Recent years have seen a rapid reduction in the summer Arctic sea ice extent. To both understand this trend and project the future evolution of the summer Arctic sea ice, a better understanding of the physical processes that drive the seasonal loss of sea ice is required. The marginal ice zone, here defined as regions with between 15 % and 80 % sea ice cover, is the region separating pack ice from the open ocean. Accurate modelling of this region is important to understand the dominant mechanisms involved in seasonal sea ice loss. Evolution of the marginal ice zone is determined by complex interactions between the atmosphere, sea ice, ocean, and ocean surface waves. Therefore, this region presents a significant modelling challenge. Sea ice floes span a range of sizes but sea ice models within climate models assume they adopt a constant size. Floe size influences the lateral melt rate of sea ice and momentum transfer between atmosphere, sea ice, and ocean, all important processes within the marginal ice zone. In this study, the floe size distribution is represented as a power law defined by an upper floe size cut-off, lower floe size cut-off, and power-law exponent. This distribution is also defined by a new tracer that varies in response to lateral melting, wave-induced break-up, freezing conditions, and advection. This distribution is implemented within a sea ice model coupled to a prognostic ocean mixed-layer model. We present results to show that the use of a power-law floe size distribution has a spatially and temporally dependent impact on the sea ice, in particular increasing the role of the marginal ice zone in seasonal sea ice loss. This feature is important in correcting existing biases within sea ice models. In addition, we show a much stronger model sensitivity to floe size distribution parameters than other parameters used to calculate lateral melt, justifying the focus on floe size distribution in model development. We also find that the attenuation rate of waves propagating under the sea ice cover modulates the impact of wave break-up on the floe size distribution. It is finally concluded that the model approach presented here is a flexible tool for assessing the importance of a floe size distribution in the evolution of sea ice and is a useful stepping stone for future development of floe size modelling.
Arctic sea ice is an important component of the climate system. The sea ice cover moderates high-latitude energy transfers between the ocean and atmosphere (Screen et al., 2013) and generates a positive feedback response to global warming via the albedo feedback mechanism (Dickinson et al., 1987; Winton, 2006, 2008). Accurate representation of the sea ice within climate models can contribute to improved projections of the climate response to present and future forcings (Vihma, 2014). On a more local scale, sea ice modelling is necessary to understand how environments within and around the Arctic are likely to develop. This is important for Arctic communities to plan for the future (Laidler et al., 2009), to enable ecologists to identify practical responses to protect vulnerable species that live in the Arctic or seasonally migrate into the region (Hauser et al., 2017; Post et al., 2009; Regehr et al., 2010), and for shipping companies to understand the potential viability of new routes in the next few decades (Aksenov et al., 2017; Ho, 2010; Smith and Stephenson, 2013).
The Arctic is currently in a state of transition (Notz and Stroeve, 2018; Stroeve and Notz, 2018). Multi-year sea ice fraction has decreased by more than 50 % with an increasing proportion of the ice cover now seasonal first-year ice (Kwok, 2018; Maslanik et al., 2007). First-year ice does not have the same surface roughness or the same mechanical or thermophysical (salinity, conductivity, permeability) properties as ice that has developed over multiple years. In particular, first-year ice is thinner and weaker (Stroeve et al., 2018) and hence more vulnerable to fracture in response to external stress (Zhang et al., 2012). Similarly, the region of the Arctic identified as the marginal ice zone (MIZ), generally defined as the region where ocean waves are able to significantly influence the dynamics of the sea ice (Strong et al., 2017), is projected to increase in extent (Aksenov et al., 2017). An alternative definition of the MIZ, and the one that will be used in the present study, is the region where the concentration of the sea ice extends between 15 % and 80 %. This definition of the MIZ is often more practical for modelling and observational studies where sea ice concentration data are more readily available than information about wave behaviour in sea ice.
Modelling the MIZ is a significant challenge due to its complexity; it is a region in which there is strong coupling between the sea ice, ocean, and atmosphere (Lee et al., 2012; McPhee et al., 1987). The sea ice cover in this region is significantly broken up and fragmented by the waves that define the MIZ (Liu et al., 1992). Wave intensity and storm frequency are projected to increase, which will strengthen wave–sea ice interactions (Casas-Prat et al., 2018; Day and Hodges, 2018). This continues a trend already observed over the past few decades (Stopa et al., 2016). Such interactions are even more prominent around Antarctica due to the dominance of seasonal sea ice in the region (Parkinson and Cavalieri, 2012) and large and increasing wave fetch (Young et al., 2011).
Floe size is a key parameter in describing the evolution of the MIZ (Rothrock and Thorndike, 1984). As sea ice floes become smaller, the available perimeter per unit area of sea ice cover increases, enhancing the lateral melt rate (Steele, 1992). Increased lateral ice melt increases the area of exposed ocean, allowing the input of more heat into the ocean mixed layer from solar insolation. Warming of the upper mixed layer also re-stratifies the ocean. These two processes increase heat available for ice melt through basal and lateral ice melting mechanisms. The former is a well-known mechanism, the albedo feedback (Curry et al., 1995). As the MIZ expands, the lateral ice melting is expected to become an increasingly significant driver of seasonal ice loss.
Currently climate models either assume a fixed and constant characteristic floe size across the Arctic cover, for all types of sea ice (Hunke et al., 2015), or they ignore floe size entirely. This approach does not allow for regional or temporal variations in floe size. Multiple sea ice processes depend on floe size. Lateral melt rate is a function of floe size; the melt rate is proportional to the perimeter per unit area of sea ice. A recent study has found that the basal melt rate may also be influenced by floe size (Horvat and Tziperman, 2018). Floe size can also impact the propagation of waves under the sea ice (Boutin et al., 2018; Meylan and Squire, 1994; Squire, 2007). The assumption of a fixed floe size also prevents sea ice models from accurately representing the impact of processes on the sea ice evolution that act via the perturbation of floe size such as lateral melting and wave-induced fragmentation of floes. Whilst these assumptions are significant, the use of a variable floe size within models will need to be justified against the increased computational cost. The most suitable modelling approach will be context dependent; for example, high-resolution regional sea ice models would be expected to require a higher complexity of floe size treatment than large-scale climate models.
There have been several observational studies aiming to characterise the floe size distribution (FSD) using techniques including satellite imagery and in situ studies (Stern et al., 2018a). FSD data are generally fitted to a power law (Rothrock and Thorndike, 1984). Values have been reported for the magnitude of the exponent of this power law ranging from 1.5 to over 3.5 between different datasets (Stern et al., 2018a). Comparing these observations is complicated by the fact that some studies report a value for the probability distribution of floe size and some for the cumulative floe size distribution. It has been recently pointed out that if a distribution adopts a power law for a probability distribution, it will have a tailing off for larger floes when plotted as a cumulative distribution (Stern et al., 2018a). Furthermore, a recent study (Stern et al., 2018b) found evidence to suggest that the exponent of the power-law FSD evolves throughout the year and is not fixed. This same study was also able to use two satellite datasets with different resolutions but operating over the same region to show floes from as small as 10 m and as large as 30 000 m follow power laws. Other studies find different values for these limits, for example Toyota et al. (2016) showed a power law extending to 1 m (using data collected in situ from a ship), whereas Hwang et al. (2017) found a tailing off from the power law around 300–400 m. As each study operates over a different spatial extent, with a different resolution and different algorithms used to extract the FSD, it is not trivial to identify whether the cut-offs in each scenario are physical or a product of limited resolution or spatial extent. Alternative approaches to a single power law have been proposed including the use of two power laws over different size ranges, with smaller floes found to have a smaller exponent (Steer et al., 2008). The Pareto distribution has also been discussed (Herman, 2010); it is analogous to a power law but with a non-constant exponent. To fully understand and characterise the FSD across the Arctic sea ice, good spatial and temporal coverage is required. Novel techniques, particularly those using autonomous platforms and robotic instruments, are enabling increased high-resolution data capture of sea ice and ocean conditions that can be used alongside time series of up to 1 m resolution FSD data obtained through remote sensing to better understand the factors driving FSD evolution (Thomson and Lee, 2017). These data could be applied within an approach analogous to that of Perovich and Jones (2014), who used aerial photography alongside simple parameterisations for lateral melting and floe fragmentation by waves, assuming the floe size cumulative distribution adopts a power law, to explore whether these processes could result in the observed changes to the FSD. There are also efforts to characterise the floe size distribution resulting from individual processes, such as laboratory analogues to the wave break-up of ice (Herman et al., 2018). Future Arctic expeditions including “Multidisciplinary drifting Observatory for the Study of Arctic Climate” (MOSAiC; Dethloff et al., 2016), planned to last 1 year within the central Arctic, should contribute to the existing FSD datasets.
Modelling studies have used contrasting approaches to represent floes as a distribution. A very simple approach is the use of a semi-empirical relationship between floe size and sea ice concentration (Lüpkes et al., 2012; Tsamados et al., 2015). Although this approach involves a simple amendment to the code and has a negligible computational cost, it is unable to respond to fragmentation processes. It will not capture the desired feedbacks during events such as storms that are expected to produce significant fragmentation of the sea ice cover. Furthermore, the parameters used within the relationship were constrained by a set of observations from a specific region and season and might not be applicable across the whole sea ice extent and full seasonal cycle.
Extending beyond using this simple dependency of floe size on sea ice concentration, Zhang et al. (2015) introduced a thickness, floe size, and enthalpy distribution. This model aims to represent the impacts on floe size of advection, thermodynamic growth, lateral melting, ice ridging, and ice fragmentation. However, the impacts of wind, current, and wave forcing are represented by an empirically parameterised floe size distribution factor. Bennetts et al. (2017) focus on the incorporation of a physically realistic wave-induced break-up model (Williams et al., 2013a, b). Bennetts et al. (2017) assume that the FSD follows a split power law, with a change in exponent at some critical diameter. The wave component of this model assumes steady-state conditions over a time step and uses a Bretschneider spectrum defined by a significant wave height and a peak period for computational efficiency and propagates it in the mean wave direction. The propagation directions are calculated from averages of the wave directions entering the neighbouring cells and weighted according to the respective wave energy. The model implementation also assumes floe sizes to be assigned to a minimum representative diameter if ice is too thin and compliant to be broken by waves. A recent study by Boutin et al. (2019) also considers the interactions between floe size and waves within the MIZ. This study includes a fully coupled ocean surface–wave model and is unique in considering the impact of momentum transfer to the sea ice from the waves via the wave radiative stress.
There has also been a significant drive to develop a physically derived prognostic floe size–thickness distribution (Horvat and Tziperman, 2015, 2017; Roach et al., 2018a). A recent approach by Roach et al. (2018a) includes the representation of five processes: new ice formation, welding of floes, lateral growth, lateral melt, and fracture by ocean surface waves. This model has the advantage that it does not involve any assumptions about the form of the distribution. Provided the model incorporates good physical representations of the processes which impact floe size, the model should respond accurately to localised extremes in behaviour (such as the large waves associated with storms) or future changes (e.g. changing wind speeds). It is also possible to model floe evolution at the floe by floe scale, for example Herman (2018) uses a discrete-element model to investigate the wave-induced behaviour of floes.
For this study, a single power law will be applied to describe the FSD
within a stand-alone sea ice model coupled to a prognostic mixed-layer model,
hereafter referred to as the WIPoFSD model (Waves-in-Ice module and Power
law Floe Size Distribution model). The distribution is defined by three
parameters:
In this study we present results to understand the thermodynamic response of the sea ice to a power-law-derived FSD and the individual impacts of wave–floe size and lateral melting–floe size interactions. Our focus will be on the impact of this FSD on the seasonal sea ice retreat and variability rather than on longer-term changes and trends.
This paper will proceed as follows: Sect. 2 describes the sea ice model
used, Sect. 2.1 describes standard model physics, and Sect. 2.2–2.4
outlines the new WIPoFSD model. Section 3 describes the modelling methodology
used including the forcing data and model domain. Section 4 describes the
results of the simulations in three sections: Sect. 4.1 looks at the
general impacts of the FSD on the sea ice, Sect. 4.2 explores the model
sensitivity to the different FSD parameters, and Sect. 4.3 looks at the
model response to a series of perturbations to the model including the
wave-in-ice set-up, floe shape parameter, lateral melt constants, and a
variable
For this study a CPOM (Centre for Polar Observation and Modelling) version
of the Los Alamos Sea Ice model v5.1.2, hereafter referred to as CICE, was
used (Hunke et al., 2015). This is a dynamic and thermodynamic
sea ice model designed for inclusion within a climate model. CICE includes a
large choice of different physical parameterisations; see
Hunke et al. (2015) for details. Section 2.1 outlines the
features pertinent to this study. Our local version also includes some
state-of-the-art parameterisations not included within the general CICE
distribution, also described in Sect. 2.1. The WIPoFSD model that we have
implemented into stand-alone CICE is adapted from an implementation developed
at the National Oceanography Centre of the UK within a coupled sea ice–ocean
framework, called the NEMO–CICE–Waves-in-Ice (WIM) model (Hosekova et al., 2015; NERSC, 2016). This approach
was originally developed to understand the impact of waves on the MIZ and
the upper ocean via the thermodynamic and dynamic response with applications
for the operational forecasting of the MIZ and large-scale coupled sea
ice–ocean global modelling, where assuming a power law is particularly
practical. The model includes the wave attenuation and floe break-up model
based on the Waves-in-Ice Model from the Nansen Environmental and Remote
Sensing Center (NERSC) Norway
(Williams et al.,
2013a, b). An overview of this scheme is given in Sect. 2.2. Floe size
is assumed to follow a single power law within the WIPoFSD model. Three new
global parameters and one tracer are required to define this power law. The
global parameters are
Within the CICE v5.1.2 model we use the incremental remapping advection scheme (Lipscomb and Hunke, 2004), an ice thickness redistribution scheme (Lipscomb et al., 2007), along with five ice thickness categories (Hunke et al., 2015). The default elastic–viscous–plastic (EVP) rheology is used (Hunke and Dukowicz, 2002) along with an ice strength formulation (Rothrock, 1975). The frictional energy dissipation parameter is set to 12. A topologically based melt pond scheme is used (Flocco et al., 2012) in conjunction with a delta-Eddington radiation scheme (Briegleb and Light, 2007). The atmospheric and oceanic neutral drag coefficients are assumed constant in time and space. An ocean heat flux formulation is used at the ice–ocean interface (Maykut and McPhee, 1995).
The rate of thermodynamic ice loss is calculated as follows:
A number of amendments are made to CICE version 5.1.2 based on recent work by Schröder et al. (2019). The maximum meltwater added to melt ponds is reduced from 100 % to 50 %. This produces a more realistic distribution of melt ponds (Rösel et al., 2012). Snow erosion, to account for a redistribution of snow based on wind fields, snow density, and surface topography, is parameterised based on Lecomte et al. (2015) with the additional assumptions described by Schröder et al. (2019). The “bubbly” conductivity formulation of Pringle et al. (2007) is also included, which results in larger thermal conductivities for cooler ice.
The full details of this module are described in Williams et al. (2013a, b), to which the reader is referred for details; here we provide an overview of the elements pertinent to our study alongside developments unique to the WIPoFSD model. The waves-in-ice module described here reproduces wave conditions near the sea ice edge within the MIZ. Local wind direction determines the direction of wave propagation with adjustments made for attenuation imposed by the sea ice cover. This is a compromise dictated by availability of forcing data, lack of observational studies, and the coarse resolution of the CICE model.
The module operates using its own internal time step defined by
We construct the wave energy spectra using
Once the wave field
The decision to use the ocean surface stress to define the primary direction of wave propagation rather than the Stokes drift direction was made because the Stokes drift direction data were not available within the sea ice field at the time of model development. The use of ocean surface stress will be sensible for wind-driven seas, but not for swell-driven seas where the Stokes drift is a more appropriate choice. Stopa et al. (2016) discuss wave climate in the Arctic between 1992 and 2014 and they find that regions exposed to the North Atlantic wave climate will be strongly influenced by swells generated within the North Atlantic Ocean. Semi-enclosed and isolated seas, e.g. Laptev and Kara seas, are more event driven and have an equal mix of wind-driven and swell-driven waves. The results presented in this study should therefore be considered in the context that the direction of wave propagation is a significant approximation. Furthermore, we are only able to represent the impacts of waves generated externally to the sea ice cover within this set-up. The choice of surface wave spread is also non-trivial. Wadhams et al. (2002) showed that a wave propagating into the MIZ could experience significant wave spreading until it was essentially isotropic. However, a distinction was found between wind seas where the isotropic state could be achieved within a few kilometres and swell seas where spreading occurs much more slowly, if at all. Wave spreading has been shown to be dependent on the wavelength. Montiel et al. (2016) found that shorter wavelengths experienced spreading and longer wavelengths did not with a transition between these two regimes defined by the maximum floe size. This is consistent with the observed behaviour of wave-driven regimens and swell-driven regimes. Using a fixed surface wave spread across a limited number of categories is a significant simplification of the rather complex spreading behaviour of waves; however it represents a balance between short wave periods that quickly achieve an isotropic state and longer wave periods that propagate much further into the MIZ before they experience significant spreading.
After advection, the attenuation of waves over each wave time step is
calculated. This will be calculated for each individual wave energy
spectrum:
After attenuation, the wave energy spectra within each grid cell are
reconstructed as a discretised function of
The floe fragmentation scheme used is identical to Williams et al. (2013a), which should be referred to for a detailed description of the
scheme. An overview of this scheme is presented here. Ice-breaking events
occur when the probability that the breaking strain amplitude,
We employ a number-weighted FSD,
Panel
The model is initiated with
It is useful here to define an additional floe size parameter,
First, Eqs. (8) and (9) can be used to give an expression for the total
sea ice area,
In our model there are four ways in which the floe size distribution can be
perturbed: lateral melt, break-up of floes by ocean waves, advection of
floes, and restoration due to freezing. Changes in
As lateral melt involves the loss of ice volume from the sides of floes, it
can be expected to reduce floe size. To represent this in the model, we set
the reduction in
Section 2.4 outlines the conditions necessary to trigger the break-up of
floes by waves. If these conditions are fulfilled,
There are three processes thought to be the main drivers of floe formation
and growth during freezing conditions: lateral growth, welding of floes, and
formation of new floes (Roach et al., 2018a). The
focus of this study is on the seasonal melt and fragmentation of sea ice
rather than the winter evolution; hence a simple floe growth restoration
scheme is used. During conditions when the model identifies frazil ice
growth,
It is worth commenting here on the limitations of the modelling approach to
floe size used in this study. The use of a power-law distribution with a
fixed exponent to describe the FSD is a valuable simplification to explore
the impact of floe size on the Arctic sea ice. The tracer
Our modified version of CICE is run over a pan-Arctic domain with a 1
All simulations are spun up between 1 January 1990 and 31 December 2004 using the standard set-up described in Sect. 2.1 with a constant floe size of 300 m (without the WIPoFSD model included). Simulations are initiated on 1 January 2005 using the output of the spin-up and evaluated for 12 years until 31 December 2016. Results are all taken from the period 2007–2016 to allow 2 years for the model to adjust to the addition of the WIPoFSD model. A reference run is also evaluated over this period using the standard set-up and a 300 m constant floe size. Figure 2 shows this model simulates the climatological monthly sea ice extent realistically for this period. All further simulations are evaluated over the same time period using the same initial model state, however with the WIPoFSD model imposed. Some simulations have additional modifications made to the model as described.
Comparison of the 2007–2016 mean cycle for the total Arctic sea ice extent simulated in the coupled CICE–prognostic mixed-layer reference set-up (marked CICE–ML, red ribbon, solid) with the results from the standard optimised CPOM CICE model (Schröder et al., 2019, marked CICE-schro, blue ribbon, small dashes) and observed sea ice extent derived from Nimbus-7 SMMR and DMSP SSM/I–SSMIS satellites using Bootstrap algorithm version 3 (Comiso, 1999, marked Observations, green ribbon, large dashes). The ribbon shows, in each case, the region spanned by the mean value plus or minus 2 times the standard deviation for each simulation. This gives a measure of the interannual variability over the 10-year period. Results show the new model performs either comparably to or better than the previous optimum set-up throughout the year. In addition, the mean CICE–ML sea ice extent falls within the interannual variability of the observations between June and December, i.e. most of the melting season, suggesting this reference state is suitable for studies focusing on this period.
Results are presented for the pan-Arctic domain with a focus on the melting season. All plots compare the mean behaviour over 10 years from 2007 to 2016 against the reference simulation, referred to as ref, which uses a constant floe size of 300 m. The results for 2005 and 2006 are discarded to allow 2 years for the model to adjust to the imposed FSD. In this study we are trying to understand the impact of the FSD and associated processes on the seasonal sea ice loss. The years 2007–2016 have been selected as the baseline for these simulations as they will capture the current climatology of the Arctic, including the record September minimum sea ice extent observed in 2012.
The WIPoFSD model introduces new parameters that can be constrained through
observations. Stern et al. (2018b) were recently able to
show a region of floe sizes could be described by power laws over a size
range from 10 to 30 000 m. This is the largest range of floe sizes that a
power law has produced a good fit to; hence these are set as the standard
values for
Difference in sea ice extent (solid, red ribbon) and volume (dashed, blue ribbon) between stan-fsd relative to ref (using a constant floe size) averaged over 2007–2016. The ribbon shows, in each case, the region spanned by the mean value plus or minus 2 times the standard deviation for each simulation. This gives a measure of the interannual variability over the 10-year period. The mean behaviour is a reduction in the sea ice extent and volume, with losses of up to 1 % and 1.2 % respectively seen in September during the period of minimum sea ice. The interannual variability shows that the impact of the WIPoFSD model with standard parameters varies significantly between years, with some years potentially showing negligible change in extent and volume and others showing a maximum reduction of over 2 %.
Figure 3 displays the percentage difference in sea ice extent and volume for
stan-fsd compared to ref. In addition, it shows the spread of twice the standard
deviation of these simulations as a measure of the interannual variability.
The impact on the pan-Arctic scale is small, with sea ice extent and volume
reductions of up to 1.2 %. The difference in sea ice area reaches a
maximum in August whereas the difference in sea ice volume peaks in
September. The differences in both extent and volume evolve over an annual
cycle, with minimum differences of
Figure 4 shows the absolute difference in the mean cumulative annual melt components between the two simulations. The plot shows lateral, basal, top, and total melt (as defined in Sect. 2.1). A large increase can be seen in the lateral melt; however the change in total melt is negligible. This is because the lateral melt increase is largely compensated by a reduction in basal melt. The top melt also shows a negligible change.
Difference in the cumulative lateral (green ribbon, dashed), basal (grey ribbon, dashed), top (red ribbon, dashed), and total (blue ribbon, solid) melts averaged over 2007–2016 between stan-fsd and ref. The ribbon shows, in each case, the region spanned by the mean value plus or minus 2 times the standard deviation for each simulation. A large increase is observed in the total lateral melt; however this is mostly compensated by a reduction in the basal melt, leading to a negligible change in total melt. A small reduction in top melt can be seen. The predicted difference in basal melt is also shown on the plot (pink ribbon, dotted); this shows the expected change in basal melt accounting only for the reduction in sea ice concentration at grid cell scale from ref to stan-fsd.
Figure 4 also shows the change in basal melt in stan-fsd only accounting for the loss of basal surface area available for melting. To explain how this is calculated, imagine for a given time step the sea ice fraction for that grid cell in the stan-fsd simulation is 0.81 and in the ref simulation it is 0.90. If this physical reduction is the only factor causing changes to the total basal melt, then the basal melt rate per unit grid cell area would also reduce by the same factor of 10 % from ref to stan-fsd. The reduction in the total basal melt volume can then be calculated for this grid cell accounting only for the reduction in sea ice fraction as the product of 0.1, the basal melt rate per unit grid cell area, and the area of the grid cell. This process can be repeated over every grid cell to obtain the total reduction in basal melt volume accounting only for reduction in sea ice concentration. The agreement (to within 1 standard deviation) between this synthetic reduction in basal melt and the actual reduction in basal melt suggests that the loss of ice area by lateral melt is sufficient to explain most of the basal melt compensation effect. Figure 5 shows the spatial distribution for the predicted reduction in basal melt from stan-fsd to ref, the actual reduction in basal melt, and the difference between the actual reduction and predicted reduction in basal melt. These map plots are presented as monthly averages for March, June, and September averaged over 2007–2016. Figure 5 shows that the predicted basal melt can capture the regional distribution of the changes in basal melt from ref to stan-fsd, not just the area-integrated quantity.
Predicted reduction in basal melt rate from stan-fsd to ref
Figure 6 explores the spatial distribution in the changes in ice extent and
volume for 3 months over the melting season, March, June, and September.
Data are shown only for regions where the sea ice cover exceeds 5 % of the
total grid cell. These results show the differences increase in magnitude
through the melting season. Although the pan-Arctic differences in extent
and volume are marginal, Fig. 6 shows distinct regional variations in sea
ice area and thickness metrics. Reductions in the sea ice concentration and
thickness are seen both within and beyond the MIZ with reductions of up to
0.1 and 50 cm observed respectively in September. Within the pack ice,
increases in the sea ice concentration of up to 0.05 and ice thickness of up
to 10 cm can be seen. In September the biggest increases in thickness are
directed along the North American coast, particularly within the Beaufort
Sea. To understand the non-uniform spatial impacts of the FSD, it is useful
to look at the behaviour of
Difference in the sea ice concentration
It has been previously discussed that the floe size parameters used within
the WIPoFSD model are poorly constrained by observations. In this section
experiments are performed using different permutations of these parameters
to assess model sensitivity to the form of the FSD. It is valuable to
consider how changes to each FSD parameter are likely to impact the
distribution: increasing the magnitude of
For the first study the
As Fig. 4 but the difference between (A) compared to
stan-fsd i.e. the impact of changing
As Fig. 6 except now the difference between (A) compared
to stan-fsd is given, i.e. the impact of changing
A further 17 sensitivity studies using different permutations of the
parameters have been completed. These are formed by varying the three key
defining parameters of the FSD shown in Fig. 1 in order to span the range of
values reported in observational studies: for
Relative change (%) in mean September sea ice extent
A total of 14 of the 17 permutations for these sensitivity studies are generated by
selecting all the different
Definitions of the parameters relating to the sensitivity studies described in Table 2.
A series of sensitivity studies have been performed to explore the behaviour of the WIPoFSD model and understand how it interacts with other model components. Table 1 defines the important parameters considered in this section and Table 2 provides a summary of the sensitivity experiments performed. The first two entries in Table 2, stan-fsd and ref, refer to a standard set-up using the standard FSD parameters described above and a constant floe size of 300 m respectively. Studies (A)–(C) are a selection of the simulations described in Sect. 4.2 to allow a comparison between model sensitivity to the parameters that define the FSD and model sensitivity to other relevant parameters and components within the WIPoFSD model. In the following section a bracketed letter will follow descriptions of sensitivity studies, which correspond to the letter assigned in Table 2.
The details of the sensitivity studies to explore the behaviour of the CICE–ML–WIPoFSD model. Parameters discussed here defined in Table 1.
Table 3 reports key metrics for the sensitivity studies described in Table 2, plus a selection of the different sensitivity studies described in
Sect. 4.2. For each experiment the September sea ice extent and volume
size are reported for both the full sea ice extent and MIZ only (taken as a
mean between 2007 and 2016), with the MIZ defined here as regions with
between 15 % and 80 % sea ice cover. In addition, the mean cumulative
lateral, basal, top, and total melts until September are reported in each
case, and the September mean
A summary of the metrics for each of the sensitivity
studies described in Table 2. Metrics are reported for sea ice extent, MIZ
extent, total sea ice volume, MIZ volume, mean
The shape of the FSD between its limiting values is defined by
The second sensitivity experiment assumes that
The results in Table 3 show imposing the time-varying
Annual variation in
Annual variation in mean
The two processes currently represented in the model that actively reduce
The results in Table 3 show that the wave–
Difference in the sea ice concentration
The floe restoration rate is the parameter,
In Fig. 13 we show the evolution of simulations stan-fsd, (F), and (K) over 2015
averaged over selected grid cells. The year 2015 has been chosen as representative over the 2007–2016 period. There are two subplots: the first gives
Daily variation in
The first-order impact of introducing a variable floe size is on the lateral
melt volume. Equation (1) shows the lateral melt volume is calculated from
several parameters beyond just floe diameter,
Table 3 shows that all four of these sensitivity studies did not produce a
large model response in terms of the overall sea ice extent and volume.
Reducing the floe shape parameter (L) produced the strongest response in the
lateral melt volume, and more generally the model metrics were more
sensitive to
The minimum ocean mixed-layer depth is a constant within the prognostic mixed-layer model required to prevent the mixed-layer depths reaching unrealistically small values. As a standard it is set to 10 m. The depth of the mixed layer is important for the strength of mixed-layer feedbacks, with a deeper mixed layer acting to damp any feedbacks via mixed-layer properties. These feedbacks include the albedo feedback mechanism and the negative feedback of increased lateral and basal melts (meltwater perturbs the mixed-layer properties towards less favourable melting conditions). Sensitivity studies are performed with the minimum mixed-layer depth both reduced to 7 m (P) and increased to 20 m (Q).
The challenge with this set of experiments is that, unlike the other sensitivity studies presented here, it acts to influence the evolution of the sea ice both via changes in the lateral melt and via the basal melt and sea ice freeze-up rates, determined by ocean properties. Experiment (P) shows a small increase in the total sea ice extent and volume and (Q) a small decrease; however both result in larger increases in the MIZ extent and volume. In comparison to other sensitivity studies, the changes in the lateral and basal melt are small, suggesting that mixed-layer feedbacks do not have a significant role in the impacts of the FSD found in stan-fsd compared to ref. It should be noted, however, that the evidence presented here is not enough to rule out the existence of multiple compensating feedback processes.
We present here a series of simulations and additional sensitivity studies completed with the newly developed WIPoFSD model to explore the impacts of a variable power-law-derived floe size distribution model on the Arctic sea ice. It is useful to consider the physical mechanisms that drive the simulation results. It was previously noted that the increase in lateral melt observed when imposing the WIPoFSD model was compensated by a loss in basal melt, resulting in a more moderate increase in the total melt. Within the model there are three possible mechanisms causing the limited basal melt. Firstly, the increase in lateral melt will correspond to a reduction in available ice area for basal melting. It is shown in Figs. 4 and 7 that this mechanism is able to explain most of the reduction in basal melt, but the difference remains large enough that further mechanisms need to be considered. The second mechanism concerns the melting potential of the ocean. If there is a large enough increase in the lateral melt to result in insufficient melting potential, both the lateral and basal melt will be reduced proportionally, as described in Sect. 2.1. A simulation (not presented) to explore this impact shows it has only a limited impact on the basal melt, and not enough to explain the observed compensation effect. The third mechanism concerns lateral melt feedback on the basal melt rate via the perturbation of mixed-layer properties. Higher freshwater release from the increase in lateral melt will lower the temperature and salinity of the ocean mixed layer, which will reduce the basal melt rate. However, the lateral melt increase also reduces the ice concentration, lowering the albedo of the ice–ocean system. This increases the absorption of shortwave solar radiation into the mixed layer, raising the temperature of the mixed layer; i.e. it has the opposite effect of the increased freshwater input. These two competing feedbacks explain the overprediction of basal melt in Fig. 4 but underprediction of basal melt in Fig. 7. The increase in total melt observed in Fig. 7 will likely correspond to a more efficient use of the available melt potential and the aforementioned albedo-feedback mechanism. The interaction between the mixed layer and FSD is further explored through the (P) and (Q) sensitivity studies where the minimum mixed-layer depth was reduced and increased respectively. These studies provide further evidence that mixed-layer feedbacks are not a leading-order effect of the FSD, given the very small perturbations of the melt component from the stan-fsd simulation. Larger changes are seen for the sea ice extent and volume metrics. However, the same mixed-layer feedbacks that change the melt rates can also independently influence the freeze-up rate of sea ice; hence it is not possible to directly attribute the changes produced by varying the minimum mixed-layer depths specifically to WIPoFSD-related feedbacks. It should also be noted that the prognostic mixed-layer model used here provides a limited representation of sea ice–ocean interactions and feedbacks. The strength of these interactions may increase within a fully coupled sea ice–ocean model (Rynders, 2017).
The series of sensitivity studies to both the floe size parameters and other
aspects of the WIPoFSD model are useful to understand the limitations of the
model. An important result is the limited sensitivity of the model to the
The sensitivity studies also give insight into the impact of waves on the
sea ice cover. In particular, the two sensitivity studies that switch off
the lateral melt–
As stated above, the model shows a strong sensitivity to the floe size
parameters with some selections of the WIPoFSD parameters showing moderate
increases in the sea ice extent and volume, and other selections driving
reductions of these values by over 50 % in September. The limited
observational data available to constrain the selected parameters is
therefore a significant challenge of this modelling approach. Furthermore, a
not insignificant model response of the order of 5 % relative to ref has been
observed to sensitivity experiment (E) performed here to explore the impacts
of the non-uniform
The WIPoFSD model used here assumes a power-law distribution with the
exponent
The reference simulation (ref) used in this study underpredicts summer sea ice concentration in the pack ice but overpredicts the concentration at the sea ice edge, consistent with other studies that use the CICE sea ice model (such as Schröder et al., 2019). An analysis of the historically forced simulations used within of the Coupled Model Intercomparison Project Phase 5 (CMIP5) found that coupled models consistently performed poorly in capturing the regional variation in sea ice concentration, showing this problem is not specific to CPOM CICE simulations (Ivanova et al., 2016). This suggests that models currently underestimate the role of the MIZ in driving the seasonal sea ice loss. The WIPoFSD model is shown here to have a non-uniform impact on the sea ice cover, with an enhancement in lateral melt and a corresponding reduction in sea ice concentration within the MIZ, as shown in Fig. 6. Whilst the changes are generally small, it shows that the use of an FSD model, either in the described form or otherwise, may be an important step towards improving the accuracy of sea ice models.
Climate model representations of sea ice currently assume that the size of floes that make up the sea ice is constant; however, observations show that floes adopt a distribution of sizes. A power law generally produces a good fit to observations of the floe size distribution (FSD), though the size range and exponent reported for this distribution can vary significantly between different studies. A power-law-derived FSD model including a waves-in-ice module (WIPoFSD) has been incorporated into the Los Alamos sea ice model coupled to a prognostic mixed-layer model, CICE–ML. In the WIPoFSD model, the FSD is defined by a lower floe size cut-off, upper floe size cut-off, and exponent. A variable FSD tracer is also introduced, which varies in response to lateral melting, wave break-up events, and freezing conditions. The lower and upper floe size cut-offs and exponent are set to fixed values. A standard set of parameters for the WIPoFSD model is identified from observations and the results of a sea ice simulation using these parameters is compared to one with a constant floe size of 300 m. Inclusion of the WIPoFSD model within CICE–ML results in increased lateral melt compensated by reductions in basal melt, resulting in only moderate impacts on the total melt. The primary mechanism by which the increased lateral melt reduces the basal melt is shown to be the reduction in available ice area for basal melt. The impact is not spatially homogeneous, with losses in sea ice area and volume dominating in the marginal ice zone (MIZ). These impacts partially correct existing model biases in the stand-alone CICE–ML model, suggesting the inclusion of an FSD is an important step forward in ensuring that models can produce realistic simulations of the Arctic sea ice.
A series of sensitivity experiments explore the limitations of the model. The model does show a strong response to a reduction in wave attenuation rate, suggesting this is an important component in understanding wave–sea ice interactions. Different selections of parameters for the FSD show a large impact on the modelled sea ice state, with some showing a moderate increase in mean September sea ice extent and volume, with others reducing these metrics by over 20 % and 50 % respectively. A newly defined parameter, effective floe size, is found to be a good predictor of model response for simulations where the lower floe size cut-off and power-law exponent are fixed. The impact of a non-uniform exponent was also explored based on observations that these parameters evolve for a given region of sea ice. Results suggest that this parameter could further enhance the differential behaviour seen between pack ice and the MIZ in response to the imposition of an FSD. These sensitivity studies also showed that the choice of WIPoFSD parameters is a source of much larger model uncertainty than other constants used within the lateral melt parameterisation, justifying the focus on developing an FSD model as a priority for improved accuracy of sea ice modelling.
Whilst the model presented here does make a major assumption that the floe size distribution adopts a power law, this is consistent with most observations. Furthermore, it has been shown that the model can easily be modified to adapt to additional findings such as the inclusion of a non-uniform exponent. This means the WIPoFSD model is a useful tool for assessing the importance of the FSD in the evolution of sea ice, particularly the seasonal retreat. Climate models require an important balance to be maintained between physical fidelity and computational expense. The simplicity of the WIPoFSD model makes it a useful stepping stone for the development of new parameterisations of floe size within climate models that can reasonably capture the physical impacts of the FSD without a large computational cost. Planned observational studies such as MOSAiC should help in the development of these novel parameterisations.
Model output used in this paper is publicly available via the
University of Reading research data archive
(
LH, with support from YA, developed the original version of the WIPoFSD model with a coupled CICE–NEMO framework. DS adapted the model into the CPOM CICE stand-alone set-up. AWB further developed the WIPoFSD model and completed the simulations and analysis under the supervision of DLF, DS, LH, JKR, and YA. DS provided additional technical support. AWB composed the paper with feedback and contributions from all authors.
The authors declare that they have no conflict of interest.
Lucia Hosekova and Yevgeny Aksenov would like to express gratitude towards Timothy D. Williams (Nansen Environmental and Remote Sensing Center, NERSC, Norway), Dany Dumont (Institut des sciences de la mer de Rimouski, Québec, 85 Canada), and Vernon A. Squire (University of Otago, New Zealand) for their kind advice on ice–wave interaction. Stefanie Rynders (National Oceanography Centre, Southampton) should receive credit for contributing to the development of the WIPoFSD model within the coupled CICE–NEMO framework alongside Lucia Hosekova and Yevgeny Aksenov. Coupled CICE–NEMO simulations were performed on the ARCHER UK National Supercomputing Service (
This research has been supported by the NERC industrial CASE studentship with the UK Met Office (grant no. NE/M009637/1), the NERC (grant nos. NE/R016690/, NE/R000654/1), the European Union Seventh Framework Programme SWARP (grant no. 607476), and the Joint UK BEIS/Defra Met Office Hadley Centre Climate Programme (grant no. GA01 101).
This paper was edited by Jennifer Hutchings and reviewed by two anonymous referees.