Evaluation of a Coupled Wave-Ice Model in the Western Arctic

,


Introduction
As the retreat of Arctic sea ice promotes increased ocean surface wave activity (Thomson & Rogers, 2014), interactions between waves and sea ice could play an elevated role in the Arctic climate system.Increasing wave heights have already been observed in the Beaufort Sea where fetch, the open water distance available for wave development, has expanded due to seasonal sea ice loss (X.L. Wang et al., 2015;Liu et al., 2016;Thomson et al., 2016;Smith & Thomson, 2016).Summer sea ice is also becoming less compact near the newly exposed, rougher seas that surround the remnant of sea ice left in the central Arctic (Martin et al., 2014;Thomson, Ackley, et al., 2018;Squire, 2020).
Interactions between waves and sea ice occur in the marginal ice zone (MIZ), the partially ice-covered region that separates interior pack ice from open ocean.We do not have direct estimates of the MIZ's location and extent because measuring waves in ice at basin scale is an ongoing challenge.While the physical significance of the dynamic MIZ stems from wave presence near the ice edge, a practicable proxy based on intermediate ice concentrations is often used to represent the MIZ.This proxy is the region with sea ice concentration (SIC) between 15% and 80% and is readily available from passive microwave satellite estimates (Comiso et al., 1997;Strong & Rigor, 2013;Strong et al., 2017).The Arctic MIZ extent, when defined as the area with 15-80% SIC, may be expanding relative to the retreating pack ice (Aksenov et al., 2017;Rolph et al., 2020), and wave-ice interactions are emerging as a leading control on seasonal sea ice and the future state of the MIZ (Thomson, Ackley, et al., 2018).
We can obtain a basic understanding of wave statistics through bulk wave characteristics, e.g., significant wave height, but full wave spectra contain additional information that becomes critical for frequency-dependent wave-ice interactions.When considering wave spectra in ice, we expect to see a narrowing of the spectral bandwidth as energy is concentrated at the low frequencies indicative of swell (Thomson et al., 2019).This narrowing occurs as waves enter the ice due to dependence of the wave-attenuation rate on frequency, where low-frequency energy is better able to survive compared to highfrequency energy (Wadhams et al., 1988;Meylan et al., 2014;Rogers et al., 2016).We do not have a comprehensive explanation for the physical processes responsible for the dissipation of wave energy in the MIZ (Meylan et al., 2018).
Swell, the low-frequency waves that have traveled outside of their original windgeneration area, can penetrate hundreds of kilometers inside the sea ice edge when the wave heights are large, according to observations from the Antarctic (Kohout et al., 2014;Li et al., 2015) where wave periods can become longer than in the Arctic Ocean.In contrast, high-frequency waves generated by local winds tend to dissipate during their first 10-20 km of travel into the sea ice field, according to Squire and Moore (1980).However, Masson and Leblond (1989) developed a model explaining how local wind waves can be generated in areas of low ice concentration and sparse ice floes.In surface buoy measurements, Smith and Thomson (2016) found support for the open water distance between floes as a control parameter for wave energy.Intense winds acting directly on sea ice, rather than on open water, can drive local wave generation even in Arctic pack ice (Johnson et al., 2021).While these studies have provided a constructive framework for studying ice-affected wind waves, we currently have a limited understanding of the impact and prevalence of locally generated, high-frequency wind waves in sea ice.
The absence thus far of wave-ice interactions in coupled climate models may explain some of the differences in Arctic sea ice between models and observations reported by several studies (e.g., Shu et al., 2020;Notz & Community, 2020).Tietsche et al. (2014) found that model errors in sea ice concentration are most severe in the MIZ, and Blanchard-Wrigglesworth et al. (2021) hypothesize that ocean waves may be responsible for the greater high-frequency variability in sea ice extent found in observations compared to CMIP models, which do not simulate wave-ice interactions.
The scarcity of observations of waves in ice continues to be an obstacle for both model evaluation and theoretical understanding.Obtaining valid measurements of wave spectra is a challenge when sea ice obscures the ocean surface.The variety of ice conditions, ranging from sparse pancake floes to extensive sheets of ice, complicates interpretation, and existing datasets sample a limited range of ocean and sea ice conditions (Collins et al., 2015).Furthermore, for any fixed location, there is a short window of time during the ice melt and growth seasons when waves in ice can be observed.Remote sensing is a promising path for extending spatial coverage and obtaining more robust waveice statistics, and recent efforts have produced estimates of wave heights in the presence of ice using satellite measurements (Ardhuin et al., 2017(Ardhuin et al., , 2019;;Stopa et al., 2018;Horvat et al., 2020).Nevertheless, basin-scale, long-term observations from remote sensing are not yet available.Multi-year in situ observations, however, are available from three recent field campaigns in the Western Arctic: the Arctic Sea State (Thomson, Ackley, et al., 2018), the Beaufort Gyre Observing System (BGOS), and the Stratified Ocean Dynamics in the Arctic (SODA) programs.These three sets of measurements are a focus of this study and are described further in section 2.2.
Here, we interpret this collection of in situ observations spanning 2012-2019 in the Beaufort Sea from subsurface moorings, supplemented by deployments of freely drifting surface buoys during wave events, measuring ocean surface waves in partial ice cover.
We compare the in situ observations with results from the Roach et al. ( 2019 In section 2, we describe the Roach et al. (2019) model and the in situ observations.We relate the methods of model-observation comparison in section 3 and present results of the comparisons in section 4. We discuss the results in section 5 and conclude in section 6.

Coupled Wave-Ice Model
We analyze results from an experiment using the Los Alamos sea ice model, CICE5 (Hunke et al., 2015) coupled to the ocean surface wave model, Wavewatch III v5.16 (The WAVEWATCH III (R) Development Group (WW3DG), 2016).To simulate wave-ice interactions, the model includes a prognostic FSD developed by Roach, Horvat, et al. (2018) and Roach et al. (2019).Floe sizes are determined by lateral growth and melt, welding of floes in freezing conditions, and the ocean surface wave spectrum through floe fracture and wave-dependent new ice formation.Attenuation of wave spectral energy in ice depends on mean floe size, ice concentration, and ice thickness based on an empirical fit to floe-scattering theory, including a supplemental attenuation term for long wavelengths (Meylan et al., 2021).Figure S1 includes illustrative values of wave attenuation coeffi- Both the sea ice model and ocean surface wave model evolve freely while forced with JRA-55 atmospheric reanalysis (Kobayashi et al., 2015;Japan Meteorological Agency, Japan, 2013) and coupled to a slab ocean model (SOM) (Bitz et al., 2012).The SOM is a single-layer model, diagnosed from the monthly climatology of a control run of the Community Climate System Model Version 4 (CCSM4), that specifies mixed-layer depths constant in time, annually periodic ocean surface currents, and an annually periodic ocean heat transport convergence, the Q f lux ; all three SOM input parameters vary in space.
The sea ice and wave models are on a displaced-pole nominal 1 • grid (gx1v6), and the size of model grid cells near observations in the Beaufort Sea is approximately 50 by 50 km.The simulation spans 1979-2019, and we analyze hourly model output over 2012-2019 in line with the period of observations.The experiment is identical to FSD-WAVEv2 in Roach et al. (2019), except we use a higher coupling frequency between the wave and sea ice components.Here, the wave and sea ice components exchange the ocean surface wave spectrum and sea ice concentration, thickness, and mean floe size every hour to better resolve short-timescale wave-ice interactions.

In Situ Observations
By aggregating sources of observations that span multiple years with generally continuous sampling, we compile a relatively large dataset to support statistical model evaluation.This dataset, denoted henceforth as BGOS-SODA, consists of two groups of subsurface moorings, spanning 2012-2019 and five locations in the central Beaufort Sea (Table 1; Figure 1).In this section, we briefly review each source of observations.The first group included in the BGOS-SODA aggregate dataset comes from the Beaufort Gyre Observing System (Krishfield et al., 2014).BGOS includes two subsurface moorings, BGOS-A and BGOS-D, with upward-looking Nortek Acoustic Wave and Current (AWAC) instruments for surface tracking.BGOS-A and BGOS-D sample every hour and began collecting measurements in 2012 and 2013, respectively.Raw data are processed following Herbers et al. (2012), Kuik et al. (1988), andThomson, Girton, et al. (2018) and converted to wave energy spectra.Data from 2012 is reported in Thomson and Rogers (2014), and a reanalysis of the same data is found in Smith and Thomson (2016).Here, we employ an extended dataset that is mostly continuous from 2012-2018 (Thomson, 2020).
The second group comes from the Stratified Ocean Dynamics in the Arctic project.
Three subsurface moorings, denoted SODA-A, SODA-B, and SODA-C, use the upwardlooking Nortek Signature Doppler profiler for acoustic surface tracking.Raw data from SODA are quality-controlled using methods comparable to the BGOS methods, producing measurements of surface wave spectra sampled every two hours.Data from the SODA moorings first appear in Brenner et al. (2021), but the wave spectra have not been previously reported.The SODA dataset spans 2018-2019.
Both sets of subsurface moorings detect surface gravity waves via altimeter measurements of surface displacement.An important nuance of the moorings is that the surface tracking simultaneously measures surface gravity waves and sea ice draft.However, the signal from surface waves can be distinguished from that of ice based on spectral characteristics.This separation is part of the quality-control process.Deformed sea ice produces a "red" spectrum with under-ice topography exhibiting peak spectral variance primarily at low frequencies (Rothrock & Thorndike, 1980), whereas the surface gravity waves tend to have peak energy in the frequency range of 0.5 to 0.05 Hz, causing sea surface displacements with distinct spectra in that range.Calm waters and smooth ice both produce flat ("white") spectra.If both ice and waves are present, moorings measure a superposition of both signals.
The processing strategies for the mooring datasets make use of these different spectral shapes to identify and separate wave signals from sea ice.The postprocessed wave datasets from BGOS and SODA exclusively contain observations where the surface gravity wave signal is sufficiently strong to be considered a wave, determined by the spectral shape and the total energy in the frequency range of ocean surface waves.If the icedraft signal is strong while the surface wave signal is weak, the instrument may be unable to produce a valid wave measurement.These instances where only ice draft is detected are excluded from the wave datasets considered here.The resulting wave dataset almost exclusively contains observations with minimal ice draft detected; when the mooring is in partial ice cover, valid wave measurements appear to come from the water between ice floes.
Separately, we include data from free-drifting surface buoys as a supplemental line of comparison.These measurements come from Surface Wave Instrument Floats with Tracking (SWIFTs) (Thomson, 2012) that were deployed for short periods of time during large wave events in the Oct-Nov 2015 Arctic Sea State campaign.The SWIFTs measure ocean surface velocities and infer wave energy spectra every hour using GPS tracking (Herbers et al., 2012).Because the SWIFTs do not sample data continuously over extended periods of time, we cannot use their results for statistical model evaluation.The surface buoy data from the SWIFTs nonetheless inform interpretation of both the model results and the BGOS-SODA observations.

Methods
A primary goal of this study is to objectively compare the in situ observations (located at specific points) and the model results (generalized over a region).We limit the model-observation comparison to the central Beaufort Sea region surrounding the observations: latitudes 72 • N to 79 • N, longitudes 165 • W to 130 • W (Figure 1).Ideally, we would focus on model results from the particular grid cells that contain the location of each observation.However, even small errors in the model ice edge position and ice concentration have substantial impacts on where waves occur in the ice, so we cannot expect the coupled model to precisely replicate the observed waves at a given location.Rather, we assess whether the general character of waves in the region is accurately represented in the model.manuscript submitted to JGR: Oceans

Distance Inside the Ice Edge
To generalize the comparison, we group observations and model results based on a calculated distance from the ice edge, denoted as ∆ dist .Following convention, the ice edge is defined as the 15% ice concentration contour, roughly separating partial ice cover from open water.∆ dist for a given location inside the ice cover is calculated purely from the ice concentration.The calculation is the Haversine distance to the nearest open water location, i.e., an ocean grid cell with SIC less than 15%.We note that the ∆ dist metric does not directly represent the distance along which wave attenuation occurs.The distance into the ice that a wave will travel before full dissipation depends on its direction of propagation, whereas this grouping by ∆ dist rather distinguishes locations based on their separation from open ocean.For simplicity, we show three ∆ dist groups: open water (SIC < 15%), 0-100 km inside the ice edge (equivalent to approximately two 50x50 km grid cells), and 100+ km inside the ice edge.We choose to group the data based on ∆ dist for three reasons: 1. Waves attenuate exponentially with distance as they enter ice cover (Squire & Moore, 1980;Wadhams et al., 1988;Meylan et al., 2018).
2. Groupings based on ∆ dist reduce dependence on replicating the true ice-edge position in the model; this enables comparison between locations that are similar in the model and the in situ observations (based on their relative ∆ dist ), rather than comparison between only the precise locations of the observations.3. Specific estimates of ice concentration from passive microwave satellite data are highly uncertain in partial ice cover, but identification of the 15% concentration contour has higher confidence based on good agreement with ice-edge positions determined by aircraft (Cavalieri et al., 1991;Fetterer, 2002;Fetterer et al., 2017).
We estimate the time-varying ∆ dist for each in situ observation using the NOAA/NSIDC Climate Data Record (CDR) of sea ice concentration, a daily satellite product derived from passive microwave observations (Fetterer et al., 2017).We regrid the satellite estimates from the native 25-km resolution to the model's nominal 1 • resolution grid before computing ∆ dist , ensuring consistency between the model and observations.This produces a ∆ dist for each in situ observation and each model grid cell in the Beaufort Sea at all points in time.

Nondimensional Scaling for Wind-Generated Ocean Waves
To support interpretation of wave statistics, we employ nondimensional scaling relations for wind-generated waves following Young (1999).These relations enable separation of wind waves from swell and provide an estimate of the implied fetch for observed wind waves in partial ice cover.We calculate the following nondimensional variables for wave energy E, frequency F , and fetch distance X: where g is the gravitational acceleration; U 10 is the 10-meter wind speed at the location of each in situ observation and model grid cell from JRA-55 reanalysis; H s is the significant wave height, defined as 4σ where σ 2 is the variance of the sea-surface height; f p is the peak frequency; and x is the fetch, i.e., the distance over which waves are generated by local winds.H s and f p are measured in situ and provided in model output.The fetch x is not measured but rather inferred for specific wind waves as described below; we refer to this variable as the implied fetch.
In the marginal sea region of the observations considered, wave generation is generally limited by fetch rather than wind duration (Hasselmann et al., 1973;Thomson & Rogers, 2014).Several studies have developed empirical estimates of power laws for E vs. X and F vs. X that describe wind-generated waves in a fetch-limited regime.Young (1999) combined these estimates into the relations E = (7.5 ± 2.0) × 10 −7 X 0.8 (2) which apply at least until reaching a fully developed limit for pure wind seas at E max = (3.6 ± 0.9) × 10 −3 and F min = 0.13 ± 0.02.Using equation (1), we reformulate these power laws in terms of the variables available from measurements and modeling, E and We identify waves that are accurately described by fetch-limited local wind generation, i.e., wind waves, as those that fall within the uncertainty bounds of the line defined by the power law in equation ( 4).If a spectrum has less energy E than predicted by the wind-wave power law for a given frequency F , and it has a wave age greater than 1, we determine that the spectrum represents swell, i.e., long-period waves produced by nonlocal winds.
Wave age c U is a nondimensional parameter defined by the ratio of the dominant phase speed c p to the wind speed U 10 , where we treat c p = g 2πfp following the deep-water limit for surface gravity waves.When the wave age exceeds 1, waves travel faster than the winds.We note that wave age can be expressed in terms of F using equation (1) such that c U = (2πF ) −1 , and wave age is greater than 1 when F is less than 1 2π .
Taking only the spectra that appear to be fetch-limited local wind waves, based on equation ( 4) and wave age as described above, we can calculate an implied fetch x corresponding to each wind-wave spectrum.This dimensional variable x is recovered by solving for the nondimensional X in equation ( 2) based on the known energy E, then using equation (1) to restore the dimension.The implied fetch is an estimate of the open water distance that would be required for local winds to generate a given wind-wave spectrum.

Significant Wave Height
We compare the significant wave height H s statistics by aggregating observations from the five BGOS and SODA moorings into a single dataset.Figure 2  We find more notable differences between the distributions in partial ice.The 0-100 km group (Figure 2b) displays a strong contrast, where the model's distribution is dominated by the smallest waves near the lower-limit of the domain, while observations show a higher prevalence of large waves.The model has only 13% of H s greater than 1.0 m, whereas 35% of observations exceed 1.0 m.
The 100+ km ∆ dist distributions differ most strongly in terms of kurtosis (Figure 2c).The model has a prominent peak at the smallest end of wave heights, paired with a thicker tail of large waves.9% of the model's H s exceed 1.0 m, whereas the 104 observations at 100+ km do not report any H s beyond that magnitude.Only 5% of observed H s exceed 0.75 m, and the distribution is relatively uniform between 0.30 and 0.75 m.We discuss how sampling biases could affect this model-observation comparison in discussion section 5.2 below, but we emphasize here that the absence of H s beyond 1.0 m in BGOS-SODA observations cannot be attributed to instrument errors.Such large wave heights exceed minimum wave height detection limits by significant margins and are reported in open water and at 0-100 km ∆ dist .The absence of H s greater than 1.0 m in the BGOS-SODA observations of Figure 2c is a robust result within the limit of our sample size.To provide some insight on differences in the distributions, we turn to the spectra.

Wave Spectra
Even if the bulk wave parameter H s appears accurately represented, the model can have significant biases in how wave energy is distributed between low and high frequencies.Inspecting the full wave spectra reveals that similar H s may have dramatically different signatures in frequency space, and these model-observations differences can highlight disagreement in wave attenuation and generation processes.Additionally, we introduce spectra from the SWIFT surface buoys as a supplemental line of comparison, recalling that SWIFTs preferentially sample significant wave events as part of experiment design.Gray shading represents the approximate BGOS-SODA detection limit and is included on all panels for ease of comparison.
The open water (SIC < 15%) spectra are generally in agreement between the moorings and the model (Figure 3a,g).We can identify the prominent spectral shape of locally developed wind waves in open water in both panels.These spectra exemplify a characteristic power-law relationship between energy and frequency (different from the nondimensionalscaling power law described in section 3.2) in the high-frequency spectral tail, i.e., the portion of the spectrum where frequency f is higher than the peak frequency f p .In open water, the spectral tail follows a consistent f −4 slope down from f p (Phillips, 1985;Thomson et al., 2013;Lenain & Melville, 2017).
In sea ice, the spectral tail is typically steeper than f −4 in observations and model results.This steeper tail has been reported in observations before (Rogers et al., 2016;Thomson et al., 2021) and is consistent with the notion that sea ice dissipates high-frequency energy most effectively.Data from the 0-100 km transition into ice, illustrated most clearly by the fan of spectral tails in the SWIFT spectra (Figure 3e) but also visible in the moorings (Figure 3b), demonstrates that waves undergo a frequency-dependent attenuation while traveling through partial ice and preferentially lose energy at the highest frequencies.The model does not show the same spread of spectral-tail slopes seen in observations, even in the first 0-100 km of partial ice (Figure 3h); all energy at high frequencies has been eliminated.A spectral shape similar to the model results, however, can be seen in some observed spectra at 0-100 km (Figure 3b,e), albeit shifted so that f p tends to be at slightly higher frequencies in observations.
Moving to 100+ km ∆ dist , we see a structural difference between the spectra in the model and those in the observations.In Figure 3i, the model shows waves retaining significant low-frequency energy far into the ice, and all of these model spectra are devoid of any high-frequency energy.On the other hand, the BGOS-SODA observations (Fig- ure 3c) have a spectral signature that is, perhaps surprisingly, reminiscent of a short-wave subset of the open water spectra.These spectral tails follow the f −4 slope, and all energy is at relatively high frequencies.The contrast between the model's low-frequency energy and the BGOS-SODA high-frequency waves suggests that there are two separate modes displayed in the spectra at 100+ km ∆ dist .The SWIFTs in Figure 3f show bimodal spectra that appear to have a swell wave group at lower frequencies concurrent with a local-wind-wave group at higher frequencies.Notably, the swell group in these bimodal spectra has higher f p and less energy compared to the model results in Figure 3i, and energy is mostly below the BGOS-SODA detection limit.

Fetch Scaling
We find that the distinction between swell and wind waves generally can be reduced to the nondimensional scaling of two bulk wave parameters, H s and f p , rather than requiring inspection of the full spectra.Whereas the H s distributions in Figure 2 compare amounts of wave energy, the distributions in Figure 4 compare how the swell and windwave modes are represented.Figure 4 applies the nondimensional scaling relations between energy and peak frequency to observations and the model, and it also includes the power law for local wind-wave generation (see section 3.2).The points that follow the power law are identified as locally generated wind waves, while points located below the line, i.e., those with less energy than predicted by the power law for a given peak frequency, and with wave age greater than 1 are identified as swell.These modes are not always well-separated because nonlocal swell and local wave generation can co-occur.
The power law captures most of the open water (Figure 4a) observations and model output, but a nonlocal component can be identified in both the model and observations that pulls some of the points below the power-law line and towards low F such that the wave age is greater than 1.This consistency between the model and observations suggests that there is not a significant bias in the prevailing wave modes in open water.waves, with the fully developed limit (Emax and Fmin) for pure wind seas denoted in red (Young, 1999).Dashed line at F = (2π) −1 indicates wave age = 1; where F < (2π) −1 , wave age > 1.
manuscript submitted to JGR: Oceans In partial ice, the results for the model become distinct from the observations.At 0-100 km ∆ dist (Figure 4b), the model immediately clusters at lower energies away from the power law, i.e, the swell mode dominates.In observations at 0-100 km, we see a spread both on and off the power-law relation.Recall that this spread, due to the combined presence of swell, local wind waves, and attenuation by the ice cover, can be seen in the mooring and SWIFT spectra (Figures 3b,e).
At 100+ km ∆ dist , separation between the model and the observations is most definite (Figure 4c).The model displays only the swell mode of lower energies with wave age greater than 1 and is removed from the wind-wave power law even more strongly than in the 0-100 km zone.The observations behave differently; they do not continue spreading away from the power law toward lower energies as seen in their 0-100 km subset.Instead, they return to clustering along the power law, indicating local wind-wave generation at 100+ km ∆ dist .The observations thus suggest that local wave generation is a significant source of wave activity far within the marginal ice zone, and this source is not captured in the model.

Discussion
While the coupled wave-ice model of Roach et al. (2019) broadly captures the range of significant wave heights in BGOS-SODA observations, comparing the shapes of the H s distributions suggests there may be substantial differences which are not apparent when considering the bulk parameter for wave energy alone.The spectral details are important given the frequency dependence of wave attenuation and floe fracture.Two key questions emerge from the spectra and nondimensional scaling at 100+ km ∆ dist : why do BGOS-SODA observations show wind waves but no swell, and why does the model show swell but no wind waves?

Wind Waves
Sea ice is known to filter out high-frequency wave energy, but BGOS-SODA observations nevertheless reveal a prevalence of high-frequency wind waves at 100+ km ∆ dist (Figure 4c).A possible explanation is that local generation of wind waves, perhaps in leads or the open water areas between sparse ice floes, occurs at significant distances inside the MIZ.In Figure 5, we calculate the implied fetch for each wind-wave spectrum in BGOS-SODA observations according to the scaling relations (as described in section 3.2).All observed wind waves at 100+ km ∆ dist could be generated by winds blowing over open water distances estimated to be less than 50 km.
Wind waves in ice are absent in model results for the central Beaufort due to multiple potential factors.First, the short implied fetch of the observed wind waves reveals that they are a sub-grid-scale process.The distance across the model grid cells, which are approximately 50 by 50 km in this region, is longer than the implied fetch for all observed wind waves at 100+ km ∆ dist (Figure 5).These short waves are sensitive to model parameters that control sub-grid-scale wave generation in partial ice.
Additionally, the model is biased high for intermediate ice concentrations (Figure 6), i.e., the 15-80% concentration range, at 100+ km ∆ dist during the summer melt season when wind waves in ice occur in observations (Figure S2).We focus on bias in the 15-80% intermediate concentration range conventionally considered part of the MIZ.We  km ∆ dist .We note that the lack of local wave generation could be partially responsible for the high concentration bias, just as the high concentrations are potentially responsible for suppressing wave generation.
Wind bias in the model could also be partially responsible.However, model winds come from atmospheric reanalysis.We believe error in the reanalysis is not a likely explanation, although we note that reanalysis does not always capture wind events in the MIZ (e.g., Brenner et al., 2020).
Are these high-frequency wind waves important for modeling wave-ice interactions?
In the Roach et al. (2019) model, waves can impact the FSD via floe fracture, described using the sub-grid-scale parameterization developed by Horvat and Tziperman (2015).
To test the importance of the observed high-frequency wind waves for floe fracture, we input the median, 75 th percentile, and maximum wave spectra, ranked by H s , from BGOS-SODA observations at 100+ km ∆ dist to the Horvat and Tziperman (2015) parameterization (computed offline).This parameterization generates realizations of the sea surface height using the ocean surface wave spectrum and computes the strain applied to sea ice floes.A statistical distribution of resulting fractured floe sizes is constructed by computing the distances where the strain field exceeds a critical value.Figure 7a shows the resulting floe size distributions that would be formed by the observed wave spectra in Figure 7c with a representative ice thickness of 0.5 m.
These results suggest that the locally generated waves at 100+ km ∆ dist tend to be strong enough to fracture sea ice: the median H s spectrum reduces 71% of the ice area to floes with radius less than 15 m.Steele (1992) found that, for floes with radius less than 15 m, lateral melt plays a critical role in Arctic summer conditions, which is when these waves appear in observations (Figure S2).Smaller floes make the dominant contribution to cumulative floe perimeter, so short wind waves in ice appear to enhance the lateral melt potential of ice floes and should be a priority for future wave-ice model development.
Note that we cannot expect model spectra to be identical to the observed spectra in partial ice because the model also represents all of the surface area where waves are damped by ice floes.A model grid cell aims to capture mean wave statistics over a partial ice region, but the in situ observations shown here appear to capture wave spectra from open water points between floes (see section 2.2 and the discussion that follows in section 5.2).We speculate that reconciling the model-observations difference in high-frequency energy does not require that model spectra become identical to those from the BGOS-SODA observations at 100+ km ∆ dist .However, the complete absence of high-frequency energy in the model spectra is striking and demands attention.

Swell
Now, we will address why BGOS-SODA observations do not show any swell at 100+ km ∆ dist while the model does.Generally, the low-frequency energy of swell experiences less dissipation than high-frequency energy during travel through partial ice cover.We anticipated that observations far inside the ice edge would preferentially show wave energy at low frequencies, similar to what we see in the model results.While large swells are relatively rare, for now, in the central Beaufort even in open water (Thomson & Rogers, 2014), the absence of low-frequency energy in BGOS-SODA observations from 100+ km ∆ dist , given its presence at 0-100 km, is conspicuous.In this section, we first consider why BGOS-SODA might not show any swell at 100+ km ∆ dist .
Could the BGOS-SODA data processing exclude swell spectra because those spectra also have a sea ice signal from under-ice topography?Recall that the subsurface BGOS-SODA measurements can represent a superposition of both ocean surface waves and sea ice draft, which each have distinct spectral shapes (see section 2.2).When sea ice is present above the moorings, the processing of the altimeter-based measurements may fail to recognize waves due to the additional signal from the ice.Therefore, the lack of swell spectra with H s greater than 0.3 m at 100 km+ ∆ dist in the BGOS-SODA observations could be partly a result of sampling bias if the swell is always coincident with a strong signal from ice.We first test this possibility by manually inspecting individual spectra in the original SODA records.We are able to find some measurements that have been excluded manuscript submitted to JGR: Oceans  3, 10, 22, 41, 70, 114, 176, 260, 370, 506, 668, and  by data processing from the wave dataset considered in this study and which have spectral shapes suggesting a combination of both sea ice and swell.However, the H s of the apparent swell in these spectra are less than 0.3 m, and the waves generally occur outside of the 100+ km ∆ dist range.While this manual inspection method is not exhaustive, it suggests there are no pervasive issues in the processing causing swell to be omitted from the data.
Could some swell be entirely hidden by the sea ice signal?If this were the case, the swell signal would be so much weaker relative to the ice signal that it would not emerge from underneath the ice's red spectrum, i.e., the swell would have no detectable spectral signature.Reprocessing of all individual spectra (including when no waves are apparent) allows us to set an upper bound on the H s of swell that may be hidden from observation based on the spectrum that is measured, which also includes the ice signal.The upper bound is determined by integrating the spectra over a frequency band associated with swell; the true H s of any hidden swell in this band must be much less than the apparent H s , i.e., the upper bound, due to how the swell spectral shape compares to a measured red spectrum.If we choose a narrow swell band of 0.08-0.125Hz based on the peak frequencies of swell in the model results, we find that 6% of the 10,283 SODA measurements that appear to be ice spectra exceed an H s upper bound of 0.3 m, corresponding to the minimum H s used throughout the analysis.It is possible that some nontrivial swell could exist hidden in these ice spectra, but we do not find any further evidence of swell with H s greater than 0.3 m in this band.
We also note that the absence of swell at 100+ km ∆ dist is supported by spectra constructed from the moorings' pressure data (not shown).These represent independent estimates of wave signals using a separate instrument on the moorings.The pressure spectra from under ice also do not report H s greater than the 0.3 m cutoff.A noteworthy supporting example comes from the 11 Oct 2015 event analyzed in Thomson et al. (2019) (see their Figure 2), which shows a swell spectrum from BGOS-A pressure data while the mooring was under ice near a major storm.In that case, the BGOS-A H s is less than 0.1 m.
We conclude that the 100+ km ∆ dist BGOS-SODA observations do not display any swell spectra because any swell that reached the moorings must have been too small to emerge with a sufficient signal.Perhaps the swell that evaded detection by the moorings resembles the swell (lower frequency) wave group in the bimodal SWIFT spectra (Figure 3f), which has energy mostly below the moorings' detection limits.Based on the recent wave climate near these moorings, large swells penetrating beyond 100 km ∆ dist in the Beaufort Sea are rare enough that they do not appear in this aggregate dataset.Finally, we consider whether the excess swell has a significant impact on floe size.
Unlike the high-frequency wind waves, which efficiently reduce floes to small sizes, the low-frequency swell has a less drastic effect.We repeat the floe-fracture test from the wind-wave discussion in section 5.1, now for swell from the model spectra at 100+ km ∆ dist using the Horvat and Tziperman (2015) parameterization (Figure 7b).This parameterization suggests that even the biased-high swell in the model fractures floes predominantly into large radius categories, with less than 1% of the ice area reduced to floe radius less than 15 m even for the maximum H s .In the case of the median H s , the swell does not cause any floe fracture.Moreover, the swell tends to occur in months of freezing conditions while new ice is forming and the ice edge is moving southward rather than melting and retreating (Figure S2).Overestimation of swell in the model is still a concern, but it appears less consequential for floe fracture and ice melt compared to the wind waves.

Conclusions
We investigate differences between the ice floes were resolved explicitly rather than parameterized based on the sea ice concentration within a grid cell, which is the scheme currently implemented in Wavewatch III.
These wind waves appear to cause substantial floe fracture and enhance lateral melt potential.Therefore, resolving or improving the parameterization of local wind-wave generation in the MIZ should be considered a priority in future model development.
On the other hand, the swell mode appears only in the model at 100+ km ∆ dist , not in the BGOS-SODA observations.Low-frequency energy appears to be overstated in the model at 100+ km ∆ dist .This swell in the model appears predominantly during the ice-growth season and has a relatively minor impact on floe fracture and melt potential compared to the wind waves.
The comparisons with observations in this study reveal important areas of development for modeling interactions between waves and sea ice.Combining multiple wave datasets to form a relatively large sample is an effective approach for model evaluation and could be replicated in other regions.However, we need more robust observations of wave spectra in sea ice across seasons at basin scale.These observations would enable stronger constraints on the physics of wave attenuation and generation in the MIZ which are critical to model development and theoretical understanding.October 2, 2021, 2:08am ) coupled sea ice-surface wave model forced with atmospheric reanalysis by evaluating wave heights, wave spectra, and the nondimensional scaling relations that can distinguish local windgenerated waves from swell.Global climate models, including the model considered in this study, have errors in ice-edge position that preclude point-by-point comparison with individual observations, so here we aggregate multiple datasets into a relatively large sample to support statistically motivated model evaluation of waves in ice in the Beaufort Sea.

Figure 1 .
Figure 1.Sea ice concentration (color shading) and corresponding ∆ dist (contour lines every 100 km from 0-500 km) at a sample, illustrative date (23 July 2018).(a) Satellite estimates of concentration with locations of in situ observations (red symbols).(b) Results from Roach et al. (2019) model with region used for comparison with observations (red box).Note that the 0-km-distance contour simultaneously denotes 15% ice concentration.
shows the combined BGOS-SODA wave height distributions in open water (SIC < 15%), 0-100 km ∆ dist , and ∆ dist > 100 km.The lower bound for H s is set at 0.3 m for the aggregate dataset to account for detection limits that vary across instruments.Model results are similarly represented as a histogram by aggregating the 2012-2019 statistics from each grid cell in the Beaufort Sea region surrounding the observations.The H s distributions have similar shapes in open water (Figure2a), but the model has more frequent large waves, with 18% of H s greater than 2.0 m compared to 9% in observations.The observations show slightly greater probability for smaller waves between 0.5 and 1.0 m.We note that sampling bias likely influences the open water comparison given that we do not control for distance outside the ice edge, i.e., all open water results are in a single group.A detailed analysis of open water results, however, is outside the scope of this study.

Figure 2 .Figure 3 .
Figure 2. Histograms and density curves for significant wave height Hs distributions in (a) open water (sea ice concentration < 15%), (b) 0-100 km ∆ dist , and (c) 100+ km ∆ dist , spanning 2012-2019 in the Beaufort Sea.In situ observations (black) are aggregated from two BGOS and three SODA moorings, and rug plots of vertical black lines along the x-axes denote exact values of individual observations.Model results (colors) are from the Roach et al. (2019) model, restricted to the Beaufort region surrounding observations.The lower bound on the domain forHs is set at 0.3 m, limiting the results to those exceeding the detection limit for all moorings considered.Note the different x-axis scale in panel (c).

Figure 4 .
Figure 4. Nondimensional scaling of wave energy vs. peak frequency grouped by distance inside the ice edge (∆ dist ).Observations shown as scatter plots (BGOS and SODA moorings as + symbols; SWIFT surface buoys as O symbols).Roach et al. (2019) model results from central Beaufort region surrounding observations shown as 2-d histograms (color shading), where the hourly mean at each model grid cell is a separate data point.(a) Open water (SIC < 15%), (b) 0-100 km ∆ dist , and (c) 100+ km ∆ dist .Only results with Hs > 0.3 m are shown.Power law (black line) with confidence intervals (shading) of E vs. F for wind-generated, fetch-limited exclude compact pack ice (SIC > 80%) because the large number of compact pack ice grid cells dominates the distribution.For the intermediate-concentration subset of grid cells, satellite estimates indicate a greater proportion of low ice concentrations compared to the model (also see Figure 1 for an illustrative example).Because wind-wave generation in Wavewatch III is scaled by a coefficient equal to the local open water fraction, the bias toward high ice concentrations excessively inhibits local wave generation at 100+

Figure 5 .
Figure 5. Histograms of implied fetch for locally generated wind waves from BGOS and SODA mooring observations.Observations located 0-100 km ∆ dist (orange) and 100+ km ∆ dist (green).Size range of Roach et al. (2019) model grid cells (approximately 50x50 km) in the vicinity of observations shown as dark shading with dashed border.

Figure 6 .
Figure 6.Histograms of intermediate (15-80%) sea ice concentrations during summer melt season (Jun-Jul-Aug) for grid cells located 100+ km ∆ dist , spanning 2012-2019 in the central Beaufort region surrounding the in situ observations.Satellite estimates (black) are from the NOAA/NSIDC Climate Data Record, and model results (green) are from the Roach et al. (2019) model.

Figure 7 .
Figure 7. (a)-(b) Histograms of predicted floe-size distributions resulting from corresponding wave spectra in (c)-(d), respectively, present at 100+ km ∆ dist , based on the Horvat and Tziperman (2015) parametrization and assuming ice thickness of 0.5 m.Floe sizes in (a)-(b) are binned into probability distributions A(r) where A(r)dr is the fraction of ice area with floe radius between r and r + dr.Plots show the probability A(ri)dri at each of the following bin centers i:3, 10, 22, 41, 70, 114, 176, 260, 370, 506, 668, and 850 m.Wave spectra represent the approx- 850 m.Wave spectra represent the approximate median (50 th percentile), 75 th percentile, and maximum based on Hs from (c) wind waves in BGOS-SODA observations and (d) swell in the Roach et al. (2019) model results, excluding spectra with Hs less than 0.3 m.Spectra in (c) have been interpolated to the frequency domain resolved in the Roach et al. (2019) model.Note that the 50 th percentile swell spectra in (d) does not cause any floe fracture and appears as a zero line in (b).

Table 1 .
Summary of In Situ Observations The model output at 100+ km ∆ dist (Figures2c, 3i, 4c) includes a number of waves exceeding the BGOS-SODA detection limit of 0.3 m H s , with maximum H s in the model reaching 1.25 m.Given that we do not see any evidence in BGOS-SODA of swells that approach the size of those in the model, this appears to suggest that the model overestimates the persistence of swell in ice, at least in the Beaufort Sea.The model's excess swell could be attributable to an open water bias that lingers as swell enters the ice, rather than the wave attenuation rate.If incident waves have energy at too-low frequencies in open water, the swell could survive at greater distances inside the MIZ.Comparison of the open water peak-frequency distributions, limited to the ice-growth season when swell is most often present in the model at 100+ km ∆ dist (FigureS3a), does not indicate any clear model bias toward low peak frequencies.However, there is an apparent bias of larger H s that could sustain the swell if that bias were present in the subset of waves that propagate into the MIZ.These explanations for the model swell are speculative, and more data is needed to support further investigation.
Roach et al. (2019)coupled wave-ice model and an aggregated dataset of recent in situ observations of waves in pack ice from the central Beaufort Sea.We group the data and model output by distance inside the ice edge, denoted ∆ dist , to enable a statistical comparison.The distributions of significant wave height are similar in open water but have more notable differences in sea ice.The model tends to have smaller H s than observations in the first 0-100 km of pack ice and greater kurtosis compared to observations beyond 100 km ∆ dist .The wave spectra and nondimensional scaling of energy and frequency illuminate different prevailing modes of waves at 100+ km ∆ dist between the model and observations.We find that observations show significant generation of local wind waves during the ice-melt season at 100+ km ∆ dist .The model lacks the resolution to generate the high-frequency wind waves that might arise if leads or open water areas between sparse