TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-11-1035-2017Brief communication: Impacts of ocean-wave-induced breakup of
Antarctic sea ice via thermodynamics in a stand-alone version of the CICE sea-ice modelBennettsLuke G.luke.bennetts@adelaide.edu.auhttps://orcid.org/0000-0001-9386-7882O'FarrellSiobhanUotilaPetterihttps://orcid.org/0000-0002-2939-7561School of Mathematical Sciences, University of Adelaide, Adelaide, SA, AustraliaCSIRO Ocean and Atmosphere, Aspendale, VIC, AustraliaFinnish Meteorological Institute, Helsinki, FinlandLuke G. Bennetts (luke.bennetts@adelaide.edu.au)3May20171131035104024November201619December201617March201731March2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/11/1035/2017/tc-11-1035-2017.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/11/1035/2017/tc-11-1035-2017.pdf
Impacts of wave-induced breakup of Antarctic sea ice on ice concentration and
volume are investigated using a modified version of the CICE sea-ice model,
run in stand-alone mode from 1979–2010. Model outputs show that during
summer wave-induced breakup
reduces local ice concentration by up to 0.3–0.4 in a vicinity of the ice
edge and total ice volume by up to a factor of 0.1–0.2.
Introduction
Speculation surrounding the impacts of ocean surface waves on sea
ice is building. In the Antarctic,
the speculation has been fuelled by findings from that
trends in ice-edge contraction (from satellite observations) are closely
correlated to trends in increasing local significant wave heights (from a
numerical model) and, conversely, trends in ice-edge expansion are correlated
to trends in decreasing significant wave heights. They attributed these
correlations to large-amplitude storm waves propagating into the ice-covered
ocean and breaking up the ice cover into relatively small floes, which are
more mobile and vulnerable to melting. This relationship can be inferred from
descriptions of the way in which waves regulate the morphology of the ice
cover in the first 10 to 100 s of kilometres in from the ice edge,
originally made by Squire, Wadhams and co-workers in the 1970s see,
for example, the review by – a region often referred to as
the marginal ice zone, although the term is not adopted in this study due to
ambiguity in its definition. suggested that incorporating
wave impacts on sea ice into climate models will empower the models to
capture sea-ice responses to climate change, for example, the regional
variability of trends in Antarctic sea-ice extent .
This study constitutes the first quantification of Antarctic sea-ice breakup by waves on ice concentration and volume.
It uses a stand-alone version of the CICE sea-ice model, modified to include wave-induced breakup,
with wave forcing provided by a Wavewatch III wave-model hindcast in ice-free grid cells close to the ice edge.
Wave energy advects into cells containing ice cover,
where models of wave-energy attenuation due to ice cover and wave-induced ice breakup are applied,
in a similar manner to the operational ice–ocean model wave–ice interaction component developed by .
CICE v4.1 is used for the study, in which floe diameters
appear in the lateral ice-melt model only, and are set to be 300 m
throughout the ice cover by default. Breakup reduces mean floe diameters
typically to 20–100 m in cells extending ∼ 100 km in from the ice
edge, beyond which the wave energy is no longer strong enough to break the
ice. When ocean temperatures are high enough to melt ice, the reduced
diameters promote lateral melt, reducing the ice concentration, which, in
turn, reduces the ice strength, so that breakup indirectly impacts both ice
concentration and volume through dynamic processes. Model outputs show that
during the summer wave-induced breakup reduces local ice concentration by up
to 0.3–0.4 and total ice volume by up to a factor of 0.1–0.2. During the
winter, the ice concentration recovers, but volume changes persist, becoming
dispersed over the inner ice pack.
Model
CICE uses an ice-thickness-distribution function g(xij,t:h)
to describe the sea-ice cover, in which xij denotes a grid cell
on the ocean surface, indexed i in longitude and j in latitude, t
denotes time and h denotes ice thickness, with g(xij,t:h)dh the fractional area of ice in cell-ij with thickness in the
interval (h,h+dh). The ice-thickness distribution is calculated as
a numerical approximation of the ice-thickness-evolution equation
∂g∂t=-∇⋅(gu)-∂∂h(fg)+ψ,
using discrete time steps with a nominal global time step Δt=1 h, a horizontal tripolar grid with a nominal resolution of one
latitudinal/longitudinal degree, and partitioning of the ice into discrete
thickness categories (five categories plus open water are used for this
study, as standard). The first term on the right-hand side of
Eq. () denotes ice advection, where u is ice
velocity, calculated via the elastic–viscous–plastic (EVP) rheology model
of . The second term denotes thermodynamic thickness
redistribution, where f is the rate of melting or freezing. The final term,
ψ, denotes mechanical redistribution due to ridging.
Waves are introduced into the model using the wave-energy-density spectrum,
S(xij,t:ω,θ), where ω and θ denote
angular frequency and wave direction, respectively. This is the standard
description of waves in oceanic general-circulation models. At the beginning
of each time step, incident spectra are prescribed in grid cells at a
latitude outside the ice cover but as close to the ice cover as possible. For
expediency, in each cell at the incident latitude, the wave field is set to
be a Bretschneider spectrum, defined by a significant wave height and a peak
period, propagating in the mean wave direction. In subsequent cells,
directions are calculated as averages of the wave directions entering the
respective cells, weighted according to the associated wave energy.
Assuming steady-state conditions over a time step,
the spatial distribution of wave energy in the ice-covered ocean is calculated according to
a discrete version of the wave-energy-balance equation
(cosθ,sinθ)⋅∇S=-αS.
The attenuation coefficient, α(xij,t:ω), is set as
α=α0≡c(α^2ω2+α^4ω4),
where the coefficients
α^2≈7.68×10-5andα^4≈4.21×10-5
(units of time2× distance-1 and
time4× distance-1, respectively), based on
the empirical model from , scaled according to the areal
concentration of sea ice on the ocean surface, c(xij,t).
In each cell, the floe-size distribution is defined by a representative floe
diameter D(xij), for consistency with the assumptions
underlying the lateral-melt model, described below. At the beginning of a
simulation, the diameters are set to the relatively large value
D(xij)=Dmx=300 m, for consistency with the value
used throughout the ice cover in existing versions of CICE. For cells in
which wave energy is non-negligible, the ice-breakup
criterion from is applied, with the diameter of the broken floes denoted
Dbk<Dmx (no wave impacts on the ice cover are
considered beyond breakup). Following wave-induced breakup, the
representative floe diameter in cell-ij is calculated as the weighted
average
D(xij)=abk(xij)Dbk(xij)+(1-abk(xij))D0(xij),
where D0 is the representative diameter in the cell at the beginning of
the time step, and abk=Lbk/Lcl, in
which Lcl is the length of the cell in the southwards
direction, and Lbk is the distance the wave spectrum propagates
southwards through the cell, whilst being attenuated according to
Eq. (), and maintains sufficient energy to cause
breakup . For cells at the outermost fringes of the
ice-covered ocean, where the ice is too thin and compliant to be broken by
waves, the floe diameters are assumed to be small and assigned the
representative diameter D=Dmn.
In cells where breakup occurs, the representative diameter of broken
floes, Dbk, is calculated by assuming the in-cell floe-size
distribution obeys a split power law, as observed by noting that alternative distributions have been postulated for the
transition from small to large floes, e.g.. The
probability-density function for the split power law, p(d), where d
denotes floe diameter, is defined by
p(d)=P0β0γ0dγ0+1ifd∈[Dmn,Dcr),whereβ0=Dmn-γ0-Dcr-γ0-1,p(d)=(1-P0)β1γ1dγ1+1ifd∈[Dcr,∞),whereβ1=Dcrγ1,
and p(d)=0 if d<Dmn. Here,
Dmn represents a minimum floe diameter, which is chosen to be
equal to the small-floe diameter; Dcr is a critical
diameter marking the transition from small to large floes found to be
in the range 15–40 m by, and γ0=1.15 and
γ1=2.5 are representative exponents for the small- and
large-floe regimes, respectively . The quantity
P0∈[0,1] weights the distribution towards small floes (large
P0) or large floes (small P0). Its value is set
as
P0=1-qDprDcrγ1,whereDpr=λ/2
is the predicted breakup diameter, equal to the distance between successive
strain maxima for a regular wave train at the dominant wavelength λ
for the spectrum S, propagating through an infinitely long, uniform floe
, so that a chosen proportion q of floe
diameters are greater than Dpr. In the uncommon event that
Dpr<Dcr then P0=0, noting that
Dcr approximates the theoretical diameter below which flexural
breakup cannot occur . The broken-floe diameter
Dbk is the mean diameter
Dbk=∫Dmn∞p(d)ddd=P0γ0β0Dmn1-γ0-Dcr1-γ0γ0-1+1-P0γ1β1Dcr1-γ1γ1-1.
The breakup model is applied at the beginning of each CICE time step,
allowing the reduced floe diameters to affect other CICE-model components.
The reduced diameters directly affect the fraction of ice that melts
laterally, rlat, via the discrete version of the model from
:
rlat=πΔtwlatμD,
which assumes floes in a given cell are identical. Here μ=0.66 is a
geometric parameter and wlat=1.6ΔT2×10-6
(units of distance × time-1) is the rate of lateral melt, in
which ΔT is the temperature difference of the sea surface above that
of the bottom of the ice (set to zero if the difference is negative). The
diameters are updated at the end of the thermodynamic routine to account for
lateral melt.
During the summer months, when the ice is weaker and towards its minimum
extent, waves cause breakup close to the coastline. The existing
thermodynamic models in CICE do not increase the diameters of these broken
floes fast enough through the winter to create a realistic seasonal cycle for
the floe-diameter distribution. Therefore, an ad-hoc floe-bonding scheme is
applied, in which the floe diameter in a given cell is doubled if the
freezing potential in that cell is positive, up to the maximum diameter
Dmx.
The representative diameter, D, is transported by (i) setting the floe
diameter to be identical for each of the different thickness categories, and
transporting the floe diameter as an area tracer for the different thickness
categories; and (ii) setting the new representative diameters to be the
diameters of the thinnest ice category (cat. 1). Step (ii) is a non-physical
simplifying assumption; tests indicate that this assumption does not affect
the concentration changes due to breakup presented in
Sect. .
Results
The model was run from 1979 to 2010 using input wave data generated by a
Wavewatch III model hindcast and atmospheric and oceanic
data from the U. S. National Centers for Environmental Prediction's Climate
Forecast System Reanalysis NCEP's CFSR,. The minimum and
critical floe diameters were set as
Dmn=5 m and Dcr=30 m, and, following breakup,
the proportion q=0.05 of floe diameters was set to be greater than the predicted breakup diameter
Dpr.
Example model outputs using
Dmn=5 m, Dcr=30 m and q=0.05. The left-hand column
(a) is representative of results in austral summer and the
right-hand column (b) of winter. The top row shows the
significant wave heights. The middle row shows the ice regions: small floes
(green), wave-broken floes (red), unbroken floes (grey) and no ice/open water
(blue). The bottom row shows the change in concentration between the
simulations without and with breakup.
Figure shows example model outputs for two dates during
1995 (i.e. a year half-way through the simulation), representative of
results in summer (1 January, left-hand panels) and winter (1 July,
right). The panels in the top row show significant wave heights, with the
sharp outer boundaries of non-zero wave heights indicating the latitudes at
which data is extracted from the wave model. This boundary is farther north
in the winter because the ice extent is greater than in the summer. The
regions of rapid wave-height decrease with respect to southward distance
indicate attenuation of wave energy due to ice cover. In the summer, packets
of wave energy are able to propagate almost to the coastline, particularly
around the Antarctic peninsula, due to reduced ice cover in that locality.
The middle row shows the extent of ice coverage, with the ice divided into
regions according to floe size. Regions of small diameter floes (green) are
identified as those cells for which D≤Dmn=5 m,
wave-broken floes (red) are the floe-size interval Dmn<D≤250 m and unbroken floes (grey) are D>250 m. The bottom row shows
the impact of the small and broken floes on ice concentration, in terms of
the difference in concentration between the simulation without breakup
(D=300 m) and the simulation with breakup, with positive values indicating
decreases in concentration due to breakup.
The Southern Ocean experiences the strongest waves during winter, as
indicated in the top row. However, the areas covered by regions of broken
ice are comparable between the two seasons (approximately 10 % smaller in
the summer for the dates shown in the middle row), as the lower summer ice
concentration allows waves to penetrate deeper into the ice-covered ocean,
relative to their incident energy. The ice is structured into approximately
uniform bands in the winter, whereas in the summer coastal effects complicate
the structure.
In the summer,
the broken ice decreases the ice concentration in a vicinity of the ice edge,
with reductions of ∼0.1 common
but with numerous pockets of 0.3–0.4 reductions apparent.
The region most impacted by breakup is estimated by the region bounded by the two black lines,
where the
outer black line denotes the first cell (with respect to each longitude) at which the ice concentration exceeds 0.1,
and the inner black line represents three cells farther in (or land if that begins before the third cell).
During the winter,
the concentration change is too small to be visible on the scale shown (order 0.01),
as the temperatures are too low to melt the broken floes.
Mean monthly ice concentrations at the ice edge (the region
bounded by the black lines in the bottom row of Fig. ),
for January (a) and July (b). Results are for the
simulation without breakup (×) and with breakup for the parameters
considered in Fig. (Dmn=5 m,
Dcr=30 m, and q=0.05, red •); smaller floes
Dmn=2.5 m, Dcr=20 m and q=0.025 (green
▾); larger floes Dmn=10 m,
Dcr=40 m and q=0.1 (green ▴); a decreased
attenuation rate α=α0/10 (grey ▾); and
an increased attenuation rate α=10α0 (grey
▴).
Figure shows mean monthly ice concentrations at the
ice edge (the region bounded by the black lines in the bottom row of
Fig. ) for each simulation year. Results are again shown
for January and July, as representations of summer and winter conditions,
respectively. Data were generated by simulations without breakup (×)
and with breakup (red •). For the summer conditions, additional
data indicate sensitivities of concentration changes to (i) the floe-size
parameters, with data given for simulations in which Dmn,
Dcr and q are decreased to Dmn=2.5 m,
Dcr=20 m and q=0.025 (green ▾) and
increased to Dmn=10 m, Dcr=40 m and q=0.1
(green ▴); and (ii) increasing or decreasing the
wave-attenuation coefficient, α, by an order of magnitude
(α=10α0, grey ▴, and α=α0/10,
grey ▾, respectively). The ranges of floe sizes and
attenuation rates are within the limits of present uncertainty.
As indicated by the bottom-right panel of Fig. , the
right-hand panel of Fig. shows that breakup has
negligible impact on ice concentration during winter. During the summer,
breakup reduces the concentration, with the mean decrease being ∼0.08
for the parameters used in Fig. (neglecting the first,
spin-up year of the simulation). Reducing the floe-size parameters increases
the impact of breakup (as smaller floes melt more rapidly than larger ones),
and increasing them reduces the impact, with the mean reductions compared to
the simulation without breakup being ∼0.11 and 0.06, respectively.
Similarly, reducing the attenuation rate increases the impact (as the waves
maintain their strength for greater distances into the ice-covered ocean),
and increasing the attenuation rate has the opposite effect – the mean
reductions are ∼0.15 and 0.04, respectively.
(a, b) Snapshots of ice volume changes per unit area
between simulations without and with breakup (D0=5 m,
Dcr=30 m and q=0.05). (c) Proportional volume
decreases on first day of month over total ice cover
(yellow ⧫)
and at ice edge (purple ∗), for 1990–1995. (d, e) Median
decrease in ice volume per degree latitude for the eastern sector (red —)
and the western sector (green —), on 1 January (e) and 1 July
(d) for all simulation years. Shaded regions show corresponding 25th
to 75th percentile ranges.
The top panels of Fig. show changes in ice volume per unit
area due to breakup, for the two dates used in Fig. ,
i.e. results representative of summer (1 January 1995, left-hand panel) and
winter (1 July 1995, right). During the summer, breakup decreases the ice
volume, particularly at the ice edge, where losses of ∼0.5 m per unit
area are common. The pattern of the decreases is strongly correlated with
the concentration decreases shown in the bottom-left panel of
Fig. . During the winter, regions of volume loss
0.1–0.3 m per unit area are visible in the interior of the ice cover (the
unbroken ice region). This contrasts with the negligible concentration losses
on the same date shown in the bottom-right panel of
Fig. . The volume losses result from summer thickness
reductions forced by dynamic processes being restored at a slower rate than
concentration. Ice advection disperses the losses over large regions.
The bottom-left panel of Fig. shows volume decreases due
to breakup as proportions of the total ice cover without breakup, over a
typical 6-year interval. The ice volumes are sums over the total ice cover
(for cells with concentrations greater than 0.1, yellow ⧫)
and cells at the ice edge (the region between the black lines, purple
∗). Seasonal cycles are evident, with, for example, peaks in both
proportions occurring in March; the peaks for the full cover are between
0.13 and 0.20, and the peaks at the ice edge are between 0.09 and 0.14. During
June and July, losses at the ice edge due to wave-induced breakup typically
contribute less than 5 % of the total volume losses, whereas during
November–March a large proportion (54–68 %) of the overall losses occur
at the ice edge, as indicated in the top-left panel of
Fig. .
The bottom-right panels of Fig. show decreases in total ice
volume per degree latitude on 1 January (bottom panel) and 1 July (top),
over the full 32 years of the simulations, in terms of the median values, and
the spread in terms of the 25th and 75th percentiles. Data are split into
losses in the eastern (red —) and western (green —) sectors of Antarctica
(as shown in the top panels of Fig. ). During the summer,
when the increased lateral melt of the reduced floe diameters impacts ice
concentration, volume losses in the two sectors are similar. However, only
the western sector carries the bulk of its volume loss into winter, as a
significant proportion of East Antarctic sea ice affected by breakup during
early–mid summer melts during February, so that the winter ice is largely
composed of new ice.
Discussion
The findings of this pilot study indicate that increased lateral sea-ice melt
over the first ∼100 km in from the ice edge, due to small wave-broken
floes, and the follow-on effects on ice dynamics, impact ice concentration
and volume in a vicinity of the edge during winter and ice volume in the
interior pack throughout the year. The coupled
ice–ocean–atmosphere model by , which includes interactions between floe
diameters, ocean circulation and ice melt, indicates that lateral melt
remains important for sea-ice evolution for floe diameters orders of
magnitude larger than the O(30 m) limit given by the model from , as
used in CICE. Presumably, therefore, integrating diameter–circulation–melt
interactions into the modified version of CICE would strengthen the impacts
of breakup. Moreover, integrating the granular floe-size-dependent
rheology of would provide a direct impact of breakup on ice dynamics.
Applying the modified CICE model in a fully coupled setting will unlock
feedbacks triggered by the breakup – for example, the reduced concentration
due to increased lateral melt releasing more oceanic heat to the atmosphere,
thus increasing upwelling of ocean heat through convection and hence
promoting further ice melt – permitting studies into influences on long-term
trends in ice concentration, volume and also extent. If further research
finds the impacts of floe-size-dependent processes to be significant, future
large-scale sea-ice models may be developed along the lines of the theories
for coupled ice-thickness and floe-size evolution outlined by
and .
The Australian Antarctic Data Centre hosts the code used
for this study at 10.4225/15/57D0EA42ED985 (Bennetts, 2016).
For input wave data see Durrant et al. (2013), and for
atmospheric and oceanic data see Saha et al. (2010).
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors thank Mark Hemer for providing advice on the wave data used and
an internal CSIRO review of the original version of the paper and the two anonymous referees and
Christopher Horvat for their valuable comments. The Australian Research
Council (DE130101571) and the Australian Antarctic Science Program (Project
4123) funded this investigation. The Academy of Finland supports PU (Contract
264358). Edited by: D. Notz
Reviewed by: two anonymous referees
ReferencesBennetts, L.: Wave-ice breakup model for inclusion in CICE Australian
Antarctic Data Centre – CAASM Metadata
(https://data.aad.gov.au/metadata/records/AAS_4123_CICE-Model), 2016,
updated 2016.
Bennetts, L. G., O'Farrell, S., Uotila, P., and Squire, V. A.: An idealised
wave–ice interaction model without subgrid spatial or temporal
discretisations, Ann. Glaciol., 56, 258–262, 2015.Durrant, T., Hemer, M., Trenham, C., and Greenslade, D.: CAWCR Wave Hindcast
1979–2010. v7, Tech. rep., CSIRO. Data Collection,
10.4225/08/523168703DCC5, 2013.Feltham, D. L.: Granular flow in the marginal ice zone, Phil. Trans. R. Soc.
Lond. A, 363, 1677–1700, 10.1098/rsta.2005.1601, 2005.Herman, A.: Sea-ice floe-size distribution in the context of spontaneous
scaling emergence in stochastic systems, Phys. Rev. E, 81, 066123,
10.1103/PhysRevE.81.066123, 2010.Horvat, C. and Tziperman, E.: A prognostic model of the sea-ice floe size and
thickness distribution, The Cryosphere, 9, 2119–2134,
10.5194/tc-9-2119-2015, 2015.Horvat, C., Tziperman, E., and Campin, J.-M.: Interaction of sea ice floe
size, ocean eddies, and sea ice melting, Geophys. Res. Lett., 43,
8083–8090, 10.1002/2016GL069742, 2016.
Hunke, E. C. and Dukowicz, J. K.: An elastic-viscous-plastic model for sea ice
dynamics, J. Phys. Oceangr., 27, 1849–1867, 1997.
Hunke, E. C. and Lipscomb, W. H.: CICE: the Los Alamos Sea Ice Model
Documentation and Software User's Manual Version 4.1 (LA-CC-06-012), Tech.
rep., T-3 Fluid Dynamics Group, Los Alamos National Laboratory, 2010.Kohout, A. L., Williams, M. J. M., Dean, S. M., and Meylan, M. H.:
Storm-induced sea-ice breakup and the implications for ice extent., Nature,
509, 604–607, 10.1038/nature13262, 2014.Meylan, M. H., Bennetts, L. G., and Kohout, A. L.: In-situ measurements and
analysis of ocean waves in the Antarctic marginal ice zone, Geophys. Res.
Lett., 41, 5046–5051, 2014.
Saha, S., Moorthi, S., Pan, H.-L., Wu, X., Wang, J., Nadiga, S.,
Tripp, P., Kistler, R., Woollen, J., Behringer, D., Liu, H.,
Stokes, D., Grumbine, R., Gayno, G., Wang, J., Hou, Y.-T.,
Chuang, H.-Y., Juang, H.-M. H., Sela, J., Iredell, M., Treadon, R.,
Kleist, D., Van Delst, P., Keyser, D., Derber, J., Ek, M., Meng,
J., Wei, H., Yang, R., Lord, S., van den Dool, H., Kumar, A.,
Wang, W., Long, C., Chelliah, M., Xue, Y., Huang, B., Schemm,
J.-K., Ebisuzaki, W., Lin, R., Xie, P., Chen, M., Zhou, S.,
Higgins, W., Zou, C.-Z., Liu, Q., Chen, Y., Han, Y., Cucurull,
L., Reynolds, R. W., Rutledge, G., and Goldberg, M.: NCEP Climate
Forecast System Reanalysis (CFSR) 6-hourly Products, January 1979 to
December 2010, 10.5065/D69K487J, 2010.
Squire, V. A., Dugan, J. P., Wadhams, P., Rotter, P. J., and Liu, A. K.: Of
ocean waves and sea ice, Annu. Rev. Fluid Mech., 27, 115–168, 1995.
Stammerjohn, S., Martinson, D. G., Smith, R. C., Yuan, X., and Rind, D.:
Trends
in Antarctic sea ice retreat and advance and their relation to El
Nino–Southern Oscillation and southern annular mode variability, J.
Geophys. Res.-Oceans, 113, C3, 2008.
Steele, M.: Sea ice melting and floe geometry in a simple ice-ocean model, J.
Geophys. Res., 94, 17729–17738, 1992.Thorndike, A. S., Rothrock, D. A., Maykut, G. A., and Colony, R.: The thickness
distribution of sea ice, J. Geophys. Res., 80, 4501,
10.1029/JC080i033p04501, 1975.Toyota, T., Haas, C., and Tamura, T.: Size distribution and shape properties
of relatively small sea-ice floes in the Antarctic marginal ice zone in late
winter, Deep-Sea Res. Pt. II, 58, 1182–1193,
10.1016/j.dsr2.2010.10.034, 2011.
Williams, T. D., Bennetts, L. G., Squire, V. A., Dumont, D., and Bertino, L.:
Towards the inclusion of wave–ice interactions in large-scale models for the
Marginal Ice Zone, ArXiV, arXiv:1203.2981v1, 2012.
Williams, T. D., Bennetts, L. G., Dumont, D., Squire, V. A., and Bertino, L.:
Wave–ice interactions in the marginal ice zone. Part 1: Theoretical
foundations, Ocean Model., 71, 81–91,
2013a.Williams, T. D., Bennetts, L. G., Dumont, D., Squire, V. A., and Bertino, L.:
Wave–ice interactions in the marginal ice zone. Part 2: Numerical
implementation and sensitivity studies along 1D transects of the ocean
surface, Ocean Model., 71, 92–101, 10.1016/j.ocemod.2013.05.011,
2013b.
Zhang, J., Schweiger, A., Steele, M., and Stern, H.: Sea ice floe size
distribution in the marginal ice zone: Theory and numerical experiments, J.
Geophys. Res., 120, 3484–3498, 2015.