itself (its thickness and strength) and not the
incident waves that decide upon the dominating floe size in MIZ, at least
during the initial stages of ice breaking at later stages, many other
factors lead to further fragmentation of ice floes, producing wide,
heavy-tailed FSDs typically observed in inner parts of
MIZ; see, e.g.,and references there. In particular, it
is worth stressing that in terms of the floe size resulting from breaking,
the results are not sensitive to the modeled attenuation rates of wave
energy (which, as already mentioned in Sect. , has been
demonstrated in model runs with artificially modified damping in bonds
connecting grains). Breaking takes place within a narrow zone of enhanced
strain close to the edge of the yet unbroken ice. Again, this is consistent
with the observational and modeling results of and , who found that breaking is likely only within a region where the secondary
ice-coupled waves contribute to the increased vertical deflection and thus
strain of the ice. The amplitude of these waves decays very fast with the
distance from the ice edge. Consequently, the probability of breaking
decreases as well, independently of the attenuation coefficient of the
gradually decaying propagating wave.
If further research confirms these results, it will have important
consequences for formulating parameterizations of wave–ice interactions for
large-scale sea ice models, so that the information on incoming waves
(especially wave steepness) is used to determine whether breaking of ice
takes place, but the maximum floe size Lmax is estimated based on
ice properties themselves. Note that, as already mentioned in the
introduction, in most parameterizations Lmax is the only variable
parameter describing the FSD; the shape of the FSD for Lo<Lmax
is assumed to be a power law with a prescribed exponent. Note also that
besides bending, a number of other wave-related processes may contribute to
floe breaking and thus to shaping the FSD, including floe–floe collisions,
overwash, and rafting. These processes are dependent on wave steepness and
thus amplitude; presumably, they modify the slope of the FSD, although no
observational data exist that would allow to formulate this dependence as a
functional relationship.
The model presented in this paper is undergoing further development as part
of a research project currently in progress. In the new version, horizontal
ice motion and ice contact mechanics will be implemented by adapting
algorithms from the DESIgn model; see, enabling us to run the model
to study floe–floe collisions and situations with significant drift and/or
surge motion of ice. At later stages of the project, it is planned to extend
the model to two horizontal dimensions (the NHWAVE model is
three-dimensional, and significant parts of the sea ice module have already
been coded for two horizontal dimensions as well). This will make it possible
to analyze how the directional width of the energy spectra of incoming
waves, as well as the angle between the wave propagation direction and the
ice edge affect the results obtained in this study. It is also worth noticing
that the code of the model can be easily extended by, e.g., replacing the
free-slip boundary conditions for velocity at the wetted surface of the ice
with other types of boundary conditions, or by including wind or other
processes already implemented in NHWAVE. It should be stressed that NHWAVE is
a very general hydrodynamic model that can be applied to a wide range of
conditions: it does not make any assumptions regarding the irrotationality of
the flow (as many sea ice–wave interaction models do) or the type of the
wave forcing. Although in the computations presented in this paper the water
depth was relatively shallow (H=10 m), deepwater waves can be simulated
without significant increase in computational costs, because the model
enables non-equally spaced σ layers, with thickness adjusted to the
vertical structure of the wave. The model also accepts a number of types of
boundary conditions, handles drying and flooding of grid cells, etc. All
these functionalities can be used in coupled wave–ice simulations, making it
a very flexible tool suitable for a wide range of conditions. A serious
limitation, however, are very high computational costs of this modeling
approach. This makes the model suitable for analyzing details of selected
processes – like in this paper – rather than for practically oriented
applications in sea ice and wave hindcasting and forecasting.