Wave–ice interactions in the marginal ice zone. Part 1: Theoretical foundations

doi:10.1016/j.ocemod.2013.05.010

Ocean Modelling 海洋建模

Volume 71, November 2013, Pages 81-91
第 71 卷，2013 年 11 月，81-91 页

https://doi.org/10.1016/j.ocemod.2013.05.010 Get rights and content 获取权利和内容

Highlights 亮点

•
A one-dimensional model for wave-ice interactionst is presented.
提出了一种一维波浪与冰相互作用的模型。
•
The main components are floe breaking parameterization and wave attenuation.
主要组成部分是冰块破碎参数化和波浪衰减。
•
Theoretical foundations for the model are laid in this paper (Part 1 of 2).
本文奠定了该模型的理论基础（共两部分，第一部分）。
•
Numerical issues are discussed and sensitivity studies in idealized and realistic settings are performed in Part 2.
在第二部分中讨论了数值问题，并在理想化和现实环境中进行了敏感性研究。

Abstract 摘要

A wave-ice interaction model for the marginal ice zone (MIZ) is reported that calculates the attenuation of ocean surface waves by sea ice and the concomitant breaking of the ice into smaller floes by the waves. Physical issues are highlighted that must be considered when ice breakage and wave attenuation are embedded in a numerical wave model or an ice/ocean model.
报告了一种边缘冰区（MIZ）的波浪-冰相互作用模型，该模型计算了海冰对海洋表面波浪的衰减以及波浪将冰块打碎成更小浮冰的过程。强调了在数值波浪模型或冰/海洋模型中嵌入冰块破裂和波浪衰减时必须考虑的物理问题。

The theoretical foundations of the model are introduced in this paper, forming the first of a two-part series. The wave spectrum is transported through the ice-covered ocean according to the wave energy balance equation, which includes a term to parameterize the wave dissipation that arises from the presence of the ice cover. The rate of attenuation is calculated using a thin-elastic-plate scattering model and a probabilistic approach is used to derive a breaking criterion in terms of the significant strain. This determines if the local wave field is sufficient to break the ice cover. An estimate of the maximum allowable floe size when ice breakage occurs is used as a parameter in a floe size distribution model, and the MIZ is defined in the model as the area of broken ice cover. Key uncertainties in the model are discussed.
本文介绍了模型的理论基础，形成了两部分系列的第一部分。波谱通过冰盖覆盖的海洋传播，遵循波能量平衡方程，该方程包括一个参数化冰盖存在所引起的波浪耗散的项。衰减率使用薄弹性板散射模型计算，并采用概率方法推导出以显著应变为标准的破碎标准。这决定了局部波场是否足以破坏冰盖。当冰块破裂发生时，最大允许冰块大小的估计被用作冰块大小分布模型中的一个参数，模型中将边界冰区（MIZ）定义为破碎冰盖的区域。讨论了模型中的关键不确定性。

Keywords 关键词

Marginal ice zone

Wave attenuation

Ice breakage

Ice/ocean modelling

边缘冰区波浪衰减冰破碎冰/海洋建模

1. Introduction 1. 引言

Access to the seasonally ice-covered seas is increasing due to the impact of climate change (see, e.g., Stephenson et al., 2011) and commercial activities there are proliferating as a result. High precision forecasts of these regions are therefore in great demand. This paper and its companion (referred to as Part 2, Williams et al., submitted for publication) is a step towards making those forecasts as accurate as practicable, by including additional physics that is currently absent in today’s ice/ocean models.
由于气候变化的影响，季节性冰覆盖海域的通行能力正在增加（参见，例如，Stephenson 等，2011），因此那里的商业活动也在不断增加。因此，对这些地区的高精度预测需求很大。本文及其伴随论文（称为第 2 部分，Williams 等，已提交出版）是朝着使这些预测尽可能准确迈出的一步，通过引入当前在今天的冰/海洋模型中缺失的额外物理学。

Improved spatial resolution has significantly enhanced how models represent the mean sea state and its variability, but it has also highlighted a number of problems that have previously remained hidden. One of them concerns the role of surface gravity waves in shaping the so-called marginal ice zone (MIZ), an important region between the open ocean and the interior pack ice where intense coupling between waves, sea ice, ocean and atmosphere occurs. The MIZ is identified visually as a collection of relatively small floes. Surface waves are the main agent responsible for ice fragmentation and, depending upon wave and sea ice properties, they can propagate long distances into the ice field and still contribute to breakage. Indeed, Prinsenberg and Peterson (2011) recorded flexural failure induced by swell propagating within multiyear pack ice during the summer of 2009, even at very large distances from the ice edge in the Beaufort Sea. (Asplin et al., 2012, further analyzed this event.) While the local sea ice there qualified as being heavily decayed by melting (Barber et al., 2009), and thus more fragile, these observations suggest that such events could occur more frequently deep within the ice pack in a warmer Arctic that is no longer protected by a durable, extensive shield of sea ice.
改善的空间分辨率显著增强了模型对平均海洋状态及其变异性的表示，但也突显出许多之前隐藏的问题。其中之一涉及表面重力波在塑造所谓的边缘冰区（MIZ）中的作用，MIZ 是开放海洋与内部浮冰之间的重要区域，在该区域内，波浪、海冰、海洋和大气之间发生强烈耦合。MIZ 在视觉上被识别为一组相对较小的冰块。表面波是导致冰块破碎的主要因素，具体取决于波浪和海冰的特性，它们可以在冰区内传播很长距离，并仍然对破碎产生影响。实际上，Prinsenberg 和 Peterson（2011）记录了 2009 年夏季在北极海中多年度浮冰内，由涌浪引起的弯曲破坏，甚至在距离冰缘非常远的地方。（Asplin 等，2012 年对此事件进行了进一步分析。）尽管当地的海冰被认为因融化而严重退化（Barber 等）。这些观察表明，在不再受到持久、广泛的海冰保护的温暖北极，类似事件可能会在冰层深处更频繁地发生。

Interactions between ocean waves and sea ice occur on small to medium scales, but they have a profound effect on the large-scale dynamics and thermodynamics of the sea ice. On a large scale the ice cover deforms in response to stresses imposed by winds and currents. It is customary to model pack ice as a uniform viscous-plastic (VP) material (Hibler, 1979, Hunke and Dukowicz, 1997), but alternatives such as the elasto-brittle rheology of Girard et al. (2010) have been proposed to account for the discrepancies in spatial and temporal scalings of ice deformations between VP model predictions and observations (Rampal et al., 2008, Girard et al., 2009). These models, however, function best when the sea ice is highly compact and sustains large internal stresses with deformation primarily along failure lines.
海洋波浪与海冰之间的相互作用发生在小到中等的尺度上，但它们对海冰的大尺度动力学和热力学有深远的影响。在大尺度上，冰盖会因风和洋流施加的应力而变形。通常将浮冰建模为均匀的粘性塑性（VP）材料（Hibler, 1979; Hunke 和 Dukowicz, 1997），但也提出了如 Girard 等人（2010）的弹脆流变学等替代模型，以解释 VP 模型预测与观测之间在冰变形的空间和时间尺度上的差异（Rampal 等人, 2008; Girard 等人, 2009）。然而，这些模型在海冰高度紧凑并承受大内部应力、变形主要沿着破裂线时效果最佳。

In contrast, floe sizes in the MIZ are generally smaller due to wave-induced ice breakage and the ice cover is therefore normally less compact, internal stresses are less important than other forcing because the ice floes are freer to move laterally, and deformations occur more fluently compared to the plastic-like, discontinuous deformation of the compact central ice pack. In this regime, internal stresses arise more from floe-floe contact forces than from any connate constitutive relation that embodies the behaviour of sea ice at large scales. Evidently, a model of the MIZ requires knowledge of how waves control the floe size distribution (FSD). Recognizing this, Shen et al., 1986, Feltham, 2005 have proposed granular-type rheologies for the MIZ that contain an explicit dependence on floe size, while others have presented direct numerical simulations of the MIZ using granular models with either a single floe diameter (e.g. Shen and Sankaran, 2004, Herman, 2011), or with floe diameters sampled from a power-law type FSD (Herman, 2013). Parameterizations for floe size-dependent thermodynamical processes have also been developed (Steele et al., 1989, Steele, 1992).
与此相比，边界冰区（MIZ）中的冰块尺寸通常较小，这主要是由于波浪引起的冰破裂，因此冰盖通常不那么紧凑，内部应力的重要性低于其他外部作用，因为冰块在横向上更自由地移动，变形相较于紧凑的中央冰盖的塑性、不连续变形更为流畅。在这种状态下，内部应力更多地来源于冰块之间的接触力，而不是任何体现大尺度海冰行为的内在本构关系。显然，MIZ 模型需要了解波浪如何控制冰块尺寸分布（FSD）。意识到这一点，Shen 等人（1986 年）、Feltham（2005 年）提出了对 MIZ 的颗粒型流变学模型，这些模型明确依赖于冰块尺寸，而其他研究则使用颗粒模型进行了 MIZ 的直接数值模拟，这些模型要么采用单一冰块直径（例如，Shen 和 Sankaran，2004 年；Herman，2011 年），要么采用从幂律型 FSD 中抽样的冰块直径（Herman，2013 年）。还开发了依赖于冰块尺寸的热力学过程的参数化（Steele 等人，1989 年；Steele，1992 年）。

The distance over which waves induce the sea ice to break, i.e. the width of the MIZ, is controlled by exponential attenuation of the waves imposed by the presence of ice-cover. The rate of wave attenuation depends on wave period and the properties of the ice cover (Squire and Moore, 1980, Wadhams et al., 1988). Wave attenuation is modeled using multiple wave scattering theory or by models in which the ice cover is a viscous fluid or a viscoelastic material. In scattering models, wave energy is reduced with distance traveled into the ice-covered ocean by an accumulation of the partial reflections that occur when a wave encounters a floe edge (Bennetts and Squire, 2012b). Scattering models are hence strongly dependent on the FSD. In viscous models (e.g. Weber, 1987, Keller, 1998, Wang and Shen, 2011a) wave energy is lost to viscous dissipation, so these models are essentially independent of the FSD. We will use an attenuation model that includes both multiple wave scattering and viscous dissipation of wave energy. This means that there is a feedback between the FSD and wave attenuation, since the amount of breaking depends on how much incoming waves are attenuated, and the amount of scattering depends on how much breaking there is.
波浪使海冰破裂的距离，即边缘冰区（MIZ）的宽度，受冰盖存在所引起的波浪指数衰减的控制。波浪衰减的速率取决于波浪周期和冰盖的特性（Squire 和 Moore，1980 年；Wadhams 等，1988 年）。波浪衰减通过多重波散射理论或将冰盖视为粘性流体或粘弹性材料的模型进行建模。在散射模型中，波浪能量在进入冰盖海洋的过程中因波浪遇到浮冰边缘时发生的部分反射的累积而减少（Bennetts 和 Squire，2012b）。因此，散射模型在很大程度上依赖于浮冰尺寸分布（FSD）。在粘性模型中（例如，Weber，1987 年；Keller，1998 年；Wang 和 Shen，2011a），波浪能量因粘性耗散而损失，因此这些模型基本上与 FSD 无关。我们将使用一个包含多重波散射和波浪能量粘性耗散的衰减模型。这意味着 FSD 与波衰减之间存在反馈，因为破碎的程度取决于入射波的衰减程度，而散射的程度则取决于破碎的程度。

The notion and importance of integrating wave-ice interactions into an ice/ocean model is not new; indeed it was broached by the third author (VAS) more than two decades ago. Since then, several authors have presented numerical models for transporting wave energy into ice-covered fluids. Masson and LeBlond (1989) were the first to incorporate the effects of ice into the wave energy transport/balance equation that had previously been only used to model waves in open water (Gelci et al., 1957, Hasselmann, 1960, WAMDI Group, 1988, Ardhuin et al., 2010). Masson and LeBlond (1989) studied the evolution of the wave spectrum with time and distance into the ice and their theory was used subsequently by Perrie and Hu (1996) to compare the attenuation occurring in the ice field with experimental data. Meylan et al. (1997) derived a similar transport equation to that of Masson and LeBlond (1989) using the work of Howells (1960), and concentrated on the evolution of the directional spectrum. While, like us, they neglected non-linearity and the effects of wind and dissipation due to wave breaking, they improved the floe model by representing the ice as a thin elastic plate rather than as a rigid body. Doble and Bidlot (in press) have also recently extended the operational wave model WAM into the ice in the Weddell Sea, Antarctica, using the attenuation model of Kohout and Meylan (2008). While this model does not allow for directional scattering, it does include the usual open-water sources of wave generation and dissipation in the same way that Masson and LeBlond, 1989, Perrie and Hu, 1996 did.
将波浪与冰相互作用整合到冰/海洋模型中的概念和重要性并不是新鲜事；实际上，第三作者（VAS）在二十多年前就提出过这个问题。从那时起，几位作者提出了将波能传输到冰覆盖流体中的数值模型。Masson 和 LeBlond（1989）是首个将冰的影响纳入波能传输/平衡方程的研究者，该方程之前仅用于模拟开水中的波浪（Gelci 等，1957 年；Hasselmann，1960 年；WAMDI 小组，1988 年；Ardhuin 等，2010 年）。Masson 和 LeBlond（1989）研究了波谱随时间和距离进入冰层的演变，他们的理论随后被 Perrie 和 Hu（1996）用于将冰区的衰减与实验数据进行比较。Meylan 等（1997）利用 Howells（1960）的研究推导出了与 Masson 和 LeBlond（1989）相似的传输方程，并集中研究了方向谱的演变。虽然他们像我们一样忽略了非线性以及由于波浪破碎引起的风和耗散效应，但他们通过将冰表示为薄弹性板而不是刚体来改进了浮冰模型。Doble 和 Bidlot（即将发表）最近还将操作波浪模型 WAM 扩展到南极韦德尔海的冰中，使用了 Kohout 和 Meylan（2008）的衰减模型。虽然该模型不允许方向散射，但它确实包括了与 Masson 和 LeBlond（1989）、Perrie 和 Hu（1996）相同的开放水域波浪生成和耗散的常规来源。

The above papers give the framework and demonstrate some implementations of wave energy transport into the sea ice, but all neglect ice breakage. In fact, it is only recently that this effect was included by Dumont et al. (2011) (hereafter referred to as DKB) in a wave transport problem. Previous papers modeling ice fracture are those by Langhorne et al., 2001, Vaughan and Squire, 2011. However, those authors only looked at general properties of the ice cover, such as the lifetimes of ice sheets and the width of the MIZ. The method used involved modeling the attenuation of an incident wave spectrum and defining probabilistic breaking criteria to decide when the strains in the ice would exceed a breaking strain. The model of DKB provides a fuller description of the resulting ice cover: it estimates the spatial variation of floe sizes throughout the entire region where breaking occurs and also allows the temporal evolution to be investigated. In addition, it considers the coupling between the breaking and the transport of wave energy.
上述论文提供了波浪能量传输到海冰中的框架，并展示了一些实现方式，但都忽略了冰的破裂。事实上，直到最近，Dumont 等人（2011 年）（以下简称 DKB）才在波浪传输问题中考虑了这一影响。之前对冰裂的建模论文包括 Langhorne 等人（2001 年）和 Vaughan 与 Squire（2011 年）。然而，这些作者仅关注冰盖的一般特性，例如冰层的寿命和边缘混合区的宽度。所使用的方法涉及建模入射波谱的衰减，并定义概率破裂标准以决定冰中的应变何时超过破裂应变。DKB 的模型提供了对结果冰盖的更全面描述：它估计了破裂发生区域内浮冰大小的空间变化，并允许研究时间演变。此外，它考虑了破裂与波浪能量传输之间的耦合。

Although the DKB model is one-dimensional, i.e. it only considers a transect of the ocean, it is theoretically generalizable to include the second horizontal dimension. Before this geometrical restriction is tackled, however, important themes have been identified for discussion and investigation, which is the purpose of this paper. Firstly, we put the work of DKB into the context of previous work on modeling wave energy in ice (Masson and LeBlond, 1989, Perrie and Hu, 1996, Meylan and Masson, 2006) and we correct their interpretation of the spectral density function. Secondly, we revise the floe-breaking criteria based on monochromatic wave amplitudes employed by DKB, and propose one that is based on wave statistics instead. Numerical issues, sensitivity analyses and model results are reserved until Part 2.
尽管 DKB 模型是一维的，即它仅考虑海洋的一个剖面，但从理论上讲，它可以推广到包括第二个水平维度。然而，在解决这一几何限制之前，已经确定了一些重要主题供讨论和研究，这也是本文的目的。首先，我们将 DKB 的工作置于之前关于冰中波浪能建模的研究背景下（Masson 和 LeBlond，1989 年；Perrie 和 Hu，1996 年；Meylan 和 Masson，2006 年），并纠正他们对谱密度函数的解释。其次，我们修订了 DKB 所采用的基于单色波幅的破冰标准，并提出一个基于波浪统计的新标准。数值问题、敏感性分析和模型结果将在第二部分中讨论。

2. Description of the waves-in-ice model
2. 冰中波浪模型的描述

2.1. Overview 2.1. 概述

Fig. 1 shows the flow of information into and out of the waves-in-ice model (WIM), whose three components, namely advection, attenuation and ice breakage, are discussed in more detail in Section 3. We briefly describe their relationship to the inputs and outputs here.
图 1 显示了波浪在冰模型（WIM）中信息的流入和流出，其三个组成部分，即平流、衰减和冰破裂，在第 3 节中进行了更详细的讨论。我们在此简要描述它们与输入和输出之间的关系。

The advection and attenuation steps depend on the group velocity, $c_{g}$ , and the attenuation coefficient, $\hat{α}$ . Both $c_{g}$ and $\hat{α}$ depend on frequency in addition to the ice properties. The advection and attenuation steps describe how the wave energy is transported into the ice-covered ocean. The WIM therefore extends contemporary external wave models (EWMs, e.g. WAM, WAVEWATCH III), which typically do not operate in ice-covered oceans. The presence of waves in ice-covered oceans causes ice breakage to occur in the MIZ, thereby altering the local FSD.
对流和衰减步骤依赖于群速度 $c_{g}$ 和衰减系数 $\hat{α}$ 。 $c_{g}$ 和 $\hat{α}$ 除了依赖于冰的特性外，还依赖于频率。对流和衰减步骤描述了波能如何被输送到冰覆盖的海洋中。因此，WIM 扩展了当代外部波模型（EWMs，例如 WAM，WAVEWATCH III），这些模型通常不在冰覆盖的海洋中运行。冰覆盖海洋中波的存在导致在边界冰区（MIZ）发生冰的破裂，从而改变了局部的冰厚分布（FSD）。

The outputs will, of course, have follow-on effects on the ice properties when they are fed back into the ice-ocean model. For instance, we use the FSD to distinguish between interior pack ice and the MIZ. Consequently, the FSD determines which ice rheology applies to different areas and thus how the ice drifts. It can also be used to change the thermodynamics of the ice by increasing melting or freezing due to the extra surface area exposed to the air and water (Steele, 1992).
这些输出当然会对冰的特性产生后续影响，当它们反馈到冰-海洋模型中时。例如，我们使用 FSD 来区分内部浮冰和边缘冰区。因此，FSD 决定了不同区域适用的冰流变学，从而影响冰的漂移。它还可以通过增加暴露于空气和水的额外表面积来改变冰的热力学，进而增加融化或冻结（Steele, 1992）。

Another important follow-on/coupling effect is the momentum/energy exchange between the waves, the ocean and the atmosphere. Even without the complicating presence of sea ice, the question of how to couple ocean models to the wave field is not yet resolved (e.g. Babanin et al., 2009, Ardhuin et al., 2008). With attendant sea ice as well, wave attenuation occurs which we include in our model by considering two processes. Part of the energy lost by the waves as they travel into an ice field is attributed to scattering. In our model the scattering process is conservative and so energy lost in this way must be reflected back into the open ocean. The proportion of reflected energy can be calculated. The remaining energy loss is parameterized in the model by adding a damping pressure, which resists particle motion at the ice-water interface (see Appendix A). The actual mechanisms responsible for this energy loss are poorly understood and inadequately parameterized at present, and further investigation will be required to balance momentum/energy in a fully coupled model. Notwithstanding, it is important to include damping in the WIM to accurately predict the distance waves travel into the ice-covered ocean, and hence the region of ice broken by the waves, i.e. the width of the MIZ.
另一个重要的后续/耦合效应是波浪、海洋和大气之间的动量/能量交换。即使没有海冰的复杂影响，如何将海洋模型与波场耦合的问题仍未解决（例如，Babanin 等，2009 年；Ardhuin 等，2008 年）。伴随海冰的存在，波浪衰减现象也会发生，我们在模型中通过考虑两个过程来包含这一现象。波浪在进入冰区时损失的部分能量归因于散射。在我们的模型中，散射过程是保守的，因此以这种方式损失的能量必须反射回开放海洋。反射能量的比例可以计算。剩余的能量损失在模型中通过添加阻尼压力进行参数化，该压力抵抗冰水界面处的粒子运动（见附录 A）。目前，导致这种能量损失的实际机制尚不清楚，参数化也不充分，进一步的研究将是必要的，以在完全耦合模型中平衡动量/能量。尽管如此，在波浪入射模型中包含阻尼是重要的，以准确预测波浪在冰盖海洋中传播的距离，从而确定波浪破坏的冰区，即边缘冰区的宽度。

2.2. Inputs and outputs 2.2. 输入和输出

The inputs to the WIM are the ice properties, the incident wave field and the initial FSD. Technically the FSD is also an ice property, but we treat it separately due to the special role it plays in the WIM.
WIM 的输入包括冰的性质、入射波场和初始 FSD。从技术上讲，FSD 也是一种冰的性质，但由于其在 WIM 中所扮演的特殊角色，我们将其单独处理。

The ice properties are all considered to vary spatially but not to vary in time. The ice concentration (c) and thickness (h) are standard variables of ice/ocean models, and so estimates for them can be easily obtained. However, the effective Young’s modulus ( $Y^{*}$ ), Poisson’s ratio (ν) and breaking strain ( $ε_{c}$ ) are non-standard and must be estimated (see Section 4.3). A value for the damping coefficient Γ, which is included to increase the attenuation of long waves as this is underpredicted by conservative scattering theory, is extracted from the attenuation measurements of Squire and Moore (1980) (see Appendix A and Section 4.2).
冰的性质被认为在空间上变化，但在时间上不变化。冰浓度（c）和厚度（h）是冰/海洋模型的标准变量，因此可以轻松获得它们的估计值。然而，有效的杨氏模量（ $Y^{*}$ ）、泊松比（ν）和破坏应变（ $ε_{c}$ ）是非标准的，必须进行估计（见第 4.3 节）。为了增加长波的衰减，阻尼系数Γ的值被提取自 Squire 和 Moore（1980）的衰减测量（见附录 A 和第 4.2 节），因为保守散射理论对其的预测不足。

The wave energy is described by the spectral density function (SDF) $S (ω, x, t)$ , where $ω = 2 π / T$ is the angular frequency and T is the wave period. (For brevity, the SDF is sometimes written $S = S (ω)$ , taking the spatial $(x)$ and temporal $(t)$ dependencies to be implicit.) The wave spectrum may be defined either in the open ocean or within the sea ice, after having undergone some attenuation. However, most EWMs only predict S inside a region known as a wave mask, which currently stops at a conservative distance from the ice edge. If $x = 0$ is the edge of the wave mask, the EWM provides the initial boundary condition for the WIM, $S (ω, 0, t) = S_{0} (ω, t)$ , where $S_{0}$ is known. The WIM advects this initial spectrum across the gap between the wave mask and the ice mask, and then into the ice-covered ocean. The wave spectrum is advected according to the energy transport equations in Section 3.1—numerical details are given in Part 2.
波能量由谱密度函数（SDF） $S (ω, x, t)$ 描述，其中 $ω = 2 π / T$ 是角频率，T 是波周期。（为简便起见，SDF 有时写作 $S = S (ω)$ ，隐含空间 $(x)$ 和时间 $(t)$ 依赖性。）波谱可以在开放海洋或海冰内定义，经过一定的衰减。然而，大多数 EWMs 仅在称为波掩模的区域内预测 S，该区域目前在离冰缘有一个保守的距离。如果 $x = 0$ 是波掩模的边缘，EWM 为 WIM 提供初始边界条件 $S (ω, 0, t) = S_{0} (ω, t)$ ，其中 $S_{0}$ 是已知的。WIM 将这个初始谱通过波掩模和冰掩模之间的间隙进行输送，然后进入冰覆盖的海洋。波谱根据第 3.1 节中的能量输送方程进行输送——数值细节在第 2 部分中给出。

The FSD is characterized by two spatially varying floe length parameters, $D_{\max} (x, t)$ and $〈 D 〉 (x, t)$ , which also evolve with time. These are the maximum floe length and average floe length, respectively. The initial FSD is generally unknown. In our experiments we assume that prior to wave-induced ice breakage all floe lengths have a large value (e.g. 500 m; the precise value turns out to be relatively unimportant). After the waves have traveled into the ice and caused ice breakage, the FSD is parameterized as in Section 4.1
FSD 的特征是两个空间变化的冰块长度参数 $D_{\max} (x, t)$ 和 $〈 D 〉 (x, t)$ ，它们也随时间演变。这分别是最大冰块长度和平均冰块长度。初始 FSD 通常是未知的。在我们的实验中，我们假设在波浪引起冰块破裂之前，所有冰块长度都具有较大值（例如 500 米；精确值相对不重要）。在波浪进入冰层并导致冰块破裂后，FSD 的参数化如第 4.1 节所述。

3. Model components 3. 模型组件

3.1. Advection and attenuation
3.1. 平流与衰减

The waves are advected according the energy balance equation, namely(1) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = R_{in} - R_{ice} - R_{other} - R_{nl},$ (Masson and LeBlond, 1989, Meylan and Masson, 2006, Ardhuin et al., 2010), where $c_{g}$ is the group velocity and $D_{t} \equiv (\partial_{t} + c_{g} \partial_{x})$ . The source terms $R_{in}, R_{ice}$ and $R_{other}$ represent respectively the wind energy input, rates of energy loss to (or due to) the ice and the total of all other dissipation sources (e.g. friction at the bottom of the sea, losses from wave breaking or white-capping, Ardhuin et al., 2010). These are all quasi-linear in S. The $R_{nl}$ term incorporates fully non-linear energy exchanges between frequencies (Hasselmann, 1962, Hasselmann, 1963).
波浪根据能量平衡方程进行输送，即 (1) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = R_{in} - R_{ice} - R_{other} - R_{nl},$ （Masson 和 LeBlond，1989 年，Meylan 和 Masson，2006 年，Ardhuin 等，2010 年），其中 $c_{g}$ 是群速度， $D_{t} \equiv (\partial_{t} + c_{g} \partial_{x})$ 。源项 $R_{in}, R_{ice}$ 和 $R_{other}$ 分别表示风能输入、能量损失（或由于冰的影响）速率以及所有其他耗散源的总和（例如，海底摩擦、波浪破碎或白顶损失，Ardhuin 等，2010 年）。这些在 S 中都是准线性的。 $R_{nl}$ 项完全包含了频率之间的非线性能量交换（Hasselmann，1962 年，Hasselmann，1963 年）。

For the WIM, we set $R_{other} = R_{nl} = 0$ and $R_{ice} = \hat{α} S$ , i.e.(2) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = - \hat{α} (ω, c, h, 〈 D 〉) S (ω; x, t) .$ The quantity $\hat{α}$ is the dimensional attenuation coefficient, given by(3) $\hat{α} = \frac{α c}{〈 D 〉},$ where α is the non-dimensional attenuation coefficient, i.e. the (average) amount of attenuation per individual floe, which is a function of ice thickness and wave period. The definition $R_{ice} = \hat{α} S$ does not allow transfer of energy between directions (via diffraction by ice floes), as done by Masson and LeBlond, 1989, Perrie and Hu, 1996, Meylan et al., 1997. This is a necessary limitation of the one-dimensional numerical model outlined in Part 2. $R_{ice}$ is quasi-linear since an S that is sufficiently large to cause breaking lowers the average floe size $〈 D 〉$ and subsequently increases $\hat{α}$ , according to (3).
对于 WIM，我们设定 $R_{other} = R_{nl} = 0$ 和 $R_{ice} = \hat{α} S$ ，即 (2) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = - \hat{α} (ω, c, h, 〈 D 〉) S (ω; x, t) .$ 。数量 $\hat{α}$ 是维度衰减系数，由 (3) $\hat{α} = \frac{α c}{〈 D 〉},$ 给出，其中α是无量纲衰减系数，即每个冰块的（平均）衰减量，它是冰厚度和波周期的函数。定义 $R_{ice} = \hat{α} S$ 不允许能量在方向之间转移（通过冰块的衍射），正如 Masson 和 LeBlond（1989 年）、Perrie 和 Hu（1996 年）、Meylan 等（1997 年）所做的那样。这是第 2 部分中概述的一维数值模型的必要限制。 $R_{ice}$ 是准线性的，因为足够大的 S 会导致破碎，从而降低平均冰块大小 $〈 D 〉$ ，并根据（3）随后增加 $\hat{α}$ 。

The effects of neglecting $R_{other}$ and $R_{nl}$ are not clear. They may be important in moving the energy across the gap between the wave and ice masks, although we note that as the resolution of the EWMs increases, this will become less of an issue. It is difficult to say how much effect these terms will have once the waves are in the ice-covered ocean, or how they should change to represent the different environment there. Masson and LeBlond, 1989, Perrie and Hu, 1996, Doble and Bidlot, in press assumed some of the effects (like wind generation) were proportional to the open water fraction, and that $R_{nl}$ was the same in the ice-covered ocean as in open water. (Polnikov and Lavrenov, 2007, recently confirmed the validity of this last assumption.) We note that by including wind generation in the ice, Perrie and Hu (1996) were able to reproduce (qualitatively at least) the observed ‘rollover’ in the effective attenuation coefficient. That is, instead of attenuation increasing monotonically with frequency, it reaches a maximum value before starting to drop again.
忽视 $R_{other}$ 和 $R_{nl}$ 的影响尚不明确。它们可能在波浪与冰面之间的能量传递中起着重要作用，尽管我们注意到，随着 EWMs 分辨率的提高，这将不再是一个问题。很难说这些项在波浪进入冰覆盖的海洋后会产生多大影响，或者它们应该如何变化以代表那里的不同环境。Masson 和 LeBlond（1989 年）、Perrie 和 Hu（1996 年）、Doble 和 Bidlot（即将发表）假设某些影响（如风生成）与开水面积的比例成正比，并且 $R_{nl}$ 在冰覆盖的海洋中与开水中相同。（Polnikov 和 Lavrenov，2007 年，最近确认了这一最后假设的有效性。）我们注意到，通过在冰中包含风生成，Perrie 和 Hu（1996 年）能够再现（至少在定性上）观察到的有效衰减系数的“翻转”。也就是说，衰减并不是随着频率单调增加，而是在达到最大值后开始再次下降。

The operator $D_{t}$ is the material derivative, or the time derivative in a reference frame moving with the wave (the Lagrangian reference frame) at the group velocity $c_{g}$ . We can also reconfigure the above problem, in between breaking events, in the Lagrangian frame, as(4a) $\frac{d x}{d t} = c_{g} (ω, x, t_{*}),$ (4b) $\frac{d}{d x} S (ω; x, t) = - \hat{α} (ω; x, t_{*}, S_{*}) S (ω; x, t),$ where $t_{*}$ is the last time ice breakage occurred at x, and $S_{*} (ω, x) = S (ω; x, t_{*})$ . Thus we have separated the problem into an advection problem and an attenuation one, and in our numerical scheme presented in Part 2, we solve (2) by alternately advecting and attenuating.
算子 $D_{t}$ 是物质导数，或在以波的群速度 $c_{g}$ 移动的参考系中时间导数（拉格朗日参考系）。我们还可以在破裂事件之间，在拉格朗日框架中重新配置上述问题，如 (4a) $\frac{d x}{d t} = c_{g} (ω, x, t_{*}),$ (4b) $\frac{d}{d x} S (ω; x, t) = - \hat{α} (ω; x, t_{*}, S_{*}) S (ω; x, t),$ ，其中 $t_{*}$ 是冰块在 x 处最后一次破裂发生的时间，以及 $S_{*} (ω, x) = S (ω; x, t_{*})$ 。因此，我们将问题分为一个平流问题和一个衰减问题，在我们在第 2 部分中提出的数值方案中，我们通过交替平流和衰减来求解 (2)。

3.2. Ice breakage 3.2. 冰块破裂

We take a probabilistic approach to define a criterion for ice breakage. It is therefore helpful to revise some relationships between the SDF (S) and different wave statistics, before defining the breaking criterion itself.
我们采用概率方法来定义冰破裂的标准。因此，在定义破裂标准之前，修订 SDF（S）与不同波动统计之间的一些关系是有帮助的。

3.2.1. Wave energy and statistics
3.2.1. 波浪能量与统计学

We assume that the sea surface elevation, $η$ , follows a Gaussian distribution, and neglect non-linear effects that cause slight asymmetry (Cartwright and Longuet-Higgins, 1956, Vaughan and Squire, 2011). The mean square sea surface elevation (vertical displacement from the mean water level), or the variance in the position of a water particle at the sea surface, $〈 η^{2} 〉 = m_{0} [η]$ , can be obtained from S via the formula(5) $m_{n} [η] = \int_{0}^{\infty} ω^{n} S (ω) d ω,$ (World Meteorological Organization, 1998). (We will also use the second spectral moment, $m_{2}$ , later on.) The significant wave height is defined by $H_{s} = 4 \sqrt{m_{0} [η]}$ .
我们假设海面高度 $η$ 遵循高斯分布，并忽略导致轻微不对称的非线性效应（Cartwright 和 Longuet-Higgins，1956 年；Vaughan 和 Squire，2011 年）。平均平方海面高度（相对于平均水位的垂直位移），或海面水粒子位置的方差 $〈 η^{2} 〉 = m_{0} [η]$ ，可以通过公式 (5) $m_{n} [η] = \int_{0}^{\infty} ω^{n} S (ω) d ω,$ 从 S 中获得（世界气象组织，1998 年）。（我们稍后还将使用第二个谱矩 $m_{2}$ 。）显著波高定义为 $H_{s} = 4 \sqrt{m_{0} [η]}$ 。

Wave heights generally follow a Rayleigh distribution, for which the probability of a wave amplitude A exceeding a certain value $A_{c}$ is approximately(6) $P (A > A_{c}) = \exp (- A_{c}^{2} / 〈 A^{2} 〉),$ (Longuet-Higgins, 1952, Longuet-Higgins, 1980), where $〈 A^{2} 〉$ denotes the mean square amplitude. If the wave spectrum has a narrow bandwidth and non-linear effects are negligible (low wave steepness), then $〈 A^{2} 〉 = 2 m_{0} [η]$ , so(7) $P (A > A_{c}) = \exp (- A_{c}^{2} / 2 m_{0} [η]) .$ The mean square displacement of the ice is approximately $〈 η_{ice}^{2} 〉 = m_{0} [η_{ice}]$ , where(8) $m_{n} [η_{ice}] = \int_{0}^{\infty} ω^{n} S (ω) W^{2} (ω) d ω .$ Here $W (ω) \approx k_{ice} | T | / k$ , where $T$ is the transmission coefficient for a wave traveling from water into ice (e.g. Williams and Porter, 2009), represents the amplitude response at each frequency of an ice floe to forcing from a wave of unit amplitude in the water surrounding it. The wave number $k (ω) = ω^{2} / g$ is the usual deep water propagating wave number, while $k_{ice} (ω)$ is the positive real root of (A.7), the dispersion relation for a section of ice-covered ocean.
波高通常遵循瑞利分布，其中波幅 A 超过某一值 $A_{c}$ 的概率约为 (6) $P (A > A_{c}) = \exp (- A_{c}^{2} / 〈 A^{2} 〉),$ （Longuet-Higgins, 1952; Longuet-Higgins, 1980），其中 $〈 A^{2} 〉$ 表示均方幅度。如果波谱具有窄带宽且非线性效应可以忽略（低波陡度），则 $〈 A^{2} 〉 = 2 m_{0} [η]$ ，因此 (7) $P (A > A_{c}) = \exp (- A_{c}^{2} / 2 m_{0} [η]) .$ 冰的均方位移约为 $〈 η_{ice}^{2} 〉 = m_{0} [η_{ice}]$ ，其中 (8) $m_{n} [η_{ice}] = \int_{0}^{\infty} ω^{n} S (ω) W^{2} (ω) d ω .$ 这里 $W (ω) \approx k_{ice} | T | / k$ ，其中 $T$ 是从水中传播到冰中的波的传输系数（例如，Williams 和 Porter, 2009），表示冰块在周围水中单位幅度波的强迫下对每个频率的幅度响应。波数 $k (ω) = ω^{2} / g$ 是通常的深水传播波数，而 $k_{ice} (ω)$ 是（A.7）的正实根，即冰盖海洋的一段的色散关系。

The probability of $A_{ice}$ exceeding a certain value $A_{c}$ is(9) $P (A_{ice} > A_{c}) = \exp (- A_{c}^{2} / 2 m_{0} [η_{ice}]),$ which is analogous to Eq. (7). In addition, we can also estimate the number of waves we expect in a given time interval $Δ t, N_{W}$ , as(10) $N_{W} = \frac{Δ t}{2 π} \sqrt{\frac{m_{2} [η_{ice}]}{m_{0} [η_{ice}]}},$ (World Meteorological Organization, 1998). (Note that factors in Eqs. (7), (10) have been corrected from their counterparts in Cartwright and Longuet-Higgins, 1956.) More precisely, this is the number of times we can expect a particle to cross its point of mean displacement in a downward direction. The quantity $N_{W}$ also defines a representative wave period(11) $T_{W} = \frac{Δ t}{N_{W}} = 2 π \sqrt{\frac{m_{0} [η_{ice}]}{m_{2} [η_{ice}]}},$ for the spectrum S at a given point and a representative (ice-coupled) wavelength of $λ_{W} = 2 π / k_{W}$ , where $k_{W} = k_{ice} (2 π / T_{ω})$ . The symbol $T_{W}$ is sometimes written $T_{m_{0, 2}}$ but we use the former to avoid clutter in our equations. Also note the factor of $2 π$ is necessary since we define the moments $m_{n}$ in terms of angular frequency $ω$ , rather than the frequency itself $(1 / T)$ .
$A_{ice}$ 超过某个值 $A_{c}$ 的概率是 (9) $P (A_{ice} > A_{c}) = \exp (- A_{c}^{2} / 2 m_{0} [η_{ice}]),$ ，这与公式 (7) 类似。此外，我们还可以估计在给定时间间隔 $Δ t, N_{W}$ 内我们期望的波数，如 (10) $N_{W} = \frac{Δ t}{2 π} \sqrt{\frac{m_{2} [η_{ice}]}{m_{0} [η_{ice}]}},$ （世界气象组织，1998）。（注意，公式 (7) 和 (10) 中的因素已从 Cartwright 和 Longuet-Higgins（1956）中的对应项进行了修正。）更准确地说，这是我们期望粒子向下穿越其平均位移点的次数。量 $N_{W}$ 还定义了在给定点的谱 S 的代表性波周期 (11) $T_{W} = \frac{Δ t}{N_{W}} = 2 π \sqrt{\frac{m_{0} [η_{ice}]}{m_{2} [η_{ice}]}},$ 和代表性（冰耦合）波长 $λ_{W} = 2 π / k_{W}$ ，其中 $k_{W} = k_{ice} (2 π / T_{ω})$ 。符号 $T_{W}$ 有时写作 $T_{m_{0, 2}}$ ，但我们使用前者以避免公式中的混乱。还要注意，因子 $2 π$ 是必要的，因为我们将矩 $m_{n}$ 定义为角频率 $ω$ ，而不是频率本身 $(1 / T)$ 。

We can also define analogous quantities for the strain, which for a thin elastic plate is defined as $ε = (h / 2) \partial_{x}^{2} η_{ice}$ . Its mean square value is $〈 ε^{2} 〉 = m_{0} [ε]$ , where(12) $m_{n} [ε] = \int_{0}^{\infty} ω^{n} S (ω) E^{2} (ω) d ω, E (ω) = \frac{h}{2} k_{ice}^{2} W (ω) .$ The latter is the approximate strain amplitude per metre of water displacement amplitude for a monochromatic wave of the form $η_{ice} = A_{ice} \cos (k_{ice} x - ω t)$ (with $A = 1$ m, so $A_{ice} = W$ m). It does not account for non-linear interactions between frequencies, which could potentially be important approaching an ice breakage event. For now we assume brittle failure of the ice, so that a linear stress–strain law applies right up to the point where the ice breaks. If we define the significant strain amplitude to be $E_{s} = 2 \sqrt{m_{0} [ε]}$ , which is two standard deviations in strain, then the probability of the maximum strain from a passing wave $E_{W}$ exceeding a breaking strain $ε_{c}$ is(13) $P_{ε} = P (E_{W} > ε_{c}) = \exp (- ε_{c}^{2} / 2 m_{0} [ε]) = \exp (- 2 ε_{c}^{2} / E_{s}^{2}) .$
我们还可以为应变定义类似的量，对于薄弹性板，其定义为 $ε = (h / 2) \partial_{x}^{2} η_{ice}$ 。其均方值为 $〈 ε^{2} 〉 = m_{0} [ε]$ ，其中 (12) $m_{n} [ε] = \int_{0}^{\infty} ω^{n} S (ω) E^{2} (ω) d ω, E (ω) = \frac{h}{2} k_{ice}^{2} W (ω) .$ 。后者是单色波形 $η_{ice} = A_{ice} \cos (k_{ice} x - ω t)$ （当 $A = 1$ m 时，因此 $A_{ice} = W$ m）每米水位位移幅度的近似应变幅度。它没有考虑频率之间的非线性相互作用，这在接近冰破裂事件时可能是重要的。目前我们假设冰的脆性破坏，因此线性应力-应变定律适用于冰破裂的点。如果我们定义显著应变幅度为 $E_{s} = 2 \sqrt{m_{0} [ε]}$ ，即应变的两个标准差，那么来自经过波的最大应变 $E_{W}$ 超过破裂应变 $ε_{c}$ 的概率为 (13) $P_{ε} = P (E_{W} > ε_{c}) = \exp (- ε_{c}^{2} / 2 m_{0} [ε]) = \exp (- 2 ε_{c}^{2} / E_{s}^{2}) .$ 。

3.2.2. Breaking criterion
3.2.2. 破坏准则

To determine whether the ice will be broken by waves, we define a critical probability threshold $P_{c}$ such that if $P_{ε} > P_{c}$ the ice will break. If it breaks, the maximum floe size is set to $D_{\max} = \max (λ_{W} / 2, D_{\min})$ where $D_{\min}$ is the size below which waves are not significantly attenuated and is set to 20 m (Kohout, 2008). These two quantities $D_{\min}$ and $D_{\max}$ determine the FSD (see Section 4.1).
为了确定冰是否会被波浪打破，我们定义一个临界概率阈值 $P_{c}$ ，如果 $P_{ε} > P_{c}$ ，冰将会破裂。如果破裂，最大浮冰尺寸设定为 $D_{\max} = \max (λ_{W} / 2, D_{\min})$ ，其中 $D_{\min}$ 是波浪不会显著衰减的尺寸，设定为 20 米（Kohout，2008）。这两个量 $D_{\min}$ 和 $D_{\max}$ 决定了浮冰尺寸分布（FSD）（见第 4.1 节）。

From (13), the criterion $P_{ε} > P_{c}$ can be written in terms of $E_{s}, ε_{c}$ and $P_{c}$ as(14) $E_{s} > E_{c} = ε_{c} \sqrt{- 2 / \log (P_{c})} .$ Thus the single parameter $E_{c}$ combines the effects of both $ε_{c}$ and $P_{c}$ . Note that $P_{c} = e^{- 2} \approx 0.14$ corresponds to the criterion of Langhorne et al. (2001), i.e. $E_{s} > ε_{c}$ , and the upper limit tested by Vaughan and Squire (2011).
从（13）中，标准 $P_{ε} > P_{c}$ 可以用 $E_{s}, ε_{c}$ 和 $P_{c}$ 表示为 (14) $E_{s} > E_{c} = ε_{c} \sqrt{- 2 / \log (P_{c})} .$ 。因此，单一参数 $E_{c}$ 结合了 $ε_{c}$ 和 $P_{c}$ 的影响。注意， $P_{c} = e^{- 2} \approx 0.14$ 对应于 Langhorne 等人（2001）的标准，即 $E_{s} > ε_{c}$ ，以及 Vaughan 和 Squire（2011）测试的上限。

The default value for $P_{c}$ that will be used in our numerical results is based on the condition for a narrow spectrum. For a monochromatic wave that produces a strain amplitude $E_{W}$ , the breaking condition would be $E_{W} > ε_{c}$ . Therefore, since $〈 ε^{2} 〉 = E_{W}^{2} / 2$ in that case, the breaking condition is $E_{s} > ε_{c} \sqrt{2}$ . This corresponds to choosing $P_{c} = e^{- 1} \approx 0.37$ in (14). We note that this value is easily changed in our model when better observational information becomes available.
在我们的数值结果中，将使用的 $P_{c}$ 的默认值是基于窄谱条件。对于产生应变幅度 $E_{W}$ 的单色波，破裂条件为 $E_{W} > ε_{c}$ 。因此，由于在这种情况下 $〈 ε^{2} 〉 = E_{W}^{2} / 2$ ，破裂条件为 $E_{s} > ε_{c} \sqrt{2}$ 。这对应于在(14)中选择 $P_{c} = e^{- 1} \approx 0.37$ 。我们注意到，当更好的观测信息可用时，这个值在我们的模型中很容易更改。

4. Model sub-components 4. 模型子组件

4.1. Floe size distribution
4.1. 浮冰大小分布

Prior to 2006, numerous researchers (e.g. Weeks et al., 1980, Rothrock and Thorndike, 1984, Matsushita, 1985, Holt and Martin, 2001, Toyota and Enomoto, 2002) made observations of floe sizes in Arctic areas. It was found that the FSD generally obeyed a power-law (Pareto) distribution, where the probability of finding a floe diameter D greater than $D_{*}$ is given by(15) $P (D > D_{*}) = P (D) = {(D_{\min} / D_{*})}^{γ} for D > D_{\min},$ where $D_{\min}$ is the minimum floe diameter. The expected value of $D^{n}$ is therefore $〈 D^{n} 〉 = - \int_{D_{\min}}^{\infty} D^{n} \partial_{D} P (D) d D = \frac{γ}{γ - n} D_{\min}^{n} .$ The fitted exponent γ was usually found to be greater than 2, which implies that the expected diameter and area are defined. However, there are problems with trying to treat small floes with the above distribution, i.e. if we try to let $D_{\min} \to 0$ . Therefore Toyota et al. (2006) investigated the FSD of small floes of diameter 1 m–1.5 km, using data obtained from the southern Sea of Okhotsk. They found that floes smaller than about 40 m still obeyed a power law, but were best fitted by a smaller value of γ (about 1.15). This regime shift was also observed in Antarctica in the late winter of 2006 and 2007 by Toyota et al. (2011), based on observations in the northwestern Weddell Sea and off Wilkes Land (around 64 $°$ S, 117 $°$ E) with a helicopter-borne digital video camera. Concurrent ice thickness measurements were also made, using a helicopter-borne electromagnetic sensor above the Weddell Sea and a video system off Wilkes land. The regime shift was consistent with the value(16) $D_{c} = {(\frac{π^{4} {Yh}^{3}}{48 ρ g (1 - ν^{2})})}^{1 / 4},$ which corresponds to the diameter below which flexural failure cannot occur (Mellor, 1986).
在 2006 年之前，许多研究者（例如 Weeks 等，1980 年；Rothrock 和 Thorndike，1984 年；Matsushita，1985 年；Holt 和 Martin，2001 年；Toyota 和 Enomoto，2002 年）对北极地区的冰块大小进行了观察。研究发现，冰块大小分布（FSD）通常遵循幂律（帕累托）分布，其中找到直径 D 大于 $D_{*}$ 的冰块的概率为 (15) $P (D > D_{*}) = P (D) = {(D_{\min} / D_{*})}^{γ} for D > D_{\min},$ ，其中 $D_{\min}$ 是最小冰块直径。因此， $D^{n}$ 的期望值为 $〈 D^{n} 〉 = - \int_{D_{\min}}^{\infty} D^{n} \partial_{D} P (D) d D = \frac{γ}{γ - n} D_{\min}^{n} .$ 。拟合的指数γ通常大于 2，这意味着期望的直径和面积是确定的。然而，试图用上述分布处理小冰块时存在问题，即如果我们试图让 $D_{\min} \to 0$ 。因此，Toyota 等（2006 年）调查了直径为 1m–1.5km 的小冰块的 FSD，使用的数据来自于鄂霍次克海南部。他们发现，直径小于约 40m 的冰块仍然遵循幂律，但更适合用较小的γ值（约 1.15）进行拟合。Toyota 等在 2006 年和 2007 年冬末也在南极观察到了这种状态转变。（2011），基于在西北韦德尔海和威尔克斯地（约 64 $°$ S，117 $°$ E）使用直升机搭载的数字摄像机进行的观察。同时还进行了冰厚度的测量，使用了在韦德尔海上空的直升机搭载电磁传感器和在威尔克斯地的录像系统。该制度转变与值 (16) $D_{c} = {(\frac{π^{4} {Yh}^{3}}{48 ρ g (1 - ν^{2})})}^{1 / 4},$ 一致，该值对应于弯曲破坏无法发生的直径（Mellor，1986）。

Toyota et al. (2011) proposed an explanation of the exponent governing the smaller floes in terms of a breaking probability $Π$ , related to γ by(17) $Π = ξ^{γ - 2} or γ = 2 + \log_{ξ} Π,$ where $Π$ is the probability that a floe will break into $ξ^{2}$ pieces. A similar explanation was suggested by Herman (2010), who proposed a generalised Lotka-Volterra model for the implementation of breaking. Such models produce distributions that are asymptotically like power-law distributions, but with better behaviour near $D = 0$ (i.e. $D_{\min}$ can be zero).
丰田等人（2011）提出了一个关于小冰块的指数的解释，该指数与破裂概率 $Π$ 有关，通过 (17) $Π = ξ^{γ - 2} or γ = 2 + \log_{ξ} Π,$ 与γ相关，其中 $Π$ 是冰块破裂成 $ξ^{2}$ 块的概率。赫尔曼（2010）提出了类似的解释，他提出了一种广义的洛特卡-沃尔泰拉模型用于破裂的实现。这些模型产生的分布在渐近上类似于幂律分布，但在 $D = 0$ 附近表现更好（即 $D_{\min}$ 可以为零）。

Note that the model of Toyota et al. (2011) always predicts $γ < 2$ , so other mechanisms are required to explain the exponent for the larger floes being greater than 2. Toyota et al. (2011) suggested herding with subsequent freezing together of floes could be one explanation. The simulations of Herman (2011) lent credibility to this as they showed that floes tended to group together in clusters, and that the diameter of these clusters obeyed power-law distributions with exponents often greater than 2 (depending on the concentration).
请注意，Toyota 等人（2011 年）的模型始终预测 $γ < 2$ ，因此需要其他机制来解释较大冰块的指数大于 2。Toyota 等人（2011 年）建议，冰块的聚集和随后的冻结可能是一个解释。Herman（2011 年）的模拟为此提供了可信度，因为它们显示冰块倾向于聚集成簇，这些簇的直径遵循幂律分布，指数通常大于 2（取决于浓度）。

We use the simpler approach of DKB, who restricted themselves to small floes and took the FSD to be over the finite interval of $D_{\min} < D < D_{\max}$ . The distribution inside was based on the ideas and parameters of Toyota et al. (2011), deriving a novel formula for the mean floe size $〈 D 〉$ . We set (as they did), the fixed values of $D_{\min} = 20$ m, $ξ = 2$ , and $Π = 0.9$ . It is important that $D_{\min}$ is not too small as $\hat{α}$ , as given by (3), will be very large when $〈 D 〉$ is small. However, Kohout and Meylan (2008) found that floes with lengths less than 20 m produced negligible scattering, so this value of $D_{\min}$ is a reasonable choice. It may also be possible to relate $Π$ to our breaking probabilities in the future.
我们采用了 DKB 的更简单的方法，他们将自己限制在小冰块上，并将 FSD 视为在有限区间 $D_{\min} < D < D_{\max}$ 内。内部的分布基于 Toyota 等人（2011）的思想和参数，推导出了一种新的平均冰块大小公式 $〈 D 〉$ 。我们设定了固定值 $D_{\min} = 20$ m， $ξ = 2$ 和 $Π = 0.9$ （与他们相同）。重要的是 $D_{\min}$ 不能太小，因为当 $〈 D 〉$ 较小时，(3)给出的 $\hat{α}$ 将会非常大。然而，Kohout 和 Meylan（2008）发现，长度小于 20m 的冰块产生的散射可以忽略不计，因此这个 $D_{\min}$ 的值是一个合理的选择。未来也可能将 $Π$ 与我们的破碎概率相关联。

4.2. Attenuation models 4.2. 衰减模型

As discussed in Section 1, attenuation models based on multiple wave scattering are closely linked to the FSD since waves encounter more floe edges after ice breakage occurs, and hence more scattering events occur. Viscosity models only depend on the concentration and are unaffected by ice breakage. We implement an attenuation model in which wave scattering is the dominant attenuation mechanism, but we also include additional attenuation provided by a particular damping model due to Robinson and Palmer (1990). Accordingly, the dimensional and non-dimensional attenuation coefficients are written, respectively,(18) $α = α^{scat} + α^{visc} and \hat{α} = {\hat{α}}^{scat} + {\hat{α}}^{visc} .$
如第 1 节所讨论的，基于多重波散射的衰减模型与 FSD 密切相关，因为在冰块破裂后，波浪会遇到更多的冰块边缘，从而发生更多的散射事件。粘度模型仅依赖于浓度，并不受冰块破裂的影响。我们实施了一个衰减模型，其中波散射是主要的衰减机制，但我们还包括了 Robinson 和 Palmer（1990）提出的特定阻尼模型所提供的额外衰减。因此，维度和无量纲衰减系数分别写为 (18) $α = α^{scat} + α^{visc} and \hat{α} = {\hat{α}}^{scat} + {\hat{α}}^{visc} .$ 。

4.2.1. Multiple wave scattering attenuation models
4.2.1. 多重波散射衰减模型

The multiple scattering model is based on linear wave theory. The model predicts the spatial profile of time-harmonic waves in a fluid domain, which has a surface that is partially covered by a large number of floes. The floes are represented by thin-elastic plates and respond to fluid motion in flexure only. The wave number for the ice-covered ocean is $k_{ice}$ and for the open ocean is k. In general $k_{ice} \neq k$ , so scattering is produced by an impedance change when a wave moves from the open ocean into a patch of ice-covered ocean, or vice versa, at a floe edge.
多重散射模型基于线性波动理论。该模型预测了流体域中时间谐波的空间分布，该流体域的表面部分被大量浮冰覆盖。浮冰被表示为薄弹性板，仅对流体运动产生弯曲响应。冰覆盖海洋的波数为 $k_{ice}$ ，而开放海洋的波数为 k。一般来说 $k_{ice} \neq k$ ，因此当波从开放海洋进入一片冰覆盖的海洋或反之，在浮冰边缘时，会因阻抗变化而产生散射。

Attenuation due to multiple wave scattering by floe edges alone is sufficient for the present investigation (Bennetts and Squire, 2012b), but extensions to scattering by other features in the ice cover, e.g. cracks and pressure ridges, are possible (see Bennetts and Squire, 2012a).
仅由于浮冰边缘的多重波散射造成的衰减对于本研究是足够的（Bennetts 和 Squire, 2012b），但也可以扩展到冰盖中其他特征的散射，例如裂缝和压力脊（见 Bennetts 和 Squire, 2012a）。

The model is confined to two-dimensional transects, i.e. one horizontal dimension and one depth dimension (see Appendix A). It cannot yet account for lateral energy leakage or directional evolution of the waves. Attenuation models capable of describing these features are being developed (Bennetts et al., 2010), but are not yet sufficiently robust to be integrated into the WIM. Even with the restriction to only one horizontal dimension, computational expense can be large as there is an infinite sum of reflections and transmissions of the wave between each pair of adjacent floe edges. In the full multiple scattering problem exponential decay is a product of localization theory, which relies on positional disorder and requires proper consideration of wave phases.
该模型仅限于二维剖面，即一个水平维度和一个深度维度（见附录 A）。它尚无法考虑横向能量泄漏或波的方向演变。能够描述这些特征的衰减模型正在开发中（Bennetts 等，2010），但尚不够稳健，无法整合到 WIM 中。即使仅限于一个水平维度，计算开销也可能很大，因为在每对相邻冰块边缘之间存在无限的波反射和传输之和。在完整的多重散射问题中，指数衰减是局部化理论的产物，该理论依赖于位置无序，并需要对波相位进行适当考虑。

Reliance on disorder implies the use of an averaging approach. The attenuation coefficient due to multiple wave scattering is hence calculated as an ensemble average of the attenuation rates produced in simulations that are randomly selected from prescribed distributions. It is natural to calculate a non-dimensional attenuation coefficient, $α^{scat}$ (i.e. per floe), for these types of problem, but this is easily mapped onto the dimensional attenuation coefficient ${\hat{α}}^{scat}$ (i.e. per meter) for use in the WIM. The distribution of floes used in the model has a large impact on the predicted attenuation and hence the width of the MIZ. This will be demonstrated using numerical results below, and the underlying reasons will discussed at that point.
依赖于无序意味着使用平均方法。因此，由于多重波散射引起的衰减系数被计算为从规定分布中随机选择的模拟所产生的衰减率的集合平均值。对于这些类型的问题，计算无量纲衰减系数 $α^{scat}$ （即每块浮冰）是自然的，但这很容易映射到用于 WIM 的有量纲衰减系数 ${\hat{α}}^{scat}$ （即每米）。模型中使用的浮冰分布对预测的衰减以及 MIZ 的宽度有很大影响。下面将通过数值结果进行演示，并将在那时讨论其潜在原因。

4.2.2. Viscosity-based attenuation models
4.2.2. 基于粘度的衰减模型

Recent model-data comparisons (Perrie and Hu, 1996, Kohout and Meylan, 2008, Bennetts et al., 2010) have shown that multiple wave scattering models give good agreement with data for mid-range periods (6–15 s quoted by Kohout and Meylan (2008)). For large periods, however, scattering is negligible and other unmodeled dissipative mechanisms are more important, although it is unclear which mechanism is dominant in this regime. Plausible candidates include secondary creep occurring when flexural strain rates are slower, and frictional dissipation at the ice-water interface. While this issue remains unresolved, the attenuation of large period waves is modeled here with the damped thin elastic plate model of Robinson and Palmer (1990) (see Appendix A). It contains a single damping coefficient Γ, which produces a drag force that damps particle oscillations at the ice-water interface.
最近的模型与数据比较（Perrie 和 Hu, 1996; Kohout 和 Meylan, 2008; Bennetts 等, 2010）显示，多重波散射模型在中等周期（Kohout 和 Meylan (2008) 引用的 6–15 秒）与数据之间有良好的一致性。然而，对于大周期，散射可以忽略不计，其他未建模的耗散机制更为重要，尽管在这一范围内尚不清楚哪种机制占主导地位。合理的候选机制包括在弯曲应变速率较慢时发生的次级蠕变，以及冰水界面的摩擦耗散。虽然这个问题尚未解决，这里使用 Robinson 和 Palmer (1990) 的阻尼薄弹性板模型对大周期波的衰减进行了建模（见附录 A）。该模型包含一个单一的阻尼系数 Γ，产生的阻力使冰水界面的粒子振荡减弱。

In practice, we solve the dispersion relation (A.7) and use the imaginary part of the damped-propagating wavenumber $K (ω, Γ) \approx k_{ice} + i δ$ (see Appendix A), and set the viscous attenuation coefficients to be(19) $α^{visc} = 2 δ 〈 D 〉 and {\hat{α}}^{visc} = 2 δ c .$ The magnitude of the damping coefficient, Γ, is set using data from the most complete single experiment on wave attenuation available at present, that of Squire and Moore (1980). More experimental data, with detailed descriptions of prevailing ice properties and wave conditions, would help to tune Γ or to compare different models of wave dissipation.
在实践中，我们求解色散关系（A.7），并使用阻尼传播波数的虚部 $K (ω, Γ) \approx k_{ice} + i δ$ （见附录 A），并将粘性衰减系数设为 (19) $α^{visc} = 2 δ 〈 D 〉 and {\hat{α}}^{visc} = 2 δ c .$ 。衰减系数Γ的大小是使用目前可用的关于波衰减的最完整单一实验数据，即 Squire 和 Moore（1980）的数据设定的。更多的实验数据，以及对现有冰层特性和波浪条件的详细描述，将有助于调整Γ或比较不同的波耗散模型。

Most other viscosity-based attenuation models take a similar but more complicated approach and model the ice as being an incompressible viscous fluid or viscoelastic medium of finite thickness, with constitutive relations involving tuned viscosity parameters. The attenuation rate from these models is also typically predicted by solving a dispersion relation and finding the analogous parameter to $δ$ .
大多数其他基于粘度的衰减模型采用类似但更复杂的方法，将冰建模为不可压缩的粘性流体或有限厚度的粘弹性介质，具有涉及调谐粘度参数的本构关系。这些模型的衰减率通常通过求解色散关系并找到与 $δ$ 相应的参数来预测。

Weber (1987) assumed that the ice was so viscous that it was in quasi-static equilibrium, with pressure and friction balancing each other out. The ocean was also given a viscosity which was tuned to roughly agree with observations. De Carolis and Desiderio (2002) developed this model further by letting the ice viscosity take a finite value. Wang and Shen (2011b) used a viscoelastic model for the sea ice, but with the underlying ocean taken to be inviscid.
韦伯（1987）假设冰的粘度如此之高，以至于处于准静态平衡状态，压力和摩擦力相互抵消。海洋的粘度也被设定为与观测结果大致一致。德卡罗利斯和德西德里奥（2002）进一步发展了这一模型，允许冰的粘度取有限值。王和沈（2011b）使用了一个海冰的粘弹性模型，但假设底层海洋是无粘性的。

An associated model in which attenuation is produced by drag due to the bottom roughness of floes was proposed by Kohout et al. (2011). This also has a drag coefficient which requires tuning. However, it is notable that the model of Kohout et al. (2011) does not predict exponential attenuation.
Kohout 等人（2011）提出了一种相关模型，其中衰减是由于浮冰底部粗糙度引起的阻力产生的。该模型也具有需要调节的阻力系数。然而，值得注意的是，Kohout 等人（2011）的模型并不预测指数衰减。

4.2.3. Comparison of two attenuation models
4.2.3. 两种衰减模型的比较

Fig. 2 shows comparisons of predictions made by two different versions of the attenuation model. The first model considered, denoted A and constructed for this paper only, uses a seemingly plausible choice for the distributions. The FSD is based on a power law discussed in Section 4.1, which was observed for small floes ( $≲$ 20–40 m) in Antarctic locations (Toyota et al., 2011). Floe separations are arbitrarily generated from an exponential distribution $P (G > g) = \exp (- g / 〈 G 〉)$ , with $〈 G 〉 = 〈 D 〉 (c^{- 1} - 1)$ and in this example the ice concentration is $c = 0.9$ , although the discussion applies equally well to any concentration. The attenuation coefficient $α = α^{scat}$ ( $α^{visc} = 0$ for this model) is calculated as the average of 100 randomly generated simulations.
图 2 显示了由两种不同版本的衰减模型所做预测的比较。考虑的第一个模型，标记为 A，仅为本文构建，使用了看似合理的分布选择。FSD 基于第 4.1 节讨论的幂律，该幂律在南极地区的小冰块（ $≲$ 20-40 米）中被观察到（Toyota 等，2011）。冰块间距是从指数分布 $P (G > g) = \exp (- g / 〈 G 〉)$ 中任意生成的，在这个例子中冰浓度为 $c = 0.9$ ，尽管讨论同样适用于任何浓度。衰减系数 $α = α^{scat}$ （对于该模型为 $α^{visc} = 0$ ）是通过 100 个随机生成的模拟的平均值计算得出的。

The second model, denoted B, is based on the recent work of Bennetts and Squire (2012b). Rather than considering spatial distributions, Bennetts and Squire (2012b) considered the wave phases as uniformly-distributed random variables and averaged over all possibilities. They argued that the model is not intended as a true replica of the MIZ, so detailed predictions about the exact distribution of wave phases cannot be relied upon. An assumption of uniformity is thus the simplest possible in the absence of a more realistic model. In this setting the attenuation coefficient may be calculated analytically rather than relying on a numerical approximation. The expression for the attenuation coefficient can be simplified further if the floes are assumed to be long, so that only the reflection produced by a single floe edge is required, and the attenuation coefficient due to scattering is then given by $α^{scat} = - 2 \log (1 - ∣ R ∣^{2})$ , where $R$ is the reflection coefficient by the edge of a semi-infinite floe of the specified thickness (calculated here using the method of Williams and Porter (2009)). Model B is also adapted to include the effect of viscous scattering (for different values of Γ), i.e.(20) $α \equiv α^{scat} + α^{visc} = - 2 \log (1 - ∣ R ∣^{2}) + 2 δ 〈 D 〉 .$
第二个模型，记为 B，基于 Bennetts 和 Squire（2012b）的最新研究。Bennetts 和 Squire（2012b）并没有考虑空间分布，而是将波相视为均匀分布的随机变量，并对所有可能性进行了平均。他们认为该模型并不旨在真实复制 MIZ，因此无法依赖于关于波相精确分布的详细预测。因此，在缺乏更现实模型的情况下，均匀性假设是最简单的选择。在这种情况下，衰减系数可以通过解析方法计算，而不是依赖数值近似。如果假设冰块较长，则衰减系数的表达式可以进一步简化，这样只需考虑单个冰块边缘产生的反射，因而由于散射引起的衰减系数为 $α^{scat} = - 2 \log (1 - ∣ R ∣^{2})$ ，其中 $R$ 是指定厚度的半无限冰块边缘的反射系数（这里使用 Williams 和 Porter（2009）的方法计算）。模型 B 还适应了粘性散射的影响（对于不同的Γ值），即 (20) $α \equiv α^{scat} + α^{visc} = - 2 \log (1 - ∣ R ∣^{2}) + 2 δ 〈 D 〉 .$ 。

Fig. 2(a) and (b) shows the attenuation coefficients produced by the different attenuation models, computed for two different ice thicknesses, and different values of the viscosity parameter (model B only). Because the B curves with $Γ = 13$ Pa s m⁻¹ include an empirical inelastic contribution, they produce the greatest attenuation for large periods. As expected from Appendix A, the damping is also less pronounced as the thickness increases. The value $Γ = 13$ Pa s m⁻¹ was fitted using the attenuation coefficients for the three largest periods of Squire and Moore (1980) (see Table 2). They were measured for thinner ( $h \sim 0.5 m$ ) Bering Sea ice, so we used $h = 0.5$ m in our tuning procedure.
图 2(a)和(b)显示了由不同衰减模型产生的衰减系数，这些系数是针对两种不同冰厚度和不同粘度参数值（仅限模型 B）计算的。由于 B 曲线与 $Γ = 13$ Pasm ⁻¹ 包含了一个经验的非弹性贡献，因此在较大周期下产生了最大的衰减。正如附录 A 所预期的，随着厚度的增加，阻尼也不那么明显。值 $Γ = 13$ Pasm ⁻¹ 是使用 Squire 和 Moore（1980）三个最大周期的衰减系数进行拟合的（见表 2）。这些数据是针对较薄的（ $h \sim 0.5 m$ ）白令海冰测量的，因此我们在调优过程中使用了 $h = 0.5$ m。

Curves corresponding to model A are markedly different from the other curves. Due to the small values of average floe length $〈 D 〉$ (in Fig. 2(a), $〈 D 〉$ is approximately 40 m, while in Fig. 2(b)–(d) it is about 64 m), the attenuation of large period waves is several orders of magnitude too small, which qualitatively contradicts the observations of Squire and Moore (1980) mentioned above. There is also some additional fine structure in the attenuation from model A for lower periods. In particular, there is an interval of periods between about 6 s and 12 s (the interval moves to higher periods as ice thickness increases), where there is much less attenuation than the other models. This has a profound effect on the ice breakage that is able to be produced by model A, as waves from that range of periods can produce very large strains if they remain unattenuated.
与模型 A 对应的曲线与其他曲线明显不同。由于平均浮冰长度 $〈 D 〉$ 的值较小（在图 2(a)中， $〈 D 〉$ 约为 40 米，而在图 2(b)–(d)中约为 64 米），大周期波的衰减小了几个数量级，这在定性上与上述 Squire 和 Moore（1980）的观察结果相矛盾。此外，模型 A 在较低周期的衰减中还有一些额外的细微结构。特别是在大约 6 秒到 12 秒之间存在一个周期区间（随着冰厚度的增加，该区间向更高的周期移动），在这个区间内衰减远低于其他模型。这对模型 A 能够产生的冰破裂产生了深远的影响，因为来自该周期范围的波如果保持不衰减，可以产生非常大的应变。

In Fig. 2(c) and (d), we show the effects of the different attenuation models on the signficant wave height $H_{s}$ and the significant strain $E_{s}$ as they travel into an ice field. As a simple example spectrum, we take the initial wave spectrum, $S_{0}$ , to be a Bretschneider spectrum, i.e.(21) $S_{0} (ω) = \frac{1.25 H_{s}^{2} T^{5}}{8 π T_{p}^{4}} e^{- 1.25 {(T / T_{p})}^{4}},$ where $T = 2 π / ω$ is the period, and $T_{p}$ is the peak period (7 s in this example). Initially $H_{s} = 1$ m, but in general, after traveling past N floes it and $E_{s}$ are given by(22) $H_{s} = 4 \sqrt{m_{0}^{(N)} [η_{ice}]}, E_{s} = 2 \sqrt{m_{0}^{(N)} [ε]},$ where(23a) $m_{0}^{(N)} [η_{ice}] = \int S_{0} (ω) W^{2} (ω) e^{- α (ω) N} d ω,$ (23b) $m_{0}^{(N)} [ε] = \int S_{0} (ω) E^{2} (ω) e^{- α (ω) N} d ω .$
在图 2(c)和(d)中，我们展示了不同衰减模型对显著波高 $H_{s}$ 和显著应变 $E_{s}$ 在进入冰区时的影响。作为一个简单的示例谱，我们将初始波谱 $S_{0}$ 设为 Bretschneider 谱，即 (21) $S_{0} (ω) = \frac{1.25 H_{s}^{2} T^{5}}{8 π T_{p}^{4}} e^{- 1.25 {(T / T_{p})}^{4}},$ ，其中 $T = 2 π / ω$ 是周期， $T_{p}$ 是峰值周期（在此示例中为 7 秒）。最初为 $H_{s} = 1$ 米，但一般来说，在经过 N 块浮冰后，它和 $E_{s}$ 由 (22) $H_{s} = 4 \sqrt{m_{0}^{(N)} [η_{ice}]}, E_{s} = 2 \sqrt{m_{0}^{(N)} [ε]},$ 给出，其中 (23a) $m_{0}^{(N)} [η_{ice}] = \int S_{0} (ω) W^{2} (ω) e^{- α (ω) N} d ω,$ (23b) $m_{0}^{(N)} [ε] = \int S_{0} (ω) E^{2} (ω) e^{- α (ω) N} d ω .$ 。

The significant effect of the FSD on the attenuation model is further illustrated in Fig. 2(c) and (d), which show how both the signficant wave height $H_{s}$ and the significant strain $E_{s}$ decay with N, the number of floes that the waves have passed. After only a small number of floes it can be seen that $H_{s}$ and $E_{s}$ for model A (chained curve) are several orders of magnitude larger than for the other two curves, which are roughly the same.
FSD 对衰减模型的显著影响在图 2(c)和(d)中进一步说明，图中展示了显著波高 $H_{s}$ 和显著应变 $E_{s}$ 如何随着 N（波浪经过的冰块数量）的增加而衰减。仅经过少量冰块后，可以看到模型 A（链式曲线）的 $H_{s}$ 和 $E_{s}$ 比其他两条曲线大几个数量级，而其他两条曲线大致相同。

We can also see that for model A, $E_{s}$ remains very close to the approximate breaking strain for the range of values of N that are plotted. Both $E_{s}$ curves produced by model B drop below $E_{c}$ after a relatively small number of floes. This suggests that the width of the MIZ, $L_{MIZ}$ , will be similarly small under either of these models but will be significantly larger for model A if strain failure is the main breakage mechanism. In fact, in simulations involving model A (not presented), we found that a 450-km transect was almost always entirely broken, when the expected range is about 50–200 km. We therefore disregard model A for the numerical results presented in Part 2, on the basis that the predicted attenuation rates are insufficient to replicate what is observed. Note that the power-law FSD model is still used for the WIM itself.
我们还可以看到，对于模型 A， $E_{s}$ 在绘制的 N 值范围内仍然非常接近近似破裂应变。模型 B 产生的两个 $E_{s}$ 曲线在相对较少的浮冰数量后降到 $E_{c}$ 以下。这表明，无论这两种模型，MIZ 的宽度 $L_{MIZ}$ 都将相对较小，但如果应变破坏是主要的破裂机制，模型 A 的宽度将显著更大。事实上，在涉及模型 A 的模拟中（未展示），我们发现 450 公里的横断面几乎总是完全破裂，而预期范围约为 50-200 公里。因此，我们在第 2 部分中呈现的数值结果中忽略模型 A，理由是预测的衰减率不足以复制观察到的现象。请注意，幂律 FSD 模型仍然用于 WIM 本身。

4.3. Ice properties 4.3. 冰的性质

Timco and O’Brien (1994) collate and analyse nearly a thousand flexural strength measurements conducted by 14 different investigators under a variety of conditions and test types, namely, in situ cantilever tests and simple beam tests with 3- or 4-point loading, to show that the flexural strength $σ_{c}$ has the following very simple dependence on brine volume fraction $υ_{b}$ :(24) $σ_{c} = σ_{0} \exp (- 5.88 \sqrt{υ_{b}}),$ where $σ_{0} = 1.76 MPa$ . This is plotted in Fig. 3(a), and shows a monotonic decrease from $σ_{0}$ as $υ_{b}$ increases. Brine volume is often a parameter in ice-ocean models but, if necessary, it can also be calculated from the ice temperature and salinity, using the formula of Frankenstein and Gardner (1967).
Timco 和 O’Brien（1994）汇总并分析了 14 位不同研究者在多种条件和测试类型下进行的近千次弯曲强度测量，即原位悬臂梁测试和 3 点或 4 点加载的简单梁测试，以表明弯曲强度 $σ_{c}$ 与盐水体积分数 $υ_{b}$ 之间存在以下非常简单的依赖关系： (24) $σ_{c} = σ_{0} \exp (- 5.88 \sqrt{υ_{b}}),$ ，其中 $σ_{0} = 1.76 MPa$ 。如图 3(a)所示，这表明随着 $υ_{b}$ 的增加，弯盐水体积通常是冰-海洋模型中的一个参数，但如果需要，也可以根据冰的温度和盐度使用 Frankenstein 和 Gardner（1967）的公式进行计算。

Flexural strength tests are normally analyzed by means of Euler–Bernoulli beam theory, in which the stress normal to the beam cross section is related to the analogous strain. In principle, therefore, to convert flexural strength into a breaking strain $ε_{c}$ for a beam of sea ice, all we require is the Young’s modulus Y for sea ice.
弯曲强度测试通常通过欧拉-伯努利梁理论进行分析，其中垂直于梁截面的应力与相应的应变相关。因此，原则上，要将弯曲强度转换为海冰的破坏应变 $ε_{c}$ ，我们所需的只是海冰的杨氏模量 Y。

In the course of a typical flexural strength test and during the recurring cyclic flexure imparted by ocean surface gravity waves, it is expected that the sea ice will experience stress levels and rates such that the total recoverable strain $ε^{T} \approx ε^{i} + ε^{d}$ , where $ε^{i}$ is the instantaneous elastic strain and $ε^{d}$ is the delayed elastic (i.e. anelastic) strain, also known as primary, recoverable creep. This suggests a variation on the instantaneous elastic Young’s modulus Y which allows for delayed elasticity to act, which is often called the effective modulus or the strain modulus and that we shall denote by $Y^{*}$ .
在典型的弯曲强度测试过程中，以及在海洋表面重力波施加的周期性弯曲作用下，预计海冰将经历应力水平和速率，使得总可恢复应变 $ε^{T} \approx ε^{i} + ε^{d}$ ，其中 $ε^{i}$ 是瞬时弹性应变， $ε^{d}$ 是延迟弹性（即非弹性）应变，也称为主要的可恢复蠕变。这表明瞬时弹性杨氏模量 Y 的一种变体，允许延迟弹性发挥作用，这通常被称为有效模量或应变模量，我们将用 $Y^{*}$ 表示。

Timco and Weeks (2010) report a linear relationship for $Y (υ_{b})$ of the form $Y = Y_{0} (1 - 3.51 υ_{b})$ , where $Y_{0} \approx 10 GPa$ is roughly the value for freshwater ice at high loading rates. But, whilst increased brine volume leads to a reduction in the effective modulus $Y^{*}$ , the data are too scattered for an empirical relationship for $Y^{*} (υ_{b})$ to be expressed. For “average” brine volumes ranging from 50 to 100 ppt ( $υ_{b} = 0.05$ to 0.1, Frankenstein and Gardner, 1967), this suggests Y will reduce to between $\sim 6$ –8 GPa.
Timco 和 Weeks（2010）报告了 $Y (υ_{b})$ 的线性关系，形式为 $Y = Y_{0} (1 - 3.51 υ_{b})$ ，其中 $Y_{0} \approx 10 GPa$ 大致是高载荷率下淡水冰的值。然而，尽管增加的卤水体积导致有效模量 $Y^{*}$ 的降低，但数据过于分散，无法表达 $Y^{*} (υ_{b})$ 的经验关系。对于“平均”卤水体积范围在 50 到 100ppt（ $υ_{b} = 0.05$ 到 0.1，Frankenstein 和 Gardner，1967），这表明 Y 将降低到 $\sim 6$ –8GPa 之间。

As we have noted above, the effect of brine volume on $Y^{*}$ is more difficult to pin down, but we believe the same kind of reduction would not be unreasonable. More challenging is determining the effect of anelasticity (delayed elasticity) on reducing Y to $Y^{*}$ . The mechanisms that achieve this power-law primary creep with no microcracking cause relaxation processes to occur during cyclical loading, so the rate of loading is important. Few data can help us here but Fig. 4 of Cole (1998) shows model predictions for the effective modulus at four loading frequencies that include those associated with surface gravity wave periods, i.e. $10^{- 2}$ – $10^{0}$ Hz (or 0.01–1 Hz), and, incidentally, the reduction in Y due to total porosity, i.e. air plus brine. The latter effects are comparable in magnitude to the reductions in Y given above; the effect of rate is about 0.5 GPa as wave period is changed from 1 s to 10 s, and about 1 GPa from 10 s to 100 s. We therefore consider a reduction of 1 GPa is reasonable in our model, and in summary we use(25a) $Y^{*} = Y_{0} (1 - 3.51 υ_{b}) - 1 GPa,$ (25b) $ε_{c} = \frac{σ_{c}}{Y^{*}} .$
正如我们上面所提到的，盐水体积对 $Y^{*}$ 的影响更难以确定，但我们认为同样程度的减少是合理的。更具挑战性的是确定非弹性（延迟弹性）对将 Y 降低到 $Y^{*}$ 的影响。实现这种无微裂纹的幂律初级蠕变的机制导致在循环加载过程中发生松弛过程，因此加载速率是重要的。这里的数据很少，但 Cole（1998）的图 4 显示了在四个加载频率下的有效模量模型预测，这些频率包括与表面重力波周期相关的频率，即 $10^{- 2}$ – $10^{0}$ Hz（或 0.01–1Hz），顺便提一下，由于总孔隙度（即空气加盐水）导致的 Y 的减少。后者的影响在数量级上与上述 Y 的减少相当；当波周期从 1 秒变化到 10 秒时，速率的影响约为 0.5GPa，而从 10 秒变化到 100 秒时约为 1GPa。因此，我们认为在我们的模型中减少 1GPa 是合理的，总结来说，我们使用 (25a) $Y^{*} = Y_{0} (1 - 3.51 υ_{b}) - 1 GPa,$ (25b) $ε_{c} = \frac{σ_{c}}{Y^{*}} .$ 。

The effective Young’s modulus and breaking strain given by Eq. (25a), (25b) are plotted as functions of brine volume fraction in Fig. 3(b) and (c). We observe that an appropriate choice of a value for the effective Young’s modulus is important from the wave modeling perspective, as the higher $Y^{*}$ becomes the more energy is reflected at each floe present and the greater the attenuation experienced by the wave train. However, because the same value of $Y^{*}$ is used to convert from flexural stress to failure strain, the analysis is self-consistent.
由方程(25a)和(25b)给出的有效杨氏模量和破坏应变在图 3(b)和(c)中作为盐水体积分数的函数绘制。我们观察到，从波动建模的角度来看，选择一个合适的有效杨氏模量值是重要的，因为 $Y^{*}$ 的值越高，每个浮冰上反射的能量就越多，波列所经历的衰减也越大。然而，由于使用相同的 $Y^{*}$ 值将弯曲应力转换为破坏应变，因此分析是一致的。

The breaking strain has a minimum value of approximately $4.8 \times 10^{- 5}$ when $υ_{b} = 0.15$ ( $Y^{*} = 3.8$ GPa). The value is approximately constant for $υ_{b} \in [0.1, 0.2]$ . It shows an increase for both higher and lower brine volumes—the less porous ice is predictably stronger, while the more porous ice is more compliant so will be able to sustain more bending before breaking. If $υ_{b} = 0.05, ε_{c} \approx 6.5 \times 10^{- 5}$ ( $Y^{*} = 7.2$ GPa), while if $υ_{b} = 0.1$ , the breaking strain drops to $ε_{c} \approx 5.0 \times 10^{- 5}$ ( $Y^{*} = 5.5$ GPa). Although lower values of $Y^{*}$ have been measured in the field, (e.g. by Marchenko et al. (2011), in the Svalbard fjords), the temporal and spatial variability of sea ice, and the origin and special character of the ice floes in the East Greenland Current, suggests it is wiser to use the value for $Y^{*}$ we have deduced, noting that it is a straightforward matter to change it.
破坏应变的最小值约为 $4.8 \times 10^{- 5}$ ，当 $υ_{b} = 0.15$ （ $Y^{*} = 3.8$ GPa）时。对于 $υ_{b} \in [0.1, 0.2]$ ，该值大致保持不变。对于更高和更低的盐水体积，它显示出增加——较少多孔的冰显然更强，而较多孔的冰则更柔韧，因此在破裂之前能够承受更多的弯曲。如果 $υ_{b} = 0.05, ε_{c} \approx 6.5 \times 10^{- 5}$ （ $Y^{*} = 7.2$ GPa），而如果 $υ_{b} = 0.1$ ，破坏应变降至 $ε_{c} \approx 5.0 \times 10^{- 5}$ （ $Y^{*} = 5.5$ GPa）。尽管在现场测得的 $Y^{*}$ 值较低（例如，Marchenko 等人（2011）在斯瓦尔巴群岛的峡湾中），但海冰的时间和空间变异性，以及东格林兰洋流中冰块的来源和特殊特征，表明使用我们推导出的 $Y^{*}$ 值更为明智，值得注意的是，改变这个值是一个简单的事情。

The final property we will need to consider in our wave modeling is Poisson’s ratio. Langleben and Pounder (1963) determined it to be $ν = 0.295 \pm 0.009$ from seismic measurements, so in most wave calculations involving ice (e.g. Fox and Squire, 1991) it is simply taken to be 0.3.
我们在波动建模中需要考虑的最后一个属性是泊松比。Langleben 和 Pounder（1963）通过地震测量确定其值为 $ν = 0.295 \pm 0.009$ ，因此在涉及冰的多数波动计算中（例如，Fox 和 Squire，1991），它通常被简单地取为 0.3。

5. Summary and discussion
5. 总结与讨论

We have set the theoretical foundations of a waves-in-ice model (WIM) in this, Part 1 of a two-part series. The WIM will provide the first link between wave models, e.g. WAM, WAVEWATCH III, and sea ice models, e.g. CICE, LIM. The primary output of the WIM is a floe size distribution (FSD), which can be used to define the marginal ice zone (MIZ) as a subregion of the ice mask. The FSD will then be available as an input for MIZ-specific dynamic and thermodynamic models in future research.
我们在这篇文章中建立了波浪-冰模型（WIM）的理论基础，这是一个两部分系列的第一部分。WIM 将首次建立波浪模型（例如 WAM、WAVEWATCH III）与海冰模型（例如 CICE、LIM）之间的联系。WIM 的主要输出是浮冰大小分布（FSD），可用于定义边缘冰区（MIZ），作为冰盖的一个子区域。FSD 将作为未来研究中 MIZ 特定动态和热力学模型的输入。

Wave-ice interactions occurring in an MIZ comprise
在边缘冰区发生的波浪与冰的相互作用包括

(i)
the attenuation of the waves due to the presence of ice cover; and,
由于冰盖的存在，波浪的衰减；并且，
(ii)
the breaking of the ice cover due to wave motion.
由于波动而导致冰盖破裂。

The WIM proposed in this work includes both components. It is a more developed version of the WIM proposed by Dumont et al. (2011), which, to our knowledge, was the first published model to combine attenuation and ice breakage.
本研究中提出的 WIM 包含两个组成部分。它是 Dumont 等人（2011）提出的 WIM 的更为完善的版本，据我们所知，这是第一个结合衰减和冰块破裂的已发表模型。

We advected the wave spectrum, S, through the ice-covered ocean using a modified version of the energy balance equation. We neglected parameterizations of dissipation due to all conventional sources, e.g. winds and white-capping, and also non-linear interactions. However, we included a new term, $R_{ice} = \hat{α} S$ , which parameterizes dissipation due to the ice cover.
我们通过修改后的能量平衡方程将波谱 S 传输通过冰盖海洋。我们忽略了由于所有传统来源（例如风和白浪）造成的耗散的参数化，以及非线性相互作用。然而，我们加入了一个新项 $R_{ice} = \hat{α} S$ ，该项参数化了由于冰盖造成的耗散。

We used an attenuation model to calculate the attenuation coefficient, $\hat{α}$ , which defines the rate of exponential decay of the waves. The multiple wave scattering, attenuation model of Bennetts and Squire (2012b) was summarized. We noted striking differences in the attenuation coefficient when using a seemingly plausible power-law FSD in the attenuation model, rather than the random wave phase model proposed by Bennetts and Squire (2012b). Furthermore, we included viscous damping to simulate the unmodeled attenuation of large period waves.
我们使用衰减模型计算了衰减系数 $\hat{α}$ ，该系数定义了波的指数衰减速率。总结了 Bennetts 和 Squire（2012b）的多波散射衰减模型。我们注意到，当在衰减模型中使用看似合理的幂律 FSD，而不是 Bennetts 和 Squire（2012b）提出的随机波相位模型时，衰减系数存在显著差异。此外，我们还加入了粘性阻尼，以模拟大周期波的未建模衰减。

We considered the attenuation coefficient to be a function of wave frequency and also to depend on the properties of the ice cover, including the FSD. The power-law FSD model of Toyota et al. (2011) was used for local regions of the ice cover in the WIM. We created a link between the FSD model and the local wave spectrum by setting the maximum floe size to be half the dominant wavelength if the wave spectrum was sufficient to cause the ice to break. Breakage would therefore abruply alter the FSD, and consequently the attenuation coefficient, in the WIM.
我们认为衰减系数是波频率的函数，并且依赖于冰盖的特性，包括浮冰尺寸分布（FSD）。我们在 WIM 的冰盖局部区域使用了 Toyota 等人（2011）的幂律 FSD 模型。我们通过将最大浮冰尺寸设定为主导波长的一半，建立了 FSD 模型与局部波谱之间的联系，前提是波谱足以导致冰层破裂。因此，破裂将突然改变 FSD，从而影响 WIM 中的衰减系数。

We outlined a criterion to determine the occurrence of ice breakage. The criterion was based on the integrated strains imposed on the ice by the passing wave spectrum. We derived a critical strain, which incorporates a critical probability and a breaking strain, above which ice breakage was applied. In the absence of experimental or theoretical data, the value of the critical probability was set according to the limit for monochromatic waves.
我们概述了一个标准，以确定冰破裂的发生。该标准基于通过波谱施加在冰上的综合应变。我们推导出一个临界应变，该应变结合了临界概率和破裂应变，超过该值时会发生冰破裂。在缺乏实验或理论数据的情况下，临界概率的值根据单色波的极限设定。

The mechanical properties of the ice cover provide important input parameters for the attenuation model and the ice breakage criterion. We formulated an expression for the breaking strain, by means of a relationship for flexural strength due to Timco and O’Brien (1994) using an Euler–Bernoulli beam model for the sea ice. Further, we also proposed the use of an effective Young’s modulus in this relationship, so that both instantaneous and delayed elasticity are incorporated, and derived an expression for this quantity.
冰盖的机械性能为衰减模型和冰破裂标准提供了重要的输入参数。我们通过 Timco 和 O’Brien（1994）提出的弯曲强度关系，利用欧拉-伯努利梁模型为海冰制定了破裂应变的表达式。此外，我们还建议在该关系中使用有效的杨氏模量，以便同时考虑瞬时和延迟弹性，并推导出该量的表达式。

The above summary highlights the presence of uncertainties in the model. These are: (i) the viscosity parameter that determines the attenuation of large period waves; (ii) the breaking strain of the ice cover; and (iii) the critical probability above which the ice will break. Sensitivity studies are therefore required with respect to these quantities, and this forms the kernel of the numerical study that follows in Part 2. An additional uncertainty in the model is the amount of wave energy lost during ice breakage. Our treatment of the energy loss is closely related to the numerical implementation of the WIM, and its discussion is therefore contained entirely in Part 2.
上述摘要强调了模型中存在的不确定性。这些不确定性包括：(i) 决定大周期波衰减的粘度参数；(ii) 冰盖的破裂应变；以及 (iii) 超过该临界概率后冰将破裂的概率。因此，需要对这些量进行敏感性研究，这构成了第 2 部分后续数值研究的核心。模型中的另一个不确定性是冰破裂过程中损失的波能量。我们对能量损失的处理与 WIM 的数值实现密切相关，因此其讨论完全包含在第 2 部分中。

The numerical implementation of the WIM itself is non-trivial and a full description of our methods are given in Part 2.
WIM 的数值实现并非简单，我们的方法的完整描述在第二部分中给出。

Acknowledgement 致谢

The work described herein is embedded within the Waves-in-Ice Forecasting for Arctic Operators project, coordinated by Nansen Environmental and Remote Sensing Center and funded by the Research Council of Norway and Total E&P Norge through the Programme for Optimal Management of Petroleum Resources (PETROMAKS). The authors acknowledge with gratitude this funding and the support of the University of Otago, New Zealand. LB acknowledges funding support from the Australian Government’s Australian Antarctic Science Grant Program (Project 4123). The authors thank Aleksey Marchenko for useful discussions, and also thank the anonymous reviewers of this and earlier revisions of this paper for their constructive criticisms.
本文所述工作嵌入于北极运营商的冰中波浪预测项目，该项目由南森环境与遥感中心协调，并由挪威研究委员会和 Total E&P Norge 通过石油资源最佳管理计划（PETROMAKS）资助。作者对这项资助以及新西兰奥塔哥大学的支持表示感谢。LB 感谢澳大利亚政府的澳大利亚南极科学资助计划（项目 4123）的资助支持。作者感谢阿列克谢·马尔琴科的有益讨论，并感谢本论文及其早期修订的匿名审稿人提出的建设性批评。

Appendix A. Thin elastic plate model with the inclusion of damping
附录 A. 考虑阻尼的薄弹性板模型

In this appendix we present the physical basis behind the dispersion relation of Robinson and Palmer (1990) (hereafter denoted RP90), which is derived by adding a damping coefficient to the usual thin elastic plate equation. Let $z = 0$ be the mean position of the ice-water interface and let $z = η_{ice}$ be the position of the interface (the z coordinate axis points upwards, and the single horizontal coordinate axis, the x-axis, points to the right). We assume that $η_{ice}$ is small enough that we can linearise about $z = 0$ . In the formulation of RP90, the thin plate equation is modified to:(A.1) $(F \partial_{x}^{4} + ρ_{ice} h \partial_{t}^{2}) η_{ice} = P |_{z = η_{ice}} - Γ \partial_{t} η_{ice},$ where F is the flexural rigidity of the plate, $ρ_{ice}$ is the ice density, h is the ice thickness, Γ is the damping coefficient and P is the water pressure. The parameter Γ contributes to a drag pressure ( $- Γ \partial_{t} η$ ) that is proportional to the particle velocity—this is usually absent from the thin plate formulation. The rigidity is given by $F = Y^{*} h^{3} / 12 (1 - ν^{2})$ , where $Y^{*}$ is the effective Young’s Modulus (see §4.3) and $ν = 0.3$ is the Poisson’s ratio.
在本附录中，我们介绍了罗宾逊和帕尔默（1990）（以下简称 RP90）色散关系背后的物理基础，该关系是通过在通常的薄弹性板方程中添加阻尼系数而推导出来的。设 $z = 0$ 为冰水界面的平均位置， $z = η_{ice}$ 为界面的位置（z 坐标轴向上，单一的水平坐标轴 x 轴向右）。我们假设 $η_{ice}$ 足够小，以便我们可以围绕 $z = 0$ 进行线性化。在 RP90 的公式中，薄板方程被修改为： (A.1) $(F \partial_{x}^{4} + ρ_{ice} h \partial_{t}^{2}) η_{ice} = P |_{z = η_{ice}} - Γ \partial_{t} η_{ice},$ ，其中 F 是板的弯曲刚度， $ρ_{ice}$ 是冰的密度，h 是冰的厚度，Γ是阻尼系数，P 是水压。参数Γ贡献了一个与粒子速度成正比的拖曳压力（ $- Γ \partial_{t} η$ ）——这在薄板公式中通常是缺失的。刚度由 $F = Y^{*} h^{3} / 12 (1 - ν^{2})$ 给出，其中 $Y^{*}$ 是有效的杨氏模量（见§4.3）， $ν = 0.3$ 是泊松比。

If we assume that the water is inviscid and incompressible and its flow is irrotational we can write the fluid particle velocity as $u = {(u, w)}^{T} = \nabla ϕ$ , where $\nabla = {(\partial_{x}, \partial_{z})}^{T}$ . The pressure P is related to $ϕ$ through the linearized Bernoulli equation, and $ϕ$ satisfies Laplace’s equation (incompressibility) and the sea floor condition for infinitely deep water:(A.2a) $P - P_{atm} = - ρ (gz + \partial_{t} ϕ),$ (A.2b) $\nabla^{2} ϕ = 0,$ (A.2c) $\lim_{z \to - \infty} \partial_{z} ϕ (x, z, t) = 0,$ where $P_{atm}$ is the atmospheric pressure, $ρ = 1025$ kg m⁻³ is the water density and $g = 9.81$ m s⁻² is the gravitational acceleration. We also need to apply a (linearized) kinematic condition at the surface:(A.3) $\partial_{t} η_{ice} = w (x, η_{ice}, t) \approx w (x, 0) = \partial_{z} ϕ (x, 0, t) .$ Thus(A.4) $\partial_{t} P |_{z = η_{ice}} = - ρ \partial_{t} (g η_{ice} + \partial_{t} ϕ (x, η_{ice}, t)) \approx - ρ (g \partial_{z} + \partial_{t}^{2}) ϕ (x, 0, t),$ which, when combined with the time-derivative of (A.1), implies that(A.5) $(F \partial_{x}^{4} + ρ (g - d \partial_{t}^{2}) + Γ \partial_{t}) \partial_{z} ϕ (x, 0, t) = - ρ \partial_{t}^{2} ϕ (x, 0, t),$ where $d = ρ_{ice} h / ρ = 0.9 h$ is the draft of the ice.
如果我们假设水是无粘性和不可压缩的，并且其流动是无旋的，我们可以将流体粒子的速度写为 $u = {(u, w)}^{T} = \nabla ϕ$ ，其中 $\nabla = {(\partial_{x}, \partial_{z})}^{T}$ 。压力 P 通过线性化的伯努利方程与 $ϕ$ 相关，并且 $ϕ$ 满足拉普拉斯方程（不可压缩性）和无限深水的海底条件： (A.2a) $P - P_{atm} = - ρ (gz + \partial_{t} ϕ),$ (A.2b) $\nabla^{2} ϕ = 0,$ (A.2c) $\lim_{z \to - \infty} \partial_{z} ϕ (x, z, t) = 0,$ ，其中 $P_{atm}$ 是大气压力， $ρ = 1025$ kgm ⁻³ 是水的密度， $g = 9.81$ ms ⁻² 是重力加速度。我们还需要在表面应用一个（线性化的）运动条件： (A.3) $\partial_{t} η_{ice} = w (x, η_{ice}, t) \approx w (x, 0) = \partial_{z} ϕ (x, 0, t) .$ 因此 (A.4) $\partial_{t} P |_{z = η_{ice}} = - ρ \partial_{t} (g η_{ice} + \partial_{t} ϕ (x, η_{ice}, t)) \approx - ρ (g \partial_{z} + \partial_{t}^{2}) ϕ (x, 0, t),$ ，当与（A.1）的时间导数结合时，意味着 (A.5) $(F \partial_{x}^{4} + ρ (g - d \partial_{t}^{2}) + Γ \partial_{t}) \partial_{z} ϕ (x, 0, t) = - ρ \partial_{t}^{2} ϕ (x, 0, t),$ ，其中 $d = ρ_{ice} h / ρ = 0.9 h$ 是冰的吃水深度。

We now look for harmonic waves that obey (A.3), (A.5) when the water depth is infinite:(A.6a) $η_{ice} (x, t) = Re [A_{ice} e^{i (κ x - ω t)}],$ (A.6b) $ϕ (x, z, t) = Re [A_{ice} \frac{ω}{i κ} e^{i (κ x - ω t) + κ z}],$ where $A_{ice}$ is the amplitude of the ice displacement, $ω = 2 π / T$ is the radial frequency (T is the wave period), and $κ$ is a complex wavenumber. A non-zero amplitude is only possible if $κ$ satisfies the dispersion relation of RP90:(A.7) $(F κ^{4} + ρ (g - d ω^{2}) - i ω Γ) κ = ρ ω^{2} .$ When $Γ = 0$ , the primary root of interest, which we denote $k_{ice}$ , is positive and real. For non-zero Γ, we denote the root closest to $k_{ice}$ by $K (ω, Γ) = {\tilde{k}}_{ice} + i δ$ , where ${\tilde{k}}_{ice}, δ > 0$ . For physical ranges of Γ ( $Γ ≲ 15$ Pa s m⁻¹) this is a unique choice, and $k_{ice} = K (ω, 0)$ .
我们现在寻找在水深为无限时遵循（A.3），（A.5）的谐波： (A.6a) $η_{ice} (x, t) = Re [A_{ice} e^{i (κ x - ω t)}],$ (A.6b) $ϕ (x, z, t) = Re [A_{ice} \frac{ω}{i κ} e^{i (κ x - ω t) + κ z}],$ ，其中 $A_{ice}$ 是冰位移的振幅， $ω = 2 π / T$ 是径向频率（T 是波周期）， $κ$ 是一个复波数。只有当 $κ$ 满足 RP90 的色散关系时，非零振幅才是可能的： (A.7) $(F κ^{4} + ρ (g - d ω^{2}) - i ω Γ) κ = ρ ω^{2} .$ 当 $Γ = 0$ 时，我们关注的主要根，我们记作 $k_{ice}$ ，是正的且真实的。对于非零Γ，我们记最接近 $k_{ice}$ 的根为 $K (ω, Γ) = {\tilde{k}}_{ice} + i δ$ ，其中 ${\tilde{k}}_{ice}, δ > 0$ 。对于Γ的物理范围（ $Γ ≲ 15$ Pasm ⁻¹ ），这是一个唯一的选择，并且 $k_{ice} = K (ω, 0)$ 。

To give us some idea of the important non-dimensional quantities we can let $L^{5} = F / (ρ ω^{2})$ , and $\bar{κ} = κ L$ . This turns (A.7) into(A.8) $({\bar{κ}}^{4} + (a - i b)) \bar{κ} = 1,$ where $a = \frac{g}{L ω^{2}} - \frac{d}{L}, b = \frac{Γ}{ρ ω L} = \frac{Γ}{ρ^{0.8} ω^{0.6} F^{0.2}} .$ The non-dimensional viscosity parameter b, which is $O (10^{- 4})$ for higher frequencies, but is slightly bigger ( $O (10^{- 3})$ ) for lower frequencies, measures the importance of the damping effects. As well as decreasing with frequency, it also decreases with thickness (h) through the rigidity F.
为了让我们对重要的无量纲量有一些了解，我们可以设定 $L^{5} = F / (ρ ω^{2})$ 和 $\bar{κ} = κ L$ 。这将(A.7)转化为 (A.8) $({\bar{κ}}^{4} + (a - i b)) \bar{κ} = 1,$ ，其中 $a = \frac{g}{L ω^{2}} - \frac{d}{L}, b = \frac{Γ}{ρ ω L} = \frac{Γ}{ρ^{0.8} ω^{0.6} F^{0.2}} .$ 。无量纲粘度参数 b，对于较高频率为 $O (10^{- 4})$ ，但对于较低频率稍大（ $O (10^{- 3})$ ），衡量了阻尼效应的重要性。除了随着频率降低外，它还通过刚度 F 随着厚度(h)降低。

Some asymptotic analysis shows that: $K (ω, Γ) = k_{ice} (1 + \frac{i b (k_{ice} L)}{4 {(k_{ice} L)}^{5} + 1}) + O (b^{2}),$ so effectively ${\tilde{k}}_{ice} \approx k_{ice}$ . Also $δ$ is approximately $O (10^{- 8} m^{- 1})$ for higher frequencies but increases to $O (10^{- 6} m^{- 1})$ for smaller frequencies. Therefore the effects of Γ can be neglected for small scale calculations such as the estimation of the strain in a single floe, or the reflection by a single ice edge. However, it is important in large scale calculations such as the attenuation by a large number of floes, so $δ$ needs to be included to produce enough attenuation of long waves (Bennetts and Squire, 2012b).
一些渐近分析表明： $K (ω, Γ) = k_{ice} (1 + \frac{i b (k_{ice} L)}{4 {(k_{ice} L)}^{5} + 1}) + O (b^{2}),$ 因此有效地 ${\tilde{k}}_{ice} \approx k_{ice}$ 。此外， $δ$ 在高频时大约为 $O (10^{- 8} m^{- 1})$ ，但在低频时增加到 $O (10^{- 6} m^{- 1})$ 。因此，对于小规模计算，如单个浮冰的应变估计或单个冰缘的反射，可以忽略 Γ 的影响。然而，在大规模计算中，如大量浮冰的衰减，Γ 是重要的，因此需要包括 $δ$ 以产生足够的长波衰减（Bennetts 和 Squire，2012b）。

Appendix B. The WIM of Dumont et al. (2011)
附录 B. Dumont 等人（2011）的 WIM

B.1. Amplitude spectrum B.1. 振幅谱

Dumont et al. (2011) (hereafter called DKB) considered small frequency intervals, $Δ ω$ wide, and set(B.1) $\frac{1}{2} A^{2} (ω) = \int_{ω - \frac{1}{2} Δ ω}^{ω + \frac{1}{2} Δ ω} S (ω^{'}) d ω^{'} \approx Δ ω S (ω) .$ This was based on the arguments that wave groups around the central frequency would separate as they traveled into the ice due to dispersion, and so the different wave groups would not interfere with each other. It was partly done in response to the numerical issue that ocean spectra produced by external wave models, if they weren’t given parametrically, would only be given at discrete values.
Dumont 等人（2011）（以下简称 DKB）考虑了宽度为 $Δ ω$ 的小频率区间，并设定了 (B.1) $\frac{1}{2} A^{2} (ω) = \int_{ω - \frac{1}{2} Δ ω}^{ω + \frac{1}{2} Δ ω} S (ω^{'}) d ω^{'} \approx Δ ω S (ω) .$ 。这是基于这样的论点：围绕中心频率的波群在进入冰层时会因色散而分离，因此不同的波群不会相互干扰。这部分是为了应对一个数值问题，即如果外部波模型没有参数化，海洋谱仅会在离散值处给出。

However, approximation (B.1) has the fundamental flaw that, as the frequency resolution tends to zero, $Δ ω \to 0$ , the amplitude also tends to zero, $A \to 0$ . Therefore, as a rough approximation, $Δ ω$ was replaced by $ω$ , i.e.(B.2) $S = \frac{1}{2 ω} A^{2} .$ This clearly causes problems when $ω$ is significantly higher than $Δ ω$ . However, we resolve the issue of the frequency resolution by considering numerical integrals of S which actually converge better as $Δ ω \to 0$ .
然而，近似（B.1）存在一个根本缺陷，即当频率分辨率趋近于零时， $Δ ω \to 0$ ，幅度也趋近于零， $A \to 0$ 。因此，作为粗略近似， $Δ ω$ 被替换为 $ω$ ，即 (B.2) $S = \frac{1}{2 ω} A^{2} .$ 。当 $ω$ 显著高于 $Δ ω$ 时，这显然会造成问题。然而，我们通过考虑 S 的数值积分来解决频率分辨率的问题，这实际上在 $Δ ω \to 0$ 时收敛得更好。

B.2. Energy transport B.2. 能量传输

Substituting (B.2) into the energy balance equation for waves in the MIZ (2) gives(B.3) $\frac{1}{c_{g}} D_{t} A = - \frac{\hat{α}}{2} A .$ This is the continuous version of the equation used by DKB to advect wave energy, so the two equations are equivalent. However, advecting S is more natural since it adds linearly, unlike $A$ .
将（B.2）代入 MIZ 中波浪的能量平衡方程（2）得到 (B.3) $\frac{1}{c_{g}} D_{t} A = - \frac{\hat{α}}{2} A .$ 。这是 DKB 用于输送波能量的方程的连续版本，因此这两个方程是等价的。然而，输送 S 更为自然，因为它是线性叠加的，不像 $A$ 。

B.3. Breaking criterion B.3. 破坏准则

The breaking criterion used by DKB in connection with the amplitude spectrum (B.2) was that the ice would break if $A (ω) > A_{c} (ω)$ where $A_{c}$ was a critical wave amplitude, applied for any of the frequencies in the range appropriate to water waves. As mentioned above, this assumed wave groups would separate in the ice, and does not allow for the possibility of constructive interference between waves of different frequencies. By integrating S over all frequency space when determining the breaking probability of Section 3.2, we allow for the latter possibility implicitly.
DKB 在与幅度谱（B.2）相关的破裂标准中使用的是，如果 $A (ω) > A_{c} (ω)$ ，则冰会破裂，其中 $A_{c}$ 是一个临界波幅，适用于水波的频率范围内的任何频率。如上所述，这假设波群会在冰中分离，并且不考虑不同频率波之间的相干干涉的可能性。在确定第 3.2 节的破裂概率时，通过对所有频率空间进行积分，我们隐含地考虑了后者的可能性。

The value used for the critical amplitude $A_{c}$ was $A_{c} = \min {A_{c}^{ε}, A_{c}^{σ}}$ . The condition $A (ω) > A_{c}^{ε}$ represents one standard deviation in the strain for the wave group centered at frequency $ω$ being greater than their breaking strain $ε_{c}$ , while the condition $A (ω) > A_{c}^{σ}$ represents one standard deviation in the stress being greater than the flexural strength $σ_{c}$ . Our breaking criterion applies the strain criterion in a different way (in order to allow for constructive interference, as discussed above), but we do not apply a stress criterion.
用于临界幅度 $A_{c}$ 的值为 $A_{c} = \min {A_{c}^{ε}, A_{c}^{σ}}$ 。条件 $A (ω) > A_{c}^{ε}$ 表示以频率 $ω$ 为中心的波群的应变大于其破坏应变 $ε_{c}$ 的一个标准差，而条件 $A (ω) > A_{c}^{σ}$ 表示应力大于弯曲强度 $σ_{c}$ 的一个标准差。我们的破坏标准以不同的方式应用应变标准（为了允许建设性干涉，如上所述），但我们不应用应力标准。

The method used by DKB to estimate the stress was intended to allow for the effects of cavitation and wetting. During cavitation, the ice floe does not follow the wave profile exactly and potentially causes a strong localized stress on the floe. However, the criterion predicts greater stress when the waves are longer than when they are shorter. This is unphysical in this regime as ice is relatively unaffected by long waves because of their low slope/curvature, normally small amplitude, and the low velocities they force surface objects to move at. As long waves also experience the least attenuation in the presence of ice cover, the stress criterion results in an unphysically wide MIZ. As a result, our parameterization does not invoke the stress criterion of DKB. However, a different method of allowing for cavitation and wetting could still be considered in the future.
DKB 用于估计应力的方法旨在考虑气蚀和润湿的影响。在气蚀过程中，冰块并不完全遵循波浪轮廓，可能会在冰块上造成强烈的局部应力。然而，该标准预测，当波浪较长时，应力会大于波浪较短时的应力。在这个范围内，这是不符合物理规律的，因为冰对长波的影响相对较小，原因在于长波的坡度/曲率较低，通常振幅较小，并且它们迫使表面物体移动的速度较低。由于长波在冰盖存在的情况下也经历最小的衰减，因此应力标准导致了一个不符合物理规律的宽广混合区（MIZ）。因此，我们的参数化并未采用 DKB 的应力标准。然而，未来仍然可以考虑采用不同的方法来考虑气蚀和润湿。

We also note that Marchenko et al. (2011) derived an ice breakage criterion based on measured sea floor water pressure during an observed breakage event. Breakage was attributed to an increase in wave amplitudes (and hence stress and strain) produced by shoaling, so that the ice would break if the water depth H was less than a certain critical depth. This critical depth agrees with the one calculated using our method (adjusted for shallow water instead of infinitely deep water) to within reasonable uncertainty limits $(\sim 11 %$ ).
我们还注意到，Marchenko 等人（2011）基于观察到的破裂事件期间测量的海底水压推导出了冰破裂标准。破裂被归因于由于浅水效应导致的波幅（因此应力和应变）增加，因此当水深 H 小于某个临界深度时，冰会破裂。这个临界深度与我们的方法计算的结果（针对浅水而非无限深水进行了调整）在合理的不确定性范围内一致。

B.3.1. Fatigue B.3.1. 疲劳

The discussion of the anelastic response of sea ice in Section 4.3 does not preclude the possibility that floes can gradually fatigue due to repeated bending imposed by passing waves. Fatigue, whether of the high-cycle type associated with elastic behavior and growth of microscopic cracks that eventually reach a critical size for fracture, or low cycle fatigue where the stress is sufficient for plastic deformation, is characterized by cumulative damage such that materials do not recover when rested, i.e. they behave inelastically as opposed to anelastically. Accordingly, the effective modulus approach described above, which includes only fully recoverable elastic deformation, cannot accommodate fatigue. There is, however, a suggestion (Langhorne et al., 1998) that an endurance limit, i.e. a value of stress for which a material will retain its integrity even when subjected to an infinite number of load cycles, exists for sea ice. This value, 0.6, was determined on stationary shore fast sea ice in McMurdo Sound, Antarctica. DKB therefore reduced their flexural strength by a factor of 0.6. We, on the other hand, have chosen not to do this because (i) the ice and wave conditions change rapidly in the MIZ so, while a stress greater than $0.6 σ_{c}$ can cause failure in principle, it may still occur at a timescale that is well beyond that associated with the local dynamics (recall that the endurance limit is for infinite time), (ii) fatigue strictly negates the use of an effective modulus, as permanent irrecoverable damage is gradually done to the sea ice either by the nucleation and propagation of cracks or by secondary and tertiary creep, and (iii) the fast ice data of Langhorne et al. (1998) show considerable scatter, which is a common feature of fatigue experiments even for simple materials. We rest content, therefore, with the expression for $Y^{*}$ defined in Eq. (25a), noting that fatigue can easily be added at a later point if results indicate that it plays a role.
第 4.3 节对海冰非弹性响应的讨论并不排除浮冰因经过波浪的反复弯曲而逐渐疲劳的可能性。疲劳，无论是与弹性行为相关的高循环疲劳，还是足以导致塑性变形的低循环疲劳，均以累积损伤为特征，使得材料在休息时无法恢复，即它们表现出非弹性而非非弹性。因此，上述有效模量方法仅包括完全可恢复的弹性变形，无法适应疲劳。然而，有研究（Langhorne 等，1998）提出，海冰存在一个耐久极限，即在经历无限次载荷循环时，材料仍能保持其完整性的应力值。这个值为 0.6，是在南极麦克默多海峡的静止岸边固定海冰上确定的。因此，DKB 将其弯曲强度降低了 0.6 倍。我们另一方面选择不这样做，因为（i）在边界冰区（MIZ）中，冰和波浪条件变化迅速，因此，尽管大于 $0.6 σ_{c}$ 的应力原则上可以导致破坏，但它可能仍会在远远超出局部动态相关的时间尺度上发生（请记住，耐久极限是针对无限时间的），（ii）疲劳严格否定了有效模量的使用，因为海冰会逐渐受到不可恢复的永久性损伤，这种损伤是通过裂缝的成核和扩展或通过二次和三级蠕变造成的，以及（iii）Langhorne 等人（1998）的快速冰数据显示出相当大的离散性，这在疲劳实验中即使对于简单材料也是一个常见特征。因此，我们对在公式（25a）中定义的 $Y^{*}$ 的表达感到满意，并注意到如果结果表明疲劳起了作用，可以在后期轻松添加。

References

Ardhuin et al., 2008
F. Ardhuin, N. Rascle, K.A. Belibassakis
Explicit wave-averaged primitive equations using a generalized Lagrangian mean
Ocean Modell., 20 (1) (2008), pp. 35-60
View PDF View article View in Scopus Google Scholar
Ardhuin et al., 2010
F. Ardhuin, E. Rogers, A.V. Babanin, J.-F. Filipot, R. Magne, A. Roland, A. van der Westhuysen, P. Queffeulou, J.-M. Lefevre, L. Aouf, F. Collard
Semiempirical dissipation source functions for ocean waves. Part I: definition, calibration, and validation
J. Phys. Oceanogr., 40 (2010), pp. 1917-1941
View in Scopus Google Scholar
Asplin et al., 2012
M.G. Asplin, R. Galley, D.G. Barber, S.J. Prinsenberg
Fracture of summer perennial sea ice by ocean swell as a result of Arctic storms
J. Geophys. Res., 117 (2012)
(C06025)
Google Scholar
Babanin et al., 2009
A.V. Babanin, A. Ganopolski, W.R.C. Phillips
Wave-induced upper-ocean mixing and climate modelling
Ocean Modell., 29 (3) (2009), pp. 189-197
View PDF View article View in Scopus Google Scholar
Barber et al., 2009
D.G. Barber, R. Galley, M.G. Asplin, R. De Abreu, K.-A. Warner, M. Pućko, M. Gupta, S.J. Prinsenberg, S. Julien
Perennial pack ice in the southern Beaufort Sea was not as it appeared in the summer of 2009
Geophys. Res. Lett., 36 (2009)
(L24501)
Google Scholar
Bennetts et al., 2010
L. Bennetts, M.A. Peter, V.A. Squire, M.H. Meylan
A three-dimensional model of wave attenuation in the marginal ice zone
J. Geophys. Res., 115 (2010)
(C12043)
Google Scholar
Bennetts and Squire, 2012a
L.G. Bennetts, V.A. Squire
Model sensitivity analysis of scattering-induced attenuation of ice-coupled waves
Ocean Modell., 45–46 (2012), pp. 1-13
View PDF View article View in Scopus Google Scholar
Bennetts and Squire, 2012b
L.G. Bennetts, V.A. Squire
On the calculation of an attenuation coefficient for transects of ice-covered ocean
Proc. R. Soc. London A, 468 (2137) (2012), pp. 136-162
Crossref View in Scopus Google Scholar
Cartwright and Longuet-Higgins, 1956
D.E. Cartwright, M.S. Longuet-Higgins
The statistical distribution of the maxima of a random function
Proc. R. Soc. London A, 237 (1209) (1956), pp. 212-232
Crossref View in Scopus Google Scholar
Cole, 1998
D.M. Cole
Modeling the cyclic loading response of sea ice
Int. J. Solids Struct., 35 (31–32) (1998), pp. 4067-4075
View PDF View article View in Scopus Google Scholar
De Carolis and Desiderio, 2002
G. De Carolis, D. Desiderio
Dispersion and attenuation of gravity waves in ice: a two-layer viscous fluid model with experimental data validation
Phys. Lett. A, 305 (2002), pp. 399-412
View PDF View article View in Scopus Google Scholar
Doble and Bidlot, in press
Doble, M.J., Bidlot, J.-R., 2013. Wavebuoy measurements at the Antarctic sea ice edge compared with an enhanced ECMWF WAM: progress towards global waves-in-ice modeling. Ocean Modell. (in press), http://dx.doi.org/10.1016/j.ocemod.2013.05.012.
Google Scholar
Dumont et al., 2011
D. Dumont, A.L. Kohout, L. Bertino
A wave-based model for the marginal ice zone including a floe breaking parameterization
J. Geophys. Res., 116 (C4) (2011), pp. 1-12
View in Scopus Google Scholar
Feltham, 2005
D.L. Feltham
Granular flow in the marginal ice zone
Philos. Trans. R. Soc. London A, 363 (2005), pp. 1677-1700
Crossref View in Scopus Google Scholar
Fox and Squire, 1991
C. Fox, V.A. Squire
Strain in shore fast ice due to incoming ocean waves and swell
J. Geophys. Res., 96 (C3) (1991), pp. 4531-4547
Google Scholar
Frankenstein and Gardner, 1967
G.E. Frankenstein, S. Gardner
Equations for determining the brine volume of sea ice from −0.5 to 22.9 °C
J. Glaciol., 6 (1967), pp. 943-944
View in Scopus Google Scholar
Gelci et al., 1957
R. Gelci, H. Cazalé, J. Vassal
Prévision de la houle. La méthode des densités spectroangulaires
Bull. Inf. Comité Central Oceanogr. d’Etude Côtes, 9 (1957), pp. 416-435
Google Scholar
Girard et al., 2010
L. Girard, D. Amitrano, J. Weiss
Failure as a critical phenomenon in a progressive damage model
J. Stat. Mech.: Theory Exp. (2010)
P01013
Google Scholar
Girard et al., 2009
L. Girard, J. Weiss, J.M. Molines, B. Barnier, S. Bouillon
Evaluation of high-resolution sea ice models on the basis of statistical and scaling properties of Arctic sea ice drift and deformation
J. Geophys. Res., 114 (2009)
(C08015)
Google Scholar
Hasselmann, 1960
K. Hasselmann
Grundgleichungen der seegangsvoraussage
Schiffstechnik, 7 (1960), pp. 191-195
Google Scholar
Hasselmann, 1962
K. Hasselmann
On the non-linear transfer in a gravity-wave spectrum. Part 1
General Theory. J. Fluid Mech., 12 (1962), pp. 481-500
View in Scopus Google Scholar
Hasselmann, 1963
K. Hasselmann
On the non-linear transfer in a gravity-wave spectrum. Part 3. Computation of the energy flux and swell-sea interaction for a Neumann spectrum
J. Fluid Mech., 15 (1963), pp. 385-398
View in Scopus Google Scholar
Herman, 2010
A. Herman
Sea-ice floe-size distribution in the context of spontaneous scaling emergence in stochastic systems
Phys. Rev. E, 81 (2010), p. 066123
Crossref View in Scopus Google Scholar
Herman, 2011
A. Herman
Molecular-dynamics simulation of clustering processes in sea-ice floes
Phys. Rev. E, 84 (2011), p. 056104
Crossref View in Scopus Google Scholar
Herman, 2013
A. Herman
Numerical modeling of force and contact networks in fragmented sea ice
Ann. Glaciol., 54 (62) (2013), pp. 114-120
View in Scopus Google Scholar
Hibler, 1979
W.D. Hibler
A dynamic thermodynamic sea ice model
J. Phys. Oceanogr., 9 (1979), pp. 815-846
Google Scholar
Holt and Martin, 2001
B. Holt, S. Martin
The effect of a storm on the 1992 summer sea ice cover of the Beaufort, Chukchi, and East Siberian Seas
J. Geophys. Res., 106 (C1) (2001), pp. 1017-1032
Google Scholar
Howells, 1960
Howells
The multiple scattering of waves by weak random irregularities in the medium
Proc. R. Soc. A, 252 (1960), pp. 431-462
Google Scholar
Hunke and Dukowicz, 1997
E.C. Hunke, J.K. Dukowicz
An elastic–viscous–plastic model for sea ice dynamics
J. Phys. Oceanogr., 27 (1997), pp. 1849-1867
View in Scopus Google Scholar
Keller, 1998
J.B. Keller
Gravity waves on ice-covered water
J. Geophys. Res., 103 (C4) (1998), pp. 7663-7670
View in Scopus Google Scholar
Kohout et al., 2011
A. Kohout, M. Meylan, D.R. Plew
Wave attenuation in a marginal ice zone due to the bottom roughness of ice floes
Ann. Glaciol., 52 (57) (2011), pp. 118-122
Crossref View in Scopus Google Scholar
Kohout and Meylan, 2008
A.L. Kohout, M.H. Meylan
An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone
J. Geophys. Res., 113 (2008), 10.1029/2007JC004434
(C09016)
Google Scholar
Langhorne et al., 1998
P.J. Langhorne, V.A. Squire, C. Fox, T.G. Haskell
Break-up of sea ice by ocean waves
Ann. Glaciol., 27 (1998), pp. 438-442
Crossref View in Scopus Google Scholar
Langhorne et al., 2001
P.J. Langhorne, V.A. Squire, C. Fox, T.G. Haskell
Lifetime estimation for a land-fast ice sheet subjected to ocean swell
Ann. Glaciol., 33 (2001), pp. 333-338
Crossref View in Scopus Google Scholar
Langleben and Pounder, 1963
M.P. Langleben, E.R. Pounder
Elastic parameters of sea ice
W.D. Kingery (Ed.), Ice and Snow, MIT Press, USA (1963), pp. 69-78
Google Scholar
Longuet-Higgins, 1952
M.S. Longuet-Higgins
On the statistical distribution of the heights of sea waves
J. Mar. Res., 11 (1952), pp. 245-266
Google Scholar
Longuet-Higgins, 1980
M.S. Longuet-Higgins
On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width
J. Geophys. Res., 85 (C3) (1980), pp. 1519-1523
Google Scholar
Marchenko et al., 2011
Marchenko, A.V., Shestov, A., Karulin, E., Morozov, E., Karulina, M., Bogorodsky, P., Muzylev, S.V., Onishchenko, D., Makshtas, A., 2011. Field studies of sea water and ice properties in Svalbard fjords. In: Proceedings of the 21st International Conference on Port and Ocean Engineering under Arctic Conditions, Montréal, Canada, pp. 148–160.
Google Scholar
Masson and LeBlond, 1989
D. Masson, P.H. LeBlond
Spectral evolution of wind-generated surface gravity waves in a dispersed ice field
J. Fluid Mech., 202 (1989), pp. 111-136
Google Scholar
Matsushita, 1985
M. Matsushita
Fractal viewpoint of fracture and accretion
J. Phys. Soc. Jpn., 54 (3) (1985), pp. 857-860
View in Scopus Google Scholar
Mellor, 1986
Mellor, M., 1986. The mechanics of sea ice. In: Untersteiner, N. (Ed.), The Geophysics of Sea Ice. pp. 165–182.
Google Scholar
Meylan et al., 1997
M. Meylan, V. Squire, C. Fox
Toward realism in modelling ocean wave behaviour in marginal ice zones
J. Geophys. Res.—Oceans, 102 (C10) (1997), pp. 22981-22991
View in Scopus Google Scholar
Meylan and Masson, 2006
M.H. Meylan, D. Masson
A linear Boltzmann equation to model wave scattering in the marginal ice zone
Ocean Modell., 11 (3–4) (2006), pp. 417-427
View PDF View article View in Scopus Google Scholar
Perrie and Hu, 1996
W. Perrie, Y. Hu
Air–ice–ocean momentum exchange. Part 1: Energy transfer between waves and ice floes
J. Phys. Oceanogr., 26 (1996), pp. 1705-1720
View in Scopus Google Scholar
Polnikov and Lavrenov, 2007
V. Polnikov, I. Lavrenov
Calculation of the nonlinear energy transfer through the wave spectrum at the sea surface covered with broken ice
Oceanology, 47 (2007), pp. 334-343
View in Scopus Google Scholar
Prinsenberg and Peterson, 2011
S.J. Prinsenberg, I.K. Peterson
Observing regional-scale pack-ice decay processes with helicopter-borne sensors and moored upward-looking sonars
Ann. Glaciol., 52 (57) (2011), pp. 35-42
Crossref View in Scopus Google Scholar
Rampal et al., 2008
P. Rampal, J. Weiss, D. Marsan, R. Lindsay, H. Stern
Scaling properties of sea ice deformation from buoy dispersion analysis
J. Geophys. Res., 113 (2008)
(C03002)
Google Scholar
Robinson and Palmer, 1990
N.J. Robinson, S.C. Palmer
A modal analysis of a rectangular plate floating on an incompressible fluid
J. Sound Vib., 142 (1990), pp. 453-460
View PDF View article View in Scopus Google Scholar
Rothrock and Thorndike, 1984
D.A. Rothrock, A.S. Thorndike
Measuring the sea ice floe size distribution
J. Geophys. Res., 89 (C4) (1984), pp. 6477-6486
View in Scopus Google Scholar
Shen et al., 1986
H.H. Shen, W.D. Hibler, M. Leppäranta
On applying granular flow theory to a deforming broken ice field
Acta Mech., 63 (1986), pp. 143-160
View in Scopus Google Scholar
Shen and Sankaran, 2004
H.H. Shen, B. Sankaran
Internal length and time scales in a simple shear granular flow
Phys. Rev. E, 70 (2004)
(051308)
Google Scholar
Squire and Moore, 1980
V. Squire, S.C. Moore
Direct measurement of the attenuation of ocean waves by pack ice
Nature, 283 (5745) (1980), pp. 365-368
Crossref View in Scopus Google Scholar
Steele, 1992
M. Steele
Sea ice melting and floe geometry in a simple ice-ocean model
J. Geophys. Res., 97 (C11) (1992), pp. 17,729-17,738
Google Scholar
Steele et al., 1989
M. Steele, J.H. Morison, N. Untersteiner
The partition of air-ice-ocean momentum exchange as a function of ice concentration, floe size, and draft
J. Geophys. Res., 94 (C9) (1989), pp. 12,739-12,750
Google Scholar
Stephenson et al., 2011
S.R. Stephenson, L.C. Smith, J.A. Agnew
Divergent long-term trajectories of human access to the Arctic
Nat. Clim. Change, 1 (2011), pp. 156-160
Crossref View in Scopus Google Scholar
Timco and O’Brien, 1994
G.W. Timco, S. O’Brien
Flexural strength equation for sea ice
Cold Regions Sci. Tech., 22 (1994), pp. 285-298
View PDF View article View in Scopus Google Scholar
Timco and Weeks, 2010
G.W. Timco, W.F. Weeks
A review of the engineering properties of sea ice
Cold Regions Sci. Tech., 60 (2010), pp. 107-129
View PDF View article View in Scopus Google Scholar
Toyota and Enomoto, 2002
Toyota, T., Enomoto, H., 2002. Analysis of sea ice floes in the Sea of Okhotsk using ADEOS/AVNIR images. In: Proc. 16th Int. Symposium on Ice, International Association for Hydro-Environment Engineering and Research, Dunedin, New Zealand, pp. 211–217.
Google Scholar
Toyota et al., 2011
T. Toyota, C. Haas, T. Tamura
Size distribution and shape properties of relatively small sea-ice floes in the Antarctic marginal ice zone in late winter
Deep-Sea Res. II, 58 (9–10) (2011), pp. 1182-1193
View PDF View article View in Scopus Google Scholar
Toyota et al., 2006
T. Toyota, S. Takatsuji, M. Nakayama
Characteristics of sea ice floe size distribution in the seasonal ice zone
Geophys. Res. Lett., 33 (2006)
(L02616)
Google Scholar
Vaughan and Squire, 2011
G.L. Vaughan, V.A. Squire
Wave induced fracture probabilities for Arctic sea-ice
Cold Regions Sci. Tech., 67 (1–2) (2011), pp. 31-36
View PDF View article View in Scopus Google Scholar
Wadhams et al., 1988
P. Wadhams, V.A. Squire, D.J. Goodman, A.M. Cowan, S.C. Moore
The attenuation rates of ocean waves in the marginal ice zone
J. Geophys. Res., 93 (C6) (1988), pp. 6799-6818
Google Scholar
WAMDI Group, 1988
WAMDI Group, 1988. The WAM model—A third generation ocean wave prediction model. J. Phys. Oceanogr. 18, 1775–1810.
Google Scholar
Wang and Shen, 2011a
R. Wang, H.H. Shen
A continuum model for the linear wave propagation in ice-covered oceans: an approximate solution
Ocean Modell., 38 (2011), pp. 244-250
View PDF View article View in Scopus Google Scholar
Wang and Shen, 2011b
R. Wang, H.H. Shen
Gravity waves propagating into an ice-covered ocean: a viscoelastic model
J. Geophys. Res., 115 (2011), 10.1029/2009JC005591
(C06024)
Google Scholar
Weber, 1987
J.E. Weber
Wave attenuation and wave drift in the marginal ice zone
J. Phys. Oceanogr., 17 (1987), pp. 2351-2361
Google Scholar
Weeks et al., 1980
W.F. Weeks, W.B. Tucker, M. Frank, S. Fungcharoen
Characteristics of surface roughness and floe geometry of sea ice over the continental shelves of the Beaufort and Chukchi Seas
R.S. Pritchard (Ed.), Sea Ice Processes and Models, University of Washington Press, Seatle (1980), pp. 300-312
Google Scholar
Williams et al., submitted for publication
Williams, T.D., Bennetts, L.G., Squire, V.A., Dumont, D., Bertino, L., 2013. Wave-ice interactions in the marginal ice zone. Part 2: numerical implementation and sensitivity studies along 1D transects of the ocean surface. Ocean Modell. (submitted for publication), http://dx.doi.org/10.1016/j.ocemod.2013.05.011.
Google Scholar
Williams and Porter, 2009
T.D. Williams, R. Porter
The effect of submergence on the scattering by the interface between two semi-infinite sheets
J. Fluids Struct., 25 (2009), pp. 777-793
View PDF View article View in Scopus Google Scholar
World Meteorological Organization, 1998
World Meteorological Organization, 1998. Guide to Wave Forecasting and Analysis, 2nd ed. WMO No. 702. World Meteorological Organization.
Google Scholar

Cited by (168) 被引用次数（168）

Wave attenuation by three-dimensional circular floating sea ice: Regular and irregular waves
三维圆形漂浮海冰对波浪的衰减：规则波和不规则波
2024, Ocean Engineering 2024，海洋工程
The evolution of ice conditions and wave patterns in polar water areas is complex, and studying the hydrodynamic response between sea ice and waves is crucial. In this study, the hydrodynamic response of sea ice in infinitely deep water is investigated by a sea ice scattering model based on linear potential flow and thin plate approximation theory. Building upon regular wave and sea ice scattering models, an irregular wave model was established and analyzed. This paper discusses the factors that influence sea ice resonance in a three-dimensional scattering model. It categorizes resonance phenomena into strong and weak based on mass and elasticity coefficients and provides a predictive function for identifying strong resonance points. Additionally, the study examines sea ice attenuation from an energy perspective. Research on sea ice in irregular waves indicates a negative correlation between the directional spreading parameter and wave attenuation. An increase in directional spreading parameter weakens the inhibitory effect on the high-frequency part of the waves. Increasing sea ice thickness enhances wave attenuation, exerting an inhibitory effect on waves in the high-frequency region.
A combined experimental and numerical approach to predict ship resistance and power demand in broken ice
一种结合实验和数值方法的预测破冰船阻力和功率需求的研究
2024, Ocean Engineering 2024，海洋工程
Despite its remoteness and hostile environmental conditions, the Arctic holds significant shipping lanes, such as the Northern Sea Route (NSR) and the Northwest Passage (NWP). Typically, merchant ships operate along these routes in summer only, when the dominating type of ice is broken ice. A challenge of operating in such ice conditions is that there is no cost- and time-efficient method for predicting the resulting ice resistance, which makes route planning difficult, among others. To address this challenge, we present and analyze two complementary approaches to predict ship resistance in broken ice, of which one is experimental and the other numerical. The experimental approach makes use of a type of non-refrigerated synthetic model ice made of polypropylene, which makes it possible to test how a ship behaves in broken ice using a conventional non-refrigerated towing tank rather than an ice tank. The numerical approach, in turn, is based on the CFD-DEM method and can be used to consider fluid effects, such as the changes in fluid velocity and ship waves, while the ship is moving ahead. Validation calculations against established empirical approaches indicate that both approaches are reasonably accurate.
A two-way coupling method for simulating wave-induced breakup of ice floes based on SPH
基于 SPH 的波浪诱发冰块破裂的双向耦合模拟方法
2023, Journal of Computational Physics
2023 年，《计算物理学杂志》
In this paper, a new numerical method for simulating the breakup of ice floes in water waves will be presented. The method is based on Smoothed Particle Hydrodynamics (SPH) but several new numerical techniques are developed to address the challenges associated with wave-ice interaction including breakup. One of them is the new interaction model for fluid-ice interaction, which is not only suitable for small density ratio but also for large density ratio. This model may also be employed by other methods for dealing with general fluid-structure integration. In addition, a technique for modelling the separation of broken ice pieces is developed. Apart from these, a treatment is proposed to rectify the problem caused by unequal time steps in simulating ice and fluid, which is necessary for achieving higher computational efficiency. The numerical method is validated by comparing the numerical results with the experimental data available in literature, which shows that the agreement between them is satisfactory. On this basis, the breakup of ice floes induced by solitary waves and focused waves are studied. To the best of knowledge of the authors, it is the first method which can simulate the ice breaking in waves by two-way coupling approach. The results for ice breaking in solitary and focused waves have not been found so far in previous publications available in literature.
Estimating the elastic modulus of landfast ice from wave observations
通过波浪观测估算陆地固定冰的弹性模量
2023, Journal of Glaciology
2023 年，《冰川学杂志》
Retrieval of sea ice drift in the Fram Strait based on data from Chinese satellite HaiYang (HY-1D)
基于中国卫星海洋（HY-1D）数据的弗拉姆海峡海冰漂移检索
2024, Cryosphere 2024，冰冻圈
In-situ observations of strong waves in ice amplitude modulation with a 12-hour period: a likely signature of complex physics governing waves in ice attenuation?
冰中强波的原位观测：12 小时周期的振幅调制——可能是控制冰中波衰减的复杂物理现象的特征？
2024, arXiv 2024，arXiv

View all citing articles on Scopus
在 Scopus 上查看所有引用的文章

View Abstract 查看摘要

[1] Ardhuin et al., 2008
F. Ardhuin, N. Rascle, K.A. Belibassakis
Explicit wave-averaged primitive equations using a generalized Lagrangian mean
Ocean Modell., 20 (1) (2008), pp. 35-60
View PDF View article View in Scopus Google Scholar

[2] Ardhuin et al., 2010
F. Ardhuin, E. Rogers, A.V. Babanin, J.-F. Filipot, R. Magne, A. Roland, A. van der Westhuysen, P. Queffeulou, J.-M. Lefevre, L. Aouf, F. Collard
Semiempirical dissipation source functions for ocean waves. Part I: definition, calibration, and validation
J. Phys. Oceanogr., 40 (2010), pp. 1917-1941
View in Scopus Google Scholar

[3] Asplin et al., 2012
M.G. Asplin, R. Galley, D.G. Barber, S.J. Prinsenberg
Fracture of summer perennial sea ice by ocean swell as a result of Arctic storms
J. Geophys. Res., 117 (2012)
(C06025)
Google Scholar

[4] Babanin et al., 2009
A.V. Babanin, A. Ganopolski, W.R.C. Phillips
Wave-induced upper-ocean mixing and climate modelling
Ocean Modell., 29 (3) (2009), pp. 189-197
View PDF View article View in Scopus Google Scholar

[5] Barber et al., 2009
D.G. Barber, R. Galley, M.G. Asplin, R. De Abreu, K.-A. Warner, M. Pućko, M. Gupta, S.J. Prinsenberg, S. Julien
Perennial pack ice in the southern Beaufort Sea was not as it appeared in the summer of 2009
Geophys. Res. Lett., 36 (2009)
(L24501)
Google Scholar

[6] Bennetts et al., 2010
L. Bennetts, M.A. Peter, V.A. Squire, M.H. Meylan
A three-dimensional model of wave attenuation in the marginal ice zone
J. Geophys. Res., 115 (2010)
(C12043)
Google Scholar

[7] Bennetts and Squire, 2012a
L.G. Bennetts, V.A. Squire
Model sensitivity analysis of scattering-induced attenuation of ice-coupled waves
Ocean Modell., 45–46 (2012), pp. 1-13
View PDF View article View in Scopus Google Scholar

[8] Bennetts and Squire, 2012b
L.G. Bennetts, V.A. Squire
On the calculation of an attenuation coefficient for transects of ice-covered ocean
Proc. R. Soc. London A, 468 (2137) (2012), pp. 136-162
Crossref View in Scopus Google Scholar

[9] Cartwright and Longuet-Higgins, 1956
D.E. Cartwright, M.S. Longuet-Higgins
The statistical distribution of the maxima of a random function
Proc. R. Soc. London A, 237 (1209) (1956), pp. 212-232
Crossref View in Scopus Google Scholar

[10] Cole, 1998
D.M. Cole
Modeling the cyclic loading response of sea ice
Int. J. Solids Struct., 35 (31–32) (1998), pp. 4067-4075
View PDF View article View in Scopus Google Scholar

[11] De Carolis and Desiderio, 2002
G. De Carolis, D. Desiderio
Dispersion and attenuation of gravity waves in ice: a two-layer viscous fluid model with experimental data validation
Phys. Lett. A, 305 (2002), pp. 399-412
View PDF View article View in Scopus Google Scholar

[12] Doble and Bidlot, in press
Doble, M.J., Bidlot, J.-R., 2013. Wavebuoy measurements at the Antarctic sea ice edge compared with an enhanced ECMWF WAM: progress towards global waves-in-ice modeling. Ocean Modell. (in press), http://dx.doi.org/10.1016/j.ocemod.2013.05.012.
Google Scholar

[13] Dumont et al., 2011
D. Dumont, A.L. Kohout, L. Bertino
A wave-based model for the marginal ice zone including a floe breaking parameterization
J. Geophys. Res., 116 (C4) (2011), pp. 1-12
View in Scopus Google Scholar

[14] Feltham, 2005
D.L. Feltham
Granular flow in the marginal ice zone
Philos. Trans. R. Soc. London A, 363 (2005), pp. 1677-1700
Crossref View in Scopus Google Scholar

[15] Fox and Squire, 1991
C. Fox, V.A. Squire
Strain in shore fast ice due to incoming ocean waves and swell
J. Geophys. Res., 96 (C3) (1991), pp. 4531-4547
Google Scholar

[16] Frankenstein and Gardner, 1967
G.E. Frankenstein, S. Gardner
Equations for determining the brine volume of sea ice from −0.5 to 22.9 °C
J. Glaciol., 6 (1967), pp. 943-944
View in Scopus Google Scholar

[17] Gelci et al., 1957
R. Gelci, H. Cazalé, J. Vassal
Prévision de la houle. La méthode des densités spectroangulaires
Bull. Inf. Comité Central Oceanogr. d’Etude Côtes, 9 (1957), pp. 416-435
Google Scholar

[18] Girard et al., 2010
L. Girard, D. Amitrano, J. Weiss
Failure as a critical phenomenon in a progressive damage model
J. Stat. Mech.: Theory Exp. (2010)
P01013
Google Scholar

[19] Girard et al., 2009
L. Girard, J. Weiss, J.M. Molines, B. Barnier, S. Bouillon
Evaluation of high-resolution sea ice models on the basis of statistical and scaling properties of Arctic sea ice drift and deformation
J. Geophys. Res., 114 (2009)
(C08015)
Google Scholar

[20] Hasselmann, 1960
K. Hasselmann
Grundgleichungen der seegangsvoraussage
Schiffstechnik, 7 (1960), pp. 191-195
Google Scholar

[21] Hasselmann, 1962
K. Hasselmann
On the non-linear transfer in a gravity-wave spectrum. Part 1
General Theory. J. Fluid Mech., 12 (1962), pp. 481-500
View in Scopus Google Scholar

[22] Hasselmann, 1963
K. Hasselmann
On the non-linear transfer in a gravity-wave spectrum. Part 3. Computation of the energy flux and swell-sea interaction for a Neumann spectrum
J. Fluid Mech., 15 (1963), pp. 385-398
View in Scopus Google Scholar

[23] Herman, 2010
A. Herman
Sea-ice floe-size distribution in the context of spontaneous scaling emergence in stochastic systems
Phys. Rev. E, 81 (2010), p. 066123
Crossref View in Scopus Google Scholar

[24] Herman, 2011
A. Herman
Molecular-dynamics simulation of clustering processes in sea-ice floes
Phys. Rev. E, 84 (2011), p. 056104
Crossref View in Scopus Google Scholar

[25] Herman, 2013
A. Herman
Numerical modeling of force and contact networks in fragmented sea ice
Ann. Glaciol., 54 (62) (2013), pp. 114-120
View in Scopus Google Scholar

[26] Hibler, 1979
W.D. Hibler
A dynamic thermodynamic sea ice model
J. Phys. Oceanogr., 9 (1979), pp. 815-846
Google Scholar

[27] Holt and Martin, 2001
B. Holt, S. Martin
The effect of a storm on the 1992 summer sea ice cover of the Beaufort, Chukchi, and East Siberian Seas
J. Geophys. Res., 106 (C1) (2001), pp. 1017-1032
Google Scholar

[28] Howells, 1960
Howells
The multiple scattering of waves by weak random irregularities in the medium
Proc. R. Soc. A, 252 (1960), pp. 431-462
Google Scholar

[29] Hunke and Dukowicz, 1997
E.C. Hunke, J.K. Dukowicz
An elastic–viscous–plastic model for sea ice dynamics
J. Phys. Oceanogr., 27 (1997), pp. 1849-1867
View in Scopus Google Scholar

[30] Keller, 1998
J.B. Keller
Gravity waves on ice-covered water
J. Geophys. Res., 103 (C4) (1998), pp. 7663-7670
View in Scopus Google Scholar

[31] Kohout et al., 2011
A. Kohout, M. Meylan, D.R. Plew
Wave attenuation in a marginal ice zone due to the bottom roughness of ice floes
Ann. Glaciol., 52 (57) (2011), pp. 118-122
Crossref View in Scopus Google Scholar

[32] Kohout and Meylan, 2008
A.L. Kohout, M.H. Meylan
An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone
J. Geophys. Res., 113 (2008), 10.1029/2007JC004434
(C09016)
Google Scholar

[33] Langhorne et al., 1998
P.J. Langhorne, V.A. Squire, C. Fox, T.G. Haskell
Break-up of sea ice by ocean waves
Ann. Glaciol., 27 (1998), pp. 438-442
Crossref View in Scopus Google Scholar

[34] Langhorne et al., 2001
P.J. Langhorne, V.A. Squire, C. Fox, T.G. Haskell
Lifetime estimation for a land-fast ice sheet subjected to ocean swell
Ann. Glaciol., 33 (2001), pp. 333-338
Crossref View in Scopus Google Scholar

[35] Langleben and Pounder, 1963
M.P. Langleben, E.R. Pounder
Elastic parameters of sea ice
W.D. Kingery (Ed.), Ice and Snow, MIT Press, USA (1963), pp. 69-78
Google Scholar

[36] Longuet-Higgins, 1952
M.S. Longuet-Higgins
On the statistical distribution of the heights of sea waves
J. Mar. Res., 11 (1952), pp. 245-266
Google Scholar

[37] Longuet-Higgins, 1980
M.S. Longuet-Higgins
On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width
J. Geophys. Res., 85 (C3) (1980), pp. 1519-1523
Google Scholar

[38] Marchenko et al., 2011
Marchenko, A.V., Shestov, A., Karulin, E., Morozov, E., Karulina, M., Bogorodsky, P., Muzylev, S.V., Onishchenko, D., Makshtas, A., 2011. Field studies of sea water and ice properties in Svalbard fjords. In: Proceedings of the 21st International Conference on Port and Ocean Engineering under Arctic Conditions, Montréal, Canada, pp. 148–160.
Google Scholar

[39] Masson and LeBlond, 1989
D. Masson, P.H. LeBlond
Spectral evolution of wind-generated surface gravity waves in a dispersed ice field
J. Fluid Mech., 202 (1989), pp. 111-136
Google Scholar

[40] Matsushita, 1985
M. Matsushita
Fractal viewpoint of fracture and accretion
J. Phys. Soc. Jpn., 54 (3) (1985), pp. 857-860
View in Scopus Google Scholar

[41] Mellor, 1986
Mellor, M., 1986. The mechanics of sea ice. In: Untersteiner, N. (Ed.), The Geophysics of Sea Ice. pp. 165–182.
Google Scholar

[42] Meylan et al., 1997
M. Meylan, V. Squire, C. Fox
Toward realism in modelling ocean wave behaviour in marginal ice zones
J. Geophys. Res.—Oceans, 102 (C10) (1997), pp. 22981-22991
View in Scopus Google Scholar

[43] Meylan and Masson, 2006
M.H. Meylan, D. Masson
A linear Boltzmann equation to model wave scattering in the marginal ice zone
Ocean Modell., 11 (3–4) (2006), pp. 417-427
View PDF View article View in Scopus Google Scholar

[44] Perrie and Hu, 1996
W. Perrie, Y. Hu
Air–ice–ocean momentum exchange. Part 1: Energy transfer between waves and ice floes
J. Phys. Oceanogr., 26 (1996), pp. 1705-1720
View in Scopus Google Scholar

[45] Polnikov and Lavrenov, 2007
V. Polnikov, I. Lavrenov
Calculation of the nonlinear energy transfer through the wave spectrum at the sea surface covered with broken ice
Oceanology, 47 (2007), pp. 334-343
View in Scopus Google Scholar

[46] Prinsenberg and Peterson, 2011
S.J. Prinsenberg, I.K. Peterson
Observing regional-scale pack-ice decay processes with helicopter-borne sensors and moored upward-looking sonars
Ann. Glaciol., 52 (57) (2011), pp. 35-42
Crossref View in Scopus Google Scholar

[47] Rampal et al., 2008
P. Rampal, J. Weiss, D. Marsan, R. Lindsay, H. Stern
Scaling properties of sea ice deformation from buoy dispersion analysis
J. Geophys. Res., 113 (2008)
(C03002)
Google Scholar

[48] Robinson and Palmer, 1990
N.J. Robinson, S.C. Palmer
A modal analysis of a rectangular plate floating on an incompressible fluid
J. Sound Vib., 142 (1990), pp. 453-460
View PDF View article View in Scopus Google Scholar

[49] Rothrock and Thorndike, 1984
D.A. Rothrock, A.S. Thorndike
Measuring the sea ice floe size distribution
J. Geophys. Res., 89 (C4) (1984), pp. 6477-6486
View in Scopus Google Scholar

[50] Shen et al., 1986
H.H. Shen, W.D. Hibler, M. Leppäranta
On applying granular flow theory to a deforming broken ice field
Acta Mech., 63 (1986), pp. 143-160
View in Scopus Google Scholar

[51] Shen and Sankaran, 2004
H.H. Shen, B. Sankaran
Internal length and time scales in a simple shear granular flow
Phys. Rev. E, 70 (2004)
(051308)
Google Scholar

[52] Squire and Moore, 1980
V. Squire, S.C. Moore
Direct measurement of the attenuation of ocean waves by pack ice
Nature, 283 (5745) (1980), pp. 365-368
Crossref View in Scopus Google Scholar

[53] Steele, 1992
M. Steele
Sea ice melting and floe geometry in a simple ice-ocean model
J. Geophys. Res., 97 (C11) (1992), pp. 17,729-17,738
Google Scholar

[54] Steele et al., 1989
M. Steele, J.H. Morison, N. Untersteiner
The partition of air-ice-ocean momentum exchange as a function of ice concentration, floe size, and draft
J. Geophys. Res., 94 (C9) (1989), pp. 12,739-12,750
Google Scholar

[55] Stephenson et al., 2011
S.R. Stephenson, L.C. Smith, J.A. Agnew
Divergent long-term trajectories of human access to the Arctic
Nat. Clim. Change, 1 (2011), pp. 156-160
Crossref View in Scopus Google Scholar

[56] Timco and O’Brien, 1994
G.W. Timco, S. O’Brien
Flexural strength equation for sea ice
Cold Regions Sci. Tech., 22 (1994), pp. 285-298
View PDF View article View in Scopus Google Scholar

[57] Timco and Weeks, 2010
G.W. Timco, W.F. Weeks
A review of the engineering properties of sea ice
Cold Regions Sci. Tech., 60 (2010), pp. 107-129
View PDF View article View in Scopus Google Scholar

[58] Toyota and Enomoto, 2002
Toyota, T., Enomoto, H., 2002. Analysis of sea ice floes in the Sea of Okhotsk using ADEOS/AVNIR images. In: Proc. 16th Int. Symposium on Ice, International Association for Hydro-Environment Engineering and Research, Dunedin, New Zealand, pp. 211–217.
Google Scholar

[59] Toyota et al., 2011
T. Toyota, C. Haas, T. Tamura
Size distribution and shape properties of relatively small sea-ice floes in the Antarctic marginal ice zone in late winter
Deep-Sea Res. II, 58 (9–10) (2011), pp. 1182-1193
View PDF View article View in Scopus Google Scholar

[60] Toyota et al., 2006
T. Toyota, S. Takatsuji, M. Nakayama
Characteristics of sea ice floe size distribution in the seasonal ice zone
Geophys. Res. Lett., 33 (2006)
(L02616)
Google Scholar

[61] Vaughan and Squire, 2011
G.L. Vaughan, V.A. Squire
Wave induced fracture probabilities for Arctic sea-ice
Cold Regions Sci. Tech., 67 (1–2) (2011), pp. 31-36
View PDF View article View in Scopus Google Scholar

[62] Wadhams et al., 1988
P. Wadhams, V.A. Squire, D.J. Goodman, A.M. Cowan, S.C. Moore
The attenuation rates of ocean waves in the marginal ice zone
J. Geophys. Res., 93 (C6) (1988), pp. 6799-6818
Google Scholar

[63] WAMDI Group, 1988
WAMDI Group, 1988. The WAM model—A third generation ocean wave prediction model. J. Phys. Oceanogr. 18, 1775–1810.
Google Scholar

[64] Wang and Shen, 2011a
R. Wang, H.H. Shen
A continuum model for the linear wave propagation in ice-covered oceans: an approximate solution
Ocean Modell., 38 (2011), pp. 244-250
View PDF View article View in Scopus Google Scholar

[65] Wang and Shen, 2011b
R. Wang, H.H. Shen
Gravity waves propagating into an ice-covered ocean: a viscoelastic model
J. Geophys. Res., 115 (2011), 10.1029/2009JC005591
(C06024)
Google Scholar

[66] Weber, 1987
J.E. Weber
Wave attenuation and wave drift in the marginal ice zone
J. Phys. Oceanogr., 17 (1987), pp. 2351-2361
Google Scholar

[67] Weeks et al., 1980
W.F. Weeks, W.B. Tucker, M. Frank, S. Fungcharoen
Characteristics of surface roughness and floe geometry of sea ice over the continental shelves of the Beaufort and Chukchi Seas
R.S. Pritchard (Ed.), Sea Ice Processes and Models, University of Washington Press, Seatle (1980), pp. 300-312
Google Scholar

[68] Williams et al., submitted for publication
Williams, T.D., Bennetts, L.G., Squire, V.A., Dumont, D., Bertino, L., 2013. Wave-ice interactions in the marginal ice zone. Part 2: numerical implementation and sensitivity studies along 1D transects of the ocean surface. Ocean Modell. (submitted for publication), http://dx.doi.org/10.1016/j.ocemod.2013.05.011.
Google Scholar

[69] Williams and Porter, 2009
T.D. Williams, R. Porter
The effect of submergence on the scattering by the interface between two semi-infinite sheets
J. Fluids Struct., 25 (2009), pp. 777-793
View PDF View article View in Scopus Google Scholar

[70] World Meteorological Organization, 1998
World Meteorological Organization, 1998. Guide to Wave Forecasting and Analysis, 2nd ed. WMO No. 702. World Meteorological Organization.
Google Scholar