In this paper we describe a waves-in-ice model (WIM), which calculates ice
breakage and the wave radiation stress (WRS). This WIM is then coupled to the
new sea-ice model neXtSIM, which is based on the elasto-brittle (EB) rheology.
We highlight some numerical issues involved in
the coupling and investigate the impact of the WRS, and of modifying the EB
rheology to lower the stiffness of the ice in the area where the ice has
broken up (the marginal ice zone or MIZ). In experiments in the absence of
wind, we find that wind waves can produce noticeable movement of the ice edge
in loose ice (concentration around 70 %) – up to 36 km, depending on the
material parameters of the ice that are used and the dynamical model used for
the broken ice. The ice edge position is unaffected by the WRS if the initial
concentration is higher (
In the presence of wind, we find that the wind stress dominates the WRS, which, while large near the ice edge, decays exponentially away from it. This is in contrast to the wind stress, which is applied over a much larger ice area. In this case (when wind is present) the dynamical model for the MIZ has more impact than the WRS, although that effect too is relatively modest. When the stiffness in the MIZ is lowered due to ice breakage, we find that on-ice winds produce more compression in the MIZ than in the pack, while off-ice winds can cause the MIZ to be separated from the pack ice.
Wave–ice interactions have received a great deal of attention
in recent years
Specifically, large parts of the Arctic are becoming, and are expected to
become, even more accessible for resource exploitation and shipping in the
summer, whereas 10 years ago they were not
Closely connected to waves in ice, but with other controlling factors apart
from waves, is the concept of floe-size distribution
On the sea-ice modelling side, there has been a lot of progress in making
sea-ice dynamics more realistic, especially in the Arctic pack.
In this paper we demonstrate the coupling of a waves-in-ice model (WIM) to neXtSIM in an idealised domain. The physical effects included in the coupling are the break-up of ice by waves, the wave radiation stress (WRS) and an additional (optional) feedback to the sea-ice model where the ice stiffness is reduced where the ice is broken (in the MIZ). We conduct experiments with waves by themselves to see the impact of the WRS on the ice edge location and also with wind to see the relative importance of the wind stress and the WRS. In addition, we carry out some simulations to see the particular effects of the rheological change.
We also highlight some general numerical issues involved with coupling wave models and sea-ice models on different grids. In addition, we carry out some theoretical reformulations of the WIM to put the ice break-up model in the context of Mohr–Coulomb failure and test the sensitivity of the MIZ width to the Young's modulus in particular, as well as the small-scale “cohesion” parameter in the WIM-breaking model. Its response to the Young's modulus was previously uninvestigated.
The ice is modelled as a thin elastic plate
The momentum balance equation we will use is the following:
We also have equations for the evolution of any conserved quantity
Like
The evolution of stress and damage from time step
Since the damage variable
Results after forcing from uniform, steady wind (with speed
14.9 m s
Initially, the concentration was relatively low, so the internal stress was
also low (see the formulae for
Let
The conditions (
Returning to Eq. (
Mohr–Coulomb envelopes have been observed on many different scales in rock
mechanics and have also been seen in ice. The parameter
This property should scale as
Cohesion values, internal friction coefficient from measured
Mohr–Coulomb failure envelopes. Also given are approximate defect sizes
deduced from these envelopes, using the scaling law (
Note that these values do not necessarily correspond to the breaking stress of ice since the measurements are not exactly taken at the point of fracture. The lab measurement (uni-axial compression test) should be closer since we know the ice did actually break and the scale of the measurement; the in situ measurements are certainly underestimations since the ice did not break, and in fact the value of 1 kPa was derived from a 3-day subset of the time series, which was bounded by the envelope with cohesion 40 kPa. That is, the lower in situ value corresponds to more remote fracturing or fracturing over a larger scale.
In their presentation of the dynamical core of the neXtSIM model (using a
resolution of approximately 10 km),
For the simulations in this paper we will use a model resolution of 4 km, so
we will test a range of cohesions from 4 to 13 kPa to be somewhat consistent
with the above choice. Also, we will discuss the ice breakage by waves (below
in Sect.
The amount of attenuation that waves in ice experience is the main factor in
determining the amount of momentum transferred to the ice. However, a
definitive confirmation of any particular physical model for this is still
lacking.
Our attenuation model is essentially model B from
As stated above, the choice of attenuation model is crucial in determining the wave radiation stress, yet physical mechanisms are still relatively uncertain. However, we can still calculate the response of the ice to waves attenuated by our model and make conclusions which should still hold for similar ranges of the WRS.
A general formulation for wave energy transport is
The scattering kernel
We use a parametric form of the FSD. We initially require that
Results for the MIZ width (not shown) with the RG approach are similar to
those with the FSD (
It is instructive to put the situation of ice breakage due to a plane wave in
the context of the discussion in Sect.
Figure
The maximum strains are produced when
The lab measurement of cohesion
When we return to our plane wave in an elastic plate, the Mohr–Coulomb criterion is equivalent to the strain criterion
When we have a spectrum of waves, the corresponding quantity to (
When the wave energy is not unidirectional, the stresses are no longer
distributed on the line
When (
Following
When we consider a complete wave spectrum, then
Schematic showing the information that passed between neXtSIM and the
WIM. Note that
Figure
In an initial, more naive implementation of the coupling,
The directional wave spectrum is remembered from the previous call, and if necessary can be updated regularly using forcing from an external model or, as in the simulations presented in this paper, using idealised (constant) wave forcing.
We can also change the dynamics of the broken ice. The default, R0 or
rheology 0, does not change the underlying EB rheology. For the alternative,
R1 or rheology 1, we increase the damage parameter
Alternative continuum approaches to MIZ dynamics are based on the idea of a
“granular temperature” (kinetic energy associated with velocity fluctuations
relative to the mean flow field). Most recently,
However, in the field of 3-D granular flows, different types of flow regimes
have also been observed. For example, the introduction of
There have also been a number of direct (discrete) numerical simulations of
collections of floes
In our results section we will partly use incident wind wave spectra based on the Bretschneider spectrum:
Since
The purpose of this section is to test sensitivity to the Young's modulus and
the small-scale cohesion, not necessarily to decide on “correct” values,
which are best determined from future observations. The experiments are
similar to those of
The Young's modulus is typically somewhere in the range of 1–10 GPa.
Variation of MIZ width
Figure
The dashed curves use fully developed seas (Pierson–Moskowitz spectra), where
The solid curves in Fig.
Variation of MIZ width
Variation of MIZ width with peak wave period and small-scale
cohesion for
Waves breaking ice in an idealised experiment (the right-hand, upper
and lower lines of the grid cells correspond to land). The wave model, based
on
This latter result (
Figure
The three values chosen are 270 kPa (approximately the flexural strength
when
Figure
The resulting MIZ width is about 50 km, which is not unrealistic. Following
Eq. (
Figure
There is a strong response to the compactness factor,
Same as
Maximum movement of the ice edge over 2 days for different pairs
However, the large-scale cohesion makes little difference in these
simulations where the ice is not failing. Part of the reason for this is that
the wave radiation stress is a compressive stress, so the stresses need to be
larger to move outside the Mohr–Coulomb envelope than if they were tensile or
shear stresses (see Fig.
Some of the runs from Fig.
Figure
Figure
To repeat what we have seen in Fig.
Panel
In this paper, we have investigated the impact of the WRS on sea-ice state
and drift in an idealised domain. While this stress can be quite large
(
Having said this, however, there are many uncertainties regarding the WRS, and
we have certainly not included all of its potential effects, especially since
the wave and ice models are not coupled to the ocean yet. For example, the
attenuation models are still uncertain (they determine the WRS), and how the
partitioning of the WRS between the ice and the ocean should be done is also
unknown. On the face of it, if less of the WRS is applied to the ice, it
should have even less effect than we find in our current paper. However,
perhaps it could then produce similar effects to those discussed and reported
by
We also highlighted the problem of numerical diffusion of
As touched on in the discussion of the WRS above, we also introduced a simple
MIZ rheology by increasing the damage where ice was broken, effectively
putting the MIZ into free drift, with the addition of the ice pressure, which
resists compression. Under compressive wind forcing this led to increased
compression in the MIZ relative to the pack ice in its vicinity. This
modification also influenced the ice flow when off-ice winds were applied to
ice that had previously been broken by swell waves. At lower wind speeds, the
MIZ was able to be move relatively freely with the wind, while the pack was
still stationary. These effects would undoubtedly be reduced in magnitude
were a rheology that represented true granular flow to be used, but could
still occur. However, it is difficult to know for certain without the
existence of such a rheology. Direct numerical simulations such as those done
by
So far we have also restricted ourselves to a simple idealised domain and
with very idealised forcings. Work to set up the current model in a
pan-Arctic domain is ongoing, and perhaps studies with forcings with more
realistic temporal and spatial variability could find that the WRS will have more
impact. In addition, the study of
These data are not publicly available. The model code is not released yet, since it is still being developed, and lacks the full documentation.
The writing of the paper and implementation of the coupling between the WIM and neXtSIM was lead by TW, with formative discussions from PR and SB guiding the progression of the writing. PR also helped with the writing itself, and in addition SB helped to implement the coupling.
The authors declare that they have no conflict of interest.
This work was primarily supported by the neXtWIM project (Norwegian Research Council grant no 244001). Earlier WIM code development was also supported by the SWARP project (EU-FP7 project 607476) and ONR Global project N62909-14-1-N010. We were also helped by discussions with Einar Ólason and Aleksey Marchenko. Finally we would like to thank our reviewers and editor for their helpful comments. Edited by: Jennifer Hutchings Reviewed by: two anonymous referees