This work analyses laboratory observations of wave energy attenuation in fragmented sea ice cover composed of interacting, colliding floes. The experiment, performed in a large (72 m long) ice tank, includes several groups of tests in which regular, unidirectional, small-amplitude waves of different periods were run through floating ice with different floe sizes. The vertical deflection of the ice was measured at several locations along the tank, and video recording was used to document the overall ice behaviour, including the presence of collisions and overwash of the ice surface. The observational data are analysed in combination with the results of two types of models: a model of wave scattering by a series of floating elastic plates, based on the matched eigenfunction expansion method (MEEM), and a coupled wave–ice model, based on discrete-element model (DEM) of sea ice and a wave model solving the stationary energy transport equation with two source terms, describing dissipation due to ice–water drag and due to overwash. The observed attenuation rates are significantly larger than those predicted by the MEEM model, indicating substantial contribution from dissipative processes. Moreover, the dissipation is frequency dependent, although, as we demonstrate in the example of two alternative theoretical attenuation curves, the quantitative nature of that dependence is difficult to determine and very sensitive to assumptions underlying the analysis. Similarly, more than one combination of the parameters of the coupled DEM–wave model (restitution coefficient, drag coefficient and overwash criteria) produce spatial attenuation patterns in good agreement with observed ones over a range of wave periods and floe sizes, making selection of “optimal” model settings difficult. The results demonstrate that experiments aimed at identifying dissipative processes accompanying wave propagation in sea ice and quantifying the contribution of those processes to the overall attenuation require simultaneous measurements of many processes over possibly large spatial domains.
This is the second part of a two-part paper in which we analyse energy attenuation of waves propagating through sea ice composed of densely packed, colliding ice floes. In the first paper
The second part of the study, described in this paper, is based on observational data from a laboratory experiment in a large ice tank, in which regular, unidirectional waves were run through ice covers composed of rectangular floes of equal size. The experiments cover a range of floe lengths and wave periods and include measurements of the vertical deflection of the ice by means of underwater pressure sensors and motion tracking methods as well as video recordings of the ice motion. The laboratory measurements, combined with results of two numerical models – a model of non-dissipative scattering by a series of floating, elastic plates and a coupled DEM–wave model described in Part 1 – are analysed in order to gain insight into processes contributing to attenuation of wave energy in fields of colliding, interacting ice floes.
Floe–floe collisions are often mentioned in the literature as one of several mechanisms contributing to wave energy dissipation in the marginal ice zone (MIZ). However, field observations of colliding ice floes are rare, and measurements directly relating collisions to dynamical processes in sea ice and underlying surface layer of the ocean are practically nonexistent. The first studies devoted to ice floe collisions, conducted in the 1980s and early 1990s, were based on measurements with accelerometers placed on the ice
As already mentioned, the ice cover analysed in this study consists of rectangular, densely packed ice floes. As the video documentation shows, the floes undergo regular collisions and, in tests with relatively high wave steepness, overwash, especially in the zone close to the ice edge, indicating dissipation mechanisms that are presumably relevant to wave attenuation. However, apart from incoming wave characteristics and the basic ice properties, the only quantitative information available is the wave amplitude at several locations along the tank. The main goal of this study is to demonstrate that the interpretation of the observed attenuation and validation of numerical models based on that type of data is problematic, as many mutually interrelated mechanisms contribute to the net attenuation. This is important because a situation in which wave amplitude data are available without additional information on dissipation is a rule rather than an exception. In particular, satellite data are increasingly used to assess wave attenuation in sea ice without complementary information on processes taking place in and under the ice. Moreover, many different models can be calibrated to reproduce observations with reasonable accuracy, especially considering large uncertainties in attenuation rates derived from measurements.
In the next section, we provide a brief description of the two models used (more details concerning the coupled DEM–wave model can be found in Part 1), followed by a description of the laboratory experiment in Sect.
As mentioned in the introduction, two very different numerical models are used in this work to aid our understanding of processes observed in the laboratory. The first model (Sect.
In order to analyse the non-dissipative attenuation processes in the set-up considered, we use the MEEM by
Notably, the context in which the MEEM model is used in this study – an ice tank of finite length, with a wave maker at the one end and a wave absorber at the other end – is almost identical to that used by
The model used in this study is described in detail in Part 1. Here we only recap its main features and summarize its behaviour.
The model consists of two coupled parts: a DEM sea ice model
Propagating, regular waves are assumed with known period
The coupled model is solved with an iterative algorithm in which the sea ice and wave modules are run in turns until a stationary wave amplitude profile
As analysed in detail in Part 1, for compact, horizontally confined sea ice, the model predicts attenuation of the form
The experiments analysed in this work were performed in the Large Ice
Model Basin (LIMB) of the Hamburg Ship Model Basin (Hamburgische Schiffbau-Versuchsanstalt – HSVA) as part of the Hydralab
A sketch of the experiment set-up is shown in Fig.
LS-WICE experiment set-up. The ice edge is located at
The water depth was
Crucially for floe motion and collisions, a floating boom was installed at the ice edge (
Each test series consisted of three groups of tests, with floe lengths
Summary of the set-up of experiments in test series 2000 and 3000 (in chronological order). The ratio
The air temperature during the tests was close to 0
The wave amplitude data of interest in this study was obtained by two methods (Fig.
The pressure sensors were located at a depth of 0.35 m and distance 0.65 m from the walls. The 12 Qualisys markers were placed along the central axis of the tank (floe row
Figure
Wavenumber
In this section we analyse the wave attenuation observed in the LS-WICE experiments based on the data from pressure sensors and from the Qualisys system – two data sources that provide a slightly different picture of the situation due to the fact that the pressure sensors have fixed positions relative to the tank and the Qualisys markers have fixed positions relative to the ice. In order to better understand the observed attenuation patterns, we supplement the data analysis with vertical deflections computed from the MEEM model described in Sect.
The amplitudes determined from measurements and computed with MEEM are shown in Supplement Figs. S1 and S2 and, for two selected tests, in Fig.
Amplitude of the vertical ice deflection along the tank in two selected tests from series 3000: 3210
Transmission coefficients computed with the MEEM model for tests from group 3000 (the results for group 2000 are similar): amplitude of the transmitted propagating component at the ice edge
Within the ice-covered region, the redistribution of wave energy among different components means that large amplitude differences can be expected within a single floe, resulting not from the overall wave attenuation but from the contribution of multiple wave modes at each floe edge – and this effect becomes stronger with increasing floe size. Figure S3, showing zoomed fragments of the tank around the location of the Qualisys markers, provides a good illustration of that variability, present both in MEEM results and, to a lesser extent, in the measured data. In consequence, the vertical deflection of the ice measured at a single point is of little value, especially if the measurement is done at a fixed position in space (as is the case with the pressure sensors in our experiment), so that the location of the sensor relative to the floes' boundaries is unknown and might change slightly during a single test. In the case of Qualisys measurements, an attempt could be made to “split” the measured amplitudes among the transmitted propagating and remaining modes, based on the corresponding amplitudes from the MEEM model (
Attenuation rates (m
A cursory inspection of Figs.
Conceptually speaking, because the attenuation takes place within the ice cover, the wave amplitude at
Attenuation rates (m
The very different results obtained with fixed and fitted
Finally, it is worth noting that the available data do not provide arguments in favour of any of the two fit types considered. Especially when
Although extrapolation of any quantitative data from the laboratory scale to the field scale has to be approached with caution, it is useful – before proceeding to the analysis of the modelling results – to relate the range of ice and wave parameters used in the experiment to the analogous full-scale conditions.
For an unscaled ice thickness of, say, 1.5 m, typical for first-year ice in many regions of the Arctic and Antarctic, and the laboratory ice thickness is
Crucially, the observed attenuation rates are realistic as well, although they tend to lie within the upper range of observed ones. For example,
In Part 1, the coupled DEM–wave model was set up with sea ice properties corresponding to those from LS-WICE series 3000, and the model sensitivity was analysed within a multi-dimensional parameter space. Here, we run the model for each test from series 3000, with floe length
With the basic settings listed above, the model was run several times for each test, with different combinations of the following four parameters: drag coefficient
Before we proceed to the analysis of the DEM simulations, it is worthwhile to note that the suitable values of
Adjustable parameters of the model set-ups discussed in Sect.
In the case of set-up B, two values of
Amplitude of the vertical ice deflection along the tank in two selected tests from series 3000: 3380
As expected from the analysis above, simulations without overwash require very high values of
The extent of overwash is, as expected, larger in runs C and D, and in some tests the model predicts overwash over the entire ice sheet – which was not observed; but, before labelling those set-ups as unrealistic, it is worth noting that the simulated dissipation due to overwash was in those cases extremely small over most of the area far from the ice edge. As can be seen from Fig.
In summary, it should be stressed that the values of
The LS-WICE results analysed in this work provide a very good example of how difficult it is to quantitatively assess wave energy attenuation in sea ice (especially in laboratory conditions, with a limited number of floes and over short distances) and to attribute the observed attenuation to individual physical processes even in a highly idealized laboratory setting. In spite of the simple geometry, regular wave forcing with small wave amplitudes, and highly uniform ice properties, several processes simultaneously modify wave propagation and dissipation, including floe collisions, floe breaking, overwash of the ice surface, production of slush, freezing between neighbouring floes and between the ice and tank walls, and possibly some others. The results of observations and of the MEEM model clearly show that the scattering model alone does not explain the observed spatial variability in wave amplitudes in fragmented ice, as the attenuation simulated with MEEM is, in most tests, very low – much lower than the observed one. Another general conclusion drawn from the data analysis is that the attenuation rates increase with increasing wave frequency. The two facts together mean that the patterns of wave attenuation observed in LS-WICE are predominantly shaped by dissipative processes and that the effectiveness of those processes in attenuating wave energy is frequency dependent.
An important aspect of the numerical part of this study is that several different combinations of the model parameters lead to reasonable agreement with observational data – even though we limited the number of adjustable parameters to three. It is very likely that even more regions of good model performance could be found within a higher-dimensional parameter space. Obviously, this ambiguity is a consequence of a large number of poorly constrained coefficients, large uncertainties in measured data, and the fact that the vertical deflection of the ice, being a combined effect of many processes, is the only validated quantity. Although some combinations of the parameters seem more “realistic” than others, it is hard to favour one set-up against the other without additional data. In particular, very high values of the drag coefficient in our “successful” set-ups are a few orders of magnitude higher than those reported in the literature, which are rarely larger than
Undoubtedly, from the point of view of analysing wave attenuation, a number of shortcomings can be listed in the LS-WICE set-up, and, based on this study, several recommendations can be formulated for future laboratory experiments designed specifically for measuring wave attenuation in fragmented sea ice (and, more generally, in other types of floating ice as well). First of all, it is crucial to locate sensors measuring the vertical deflection of the ice, acceleration of ice floes, and possibly other quantities, in the zone of the strongest attenuation close to the ice edge. At the same time, it is important not to limit the observations to that zone, as the attenuation further down-wave is very likely much weaker and cannot be extrapolated from that observed at the ice edge. Moreover, distributing sensors over a possibly large distance is necessary if the suitability of alternative theoretical attenuation curves is to be tested. The LS-WICE data, as demonstrated in this work, are clearly not sufficient for that purpose. It must be remembered, however, that this requirement is easy to formulate but very hard to fulfil in a wave tank, as its length limits the possible number of wavelengths. Furthermore, as discussed at the beginning of this section, LS-WICE shows that, although it would be desirable to design experiments eliminating all other dissipation mechanisms except the one of interest, this goal is very hard, if not impossible, to achieve. Some factors present in LS-WICE, e.g. freezing to the side walls, can be eliminated with some effort (though it is not straightforward in an ice tank several tens of metres long), but the influence of other factors has to be accepted and, as they are impossible to eliminate, quantified. In particular, overwash is very difficult to eliminate in laboratory conditions due to small thickness and therefore very low freeboard of the ice so that, apart from recording overwash presence and extent by video, assessment of its thickness is desirable, enabling formulation of more advanced parameterizations than the primitive one proposed in this study.
Finally, as already mentioned in the discussion in Part 1, integrating scattering effects in the DEM presented here is a major challenge that must be addressed to make the model suitable for analysing mutual relationships between non-dissipative and dissipative processes contributing to wave energy attenuation. Obviously, attenuation in real sea ice is not a simple superposition of individual processes that can be considered independently of each other, as in the present study.
The code of the DESIgn model is freely available at
The supplement related to this article is available online at:
All authors contributed to planning of the research and to the discussion and analysis of the results. SC performed the analysis of experimental data and the MEEM simulations. AH performed the numerical simulations and wrote the text.
The authors declare that they have no conflict of interest.
The authors would like to thank the Hamburg Ship Model Basin (HSVA), especially the ice tank crew, for the hospitality, technical and scientific support, and the professional execution of the test programme in the research infrastructure ARCTECLAB. We are also very grateful to two anonymous reviewers for very insightful and constructive comments on the draft of this paper.
The development of the numerical model used in this work has been financed by the Polish National Science Centre research grant no. 2015/19/B/ST10/01568 (“Discrete-element sea ice modeling – development of theoretical and numerical methods”). The co-authors Sukun Cheng and Hayley H. Shen are supported in part by ONR grant no. N00014-17-1-2862. The laboratory work described in this publication was supported by the European Community’s Horizon 2020 programme through the grant to the budget of the Integrated Infrastructure Initiative Hydralab
This paper was edited by Lars Kaleschke and reviewed by two anonymous referees.