Modelling ocean waves in ice-covered seas

Applied Ocean Research 83 (2019) 30–36

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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Modelling ocean waves in ice-covered seas

T

Hayley H. Shen Dept. Civil & Environmental Engineering, Clarkson University, Potsdam, NY 13699, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Sea ice Ocean waves Models

Over the past decade there has been a rapid growth of interest in wave propagation through ice covers. This paper summarizes the author’s observation of the modeling efforts on this topic. Models can be theory-based, data-driven, or a combination of the two. A pure data-driven model relies on a large amount of observations and is only becoming available recently. Theory-based models on the other hand have a long history. They are always a simplified version of the reality. As our knowledge grows, theories become more complicated. A theory for waves-in-ice that captures all possible processes does not exist. However, when integrated with observation through calibration, these combined theory + data-based models may be used with some confidence. In this paper, different models, their basic concepts, their calibration and validation are discussed. The present theorybased models do not have the correct spectral attenuation trend as observed from field or laboratory experiments. Hence, through calibration they may fit different parts of the wave spectra but not all. Pure data-driven models can reproduce the correct trend, but its dependability outside the situation where the data are collected is uncertain. In addition to offering tools to forecast waves-in-ice, these model building and validating efforts point to missing mechanisms that should be carefully studied. Despite the many challenges towards building a satisfactory general waves-in-ice model, significant progress has been made for models that work reasonably well in the marginal ice zone. We anticipate much more data will become available in the coming years to help us improve the existing models.

1. Introduction Modelling wave propagation through ice covers can be traced back to 1886 [1]. At that time analytical methods were the only tools for mathematical models, hence an ice cover was idealized as a continuous sheet that was thin, uniform, purely elastic, and floating over inviscid water. This research topic remained obscure in the broad ocean research field, until the interest in the polar regions increased from the 1970s. The amount of studies on wave propagation through ice covers has been steadily increasing in the recent decades. With the rapid decline of Arctic ice, wave propagation through ice covers has become a prominent topic in both fundamental and applied research. The awareness of the feedback effect from increased waves due to expanding fetch, leading to ice break up and enhanced ice reduction, as well as the opportunities of shipping and other engineering developments in the Arctic, has accelerated the research pace of this topic. Several reviews, books and encyclopedia entries have been written [2–8]. Still, the speed of research output has far outpaced that of the reviews. It is a daunting task to comprehensively review all studies that have contributed to modelling waves through ice covers. In this article, the

E-mail address: [email protected]. https://doi.org/10.1016/j.apor.2018.12.009 Received 12 November 2018; Accepted 10 December 2018 0141-1187/ © 2018 Published by Elsevier Ltd.

author will focus on aspects of these models aimed at their applications in wave forecasting. Cavaleri et al. [9] provided a thorough review of wave modelling for open water. The history of modelling waves in open water is much longer, with orders of magnitude greater effort invested, than in ice-covered seas. In many ways, the development of these open water wave models may be viewed as a roadmap for waves in icecovered seas. As summarized in their review, Cavaleri et al. stated “On one hand we deal with a very complex physical process where physics, from fundamental principles till very practical problems, plays a dominant role. On the other hand the subject is highly in demand for its very wide applications, with a continuous push by the market forces to improve the quality of the results.” Modelling waves in ice-covered seas is facing the same situation. In this paper, the mathematical framework will be described, a number of theories will be given. These theories are calibrated with laboratory and field data. Validation of these calibrated models is becoming feasible with increasing field programs and remotely sensed data. Findings from these efforts will be discussed. As the modelling effort is far from complete, an author’s outlook on future development is given at the end.

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2. A look at the ice covers Ocean wave propagation through ice covers (abbr. waves-in-ice) belongs to a broader field of wave propagation through layered materials. This process is governed by each of the layers and their interfaces. For waves-in-ice, we need to look at these ice covers first, which is less understood than the water body underneath. Three basic descriptors are commonly used for an ice cover: thickness, concentration, and type. Ice thickness is further refined to include a distribution of different categories, with a related partial concentration within each category. Definition of the sea ice types has evolved since the early observations from the Inuit. Here we mean not the ice crystal types but the large scale surface composition, referred to as the ice cover morphology. A unified definition is now available to meet the needs of increasing multinational-multiscale-multisensor field programs. According to the Arctic Shipborne Sea Ice Standardization Tool (ASSIST) [10], there are fourteen different sea ice types from frazil to fast ice. A library of photo images corresponding to each of these types may be found at [11,12]. These ice types are for ship-based visual observation, with a range of 1–10 km. Satellites have a much larger range, but their abilities to differentiate ice types are limited by the wavelengths and resolution of the sensors. Using these remote sensing images, it is possible to differentiate multi-year ice from the rest of ice types, floes that are larger than the sensor pixel size, and large leads and ridges. Many other ice types including grease and pancake ice are difficult to detect. New algorithms need to be developed to overcome such challenges [13]. Regions of ice covers may be roughly classified, by their proximity to the open sea, into the marginal ice zone (MIZ) and the central pack (for the Arctic) or the land fast ice (for the Antarctic). In the subsequent discussions, we focus on the Arctic only. Fig. 1 shows a Landsat 8 image taken on 22 October 2015 in the Beaufort Sea [14]. It covers the MIZ and the open water. The area of the image is 180 km by 185 km. In this image, the upper part shows consolidated ice distributed with long and short leads. Near the dark open water at the bottom of the image the ice cover becomes diffused. The very bottom of the image is cloud cover. It is impossible to clearly discern the ice morphology from such an image. The black polygon in Fig. 2 corresponds to the area where this image was taken. In the map two ice edges defined by the 15% ice

Fig. 2. Map containing the Landsat 8 image in Fig. 1, 15% ice concentration contours, and locations of in situ ice conditions. The blue contour and blue circle correspond to 14 Oct. 2015. The black contour and black circle correspond to 22 Oct. 2015.

concentration contours are also shown. The blue contour corresponds to the ice edge on 14 October 2015 and the black contour corresponds to 22 October on the same day as the Landsat 8 image shown. The two circles mark the locations where photos of the in situ ice condition were obtained from a ship during a field study [15–17]. One photo was taken at the location of the blue circle on 14 Oct, 2015. The other was taken on 22 Oct, 2015 at the same time as the Landsat 8 image but far away. It is difficult to find simultaneous and co-located remote sensing images with in situ ice observations. Nevertheless, from these visual images it is clear that near the ice edge the ice morphology differs significantly from the interior ice cover. Furthermore, the photo corresponding to the black circle reveals that the ice cover had gone through a pancake formation process, then refroze into a solid sheet. It is also worth noting that although the blue circle lies south of the 15% ice concentration contour based on the once-a-day AMSR2 data (The Advanced Microwave Scanning Radiometer 2) [18], on the same day at the time when the photo was taken the surface was nearly fully covered with pancake ice. The field condition during that day was rapidly changing as reported from the field notes [15,16]. From both the ship-based and satellite-based observations, it is evident that ice covers are spatially heterogeneous and change dynamically under thermal and mechanical processes. Apparently different types of ice may affect wave propagation differently. We do not expect that any simple idealized model can fit all different ice types. Still, based on the experience of many other similar problems of waves in composite materials, we are optimistic that solutions may be obtained to capture some first order characteristics of waves-in-ice. 3. Inclusion of ice effect in ocean wave models The mathematical basis for global wave models such as WAVEWATCHIII® (WW3) is the radiative transfer equation [19]. This equation governs the evolution of the directional spectral energy density F (x , t , σ , θ) . To simplify the subsequent discussion, deep water condition is assumed herein:

∂N S + ∇x ∙x˙ N = ∂t σ

(1)

x˙ = cg + U

(2) (3)

S = Sin + Sds + Snl + Sice F σ

is the action density where bold-face quantities are vectors, N = spectrum of the angular frequency σ = 2πf , f is the frequency, cg is the group velocity, U is the current. The source term S includes: the wind input Sin , dissipation (white-capping and turbulence) Sds , nonlinear transfer between frequencies Snl , and ice damping Sice . The first two source terms, Sin + Sds , are still under intense research even in open sea. Whether they need to be modified in ice-covered condition has not been studied. At present, the global wave model WW3 treats the ice-covered

Fig. 1. A Landsat 8 image taken on 22 Oct., 2015. Image ID LC08_L1GT_077009_20151022_20170225_01_T2. The location of this image is shown in Fig. 2. 31

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attenuation in this case came from the effective viscosity in the water body. The predicted change of dispersion from open water under field conditions was negligible. Liu and Mollo-Christensen [34] took another approach by assuming that the ice cover was elastic, floating over a turbulent water body described by an eddy-viscosity, which was the same concept as the effective viscosity as in [33]. However, the elastic deformation of the ice cover changed the dispersion as in the thin elastic plate theory [1]. Both [33,34] attribute wave attenuation entirely to the eddy viscosity in the water. The resulting rate of attenuation in both cases behaves as σ 3.5 . Aimed specifically at modeling grease ice, which is a water-ice mixture with a slurry consistency, a model which considered a viscous layer over inviscid water was developed [35]. In addition to ignoring the dissipation from the water body, there is also an important difference between this model and that in [33]. In the viscous layer model [35] the full momentum equation for the ice cover is retained, removing the assumption that the surface layer is inextensible. This viscous layer theory showed a slight deviation of the dispersion relation from open water, until the viscosity or wave frequency became extremely large. In addition, attenuation was entirely from the ice cover, increasing with wave frequency as σ7 . The thin elastic theory [1] was expanded to include the viscoelastic effect by allowing the elastic modulus to be a complex number, but kept the assumptions as in the thin plate theory [36]. The viscous layer theory [35] was expanded to include elastic effects, but allowed the floating layer to deform according to an isotropic linear Voigt constitutive law [37]. Both of these viscoelastic theories attribute the attenuation entirely to the ice cover with an inviscid underlying water body. These two theories produced dispersion relations that significantly differ from open water. As shown in [38], the attenuation rate in one behaves as σ 11 [36] and as σ7 in the other [35,37]. There is one recent theory that combines both dissipation from a viscoelastic ice layer and a viscous water body [39], which shows a σ 2 ∼ 3.5 trend for attenuation at low frequencies, depending on the equivalent elasticity of the ice layer. Key findings of the various theories developed for the dissipative ice source term are summarized below.

region by modifying these two terms to (1 − C )(Sin + Sds ) where C is the ice concentration. Conceptually this means that in a partially icecovered region, the wind input and dissipation are unaffected over the open water portion, while over the ice-covered portion both effects completely shut down. In the same vein, let Sice be the damping rate in a fully ice-covered region, in partially ice-covered region the ice source term is scaled as CSice . Very little is known at present to challenge these simple scaling principles. In this paper, the focus is on the ice source term Sice and the group velocity cg which is affected by the dispersion relation of waves in ice. It should also be noted that there have been no published discussions on whether cg should be scaled with ice concentration, though this issue no doubt has been on researchers’ minds. The ice source term has been divided into a conservative directional redistribution due to scattering, Sice, s , and a non-conservative dissipation, Sice, d [20–22]:

Sice, s = −cg αs (x , t , k , θ) F (x , t , k , θ) + cg

∫0

2π

Sσ (x , t , k , θ , θ′) F (x , t , k , θ) dθ′

Sice, d = −cg αd (x , t , k ) F (x , t , k )

(4) (5)

The integral term in Eq. (4) represents the directional redistribution from θ , the direction of k , to θ′. The scattering and dissipative attenuation coefficients are αs and αd , respectively. The wave number k is related to the angular frequency σ through a dispersion relation, which is one of the central questions for waves-in-ice. There is a large body of studies on scattering and the associated attenuation [23]. The mathematical foundation was laid out in the 1960s [24,25], and extensively expanded in a series of studies began with the seminal papers of Fox and Squire [26–28]. Scattering due to floes of semi-infinite, finite, circular or arbitrary shapes, with or without submergences, across leads or thickness variations, have all been investigated. The science for the scattering process is mature. Some key findings are summarized below.

• The scattering attenuation of forward-going wave energy from a • • •

• Wave dissipation has been solved as from the water body only [33,34] or from the ice cover only [35–37]. • Theories considering dissipation from both ice and water body in-

two-dimensional array of rectangular floes is similar to a three-dimensional field of circular floes of the same thickness and with diameter equal to the width of the rectangular floes [29]. The scattering attenuation is exponential with distance. The exponent drops with decreasing wave frequency [30]. The attenuation is maximum when the floe’s width is similar to the wavelength. For floes less than 1/10 of the wavelength, scattering attenuation is negligible [30]. The attenuation due to leads is much less than due to floes or ridges of the same horizontal dimension [31].

• •

Based on these findings, scattering induced attenuation depends on the floe size in the MIZ. As the floes become larger and freeze together to form a semi-continuous cover, scattering becomes controlled by the leads distribution, until near the central ice pack where ridges become prominent to take over the scattering role. Associated with attenuation, scattering redistributes the forward going wave energy into other directions. Consequently, the energy spectrum gradually becomes isotropic as the waves propagate further into the ice cover. Mathematically, this process may be efficiently modeled as a diffusion process [32], with the diffusivity determined by the relative floe size to wavelength. In the current global wave model WW3 [19], scattering is implemented as a distributive source term as shown in Eq. (4), with a simplification that isotropic re-distribution is achieved regardless of the wavelength and floe size. The physical basis for the dissipative attenuation Sice, d is much less developed. The earliest theory assumed that ice was a high viscous material over a viscous water [33]. This theory produced simple analytical solution when the ice viscosity approached infinity, thus the ice cover became inextensible and the dissipation within vanished. The

clude viscous effects only [33] and viscoelastic effects from ice and eddy viscosity effect from water [39]. Theories based on pure viscosity do not show significant change of the dispersion relation from open water, until the viscosity or frequency become very high. With elasticity, wavelength may shorten or lengthen, depending on whether the elasticity contribution is less or greater than the buoyancy effect of the ice cover [40,41]. All theories show the attenuation rate depends on ice thickness and wave frequency, but the functional dependence differ widely among different theories [38].

4. Model calibration All of the above models for Sice, d contain parameters. Two of them can use independently determined parameters: the pure viscous layer model and the eddy-viscosity model. The pure viscous layer model has been applied to grease ice in the laboratory. Direct measurement of the viscosity of grease ice using a viscometer has not been done successfully. By propagating monochromatic waves through a grease ice layer, and measuring the resulting wavenumber and attenuation, the viscosity of the grease ice was inversely determined [42]. In the field, pure grease ice cover is found under wind waves in the leads and polynyas [43–45], between large floes, or at the onset of freezing in the open water [17,46]. Whether this model can be used for other ice types is unclear. Comparisons between measured spectral damping in a pancake ice field and theoretical predictions from a pure viscous theory showed increasing differences at high frequencies [47], hence there is doubt that 32

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Table 1 Ice conditions in each of the wave experiments. Wave experiment

h (m)

C

Ma

Elastic parameter (Pa)

Viscous parameter (m2/s)

3 6 7

[0.01, 0.03] [0.01, 0.04] [0.03, 0.07]

[0.15, 0.55] [0.25, 0.5] [0.15, 0.6]

0/0.59/0.41/0 0/0.43/0.27/0.29 0/0.23/0.13/0.64

10−5.6 or 103 104 10−5.7 or 104

100.9 or 100.4 10° 101 or 100.8

a

The morphology M is defined by the combinations of ice types in the study area, recorded hourly by an ice observer following the ASSIST protocol [58]. The

mean values from the collected hourly data are given in the order of

¯ ¯ Cct 1 Cct 2 ¯ / Tc ¯ Tc

/

¯ ¯ Cct 3 Cct 4 ¯ / Tc ¯ Tc

, where cti is the partial concentration of category i and Tc the total ice

concentration. These categories from i = 1 to 4 are, respectively, thick/ pancake/thin/other. “Thick” ice includes second and multi-year, and fast ice. “Thin” ice includes frazil, grease, shuga, slush and nilas. “Other” ice includes grey, first year and brash ice.

such theory is applicable to pancake ice fields. The eddy-viscosity model can use measured turbulence data under the ice cover [34], more contemporary measurements can be found in [46,48]. The viscoelastic models are presently rheological, which means the parameters can only be determined by the model’s response to waves. Although these parameters no doubt are products of physical processes that actually exist, deriving these parameters from the first principles is a challenging task. Not only that we are facing an inherently complex mechanical problem with fluid-solid interactions, we also have the additional difficulty of not knowing all the relevant processes. As our observations become more sophisticated, more of those complex processes are identified. For example, ice floe interactions were known many years ago [49], but overwash of water spilling on top of ice floes in high waves is a recent discovery [50]. Both processes dissipate wave energy, whether they can be parameterized into an equivalent viscosity is another question. Even if we can reason that they are rate dependent processes, hence should be modeled as a viscous term, there is no reason why they should yield a constant viscosity where the energy dissipation scales with the square of the velocity gradient. So far only the viscosity of grease ice has been theoretically derived from fluid mechanics principles for a suspension [51]. For other types of ice, we can only use data to calibrate these parameters. Provided that these calibrated parameters are invariant if the ice type is fixed, its utility for wave modeling is justified. Such approach has been carried out for different types of ice. For the thin-elastic plate model, seismic tests were used to determine the effective elastic modulus of an ice sheet [52]. This type of inverse method is commonly used in material testing both large and small scale. But for ice covers consisting agglomerates of floes, cracks and ridges, data scatter is significant. For the pure viscous layer model, as mentioned earlier, laboratory test of monochromatic wave propagation through grease ice was used to measure the effective viscosity [42,53]. Encouragingly, the inversely determined viscosity is on the same order of magnitude as from the theory of suspension [51]. For the viscoelastic layer model, laboratory data calibration has been used to determine the equivalent viscosity and elasticity of grease/pancake and fragmented ice covers [54,55]. Improper application of these models can lead to spurious results. As shown in [56,57], when using a thin-elastic plate model with vanishing elasticity (i.e. the mass-loading model) to interpret wave spectra measured from SAR images over a pancake ice field, unreasonably higher ice thickness than the in situ data was obtained. While using a viscous layer model, the same wave data produced a much more reasonable ice thickness. More comprehensive and systematic calibration of the viscoelastic models has been conducted recently [58] using data from the ChukchiBeaufort Sea and the western Arctic Ocean. The ice covers were dominated by pancake ice, mixed with a variety of grease, brash, and occasionally larger floes. Several extensive experiments with different degrees of wave intensity were conducted using arrays of drifting buoys [15–17]. Model calibration was conducted for three of these experiments which had long duration of record. Directional wave spectra from pairs of buoys were used to determine the spectral attenuation. Wind input and nonlinear interactions between each pair of buoys were

removed to isolate the dissipation due to ice covers from the total attenuation. The wavenumber through these ice covers was measured for one of the wave experiments using the marine radar signal and from the buoys’ heave-pitch-roll correlation [59]. The data from marine radar covered wave spectra up to 0.32 Hz. They showed a small (< 2%) deviation from the open water dispersion. Buoy results, however, suggested that wavenumber might increase with frequency in the 0.3 to 0.5 Hz range. Assuming this wavenumber change was negligible, optimization method was used to calibrate the viscoelastic parameters by minimizing the difference between the model and measured wave number and attenuation over the range of frequency between 0.057 and 0.49 Hz from hundreds of paired buoy records. In minimizing the difference of attenuation, a weighting factor was used to emphasize the high frequency waves observed in this field experiment. Results of this calibration are summarized in Table 1. There was significant data scatter in the attenuation, hence the resulting calibrated parameters also scattered. In Table 1 the mean values of the equivalent elasticity and viscosity of the ice covers are shown. In this table the ice conditions are specified by three variables: ice thickness, h , concentration, C , and morphology, M . Ice thickness and concentration data are from, respectively, the SMOS (Soil Moisture and Ocean Salinity) [60] and the AMSR2 [18] satellite. The morphology is defined by the partial concentrations of four categories of ice: thick/ pancake /thin/other using the definition following the ASSIST protocol [61]. The calibrated mean equivalent elasticity and viscosity parameters varied among different wave experiments (thus different ice conditions). For wave experiments 3 and 7, the calibrated parameters clustered in two groups with high and low equivalent elasticity, and corresponding equivalent viscosity, both sets of values are given in Table 1. The cause of such distinct clusters has not been identified. This study shows that it is difficult to pinpoint the value of the model parameters that would reproduce the observation exactly. The ice covers were dynamically changing, as can be seen from the range of ice thickness and concentration in Table 1. From the hourly ice record taken during the field experiment [15–17], the ice morphology varied even more than the two photos shown in Fig. 2. It is thus more realistic to expect that a calibration study using field data would give only order of magnitude estimates for the model parameters. There is another method to parameterize wave propagation in ice that bypasses any rheological models. In this method field data are directly used to relate the observed attenuation to ice thickness and wave period [62]. In [62], long buoy records obtained in an Antarctic pancake ice zone were analyzed to get the spectral attenuation α (f ) from f = 0.04 to 0.2 Hz A simple function is then used to fit the attenuation data

α (f ) = af 2 + bf 4

(6)

with a= 2.12 × 10−3 (s2/m) b= 4.59 × 10−2 (s4/m). (Similar formula has also been proposed [63], except that they used the scattering theory results from [30] hence strictly speaking it is for the Sice, s but not the Sice, d term.) Another purely data-driven model with an innovative application of WW3 was presented in [64]. The method was demonstrated by using the same buoy data from wave experiment 3 shown in Table 1. 33

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function, Eq. (6), with data from the Antarctic, with pancake floes 2 to 3 m in size nearer the ice edge and 10 to 20 m further in, and thickness from 0.5 to 1 m.

In this method, an exponential dissipation for each frequency component was obtained by matching the measured wave energy at that frequency with a hindcast simulation using WW3. In this way, the “perfect” spectral attenuation model was determined for each buoy record. A step function was then defined from the mean of the resulting spectral attenuation data. The resulting empirical formula is not expected to be universally applicable to different ice types, as the attenuation data from much different ice fields [65] were significantly different from the empirical formula obtained in [64].

6. Response of wave spectra to ice models Section 3 mentioned that the spectral attenuation from different models for Sice, d produced different trend, with a wide range from σ 3.5 (eddy viscosity model) to σ 11 (thin viscoelastic plate model). In [38] other hypothetical viscous models were shown to produce σ 2 or σ 3 behavior. The moment we pick any of such models, we are limited by the model’s attenuation trend. If the trend deviates from that of the measured data, our best-fit model can only fit part of the observed spectral attenuation. This is what has been observed using calibrated data from Table 1 to calculate the spectral attenuation over the entire frequency range. The σ7 trend of the viscoelastic layer model dropped off too quickly at low frequencies than the observed attenuation, which followed rougly σ 2 for frequencies below 0.25 Hz [47,64]. Since the weighting factor emphasized the high frequency part, the best-fit model deviated from the observed attenuation at low frequencies. By changing the weighting factor, different parts of the spectral attenuation could be fit with different calibration but not the whole spectrum [38,58 in Supporting information Fig. S14]. Therefore, when validating a model, comparing only the resulting mean period and the significant waveheight may be sufficient for many applications, such comparison does not truly support the model itself as fully capturing the physical processes. There is another issue in validating ice models for the general wave energy transfer defined in Eqs. (1)–(3). That is, we still do not know what other source terms might change in the presence of a partial or full ice cover. When we isolate the ice contribution Sice, d from the field data that include effects from all sources, we rely on the knowledge of other contributions. At present we can only assume that the simple scaling by ice concentration shown in the beginning of Section 3 works. To avoid this uncertainty, laboratory studies can help, where it is possible to remove the wind input, dissipation from wave breaking, and nonlinear transfer among different frequency components. A few laboratory studies have been conducted with grease/pancake and fragmented ice floes [54,55], and with polydimethylsiloxane (PDMS), a viscoelastic polymer material [69,70]. These experiments showed that the viscoelastic dispersion relation appeared to agree with measurements, but the attenuation did not. Just like the study using field data [47,58] the measured attenuation in the laboratory with real ice was above the calibrated model results at low frequencies. With PDMS, the measured attenuation was above the theoretical results at all frequencies [70]. In the PDMS case, since the viscoelastic parameters of the floating cover were directly measured with a high precision rheometer, no calibration was needed. The high attenuation thus suggests additional damping mechanisms not within the floating cover itself. Since the most obvious sources of additional dissipation such as wall friction and overwash were ruled out in that experiment, the highly suspected mechanism was the boundary layer under the cover [70]. No measured data is presently available to verify this hypothesis. Under an ice cover, the roughness factor is different from that of a carefully prepared PDMS sheet, hence the boundary layer structure may be quite different. Thus there is no reason to expect that there is a simple scaling law that could translate the laboratory data to field scale. As reported in [64], a comparison of attenuation data from field and laboratory experiments showed that the spectral attenuation level of a laboratory study clearly exceeded that of extrapolated curves from the field experiments, both under the grease/ pancake ice type but with different scales. This discrepancy could mean nonlinear effect relating to wave amplitude as suggested in [64], or different scaling laws for the boundary layer effect in the laboratory and in the field. Nevertheless, even though we cannot directly extrapolate the results, laboratory studies allow us to isolate mechanisms so that clear understanding of each process may be obtained. It is these

5. Model validation Model validation requires data. To do so we need to apply the calibrated model to cases with similar ice conditions but were not used in the calibration. If the model results agree with the measured data, the model is validated, until challenged by additional data. To the author’s knowledge, the eddy viscosity model has not been validated, although many independent direct measurements of the eddy viscosity have been reported. Since the eddy viscosity is a phenomenological parameter strongly associated with the flow condition, it changes with the wave and current, as well as the motion and bottom roughness of the ice cover itself. Some historical data exist for the central Arctic, where the Ekman layer under a large ice floe was measured and the eddy viscosity was inferred to be on the order of 10−3 m2/s [66]. Higher values on the order of 10−2 m2/s have also been reported from the Antarctic ice cover [67] and in more recent studies in the Barents Sea and a fjord near Svalbard [46,48]. These data indicate the dependence of this parameter on ice and the hydrodynamic conditions. To understand this dependence requires direct measurement of the boundary layer structure under an ice cover in a wave field. Such study is becoming feasible recently. Results from these studies show definitive change of turbulence dissipation under ice covers from the open water condition [64,68]. In open water most of the turbulence dissipation happens near the surface, while under ice covers the surface dissipation is suppressed but significantly increased deeper down. It is hopeful that with this type of measurements in both field and laboratory studies, we will be able to build a physics-based model for dissipation under the ice cover. The pure viscous and pure elastic theories may be considered as special cases of the viscoelastic layer theory [37]. In the previous section calibrations of the elastic and viscous parameters are shown for three wave experiments. Using the mean values of these two parameters to perform a hindcast with WW3, results of the significant waveheight and mean period at two other locations far from the buoys used for calibration showed reasonable agreement over a three-day period with the measured data [58]. However, until we can confidently establish that dissipation from the water body in this case was much less than that from the ice cover, this validation is tentative. Furthermore, matching only the significant waveheight and the mean period does not mean that the model also reproduces the whole wave spectrum. A thorough validation study was reported on one of the pure datadriven models using the same dataset from wave experiment 3 in Table 1 [64]. In which the authors first constructed a 10-step function in ten frequency bins for Sice, d using wave experiment 3. They then implemented this model and several other waves-in-ice models to perform a one-month hindcast using WW3 covering many wave experiments. They compared the measured data with the results from WW3. The comparisons included: the mean wave period, significant waveheight, the fourth moment of the energy spectrum, and the statistical variations of these comparisons. The fourth moment of the energy spectrum is defined as ∫ E (f ) f 4 df , where E = ∫ Fdθ is the scalar energy density. These comparisons were made for each buoy and a subsurface moored wave and current meter. Results of these comparisons show that the calibrated model may be used in the same region with similar ice types. A surprising finding was that the 10-step function using extensive data from the Arctic, with pancakes of size below 1 m and thickness 0.1 to 0.3 m, was very similar to the simpler binomial 34

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the field, empirical formulas need to be established. These formulas can quickly make use of the measured data to produce models that will have practical utility. Finally, hindcast and forecast exercises using various models need to be performed to test their basin-wide behavior. Sensitivity of these results to various models will help us identify and prioritize processes that are more important than others.

understandings that will allow us to build reliable theories that can work in different scales. 7. Outlook for future progress The above discussion summarizes what the author has observed over the past decade of the waves-in-ice development. As in any field of science, at first there are imaginations that help forming theories. Then various theoretical predictions are put against observations. Discrepancies among theories, and between theories and observations, propel improvements. Practical needs accelerate this process. Waves-inice is a multi-process and multi-scale problem. At the scale of the wavelength, changes of momentum transfer from wind to water due to the presence of an ice cover and its type are unknown. Wave breaking mechanisms may also change. The most obvious is the absence of capillary waves when there is just a thin layer of frazil ice on the surface. The presence of ice modifies the Sin + Sds in Eq. (2), in addition to the Sice effect. For the MIZ, at the scale of a wavelength, identified processes include floe-floe interactions [49,71], overwash [50], and losses due to ice-water friction [64,68]. At the scale over a large number of wavelengths, the small scale processes may be homogenized to form a continuum model. From what have been observed, variability of these model parameters is significant. Hence a statistical approach seems necessary to constrain the reliability of these model parameters. It is at this scale, a practical model may be built for the polar wave forecasts. The calibration studies so far do not produce a conclusive picture of how ice types are related to their model parameters. The only exception is grease ice, with which pretty consistent values of 10−3 ∼ 10−2 m2/s have been obtained from theory and direct measurements. Using the viscoelastic layer model [37], the inversely determined elastic parameter are found to lie broadly in the range of 0 ∼ 105 Pa and the viscous parameter lies in 10−3 ∼ 10 m2/s. All of these values have been obtained from wave data collected in grease/pancake ice with different thickness and diameter, in the laboratory [54,55] and in the field [47,58]. It begs the question: what causes the spread of these parameters? At present, the author is pretty convinced that it is insufficient to blame all dissipation mechanisms to the ice cover. The water body under the ice cover needs to be thoroughly investigated. So, where do we go from here as far as building a waves-in-ice model that can reliably predict the dispersion and attenuation over the entire wave spectrum? The answer is universal in any field of research: we need more observations and we need to improve the theories. The existing theories starting from more than a century ago are all correct, partially. Ice covers are like any other materials. That is, when they deform under the wave forcing, they store part of the energy and dissipate the rest. We may further decompose the dissipative process into anelastic or viscous. Following this thought, we may be able to expand the existing theories using the simplest possible model that allows each of these processes be formulated in its basic form. For example, a laboratory experiment performed nearly two decades ago [72], showed that the equivalent elasticity of an array of elastic floes was much less than the intrinsic elasticity of the floating material. This same phenomenon must also exist in the field with finite ice floes. How we should modify the elasticity of a sheet of continuous ice to a field of individual floes, even though they are intrinsically the same ice? Further, ice covers float on water, its motion and deformation must impact the water body. Hence the hydrodynamics of the water body cannot be dismissed on the outset. How to best parameterize the water body without overly increasing the complexity of the model? Answering these fundamental questions appears to be an important next step for model improvement. We are at a time when the practical needs for waves-in-ice models are acute. In the next decade we anticipate a rapidly growing body of data from in situ and remote sensing sources, laboratory studies that will help us understand the processes involved, and implementations of these understanding into theories. While we wait for the maturing of

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