Wave–ice interactions in the marginal ice zone. Part 2: Numerical implementation and sensitivity studies along 1D transects of the ocean surface

doi:10.1016/j.ocemod.2013.05.011

Ocean Modelling 海洋建模

Volume 71, November 2013, Pages 92-101
第 71 卷，2013 年 11 月，92-101 页

https://doi.org/10.1016/j.ocemod.2013.05.011 Get rights and content 获取权利和内容

Highlights 亮点

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A one-dimenstional model for wave–ice interactions was presented in part 1 of this paper.
本文第一部分提出了一种一维波浪与冰相互作用的模型。
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The main components are summarised here.
主要组成部分在此总结。
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The numerical schemes and discretizations used to solve the problems are presented and discussed.
用于解决这些问题的数值方案和离散化方法被提出并讨论。
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Sensitivity studies in idealized and realistic settings are performed.
在理想化和现实环境中进行敏感性研究。

Abstract 摘要

The theoretical foundation of a wave–ice interaction model is reported in Part 1 of this study. The model incorporates attenuation of ocean surface waves by sea ice floes and the concomitant breaking of the floes by waves that determines the structure of the marginal ice zone (MIZ). A numerical implementation of the method is presented here. Convergence of the numerical method is demonstrated, as temporal and spatial grids are refined. A semi-analytical method, which does not require time-stepping, is also developed to validate the numerical results, when dispersion is neglected. The wave energy lost during ice breakage is parameterized, as part of the numerical method. Sensitivity studies are conducted in relation to the energy loss and also dispersive effects, the choice of the attenuation model, the properties of the wave field, and sea ice properties such as concentration, thickness and breaking strain. Example simulations intended to represent conditions in the Fram Strait in 2007, which exploit reanalyzed wave and ice model data, are shown to conclude the results section. These are compared to estimates of MIZ widths based on a concentration criteria, and obtained from remotely-sensed passive microwave images.
本研究的第一部分报告了波浪与冰相互作用模型的理论基础。该模型考虑了海冰浮块对海洋表面波浪的衰减以及波浪对浮块的破坏，这决定了边缘冰区（MIZ）的结构。这里展示了该方法的数值实现。随着时间和空间网格的细化，数值方法的收敛性得到了验证。还开发了一种半解析方法，在忽略色散的情况下，该方法不需要时间步进，以验证数值结果。冰破裂过程中损失的波能被参数化，作为数值方法的一部分。进行了敏感性研究，涉及能量损失和色散效应、衰减模型的选择、波场的特性以及海冰的特性，如浓度、厚度和破裂应变。最后，展示了旨在代表 2007 年弗拉姆海峡条件的示例模拟，这些模拟利用了重新分析的波浪和冰模型数据，以总结结果部分。这些与基于浓度标准的 MIZ 宽度估计进行比较，并且这些估计是通过遥感被动微波图像获得的。

Keywords 关键词

Wave–ice interactions

Wave attenuation

Ice breakage

Floe size distribution

Marginal ice zone

波浪与冰的相互作用波浪衰减冰的破裂冰块大小分布边缘冰区

1. Introduction 1. 引言

Predictions of wave and ice conditions in the marginal ice zone (MIZ) are becoming increasingly important in the era of climate change and enhanced access to the Arctic Ocean. However, contemporary sea ice models do not contain information on floe sizes, and contemporary wave models generally do not extend into the ice-covered ocean. Modeling the interactions of ocean surface waves with sea ice is necessary to rectify these conspicuous omissions, because (i) floe sizes in the MIZ are far smaller than those in the ice interior due to wave-induced ice breakage (Toyota et al., 2006), and (ii) the presence of the ice-cover strongly attenuates the waves (Wadhams et al., 1988), acting as a low-pass filter, and hence is a necessary additional consideration when modeling the transport of wave energy in the MIZ.
在气候变化和对北极海洋的增强访问时代，边缘冰区（MIZ）波浪和冰情的预测变得越来越重要。然而，当前的海冰模型不包含浮冰大小的信息，而现代波浪模型通常不延伸到冰覆盖的海洋中。建模海洋表面波浪与海冰的相互作用是必要的，以纠正这些明显的遗漏，因为（i）由于波浪引起的冰破裂，MIZ 中的浮冰大小远小于冰内部的浮冰大小（Toyota 等，2006），以及（ii）冰盖的存在强烈衰减波浪（Wadhams 等，1988），充当低通滤波器，因此在建模 MIZ 中波浪能量的传输时，必须考虑这一额外因素。

Part 1 of this investigation (Williams et al., 2013) describes a waves-in-ice model (WIM) that extends the work of Dumont et al. (2011) (hereafter referred to as DKB). The WIM provides predictions of (i) the ice floe size distribution (FSD) resulting from wave-induced flexural breakage of the ice cover, and (ii) the wave spectrum within the ice cover. The model includes two interrelated sub-components. First, a wave attenuation model that calculates the proportion of wave energy that is reflected by floe edges, and lost to dissipative processes, as a function of the number of ice floes encountered along the propagation path. Second, an ice breakage model that decides when the strain imposed by the passing waves on the ice cover is sufficient to cause fracture and how the resulting FSD evolves.
本研究的第一部分（Williams et al., 2013）描述了一种波浪-冰模型（WIM），该模型扩展了 Dumont et al.（2011）的工作（以下简称 DKB）。WIM 提供了以下预测：（i）由于波浪引起的冰盖弯曲破裂而导致的冰漂浮体大小分布（FSD），以及（ii）冰盖内的波谱。该模型包括两个相互关联的子组件。首先是一个波浪衰减模型，该模型计算沿传播路径遇到的冰漂浮体数量对波浪能量的反射比例和因耗散过程而损失的能量。其次是一个冰破裂模型，该模型决定波浪对冰盖施加的应变何时足以导致破裂，以及由此产生的 FSD 如何演变。

The FSD provided by the WIM will allow floe-size-dependent processes to be modeled in the MIZ. The smaller floe sizes in the MIZ are potentially important for thermodynamic exchanges such as lateral melting between the atmosphere, ice and ocean; dynamic exchanges, e.g., form and skin drag coefficients; and rheology, i.e., how horizontal stresses relate to deformation rates. Floe-size-dependent thermodynamic and dynamic models have been developed (e.g., Shen et al., 1986, Steele et al., 1989, Feltham, 2005), but can only be tested in fully coupled models once a floe size parameter is incorporated in sea ice models.
WIM 提供的 FSD 将允许在 MIZ 中对依赖于冰块大小的过程进行建模。MIZ 中较小的冰块大小在热力学交换中可能具有重要意义，例如大气、冰和海洋之间的侧向融化；动态交换，例如形状和表面摩擦系数；以及流变学，即水平应力与变形速率之间的关系。已经开发了依赖于冰块大小的热力学和动态模型（例如，Shen 等，1986 年；Steele 等，1989 年；Feltham，2005 年），但只有在将冰块大小参数纳入海冰模型后，才能在完全耦合模型中进行测试。

In this paper, we place the model theory of Part 1 into a discrete spatial and temporal framework for the purpose of numerical calculations. As part of the numerical scheme, we propose a method to simulate the wave energy lost during ice breakage. Semi-analytical schemes are devised for two limiting cases of wave energy loss. These two schemes neglect dispersion, which is shown to have a negligible affect on the FSD.
在本文中，我们将第一部分的模型理论置于离散的空间和时间框架中，以便进行数值计算。作为数值方案的一部分，我们提出了一种模拟冰块破裂过程中波浪能量损失的方法。针对波浪能量损失的两个极限情况，我们设计了半解析方案。这两个方案忽略了色散，结果表明色散对 FSD 的影响可以忽略不计。

Here, we consider one-dimensional transects of the ocean surface only, although the full numerical algorithm can be generalized to two-dimensional ocean surfaces. The one-dimensional restriction, however, provides a convenient setting to test the sensitivities of the WIM to the key numerical and physical parameters. Idealized incident wave and ice conditions are used to investigate the influence of the grid size, time step, the wave damping parameter, wave energy lost during ice breakage and breaking strain on the FSD produced by the WIM. Numerical experiments are also conducted with ‘realistic’ input data that represent the Fram Strait in 2007. In the absence of measured FSD data to validate the WIM, we compare our MIZ width predictions, i.e., the length of the interval of ice cover broken by waves, with MIZ widths based on a concentration criteria, using AMSR-E satellite data.
在这里，我们仅考虑海洋表面的单维横断面，尽管完整的数值算法可以推广到二维海洋表面。然而，单维限制提供了一个方便的环境来测试 WIM 对关键数值和物理参数的敏感性。使用理想化的入射波和冰条件来研究网格大小、时间步长、波衰减参数、冰破裂过程中波能量损失和破裂应变对 WIM 产生的 FSD 的影响。还进行了数值实验，使用代表 2007 年弗拉姆海峡的“现实”输入数据。由于缺乏测量的 FSD 数据来验证 WIM，我们将我们的 MIZ 宽度预测（即波浪破坏的冰盖区间的长度）与基于浓度标准的 MIZ 宽度进行比较，使用 AMSR-E 卫星数据。

2. Statement of the problem
2. 问题陈述

We consider a one-dimensional transect $x \in [0, X_{2}]$ of spatially varying ice concentration $c (x)$ and thickness $h (x)$ . We typically use $X_{2} = 450$ km. The transect is discretized into $N_{x}$ grid cells with uniform widths $Δ x = X_{2} / N_{x}$ . The ice edge is located at $x = X_{1}$ such that the open water region is $[0, X_{1}]$ and the ice-covered region is $[X_{1}, X_{2}]$ (see Fig. 1). In our idealized simulations, we use an exponential thickness profile of the form(1a) $h (x) = \{\begin{matrix} 0 & for 0 < x < X_{1}, \\ h_{\infty} (0.1 + 0.9 (1 - e^{- (x - X_{1} / X_{h})})) & for X_{1} < x < X_{2}, \end{matrix}$ and a uniform concentration(1b) $c (x) = \{\begin{matrix} 0 & for 0 < x < X_{1}, \\ c_{\infty} & for X_{1} < x < X_{2} . \end{matrix}$ For realistic simulations, the concentration and thickness profiles are taken from the TOPAZ operational forecasting system (Sakov et al., 2012). The parameter $X_{h}$ in (1a) was chosen to be 60 km to approximate TOPAZ thickness outputs. Table 1 lists the default values of all parameters.
我们考虑一个一维横断面 $x \in [0, X_{2}]$ ，其冰浓度 $c (x)$ 和厚度 $h (x)$ 在空间上变化。我们通常使用 $X_{2} = 450$ 公里。该横断面被离散化为 $N_{x}$ 个具有均匀宽度 $Δ x = X_{2} / N_{x}$ 的网格单元。冰缘位于 $x = X_{1}$ ，使得开水区域为 $[0, X_{1}]$ ，冰覆盖区域为 $[X_{1}, X_{2}]$ （见图 1）。在我们的理想化模拟中，我们使用形式为 (1a) $h (x) = \{\begin{matrix} 0 & for 0 < x < X_{1}, \\ h_{\infty} (0.1 + 0.9 (1 - e^{- (x - X_{1} / X_{h})})) & for X_{1} < x < X_{2}, \end{matrix}$ 的指数厚度剖面和均匀浓度 (1b) $c (x) = \{\begin{matrix} 0 & for 0 < x < X_{1}, \\ c_{\infty} & for X_{1} < x < X_{2} . \end{matrix}$ 。对于现实模拟，浓度和厚度剖面取自 TOPAZ 操作预报系统（Sakov 等，2012）。在（1a）中参数 $X_{h}$ 被选择为 60 公里，以近似 TOPAZ 厚度输出。表 1 列出了所有参数的默认值。

Table 1. Default model parameters.
表 1. 默认模型参数。

Quantity 数量	Symbol 符号	Value 价值
Ice thickness 冰厚度	$h_{\infty}$	1–4 m 1–4 米
Ice concentration 冰浓度	$c_{\infty}$	0.75
Water density 水密度	$ρ$	1025 kg m⁻³ 1025 千克米 ⁻³
Ice density 冰的密度	$ρ_{ice}$	922.5 kg m⁻³ 922.5 千克米 ⁻³
Gravitational acceleration 重力加速度	g	9.81 m s⁻² 9.81 毫秒 ⁻²
Brine volume fraction 盐水体积分数	$υ_{b}$	0.1
Incident significant wave height 事件显著波高	$H_{s}$	3 m 3 米
Incident peak period 事件高峰期	$T_{p}$	6–10 s 6–10 秒
Minimum floe size in FSD FSD 中的最小浮冰尺寸	$D_{\min}$	20 m 20 米
FSD cut-off length FSD 截断长度	$D_{unif}$	200 m 200 米
Initial value of $D_{\max}$ 初始值为 $D_{\max}$	$D_{init}$	500 m 500 米
Fragility 脆弱性	$Π$	0.9
Number of broken pieces 破碎片数	$ξ$	2
Number of spatial grid cells 空间网格单元的数量	$N_{x}$	91
Grid size 网格大小	$Δ x$	5 km 5 公里
Time step 时间步长	$Δ t$	400 s 400 秒
Number of spectral components 光谱成分数量	$N_{ω}$	31
Minimum wave period 最小波周期	$T_{30} = 2 π / ω_{30}$	2.5 s 2.5 秒
Maximum wave period 最大波周期	$T_{0} = 2 π / ω_{0}$	23.8 s 23.8 秒
Spectral resolution 光谱分辨率	$Δ ω$	7.5 × 10⁻² s⁻¹ 7.5×10^{0} s^{1}
Breaking probability threshold 突破概率阈值	$P_{c}$	$e^{- 1} \approx 0.37$
Flexural strength 弯曲强度	$σ_{c}$	0.27 GPa
Effective Young’s modulus 有效的杨氏模量	$Y^{*}$	5.5 GPa
Breaking strain 断裂强度	$ε_{c}$	$4.99 \times 10^{- 5}$
Breaking significant strain 打破重大压力	$E_{c} = ε_{c} \sqrt{2}$	$7.05 \times 10^{- 5}$
Viscous damping parameter 粘性阻尼参数	Γ	$13.0$ Pa s m⁻¹

The wave energy is described by the spectral density function $S (ω; x, t)$ , where $ω = 2 π / T$ is the angular frequency and T is the wave period. The wave spectrum is defined in both the open ocean and the ice-covered ocean, after having undergone some attenuation. The incident wave spectrum is prescribed at $x = 0$ . Because data obtained from operational wave models are usually given parametrically in terms of the significant wave height $H_{s}$ and the peak period $T_{p}$ (Ochi, 1998, WMO, 1998), we use the Bretschneider two-parameter spectrum (Bretschneider, 1959), i.e.,(2) $S (ω; 0, t) = S_{B} (ω; T_{p}, H_{s}) = \frac{1.25 H_{s}^{2} T^{5}}{8 π T_{p}^{4}} e^{- 1.25 {(T / T_{p})}^{4}} .$ Note that in the realistic experiments $H_{s}$ and $T_{p}$ evolve in time causing the incident wave spectrum to be temporally dependent. It may be possible to obtain more detailed incident wave spectra in the future—for example spectra with a parameterization of swell as well as wind waves, or the full frequency and directional spectrum.
波能量由谱密度函数 $S (ω; x, t)$ 描述，其中 $ω = 2 π / T$ 是角频率，T 是波周期。波谱在开放海洋和冰覆盖海洋中定义，经过一定的衰减。入射波谱在 $x = 0$ 处规定。由于从操作波模型获得的数据通常以显著波高 $H_{s}$ 和峰值周期 $T_{p}$ 的参数形式给出（Ochi, 1998, WMO, 1998），我们使用 Bretschneider 双参数谱（Bretschneider, 1959），即 (2) $S (ω; 0, t) = S_{B} (ω; T_{p}, H_{s}) = \frac{1.25 H_{s}^{2} T^{5}}{8 π T_{p}^{4}} e^{- 1.25 {(T / T_{p})}^{4}} .$ 。请注意，在实际实验中， $H_{s}$ 和 $T_{p}$ 随时间演变，导致入射波谱具有时间依赖性。未来可能获得更详细的入射波谱，例如包含涌浪和风浪的参数化谱，或完整的频率和方向谱。

The FSD is characterized by two spatially varying floe length parameters $D_{\max} (x, t)$ and $〈 D 〉 (x, t)$ , the maximum floe length and average floe length, respectively, which also evolve with time. The detailed parameterization of the FSD is presented in Section 4.1 of Part 1.
FSD 的特征是两个空间变化的浮冰长度参数 $D_{\max} (x, t)$ 和 $〈 D 〉 (x, t)$ ，分别是最大浮冰长度和平均浮冰长度，这些参数也随时间变化。FSD 的详细参数化在第一部分的第 4.1 节中介绍。

3. Theoretical preliminaries
3. 理论前提

In this section we recap key definitions and ideas from Part 1.
在本节中，我们回顾第一部分的关键定义和概念。

3.1. Wave statistics 3.1. 波浪统计

Let the displacement of the (horizontal) air-ice interface be $η_{ice} (x, t)$ . Assuming the ice can be represented by a thin plate model, the horizontal strain in the plane of the wave is(3) $ε = \frac{h}{2} \partial_{x}^{2} η_{ice},$ where h is the ice thickness. The main statistics we are interested in are the mean square values of these quantities, $〈 η_{ice}^{2} 〉$ and $〈 ε^{2} 〉$ . These give us the significant wave height $H_{s}$ and the significant strain $E_{s}$ :(4) $H_{s} = 4 〈 η_{ice}^{2} 〉^{1 / 2}, E_{s} = 2 〈 ε^{2} 〉^{1 / 2} .$ The dominant wave period $T_{W}$ also plays a role as it defines a dominant wavelength $λ_{W}$ that, if breaking occurs, determines the maximum lengths of the consequent broken floes. It, like $〈 η_{i ce}^{2} 〉$ and $〈 ∊^{2} 〉$ , is defined in terms of integrals involving the wave spectrum S (see equations 5 and 12 of Part 1).
设（水平）空气-冰界面的位移为 $η_{ice} (x, t)$ 。假设冰可以用薄板模型表示，波面内的水平应变为 (3) $ε = \frac{h}{2} \partial_{x}^{2} η_{ice},$ ，其中 h 为冰的厚度。我们感兴趣的主要统计量是这些量的均方值 $〈 η_{ice}^{2} 〉$ 和 $〈 ε^{2} 〉$ 。这些给出了显著波高 $H_{s}$ 和显著应变 $E_{s}$ ： (4) $H_{s} = 4 〈 η_{ice}^{2} 〉^{1 / 2}, E_{s} = 2 〈 ε^{2} 〉^{1 / 2} .$ 。主导波周期 $T_{W}$ 也起着作用，因为它定义了主导波长 $λ_{W}$ ，如果发生破裂，则决定了随之而来的破碎浮冰的最大长度。它与 $〈 η_{i ce}^{2} 〉$ 和 $〈 ∊^{2} 〉$ 一样，是通过涉及波谱 S 的积分定义的（见第 1 部分的方程 5 和 12）。

As discussed in Appendix A of Part 1, we assume the displacement due to a wave with the single frequency ω follows a sinusoidal profile(5) $η_{ice} (x, t) = Re [A_{ice} e^{i (κ x - ω t)}],$ where $κ (ω, Γ)$ satisfies the dispersion relation for ice-covered water, as follows(6) $(F κ^{4} + ρ (g - d ω^{2}) - i ω Γ) κ = ρ ω^{2} .$ In (6) F is the flexural rigidity of the ice, $ρ_{ice}$ 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 ( $- Γ \partial_{t} η_{ice}$ ) that is proportional to the particle velocity and which 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 and $ν = 0.3$ is the Poisson’s ratio.
如第 1 部分附录 A 所讨论，我们假设单频率ω引起的位移遵循正弦波形 (5) $η_{ice} (x, t) = Re [A_{ice} e^{i (κ x - ω t)}],$ ，其中 $κ (ω, Γ)$ 满足冰盖水体的色散关系，如下所示 (6) $(F κ^{4} + ρ (g - d ω^{2}) - i ω Γ) κ = ρ ω^{2} .$ 。在(6)中，F 是冰的弯曲刚度， $ρ_{ice}$ 是冰的密度，h 是冰的厚度，Γ是阻尼系数，P 是水压。参数Γ贡献于一个与粒子速度成正比的阻力压力（ $- Γ \partial_{t} η_{ice}$ ），而这种压力通常在薄板公式中缺失。刚度由 $F = Y^{*} h^{3} / 12 (1 - ν^{2})$ 给出，其中 $Y^{*}$ 是有效杨氏模量， $ν = 0.3$ 是泊松比。

Let $k_{ice} (ω) = K (ω, 0)$ be the real positive root of (6) when $Γ = 0$ . When $Γ > 0$ , this root becomes complex, and $κ \approx k_{ice} (ω) + i δ (ω, Γ)$ , where $δ > 0$ is small enough to be ignored on small-scale computations, and only makes a significant contribution to large scale wave attenuation. Also let(7) $W (ω) = \frac{{gk}_{ice}}{ω^{2}} | T |, E (ω) = \frac{h}{2} k_{ice}^{2} W (ω);$ W is a factor that approximately converts the wave amplitude in open water, A, to the wave amplitude in ice, i.e., $A_{ice} \approx WA$ , and $T$ is the transmission coefficient for a wave traveling from a region of open water into an ice-covered region (Williams and Porter, 2009), which depends on both $ω$ and the ice properties involved in (6). Similarly the strain amplitude is $E_{W} \approx EA$ .
设 $k_{ice} (ω) = K (ω, 0)$ 为(6)的实正根，当 $Γ = 0$ 时。当 $Γ > 0$ 时，该根变为复数，并且 $κ \approx k_{ice} (ω) + i δ (ω, Γ)$ ，其中 $δ > 0$ 足够小以在小规模计算中被忽略，仅在大规模波衰减中产生显著贡献。还设 (7) $W (ω) = \frac{{gk}_{ice}}{ω^{2}} | T |, E (ω) = \frac{h}{2} k_{ice}^{2} W (ω);$ W 是一个因子，近似将开水中的波幅 A 转换为冰中的波幅，即 $A_{ice} \approx WA$ ，而 $T$ 是从开水区域传播到冰覆盖区域的波的传输系数（Williams 和 Porter，2009），它依赖于 $ω$ 和(6)中涉及的冰的性质。类似地，应变幅度为 $E_{W} \approx EA$ 。

We now define the following integrals over frequency:(8a) $m_{n} [η_{ice}] = \int_{0}^{\infty} S (ω) ω^{n} W^{2} (ω) d ω,$ (8b) $m_{n} [ε] = \int_{0}^{\infty} S (ω) ω^{n} E^{2} (ω) d ω .$ These integrals can be used to determine the expected response to a given wave field in a way that allows for the possibility of constructive and destructive interference between frequencies. Our main quantities of interest are then given by(9) $〈 η_{ice}^{2} 〉 = m_{0} [η_{ice}], 〈 ε^{2} 〉 = m_{0} [ε], T_{W} = 2 π \sqrt{\frac{m_{0} [η_{ice}]}{m_{2} [η_{ice}]}} .$ If $k_{W} = k_{ice} (2 π / T_{W})$ is the real part of the wavenumber κ when the period is $T_{W}$ , the dominant wavelength referred to earlier is given by $λ_{W} = 2 π / k_{W}$ . The probability of the strain amplitude exceeding the breaking strain $∊_{c}$ is(10) $P_{ε} \equiv P (E_{W} > ε_{c}) = \exp (\frac{- ε_{c}}{2 〈 ε^{2} 〉}) .$
我们现在定义以下频率上的积分： (8a) $m_{n} [η_{ice}] = \int_{0}^{\infty} S (ω) ω^{n} W^{2} (ω) d ω,$ (8b) $m_{n} [ε] = \int_{0}^{\infty} S (ω) ω^{n} E^{2} (ω) d ω .$ 这些积分可以用来确定对给定波场的预期响应，以允许频率之间发生建设性和破坏性干涉。我们主要关注的量由 (9) $〈 η_{ice}^{2} 〉 = m_{0} [η_{ice}], 〈 ε^{2} 〉 = m_{0} [ε], T_{W} = 2 π \sqrt{\frac{m_{0} [η_{ice}]}{m_{2} [η_{ice}]}} .$ 给出。如果 $k_{W} = k_{ice} (2 π / T_{W})$ 是波数κ的实部，当周期为 $T_{W}$ 时，之前提到的主导波长由 $λ_{W} = 2 π / k_{W}$ 给出。应变幅度超过破坏应变 $∊_{c}$ 的概率为 (10) $P_{ε} \equiv P (E_{W} > ε_{c}) = \exp (\frac{- ε_{c}}{2 〈 ε^{2} 〉}) .$ 。

3.2. Breaking criterion 3.2. 破坏标准

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 and Meylan, 2008). These two quantities, $D_{\min}$ and $D_{\max}$ , determine the FSD (see Section 4.1 of Part 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 和 Meylan，2008）。这两个量 $D_{\min}$ 和 $D_{\max}$ 决定了浮冰尺寸分布（FSD）（见第 1 部分第 4.1 节）。

From (10), the criterion $P_{ε} > P_{c}$ can be written in terms of $E_{s}$ , $ε_{c}$ and $P_{c}$ as(11) $E_{s} > E_{c} = ε_{c} \sqrt{- 2 / \log (P_{c})} .$ Thus the single parameter $E_{c}$ combines the effects of both $ε_{c}$ and $P_{c}$ . Consequently, testing the sensitivity of the WIM to $E_{c}$ allows for the combined effects of our choice of $P_{c}$ and also of uncertainties in the breaking strain $ε_{c}$ , which are considerable. Note that if $P_{c} = e^{- 1} \approx 0.37$ , the breaking criterion becomes $E_{s} > ε_{c} \sqrt{2}$ , which is the same as for a monochromatic wave.
从（10）中，标准 $P_{ε} > P_{c}$ 可以用 $E_{s}$ 、 $ε_{c}$ 和 $P_{c}$ 表示为 (11) $E_{s} > E_{c} = ε_{c} \sqrt{- 2 / \log (P_{c})} .$ 。因此，单一参数 $E_{c}$ 结合了 $ε_{c}$ 和 $P_{c}$ 的影响。因此，测试 WIM 对 $E_{c}$ 的敏感性可以考虑我们选择的 $P_{c}$ 的综合影响，以及破裂应变 $ε_{c}$ 的不确定性，这些不确定性是相当大的。请注意，如果 $P_{c} = e^{- 1} \approx 0.37$ ，破裂标准变为 $E_{s} > ε_{c} \sqrt{2}$ ，这与单色波的情况相同。

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

Following DKB and Toyota et al., 2011, we use a fractal breaking model that predicts the FSD from $D_{\max}$ and $D_{\min}$ . We assume floes that break produce $ξ^{2}$ pieces, and that the fragility of the floes (the probability that a floe will break) is fixed at Π. We use $ξ = 2$ and $Π = 0.9$ .
根据 DKB 和丰田等人（2011）的研究，我们使用一种分形破碎模型来预测从 $D_{\max}$ 到 $D_{\min}$ 的 FSD。我们假设破碎的冰块产生 $ξ^{2}$ 个碎片，并且冰块的脆弱性（冰块破裂的概率）固定为Π。我们使用 $ξ = 2$ 和 $Π = 0.9$ 。

We determine the mean floe size from the formula of DKB:(12) $〈 D 〉 = \frac{\sum_{m = 1}^{M} {(ξ Π)}^{m}}{\sum_{m = 1}^{M} {(ξ^{2} Π)}^{m}}, M = ⌊\log_{ξ} (D_{\max} / D_{\min})⌋,$ where $⌊ \cdot ⌋$ denotes rounding down to the nearest integer. The FSD is discussed in more detail in Section 4.1 of Part 1.
我们通过 DKB 的公式确定平均冰块大小： (12) $〈 D 〉 = \frac{\sum_{m = 1}^{M} {(ξ Π)}^{m}}{\sum_{m = 1}^{M} {(ξ^{2} Π)}^{m}}, M = ⌊\log_{ξ} (D_{\max} / D_{\min})⌋,$ ，其中 $⌊ \cdot ⌋$ 表示向下取整到最接近的整数。冰块大小分布（FSD）在第一部分的第 4.1 节中进行了更详细的讨论。

4. Wave energy transport in the MIZ
4. 边界冰区的波能传输

4.1. Continuous equations
4.1. 连续方程

The energy balance equation for waves in the ice-covered ocean is(13) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = R_{in} - R_{ice} - R_{other} - R_{nl},$ (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 sea 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).
冰盖海洋中波浪的能量平衡方程为 (13) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = R_{in} - R_{ice} - R_{other} - R_{nl},$ (Masson 和 LeBlond, 1989; Meylan 和 Masson, 2006; Ardhuin 等, 2010)，其中 $c_{g}$ 是群速度， $D_{t} \equiv (\partial_{t} + c_{g} \partial_{x})$ 。源项 $R_{in}$ 、 $R_{ice}$ 和 $R_{other}$ 分别代表风能输入、向海冰的能量损失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）。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; Hasselmann, 1963）。

In the present study we consider only wave attenuation caused by the presence of ice cover. Our simplified equation is therefore(14) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = - R_{ice} \approx - \hat{α} (ω, x, t) S (ω; x, t),$ where $\hat{α}$ is the dimensional attenuation coefficient, i.e., the rate of exponential attenuation per meter. We model the wave attenuation as being the sum of linear wave scattering at floe edges and a viscosity term; this is discussed in detail in Section 4.2 of Part 1. The attenuation coefficient is not explicitly dependent on S, but changes suddenly when the wave energy (or more specifically the significant strain $E_{s}$ ) becomes large enough to cause ice breakage. This added subtlety is unique to our model and its predecessor DKB.
在本研究中，我们仅考虑由冰盖存在引起的波衰减。因此，我们的简化方程为 (14) $\frac{1}{c_{g}} D_{t} S (ω; x, t) = - R_{ice} \approx - \hat{α} (ω, x, t) S (ω; x, t),$ ，其中 $\hat{α}$ 是维度衰减系数，即每米的指数衰减速率。我们将波衰减建模为浮冰边缘的线性波散射与一个粘度项的总和；这一点在第一部分的第 4.2 节中有详细讨论。衰减系数并不明确依赖于 S，但当波能量（或更具体地说，显著应变 $E_{s}$ ）变得足够大以导致冰破裂时，会突然变化。这一附加的细微之处是我们模型及其前身 DKB 所独有的。

As discussed in Section 3.1 of Part 1, Eq. (14) represents advection of S at the group velocity $c_{g}$ followed by its attenuation using $\hat{α}$ . This can be seen by considering the above problem, between breaking events, in the Lagrangian frame. The resulting expressions are(15a) $\frac{d x}{d t} = c_{g} (ω, x, t_{*}) and$ (15b) $\frac{d}{d x} S (ω; x, t) = - \hat{α} (ω; x, t_{*}, S_{*}) S (ω; x, t),$ where $t_{*}$ is the last time breaking occurred at x, and $S_{*} (ω, x) = S (ω; x, t_{*})$ . Thus, we have separated the problem into an advection problem (in which we solve $D_{t} S = 0$ ) and an attenuation one. In our numerical scheme presented in the next section, we solve (14) by alternately advecting and attenuating.
如第 1 部分第 3.1 节所讨论的，方程(14)表示以群速度 $c_{g}$ 对 S 的平流，随后使用 $\hat{α}$ 进行衰减。通过考虑上述问题，在破裂事件之间，在拉格朗日框架中可以看出这一点。得到的表达式为 (15a) $\frac{d x}{d t} = c_{g} (ω, x, t_{*}) and$ (15b) $\frac{d}{d x} S (ω; x, t) = - \hat{α} (ω; x, t_{*}, S_{*}) S (ω; x, t),$ ，其中 $t_{*}$ 是 x 处最后一次破裂发生的时间， $S_{*} (ω, x) = S (ω; x, t_{*})$ 。因此，我们将问题分为一个平流问题（在其中我们求解 $D_{t} S = 0$ ）和一个衰减问题。在下一节中提出的数值方案中，我们通过交替平流和衰减来求解(14)。

4.2. Full numerical implementation
4.2. 完整的数值实现

Let us discretize our space, time and frequency variables using $\begin{matrix} Space : x_{j} = j Δ x (j = 0, 1, \dots, N_{x}), \\ Time : t_{n} = n Δ t (n = 0, 1, \dots, N_{t}), \\ Frequency : ω_{r} = ω_{0} + r Δ ω (r = 0, 1, \dots, N_{ω}) . \end{matrix}$ We choose $ω_{0}$ and $Δ ω$ so that 31 periods between 2.5 s and 25 s are included. For all temporal indices n, spatial indices j and frequency indices r, we use the shorthand notations $\begin{matrix} c_{j} = c (x_{j}), h_{j} = h (x_{j}), \\ c_{g, r} = c_{g} (ω_{r}), S_{j, r}^{n} = S (ω_{r}, x_{j}, t_{n}), \\ D_{j}^{n} = D_{\max} (x_{j}, t_{n}), 〈D_{j}^{n}〉 = 〈 D 〉 (x_{j}, t_{n}) . \end{matrix}$ The Courant number is ${(C)}_{r} \equiv c_{g, r} Δ t / Δ x \in (0, 1]$ (for $r = 0, 1, \dots, N_{ω}$ ). It represents the proportion of one grid cell a wave of a given frequency travels in one time step.
让我们使用 $\begin{matrix} Space : x_{j} = j Δ x (j = 0, 1, \dots, N_{x}), \\ Time : t_{n} = n Δ t (n = 0, 1, \dots, N_{t}), \\ Frequency : ω_{r} = ω_{0} + r Δ ω (r = 0, 1, \dots, N_{ω}) . \end{matrix}$ 对我们的空间、时间和频率变量进行离散化。我们选择 $ω_{0}$ 和 $Δ ω$ ，以便包含 2.5 秒到 25 秒之间的 31 个周期。对于所有时间索引 n、空间索引 j 和频率索引 r，我们使用简写符号 $\begin{matrix} c_{j} = c (x_{j}), h_{j} = h (x_{j}), \\ c_{g, r} = c_{g} (ω_{r}), S_{j, r}^{n} = S (ω_{r}, x_{j}, t_{n}), \\ D_{j}^{n} = D_{\max} (x_{j}, t_{n}), 〈D_{j}^{n}〉 = 〈 D 〉 (x_{j}, t_{n}) . \end{matrix}$ 。Courant 数为 ${(C)}_{r} \equiv c_{g, r} Δ t / Δ x \in (0, 1]$ （对于 $r = 0, 1, \dots, N_{ω}$ ）。它表示给定频率的波在一个时间步长内穿过一个网格单元的比例。

For $j = 1, \dots, N_{x}$ , we also let(16) $W_{j, r} = W (ω_{r}), E_{j, r} = E (ω_{r}),$ observing that we will need these to approximate the integrals (8a), (8b). Note that W and E have an implicit dependence on the ice properties, which is why $W_{j, r}$ and $E_{j, r}$ depend on the index of the grid cell as well as the frequency index.
对于 $j = 1, \dots, N_{x}$ ，我们还让 (16) $W_{j, r} = W (ω_{r}), E_{j, r} = E (ω_{r}),$ ，观察到我们需要这些来近似积分(8a)、(8b)。注意，W 和 E 对冰的性质有隐含的依赖关系，这就是为什么 $W_{j, r}$ 和 $E_{j, r}$ 依赖于网格单元的索引以及频率索引。

Following DKB, our numerical implementation (which we call N1) proceeds as follows.
根据 DKB，我们的数值实现（称为 N1）如下进行。

1.
Initialization. For $r = 0, 1, \dots, N_{ω}$ :
初始化。对于 $r = 0, 1, \dots, N_{ω}$ :
We initialise the problem by setting the incident wave spectrum and initial FSD to:(17a) $S_{j, r}^{0} = \{\begin{matrix} S_{B} (ω_{r}; T_{p}, H_{s}) & for j = 0, 1, 2, \\ 0 & for j = 3, 4, \dots, N_{x}, \end{matrix}$ (17b) $and D_{j}^{0} = 〈D_{j}^{0}〉 = \{\begin{matrix} D_{init} & if c_{j} > 0, \\ 0 & if c_{j} = 0 . \end{matrix}$ Here $D_{init}$ is an arbitrarily chosen (relatively large) value. By invoking (17b) at $j = 0, 1, 2$ , we can apply (2) via the Neumann condition $\partial_{x} S (ω, 0, t) = 0$ during the advection step. Note that this implies $\partial_{t} S (ω, 0, t) = 0$ , since the advection equation is $D_{t} S = 0$ . We need three points initially constant, as we advect S using a second order method.
我们通过设置入射波谱和初始 FSD 来初始化问题： (17a) $S_{j, r}^{0} = \{\begin{matrix} S_{B} (ω_{r}; T_{p}, H_{s}) & for j = 0, 1, 2, \\ 0 & for j = 3, 4, \dots, N_{x}, \end{matrix}$ (17b) $and D_{j}^{0} = 〈D_{j}^{0}〉 = \{\begin{matrix} D_{init} & if c_{j} > 0, \\ 0 & if c_{j} = 0 . \end{matrix}$ 这里 $D_{init}$ 是一个任意选择的（相对较大）值。通过在 $j = 0, 1, 2$ 处调用(17b)，我们可以在平流步骤中通过 Neumann 条件 $\partial_{x} S (ω, 0, t) = 0$ 应用(2)。请注意，这意味着 $\partial_{t} S (ω, 0, t) = 0$ ，因为平流方程是 $D_{t} S = 0$ 。我们需要三个初始常数点，因为我们使用二阶方法平流 S。
2.
Time integration. For $n = 1, 2, \dots, N_{t}$ :
时间积分。对于 $n = 1, 2, \dots, N_{t}$ :
For $r = 0, 1, \dots, N_{ω}$ : 对于 $r = 0, 1, \dots, N_{ω}$ ：
(i) Advection. In our integration we alternate between advection and attenuation. The advection is done by solving the equation $D_{t} S = 0$ using the Lax-Wendroff scheme (a second order direct space–time method) with Superbee flux limiting (Roe, 1986) and a Neumann boundary condition, as mentioned above. The scheme is stable for Courant number $C_{r} \in (0, 1]$ and has very little numerical diffusion for $0.1 ≲ C_{r} < 1$ . We perform the advection over the whole domain in one step, mapping $S_{j, r}^{n - 1}$ onto an unattenuated intermediate spectrum ${\hat{S}}_{j, r}^{n}$ ( $\forall j = 1, 2, \dots, N_{x}$ ).
（i）平流。在我们的积分中，我们在平流和衰减之间交替进行。平流是通过使用 Lax-Wendroff 方案（一个二阶直接时空方法）解决方程 $D_{t} S = 0$ 来完成的，该方案采用 Superbee 通量限制（Roe，1986）和上述的 Neumann 边界条件。该方案在 Courant 数 $C_{r} \in (0, 1]$ 下是稳定的，并且对于 $0.1 ≲ C_{r} < 1$ 几乎没有数值扩散。我们在一个步骤中对整个区域进行平流，将 $S_{j, r}^{n - 1}$ 映射到未衰减的中间谱 ${\hat{S}}_{j, r}^{n}$ （ $\forall j = 1, 2, \dots, N_{x}$ ）。
For $j = 1, 2, \dots, N_{x}$ : 对于 $j = 1, 2, \dots, N_{x}$ ：
We now apply attenuation and the subsequent integration over frequency locally, i.e., we consider each cell separately. We reset $m_{0} [η_{ice}] = m_{2} [η_{ice}] = m_{0} [ε] = 0$ , and these integrals are calculated cumulatively as we loop through the frequencies. For $r = 0, 1, \dots, N_{ω}$ :
我们现在在局部应用衰减和随后的频率积分，即我们分别考虑每个单元。我们重置 $m_{0} [η_{ice}] = m_{2} [η_{ice}] = m_{0} [ε] = 0$ ，这些积分在我们遍历频率时累积计算。对于 $r = 0, 1, \dots, N_{ω}$ ：
(ii) Attenuation. We calculate the attenuation coefficient and the attenuated wave spectrum to be(18a) ${\hat{α}}_{j, r}^{n} = \frac{α_{j, r} c_{j}}{〈D_{j}^{n - 1}〉} and$ (18b) $S_{j, r}^{n} = {\hat{S}}_{j, r}^{n} \exp (- {\hat{α}}_{j, r}^{n} c_{g, r} Δ t),$ where $α_{j, r} = α (x_{j}, ω_{r})$ is the non-dimensional attenuation coefficient (cf. Section 3.1 of Part 1).
（ii）衰减。我们计算衰减系数和衰减波谱为 (18a) ${\hat{α}}_{j, r}^{n} = \frac{α_{j, r} c_{j}}{〈D_{j}^{n - 1}〉} and$ (18b) $S_{j, r}^{n} = {\hat{S}}_{j, r}^{n} \exp (- {\hat{α}}_{j, r}^{n} c_{g, r} Δ t),$ ，其中 $α_{j, r} = α (x_{j}, ω_{r})$ 是无量纲衰减系数（参见第 1 部分第 3.1 节）。
(iii) Integration over frequency. The integrals over frequency are approximated using Simpson’s rule, i.e.,(19) $\int_{0}^{\infty} f (ω) d ω \approx \int_{ω_{0}}^{ω_{N_{ω}}} f (ω) d ω \approx \sum_{r = 0}^{N_{ω}} w_{r} f (ω_{r}) .$ Thus we can update the integrals we need as the r loop proceeds:(20a) $m_{0} [η_{ice}] = m_{0} [η_{ice}] + w_{r} S_{j, r}^{n} W_{j, r}^{2};$ (20b) $m_{2} [η_{ice}] = m_{2} [η_{ice}] + w_{r} ω_{r}^{2} S_{j, r}^{n} W_{j, r}^{2};$ (20c) $and m_{0} [ε] = m_{0} [ε] + w_{r} S_{j, r}^{n} E_{j, r}^{2} .$
(iii) 频率上的积分。频率上的积分使用辛普森法则进行近似，即， (19) $\int_{0}^{\infty} f (ω) d ω \approx \int_{ω_{0}}^{ω_{N_{ω}}} f (ω) d ω \approx \sum_{r = 0}^{N_{ω}} w_{r} f (ω_{r}) .$ 因此我们可以在 r 循环进行时更新所需的积分： (20a) $m_{0} [η_{ice}] = m_{0} [η_{ice}] + w_{r} S_{j, r}^{n} W_{j, r}^{2};$ (20b) $m_{2} [η_{ice}] = m_{2} [η_{ice}] + w_{r} ω_{r}^{2} S_{j, r}^{n} W_{j, r}^{2};$ (20c) $and m_{0} [ε] = m_{0} [ε] + w_{r} S_{j, r}^{n} E_{j, r}^{2} .$
(iv) Floe breaking. Having completed the frequency integration, the significant strain, $E_{s}$ and the dominant period $T_{W}$ is obtained from (4), (9) following Section 3.2 of Part 1. If $E_{s} > E_{c} = \sqrt{2} ε_{c}$ , the ice breaks, and we reduce the maximum floe size to $D_{j}^{n} = \max {D_{\min}, \min {λ_{W} / 2, D_{j}^{n - 1}}}$ , where $λ_{W} = 2 π / k_{ice} (2 π / T_{W})$ is the wavelength corresponding to $T_{W}$ and the ice properties in the cell. We then calculate the new average floe size $〈 D_{j}^{n} 〉$ from (12).
（四）冰块破裂。在完成频率积分后，从第 1 部分第 3.2 节的（4）、（9）中获得显著应变 $E_{s}$ 和主导周期 $T_{W}$ 。如果 $E_{s} > E_{c} = \sqrt{2} ε_{c}$ ，冰块破裂，我们将最大冰块大小减少到 $D_{j}^{n} = \max {D_{\min}, \min {λ_{W} / 2, D_{j}^{n - 1}}}$ ，其中 $λ_{W} = 2 π / k_{ice} (2 π / T_{W})$ 是与 $T_{W}$ 和单元中的冰属性对应的波长。然后，我们从（12）计算新的平均冰块大小 $〈 D_{j}^{n} 〉$ 。
3.
Define the MIZ. At the end of the integration, the point $x_{j}$ is defined to be inside the MIZ if the corresponding cell contains ice and if ice breakage has occurred in that cell, i.e., if $0 < D_{j}^{N_{t}} < D_{init}$ ( $j = 0, 1, \dots, N_{x}$ ). The MIZ width, $L_{MIZ}$ , is then the distance from the ice edge to the last point in the MIZ, which includes any internal polynyas. We also define $D_{MIZ}$ as the maximum floe size in this region.
定义 MIZ。在积分结束时，如果相应的单元格包含冰并且该单元格内发生了冰破裂，即如果 $0 < D_{j}^{N_{t}} < D_{init}$ ( $j = 0, 1, \dots, N_{x}$ )，则点 $x_{j}$ 被定义为在 MIZ 内部。MIZ 宽度 $L_{MIZ}$ 是从冰缘到 MIZ 中最后一个点的距离，该点包括任何内部开阔水域。我们还将 $D_{MIZ}$ 定义为该区域内的最大浮冰大小。

C_{r} = 1

the waves travel one grid cell every time step and hence do not experience any attenuation from any ice they break, as the broken ice is always behind them. However, if

C_{r} < 1

, waves travel less than a grid cell per time step and must therefore pass through a proportion of this broken ice before escaping the cell. This is because we use a well-mixed grid cell, as opposed to a partial grid cell. The proportion of broken ice the wave must pass through in a grid cell is

1 - C_{r}

, i.e., it increases as the Courant number

C_{r}

decreases. In our numerical results we will show that the FSD is insensitive to the exact amount of broken ice the waves travel through if the maximum Courant number

C = \max {C_{r} | r = 0, 1, \dots, N_{ω}}

is less than approximately

C ≲ 0.7

. This represents an equilibrium between the wave field and the FSD, which will be discussed in Section 4.3. In addition, while the scheme depends on the initial floe size

D_{init}

for

C \approx 1

, it does not in the limit

C \to 0

, and therefore for

C ≲ 0.7

.
如果

C_{r} = 1

波浪每个时间步长移动一个网格单元，因此不会受到它们所破坏的任何冰的衰减，因为破碎的冰总是在它们的后面。然而，如果

C_{r} < 1

，波浪每个时间步长移动的距离少于一个网格单元，因此必须通过一部分破碎的冰才能逃离该单元。这是因为我们使用的是一个充分混合的网格单元，而不是部分网格单元。波浪在一个网格单元中必须穿过的破碎冰的比例是

1 - C_{r}

，即随着 Courant 数

C_{r}

的减小而增加。在我们的数值结果中，我们将表明，如果最大 Courant 数

C = \max {C_{r} | r = 0, 1, \dots, N_{ω}}

小于大约

C ≲ 0.7

，则 FSD 对波浪穿过的破碎冰的确切数量不敏感。这代表了波场与 FSD 之间的平衡，将在第 4.3 节中讨论。此外，虽然该方案依赖于初始浮冰大小

D_{init}

用于

C \approx 1

，但在极限情况下

C \to 0

并不依赖，因此对于

C ≲ 0.7

。

4.3. Semi-analytical schemes
4.3. 半解析方案

The N1 scheme described in the previous section is a general numerical implementation of the WIM that is applicable to any ice and wave conditions. In particular, it can deal with wave dispersion (wave speed dependent on frequency), and it is generalizable to two horizontal dimensions. However, if we neglect dispersion we can derive semi-analytic methods for the $C \to 1$ and the $C \to 0$ limits. The purpose of doing this is twofold: (i) to check our numerical method; and (ii) to produce a much faster algorithm to determine MIZ width, as the frequency loop is only inside a single spatial loop instead of being within both spatial and temporal loops (as in the N1 algorithm). Of course, if we wish to know the wave spectrum at a particular time in the ice—for example, if we wish to know when a group of large waves will reach a certain point, dispersive effects must be considered. Notwithstanding, it will be shown in Section 5.1 that the predicted FSD is insensitive to the effects of wave dispersion. Finally, we note that generalizing semi-analytical methods to the two-dimensional situation is challenging and that the numerical model is necessary to overcome the added complexity of the extra dimension. When we set $C_{r} = 1$ for all r in the N1 scheme, all of the ice breakage is caused by the lead waves as they always travel through unbroken ice. The waves do not suffer additional attenuation due to floes that have been freshly broken. Accordingly, in this situation it is possible to calculate the breaking penetration, and hence the width of the MIZ, just by considering the attenuation of the lead waves (referred to hereinafter using the superscript ‘lw’). We denote the semi-analytic method that reproduces the $C_{r} \to 1$ ( $r = 0, 1, \dots, N_{ω}$ ) limiting case by A1. This method is essentially the same as that of Vaughan and Squire, 2011. The wave spectrum when the lead wave is at a given position x is given explicitly by(21) $S^{lw} (ω, x) = S (ω, 0, 0) \exp (- \int_{0}^{x} \hat{α} (ω, x^{'}, 0) d x^{'}) .$ We can also calculate the moments for the lead wave(22a) $m_{ε}^{lw} (x) = \int_{0}^{\infty} S^{lw} (ω, x) E^{2} (ω) d ω and$ (22b) $m_{n}^{lw} (x) = \int_{0}^{\infty} ω^{n} S^{lw} (ω, x) W^{2} (ω) d ω,$ which give us the significant strain and the dominant wave period: $E_{s}^{lw} (x) = 2 \sqrt{m_{ε}^{lw} (x)}, and T_{W}^{lw} (x) = 2 π \sqrt{\frac{m_{0}^{lw} (x)}{m_{2}^{lw} (x)}} .$ We can then find the width of the MIZ, i.e., the distance over which the ice cover is broken, $L_{MIZ}$ , by solving $E_{s}^{lw} (L_{MIZ}) = E_{c}$ . In practice, we still discretize the problem as before to calculate the integral (21) in which $\hat{α}$ varies spatially, but we no longer have to consider the time dimension. The FSD is calculated as a function of x from the wavelength corresponding to $T_{W}^{lw} (x)$ .
在前一节中描述的 N1 方案是 WIM 的一种通用数值实现，适用于任何冰和波浪条件。特别是，它可以处理波浪色散（波速依赖于频率），并且可以推广到两个水平维度。然而，如果我们忽略色散，我们可以推导出 $C \to 1$ 和 $C \to 0$ 极限的半解析方法。这样做的目的有两个：（i）检查我们的数值方法；（ii）生成一个更快的算法来确定 MIZ 宽度，因为频率循环仅在一个空间循环内，而不是在空间和时间循环内（如 N1 算法中）。当然，如果我们希望知道冰中某一特定时间的波谱——例如，如果我们希望知道一组大波何时会到达某个点，则必须考虑色散效应。尽管如此，在第 5.1 节中将表明，预测的 FSD 对波浪色散的影响不敏感。最后，我们注意到将半解析方法推广到二维情况是具有挑战性的，并且数值模型是克服额外维度所带来的复杂性的必要条件。当我们在 N1 方案中为所有 r 设置 $C_{r} = 1$ 时，所有的冰破裂都是由引导波引起的，因为它们总是穿过未破裂的冰。波浪不会因为新近破裂的浮冰而遭受额外的衰减。因此，在这种情况下，仅通过考虑引导波的衰减（以下简称使用上标“lw”）就可以计算破裂渗透深度，从而得出 MIZ 的宽度。我们将再现 $C_{r} \to 1$ （ $r = 0, 1, \dots, N_{ω}$ ）极限情况的半解析方法称为 A1。该方法本质上与 Vaughan 和 Squire 在 2011 年的方法相同。当引导波位于给定位置 x 时，波谱明确表示为 (21) $S^{lw} (ω, x) = S (ω, 0, 0) \exp (- \int_{0}^{x} \hat{α} (ω, x^{'}, 0) d x^{'}) .$ 。我们还可以计算引导波的矩 (22a) $m_{ε}^{lw} (x) = \int_{0}^{\infty} S^{lw} (ω, x) E^{2} (ω) d ω and$ (22b) $m_{n}^{lw} (x) = \int_{0}^{\infty} ω^{n} S^{lw} (ω, x) W^{2} (ω) d ω,$ ，这给我们提供了显著的应变和主导波周期： $E_{s}^{lw} (x) = 2 \sqrt{m_{ε}^{lw} (x)}, and T_{W}^{lw} (x) = 2 π \sqrt{\frac{m_{0}^{lw} (x)}{m_{2}^{lw} (x)}} .$ 。然后，我们可以通过求解 $E_{s}^{lw} (L_{MIZ}) = E_{c}$ 来找到 MIZ 的宽度，即冰盖破裂的距离 $L_{MIZ}$ 。在实践中，我们仍然像以前一样对问题进行离散化，以计算积分（21），其中 $\hat{α}$ 在空间上变化，但我们不再需要考虑时间维度。FSD 作为与 $T_{W}^{lw} (x)$ 对应的波长的 x 的函数进行计算。

The precise A1 algorithm proceeds as follows.
精确的 A1 算法如下进行。

1.
Initialization. For $r = 0, 1, \dots, N_{ω}, j = 0, 1, \dots, N_{x}$ :
初始化。对于 $r = 0, 1, \dots, N_{ω}, j = 0, 1, \dots, N_{x}$ :
We set the incident wave spectrum and initial FSD to be(23a) $S_{0, r}^{lw} = S_{B} (ω_{r}; T_{p}, H_{s}),$ (23b) $and D_{j}^{0} = 〈D_{j}^{0}〉 = \{\begin{matrix} D_{init} & if c_{j} > 0, \\ 0 & if c_{j} = 0 . \end{matrix}$
我们将入射波谱和初始 FSD 设置为 (23a) $S_{0, r}^{lw} = S_{B} (ω_{r}; T_{p}, H_{s}),$ (23b) $and D_{j}^{0} = 〈D_{j}^{0}〉 = \{\begin{matrix} D_{init} & if c_{j} > 0, \\ 0 & if c_{j} = 0 . \end{matrix}$
2.
Propagation of the lead waves. For $j = 1, 2, \dots, N_{x}$ : Reset the following integrals to zero: $m_{0}^{lw} [η_{ice}] = m_{2}^{lw} [η_{ice}] = m_{0}^{lw} [ε] = 0$ .
引导波的传播。对于 $j = 1, 2, \dots, N_{x}$ ：将以下积分重置为零： $m_{0}^{lw} [η_{ice}] = m_{2}^{lw} [η_{ice}] = m_{0}^{lw} [ε] = 0$ 。
For $r = 0, 1, \dots, N_{ω}$ : 对于 $r = 0, 1, \dots, N_{ω}$ ：
(i) Advection. The waves move from one grid cell to the next without the effects of time-stepping and the Courant number $C$ :(24) ${\hat{S}}_{j, r}^{lw} = S_{j - 1, r}^{lw} .$
(i) 平流。波动在网格单元之间移动，而不受时间步进和库朗数 $C$ : (24) ${\hat{S}}_{j, r}^{lw} = S_{j - 1, r}^{lw} .$ 的影响。
(ii) Attenuation. We calculate the dimensional attenuation coefficient from the initial FSD, so breaking effects do not influence the transmission of the waves. The energy S is also reduced accordingly at this point.(25a) ${\hat{α}}_{j, r} = \frac{α_{j, r} c_{j}}{〈D_{j}^{0}〉}, and$ (25b) $S_{j, r}^{lw} = {\hat{S}}_{j, r}^{lw} \exp (- {\hat{α}}_{j, r} Δ x) .$
（ii）衰减。我们从初始的 FSD 计算维度衰减系数，因此破坏效应不会影响波的传播。此时，能量 S 也相应减少。 (25a) ${\hat{α}}_{j, r} = \frac{α_{j, r} c_{j}}{〈D_{j}^{0}〉}, and$ (25b) $S_{j, r}^{lw} = {\hat{S}}_{j, r}^{lw} \exp (- {\hat{α}}_{j, r} Δ x) .$
(iii) Integration over frequency. We update the integrals that we need(26a) $m_{0}^{lw} [η_{ice}] = m_{0}^{lw} [η_{ice}] + w_{r} S_{j, r}^{lw} W_{j, r}^{2},$ (26b) $m_{2}^{lw} [η_{ice}] = m_{2}^{lw} [η_{ice}] + w_{r} ω_{r}^{2} S_{j, r}^{lw} W_{j, r}^{2},$ (26c) $and m_{0}^{lw} [ε] = m_{0}^{lw} [ε] + w_{r} S_{j, r}^{lw} E_{j, r}^{2} .$
(iii) 频率上的积分。我们更新所需的积分 (26a) $m_{0}^{lw} [η_{ice}] = m_{0}^{lw} [η_{ice}] + w_{r} S_{j, r}^{lw} W_{j, r}^{2},$ (26b) $m_{2}^{lw} [η_{ice}] = m_{2}^{lw} [η_{ice}] + w_{r} ω_{r}^{2} S_{j, r}^{lw} W_{j, r}^{2},$ (26c) $and m_{0}^{lw} [ε] = m_{0}^{lw} [ε] + w_{r} S_{j, r}^{lw} E_{j, r}^{2} .$
(iv) Floe breaking. Having finished the frequency (r) loop, we can calculate $E_{s}^{lw}$ and $T_{W}^{lw}$ from (26c). If $E_{s}^{lw} > E_{c}$ then the ice breaks, giving a maximum floe size $D_{j}^{lw} = \max {D_{\min}, \min {λ_{W}^{lw} / 2, D_{j}^{0}}}$ , where $λ_{W}^{lw}$ is the wavelength corresponding to $T_{W}^{lw}$ and the thickness $h_{j}$ . Calculate the new average floe size $〈 D_{j}^{lw} 〉$ .
（iv）冰块破裂。完成频率（r）循环后，我们可以从（26c）计算 $E_{s}^{lw}$ 和 $T_{W}^{lw}$ 。如果 $E_{s}^{lw} > E_{c}$ ，那么冰会破裂，产生最大冰块大小 $D_{j}^{lw} = \max {D_{\min}, \min {λ_{W}^{lw} / 2, D_{j}^{0}}}$ ，其中 $λ_{W}^{lw}$ 是与 $T_{W}^{lw}$ 和厚度 $h_{j}$ 相对应的波长。计算新的平均冰块大小 $〈 D_{j}^{lw} 〉$ 。
3.
Define the MIZ. When the lead waves have left the domain, i.e., after the j loop has been completed, we can define the MIZ as in the N1 scheme.
定义混合层（MIZ）。当主波浪离开该区域，即在 j 循环完成后，我们可以按照 N1 方案定义混合层。

We denote the scheme that approximates the FSD in the

C \to 0

limit by A0. It is produced by reversing the order in which we apply breaking and attenuation in the A1 scheme. More precisely, we move the attenuation loop over r (A1 algorithm, step 2.ii) to after the breaking step (A1 algorithm, step 2.iv) and replace (25a) with(27)

{\hat{α}}_{j, r} = \frac{α_{j, r} c_{j}}{〈 D_{j}^{lw} 〉} .

我们用 A0 表示在

C \to 0

极限下近似 FSD 的方案。它是通过反转在 A1 方案中应用破坏和衰减的顺序而产生的。更准确地说，我们将衰减循环（A1 算法，步骤 2.ii）移动到破坏步骤（A1 算法，步骤 2.iv）之后，并用 (27)

{\hat{α}}_{j, r} = \frac{α_{j, r} c_{j}}{〈 D_{j}^{lw} 〉} .

替换 (25a)。

Under the A1 scheme, the lead waves travel through the ice relatively unhindered, leaving broken ice in their path. The energy they lose is due to viscous damping and scattering at the relatively few floe edges they meet on their way, which is inversely proportional to the initial floe size $D_{init}$ . Under the A0 scheme, the waves at a certain point have the same energy as if they had to travel through all the broken ice they produce. Therefore, the wave spectrum inside the broken ice is the result of an equilibrium between attenuation and breaking and is more stable.
在 A1 方案下，主波在冰层中相对不受阻碍地传播，留下破碎的冰块。它们损失的能量是由于粘性阻尼和在其路径上遇到的相对较少的浮冰边缘的散射，这与初始浮冰的大小成反比 $D_{init}$ 。在 A0 方案下，某一点的波浪能量与它们必须穿过所有产生的破碎冰块时的能量相同。因此，破碎冰层内的波谱是衰减与破碎之间平衡的结果，并且更加稳定。

An issue that is related to the two limiting cases is the amount of energy lost due to ice breakage. The A1 FSD is one extreme in which no energy is lost during this process. The A0 FSD is another critical point where the result of the amount of energy being lost is the same as the attenuation loss due to propagating through any broken floes that the waves themselves produce. Note that, if even more wave energy than this is lost, the MIZ due to the lead waves will initially be much narrower than the A0 MIZ, but following waves will gradually extend it towards the A0 limit. If less than this amount is lost during ice breakage, then we will be able to tell how sensitive the FSD is to the exact amount by testing the sensitivity of the N1 results to the $C$ parameter, which moves the N1 FSD between the A1 and A0 limits.
与这两个极限情况相关的一个问题是由于冰破裂而损失的能量量。A1 FSD 是一个极端情况，在这个过程中没有能量损失。A0 FSD 是另一个关键点，在这个点上，损失的能量量与波浪通过任何破碎冰块所产生的衰减损失相同。请注意，如果损失的波浪能量超过这个量，由于引导波，MIZ 最初会比 A0 MIZ 窄得多，但后续波浪会逐渐将其延伸到 A0 限制。如果在冰破裂过程中损失的能量少于这个量，那么我们将能够通过测试 N1 结果对 $C$ 参数的敏感性来判断 FSD 对确切损失量的敏感程度，该参数使 N1 FSD 在 A1 和 A0 限制之间移动。

5. Results 5. 结果

Table 1 lists the default model parameter values used in all simulations, unless otherwise specified. Attenuation model B from (Bennetts and Squire, 2012) is used with the default value of the viscous damping parameter, $Γ = 13$ Pa s m⁻¹. In the idealized simulations of Sections 5.1 Sensitivity to the Courant number, dispersion and horizontal resolution, 5.2 Sensitivity to wave attenuation and ice properties we use the thickness and concentration profiles of (1a), (1b) (also see Fig. 1).
表 1 列出了所有模拟中使用的默认模型参数值，除非另有说明。使用了（Bennetts 和 Squire，2012）的衰减模型 B，粘性阻尼参数的默认值为 $Γ = 13$ Pasm ⁻¹ 。在第 5.1 节“对 Courant 数、色散和水平分辨率的敏感性”、第 5.2 节“对波衰减和冰特性的敏感性”的理想化模拟中，我们使用了（1a）、（1b）的厚度和浓度剖面（另见图 1）。

5.1. Sensitivity to the Courant number, dispersion and horizontal resolution
5.1. 对库朗数、色散和水平分辨率的敏感性

Fig. 2 presents the model sensitivity of the different numerical schemes to the Courant number in the case where dispersion is neglected, i.e., $C_{r} = C$ $\forall r = 0, 1, \dots, N_{ω}$ . Fig. 2(a) shows values of $D_{\max}$ along a transect. Results are produced by the numerical scheme N1, in which the waves travel through proportions 0, 0.01, 0.05, 0.1 and 0.3 of the ice they break (i.e., with Courant numbers $C = 1$ , 0.99, 0.95, 0.9 and 0.7). Results are compared to those obtained by the two semi-analytical schemes A1 and A0. N1 and A1 agree exactly when $C = 1$ while N1 and A0 agree as $C$ decreases.
图 2 展示了在忽略色散的情况下，不同数值方案对 Courant 数的模型敏感性，即 $C_{r} = C$ $\forall r = 0, 1, \dots, N_{ω}$ 。图 2(a)显示了沿横断面的 $D_{\max}$ 值。结果由数值方案 N1 生成，其中波浪通过它们破坏的冰的比例为 0、0.01、0.05、0.1 和 0.3（即，Courant 数为 $C = 1$ 、0.99、0.95、0.9 和 0.7）。结果与两个半解析方案 A1 和 A0 获得的结果进行了比较。当 $C = 1$ 时，N1 和 A1 完全一致，而当 $C$ 减小时，N1 和 A0 一致。

Fig. 2(b) shows a snapshot in time of the significant wave height as the waves travel further into the ice. We only show results for N1 with $C = 1$ and 0.9, and the semi-analytical schemes A1 and A0 in this case. We see that $H_{s}$ under the A1 scheme decreases slowly and smoothly as the lead waves travel into the ice, only being attenuated by unbroken ice. In contrast, under the A0 scheme, when the lead waves must travel through all the ice that they break, the significant wave height decreases rapidly due to the broken ice, until about $x = 110$ km. This represents the end of the A0 MIZ, where the waves reach unbroken ice and $H_{s}$ drops less rapidly.
图 2(b)显示了波浪在进一步进入冰层时显著波高的瞬时快照。我们仅展示 N1 在 $C = 1$ 和 0.9 下的结果，以及在这种情况下的半解析方案 A1 和 A0。我们看到在 A1 方案下， $H_{s}$ 随着引导波进入冰层而缓慢平稳地减少，仅受到未破碎冰层的衰减。相比之下，在 A0 方案下，当引导波必须穿过它们所破坏的所有冰层时，显著波高由于破碎冰层而迅速下降，直到大约 $x = 110$ 公里。这代表了 A0 MIZ 的结束，波浪到达未破碎冰层， $H_{s}$ 的下降速度减缓。

Under the N1 scheme with $C = 1$ , we can see that, as expected, the lead wave (the right-most circle) tracks the A1 curve exactly. However, the following waves have heights that are several orders of magnitude smaller. Inside the A0 MIZ, $H_{s}$ for these waves tracks the A0 curve, but drops below it outside this region. This is because the lead wave with this Courant number is still able to break ice outside the A0 MIZ, so the following waves are still traveling through broken ice.
在 N1 方案下， $C = 1$ ，我们可以看到，正如预期的那样，主波（最右侧的圆圈）完全跟踪 A1 曲线。然而，随后的波浪高度则小了几个数量级。在 A0 MIZ 内部， $H_{s}$ 对于这些波浪跟踪 A0 曲线，但在该区域外则低于它。这是因为具有此 Courant 数的主波仍然能够在 A0 MIZ 外破冰，因此随后的波浪仍然在破碎的冰中传播。

When $C$ drops to 0.9, the wave heights under the N1 scheme follow the A0 ones almost exactly. Only the two right-most points (black dots) drop below the A0 curve as numerical error from the advection algorithm begins to take effect. The A0 wave heights thus represent a kind of steady-state or equilibrium solution.
当 $C$ 降至 0.9 时，N1 方案下的波高几乎完全跟随 A0 方案。只有最右侧的两个点（黑点）低于 A0 曲线，因为平流算法的数值误差开始显现。因此，A0 波高代表了一种稳态或平衡解。

In addition to the above, the following conclusions can be inferred from the results. First, the significant wave heights predicted by the N1 scheme for all Courant numbers agree for the interval in which they share broken ice, i.e., before the edge of the MIZ under the A0 scheme. Second, $D_{MIZ}$ is not sensitive to the Courant number for the N1 scheme, but $L_{MIZ}$ in the A1 and A0 limits consistently differs by a factor of about 1.6. However, the MIZ width rapidly drops to the A0 value even for N1 with $C = 0.9$ . That is, below a certain value, $L_{MIZ}$ is insensitive to $C$ . As we expect that a significant amount of wave energy will be lost during the breaking process, this indicates little sensitivity to the precise quantity lost, explored here by varying the Courant number.
除了上述内容外，还可以从结果中推断出以下结论。首先，N1 方案预测的所有 Courant 数的显著波高在它们共享破碎冰的区间内是一致的，即在 A0 方案下 MIZ 边缘之前。其次，对于 N1 方案， $D_{MIZ}$ 对 Courant 数不敏感，但在 A1 和 A0 极限下， $L_{MIZ}$ 始终相差约 1.6 倍。然而，即使对于 N1，MIZ 宽度也会迅速降至 A0 值， $C = 0.9$ 。也就是说，低于某个值时， $L_{MIZ}$ 对 $C$ 不敏感。由于我们预计在破碎过程中会损失大量波能，这表明对损失的精确数量的敏感性很小，这里通过改变 Courant 数进行了探讨。

In Fig. 3(a) and (b) we further investigate the sensitivity of N1 to the Courant number as a proxy for energy loss. Results are shown for both the maximum floe size, $D_{MIZ}$ , and the width of the MIZ $L_{MIZ}$ as functions of the peak period, for a maximum ice thickness of $h_{\infty} = 4$ m. It is again evident that the results of N1 converge rapidly to those of A0, as the Courant number decreases.
在图 3(a)和(b)中，我们进一步研究了 N1 对 Courant 数的敏感性，作为能量损失的代理。结果显示了最大浮冰尺寸 $D_{MIZ}$ 和 MIZ 宽度 $L_{MIZ}$ 作为峰值周期的函数，最大冰厚为 $h_{\infty} = 4$ 米。再次明显的是，随着 Courant 数的降低，N1 的结果迅速收敛到 A0 的结果。

In Fig. 3(c) and (d) we test the effect of allowing dispersion. Results are presented for maximum Courant numbers $C = 1$ and 0.1. The semi-analytical A0 scheme is also shown, and both N1 curves lie almost exactly upon it. Thus $D_{MIZ}$ and $L_{MIZ}$ display very little sensitivity to dispersion and when it is included the results are essentially independent of $C$ . As noted in Section 4.3, this is an extremely useful result for computational efficiency in the later results of this paper.
在图 3(c)和(d)中，我们测试了允许色散的影响。结果呈现了最大 Courant 数 $C = 1$ 和 0.1 的情况。半解析 A0 方案也被展示，两个 N1 曲线几乎完全重合。因此， $D_{MIZ}$ 和 $L_{MIZ}$ 对色散的敏感性非常小，当包括色散时，结果基本上独立于 $C$ 。正如在第 4.3 节中所提到的，这对于本文后续结果的计算效率是一个极其有用的结果。

Two key conclusions can be drawn from Fig. 2, Fig. 3. First, the numerical scheme is not very sensitive to the energy lost during ice breakage (parameterized by the Courant number) with the current floe breaking parameterization. Second, dispersion is not necessary to calculate the FSD. Consequently, it is valid to use the numerically efficient A0 scheme to test the sensitivity of the model to the ice properties (Section 5.2) and for the realistic simulations presented in Section 5.3.
从图 2 和图 3 可以得出两个关键结论。首先，当前的冰块破碎参数化下，数值方案对冰块破碎过程中能量损失（由 Courant 数参数化）并不十分敏感。其次，计算冰块大小分布（FSD）并不需要考虑色散。因此，使用数值效率高的 A0 方案来测试模型对冰属性的敏感性（第 5.2 节）以及进行第 5.3 节中呈现的真实模拟是有效的。

The final numerical issue that we investigate is the spatial resolution. Fig. 4 shows the convergence of the two numerical schemes as the default grid size $Δ x = 5$ km is reduced ten-fold. The MIZ width $L_{MIZ}$ converges a lot faster with $C = 1$ than with $C = 0.7$ . However, the latter only overestimates $L_{MIZ}$ by about one or two grid cells, so using $Δ x = 5$ km will not produce significant inaccuracies. High resolution ice-ocean models generally have grid sizes of about 2–4 km, while coarser models use approximately 10–20 km. In both cases the errors should be well below the noise level as the ice edge can be incorrectly located by as much as 40 km in contemporary models.
我们研究的最后一个数值问题是空间分辨率。图 4 显示了当默认网格大小 $Δ x = 5$ 公里减少十倍时，两种数值方案的收敛情况。与 $C = 0.7$ 相比，MIZ 宽度 $L_{MIZ}$ 在 $C = 1$ 下收敛得更快。然而，后者仅将 $L_{MIZ}$ 高估约一到两个网格单元，因此使用 $Δ x = 5$ 公里不会产生显著的不准确性。高分辨率冰-海模型的网格大小通常约为 2-4 公里，而粗糙模型则使用大约 10-20 公里。在这两种情况下，误差应远低于噪声水平，因为在当代模型中，冰缘的位置可能错误地偏移多达 40 公里。

5.2. Sensitivity to wave attenuation and ice properties
5.2. 对波衰减和冰属性的敏感性

We first revisit Fig. 3, Fig. 4 to investigate the sensitivity of MIZ width to the ice thickness. Comparing the red A0 curves in Figs. 3(d) and 4(b), we see that $L_{MIZ}$ for $h_{\infty} = 2$ , 3 and 4 m is respectively about 15, 17 and 25 km when $T_{p} = 6$ s, and about 48, 55 and 75 km when $T_{p} = 10$ s. Thus doubling the thickness increases the MIZ width by a factor of approximately 1.6. Thickness observations are much more difficult to obtain than measurements of properties such as concentration, so ice models rarely assimilate thickness. As a result model predictions for thickness can be quite inaccurate. Accordingly, the high sensitivity of our results to thickness is of potential concern. Notwithstanding, the realistic simulations presented in the following section do not show such high variability with changes to thickness.
我们首先回顾图 3 和图 4，以研究 MIZ 宽度对冰厚的敏感性。比较图 3(d)和图 4(b)中的红色 A0 曲线，我们看到当 $T_{p} = 6$ 秒时， $h_{\infty} = 2$ 、3 和 4 米的 $L_{MIZ}$ 分别约为 15、17 和 25 公里，而当 $T_{p} = 10$ 秒时则约为 48、55 和 75 公里。因此，厚度加倍使 MIZ 宽度增加了大约 1.6 倍。与浓度等属性的测量相比，厚度观测要困难得多，因此冰模型很少同化厚度。因此，厚度的模型预测可能相当不准确。因此，我们的结果对厚度的高度敏感性可能令人担忧。尽管如此，下一节中呈现的现实模拟并未显示出随着厚度变化而产生如此大的变异性。

Figs. 5(a) and (b) show the effect of varying the damping parameter Γ in attenuation model B on the width of the MIZ predicted by our WIM. As expected, without the extra damping, i.e., when $Γ = 0$ , waves can penetrate further into the ice-covered ocean and cause more ice breakage. The change is most pronounced for large values of the incident peak period. This is because scattering dominates the attenuation rate for small to medium values of wave period. The largest sensitity of the width of the MIZ to Γ is for the simulation with thinner ice, shown in panel (a). This is because the flexural rigidity F in (6) is proportional to $h^{3}$ , so it quickly begins to dominate Γ at larger thicknesses, reducing the damping effect (also see the discussion in Appendix A of Part 1).
图 5(a)和(b)显示了在衰减模型 B 中变化阻尼参数Γ对我们 WIM 预测的 MIZ 宽度的影响。如预期的那样，在没有额外阻尼的情况下，即当 $Γ = 0$ 时，波浪可以更深入地穿透冰覆盖的海洋，导致更多的冰破裂。对于入射峰周期较大的情况，这种变化最为明显。这是因为在小到中等波周期的情况下，散射主导了衰减率。MIZ 宽度对Γ的最大敏感性出现在图(a)中较薄冰的模拟中。这是因为公式(6)中的弯曲刚度 F 与 $h^{3}$ 成正比，因此在较大厚度时，它迅速开始主导Γ，减少了阻尼效应（另见第 1 部分附录 A 中的讨论）。

We further note that the prediction of $L_{MIZ}$ for larger values of Γ is less sensitive to changes in thickness. In these results doubling the thickness roughly halves the MIZ width when $Γ = 0$ , but only reduces it by a factor of approximately 1.6 when $Γ = 13$ Pa s m⁻¹.
我们进一步注意到，对于较大值的Γ， $L_{MIZ}$ 的预测对厚度变化的敏感性较低。在这些结果中，当 $Γ = 0$ 时，厚度加倍大约使 MIZ 宽度减半，但当 $Γ = 13$ Pasm ⁻¹ 时，仅减少约 1.6 倍。

Fig. 5(c) shows the effect of changing the ice concentration on the width of the MIZ. Doubling the concentration, for example, doubles the number of floe edges and thus doubles the attenuation coefficient. We may, therefore, expect this to cause $L_{MIZ}$ to change by a factor of a half. However, the drop in $L_{MIZ}$ in going from $c_{\infty} = 0.25$ to $c_{\infty} = 0.5$ is approximately 25% rather than 50%, and in going from $c_{\infty} = 0.25$ to $c_{\infty} = 0.75$ is approximately 50%, rather than 66%. The results therefore do not behave as simply as one can anticipate for a single monochromatic wave. In reality it represents the combined action of non-linear effects arising by considering a wave spectrum and feedback between attenuation and ice breakage.
图 5(c)显示了冰浓度变化对 MIZ 宽度的影响。例如，浓度加倍会使浮冰边缘的数量加倍，从而使衰减系数加倍。因此，我们可以预期这会导致 $L_{MIZ}$ 变化为原来的二分之一。然而，从 $c_{\infty} = 0.25$ 到 $c_{\infty} = 0.5$ 的 $L_{MIZ}$ 下降约为 25%，而不是 50%；从 $c_{\infty} = 0.25$ 到 $c_{\infty} = 0.75$ 的下降约为 50%，而不是 66%。因此，结果并不像人们对单一单色波的预期那样简单。实际上，它代表了通过考虑波谱和衰减与冰破裂之间的反馈所产生的非线性效应的综合作用。

In Fig. 5(d), we test the sensitivity to the breaking strain parameter $E_{c}$ . This parameter incorporates the effect of the absolute breaking strain $ε_{c}$ , the probability threshold $P_{c}$ directly, and implicitly the incident significant wave height $H_{s}$ . Since the Bretschneider spectrum is proportional to $H_{s}^{2}$ , the significant strain will be approximately proportional to $H_{s}$ . Thus doubling $H_{s}$ will have about the same effect as halving the breaking strain $E_{c}$ .
在图 5(d)中，我们测试了对破坏应变参数 $E_{c}$ 的敏感性。该参数结合了绝对破坏应变 $ε_{c}$ 、概率阈值 $P_{c}$ 的直接影响，以及事件显著波高 $H_{s}$ 的隐含影响。由于 Bretschneider 谱与 $H_{s}^{2}$ 成正比，显著应变将大致与 $H_{s}$ 成正比。因此，将 $H_{s}$ 加倍的效果大约与将破坏应变 $E_{c}$ 减半的效果相同。

Choosing $P_{c} = 0.01$ , $e^{- 2} \approx 0.1$ , $e^{- 1} \approx 0.37$ , or $e^{- 2 / 9} \approx 0.8$ (respectively) makes $β_{ε} = E_{c} / E_{c}^{(0)} = 0.47$ , 0.7, 1, or 2.1, where $E_{c}^{(0)} = ε_{c} \sqrt{2} \approx 7.05 \times 10^{- 5}$ is our default value, which is consistent with the limit for monochromatic waves (see Section 3.2.2 of Part 1). Testing values of $β_{ε}$ between $1 / 3$ and 3 should cover most reasonable variations in $P_{c}$ , and also our uncertainties in the values of $ε_{c}$ and $H_{s}$ . This range gives variations of about 50%. Again though, when we move to more realistic tests where the different variables interact in more complicated ways, there is generally a lot less variation with $β_{ε}$ than is observed here.
选择 $P_{c} = 0.01$ 、 $e^{- 2} \approx 0.1$ 、 $e^{- 1} \approx 0.37$ 或 $e^{- 2 / 9} \approx 0.8$ （分别）使得 $β_{ε} = E_{c} / E_{c}^{(0)} = 0.47$ 、0.7、1 或 2.1，其中 $E_{c}^{(0)} = ε_{c} \sqrt{2} \approx 7.05 \times 10^{- 5}$ 是我们的默认值，这与单色波的限制是一致的（见第一部分第 3.2.2 节）。测试 $β_{ε}$ 在 $1 / 3$ 和 3 之间的值应覆盖大多数合理的 $P_{c}$ 变化，以及我们对 $ε_{c}$ 和 $H_{s}$ 值的不确定性。这个范围给出的变化大约为 50%。然而，当我们转向更现实的测试时，不同变量以更复杂的方式相互作用，通常 $β_{ε}$ 的变化要比这里观察到的少得多。

5.3. Realistic experiments in the Fram Strait
5.3. 在弗拉姆海峡的现实实验

Here we repeat some of the sensitivity studies in simulations using the A0 scheme with realistic wave forcings, ice concentrations and ice thicknesses along a transect of the Fram Strait during 2007. Fig. 6 shows a map of the area and the location of the transect. It also shows the location of the grid cell where wave forcing data was extracted from the WAM ERA-Interim reanalysis. Fig. 7(a) shows a time series of this wave forcing, while Figs. 7(b) and (c) show, respectively, ice concentrations and thicknesses obtained from a TOPAZ reanalysis (Sakov et al., 2012) in which concentration data derived from the Ocean and Sea Ice Satellite Application Facility (OSI SAF, met.no) have been assimilated. On average, the modeled ice edge is 45 km west of the ice edge observed by AMSR-E (University of Bremen) and determined from the analysis of Kloster and Sandven (2011), which is plotted as a solid line in Fig. 7(b). This discrepancy is well within the uncertainties and resolution of the model (TOPAZ has a resolution of about 13 km) and the resolution of the AMSR-E analysis. (Kloster and Sandven, 2011, divided the transect from 15°W to 5°E into bins with widths of about 21.2 km, i.e. 1 degree in longitude, and analyzed them for ice concentration.) The internal concentrations from the model and the data also compare well.
在这里，我们重复了一些使用 A0 方案进行的敏感性研究，这些模拟采用了 2007 年弗拉姆海峡横断面上的现实波浪强迫、冰浓度和冰厚度。图 6 显示了该区域的地图以及横断面的位置。它还显示了从 WAM ERA-Interim 再分析中提取波浪强迫数据的网格单元的位置。图 7(a)显示了该波浪强迫的时间序列，而图 7(b)和(c)分别显示了从 TOPAZ 再分析（Sakov 等，2012）中获得的冰浓度和厚度，其中浓度数据来自海洋和海冰卫星应用设施（OSI SAF，met.no）的同化。平均而言，模拟的冰缘位于 AMSR-E（不来梅大学）观测到的冰缘以西 45 公里，该冰缘由 Kloster 和 Sandven（2011）分析确定，并在图 7(b)中以实线绘制。这个差异在模型的不确定性和分辨率范围内（TOPAZ 的分辨率约为 13 公里）以及 AMSR-E 分析的分辨率范围内。（Kloster 和 Sandven，2011）将从 15°W 到 5°E 的横断面划分为宽约 21.2 公里的区间。 1 度经度，并对冰浓度进行了分析。）模型和数据的内部浓度也比较一致。

Also plotted (dashed line) in Fig. 7(b) is an estimate for the inner edge of the MIZ, determined from the same AMSR-E concentrations using the criterion that $c < 0.9$ corresponds to the MIZ. While this is a different criterion from the floe size criterion, where we define the MIZ by whether the ice is broken or not—see step 3 of the N1 algorithm in Section 4.2 we use in this paper—it provides a rough estimate of the accuracy of the predictions obtained from our WIM (Strong, 2012).
在图 7(b)中，虚线绘制的是 MIZ 内缘的估计值，该值是根据相同的 AMSR-E 浓度确定的，使用的标准是 $c < 0.9$ 对应于 MIZ。虽然这与浮冰大小标准不同，我们在该标准中通过冰是否破碎来定义 MIZ——请参见本论文第 4.2 节中 N1 算法的第 3 步——但它提供了我们从 WIM（Strong, 2012）获得的预测准确性的粗略估计。

The mean ice thickness is roughly 0.8 m, creeping up towards 2 m in the summer, which is thicker due to greater movement south of multi-year ice from the Arctic Ocean at that time. According to Widell et al., 2003, these ice thicknesses are probably too low, so we have also run simulations in which the ice thicknesses are multiplied by a factor $β_{h} = 1.75$ or 2.5 in order to get closer to observations. We observe that the Fram Strait is a particularly challenging MIZ to model, as it is made up of sea ice that is continuously being channeled out of the Arctic Basin, locally-growing sea ice in winter, and liberated land fast ice that can include sikussak. It is also baroclinically unstable so its edge is often characterized by the presence of many eddies and meanders.
平均冰厚约为 0.8 米，夏季逐渐增厚至 2 米，这主要是由于当时来自北冰洋的多年冰向南移动所致。根据 Widell 等人（2003）的研究，这些冰厚度可能过低，因此我们还进行了模拟，将冰厚度乘以 0 或 2.5 的因子，以更接近观测结果。我们观察到，弗拉姆海峡是一个特别具有挑战性的边界冰区（MIZ），因为它由不断从北极盆地流出的海冰、冬季局部生长的海冰以及可能包含 sikussak 的解冻陆地固定冰组成。它也是一个重力不稳定的区域，因此其边缘常常以许多涡旋和曲流为特征。

Fig. 8 shows the results of numerical experiments using the model outlined in this work. Results for the expected ice breakage are calculated daily using either the TOPAZ thicknesses or the increased thicknesses, the TOPAZ concentrations and the ERA-Interim waves. Floe sizes are re-initialized for each model pass to be uniformly $D_{init} = 500$ m long. An extension to include a memory in each cell of $D_{\max}$ and a gradual refreezing was rejected on the basis that we are unable to embed the more important effects of ice movement due to winds and current into this one-dimensional experiment.
图 8 显示了使用本研究中概述的模型进行的数值实验结果。预期冰块破裂的结果是每天计算的，使用的是 TOPAZ 厚度或增加的厚度、TOPAZ 浓度和 ERA-Interim 波浪。每次模型运行时，冰块大小被重新初始化为均匀的 $D_{init} = 500$ 米长。由于我们无法将风和洋流引起的冰运动的更重要影响嵌入到这个一维实验中，因此拒绝了在每个单元中包含 $D_{\max}$ 的记忆和逐渐再冻结的扩展。

Figs. 8(a) and (c) show the results when attenuation model B is used with $Γ = 13$ Pa s m⁻¹. The variations are systematic in that increasing the thickness or breaking strain makes the MIZ narrower. For the winter months, MIZ widths estimated from our ice breakage model are about half the widths determined from the AMSR-E-measured concentrations. In the summer, when the wave heights are much lower and the ice is thicker, there is a lot less ice breakage, whereas the concentration criterion defines the MIZ as being much larger than the winter. This could be due to the more dilute ice being able to spread out even further in response to stresses from off-ice winds and currents. The neglected effect of ice advection would thus become more important in this period as well.
图 8(a)和(c)显示了使用衰减模型 B 与 $Γ = 13$ Pasm ⁻¹ 时的结果。变化是系统性的，增加厚度或断裂应变会使 MIZ 变得更窄。在冬季，我们的冰破坏模型估计的 MIZ 宽度约为 AMSR-E 测量浓度确定的宽度的一半。在夏季，当波高明显降低且冰层更厚时，冰破坏显著减少，而浓度标准则将 MIZ 定义为比冬季大得多。这可能是由于更稀薄的冰能够在离岸风和洋流的应力作用下进一步扩展。因此，冰的平流效应在这一时期变得更加重要。

Fig. 8(a) shows the variation of the MIZ width with thickness. There is more variation in the summer when the ice is already very thick. However, in the winter, the variations are much reduced, both in comparison with the summer variability and the idealized results of the previous section.
图 8(a)显示了 MIZ 宽度与厚度的变化。在夏季，当冰层已经非常厚时，变化更为明显。然而，在冬季，与夏季的变化性以及前一节的理想化结果相比，变化大大减少。

Fig. 8(c) shows that the MIZ width responds to variations in the breaking strain in a similar way that it did to thickness variations. Again, there is significantly more variation in the summer, but the winter results are much less sensitive than they would be expected to be from idealized experiments.
图 8(c)显示，MIZ 宽度对破裂应变的变化反应与其对厚度变化的反应类似。同样，夏季的变化显著更多，但冬季的结果比理想化实验所预期的要敏感得多。

Fig. 8(b) shows that the biggest source of variability comes from the choice of viscosity parameter Γ. When $Γ = 0$ , maxima in the winter MIZ widths (reflecting days with strong incident wave fields) often reach about 0.8–1.0 times the AMSR-E widths, but are generally about one half to one third of them. The noise in the curves reflects the day-to-day variations in the incident wave fields. In the summer, all three values of Γ predict very low widths due to the weak incident waves. With Γ increased to 6.5 Pa s m⁻¹, the values of $L_{MIZ}$ drop and become much closer to the widths produced by using $Γ = 13$ Pa s m⁻¹. This behaviour was also observed in the previous section, where results were variable with Γ when low values were used, but were more stable for Γ in the range 6.5–13 Pa s m⁻¹. The default value of 13 Pa s m⁻¹ was chosen to make the attenuation of long waves match the measurements of Squire and Moore, 1980, as they can not be fully explained by present scattering theory. While low frequency measurements can have more noise in them, and more experiments to confirm these attenuation results are necessary, the stability of our results over the correct order of magnitude is encouraging.
图 8(b)显示，变异性的最大来源来自于粘度参数Γ的选择。当 $Γ = 0$ 时，冬季 MIZ 宽度的最大值（反映强烈入射波场的天数）通常达到 AMSR-E 宽度的约 0.8-1.0 倍，但一般约为其一半到三分之一。曲线中的噪声反映了入射波场的日常变化。在夏季，由于入射波较弱，所有三种Γ值预测的宽度都非常低。当Γ增加到 6.5Pasm ⁻¹ 时， $L_{MIZ}$ 的值下降，并且变得更接近使用 $Γ = 13$ Pasm ⁻¹ 所产生的宽度。这种行为在前一节中也有观察到，当使用低值时，结果随着Γ的变化而变化，但在Γ范围为 6.5-13Pasm ⁻¹ 时则更为稳定。选择默认值 13Pasm ⁻¹ 是为了使长波的衰减与 Squire 和 Moore（1980）的测量结果相匹配，因为这些结果无法完全用现有的散射理论解释。虽然低频测量可能会有更多的噪声，并且需要更多的实验来确认这些衰减结果，但我们结果在正确数量级上的稳定性令人鼓舞。

Our results do, however, suggest that a (modeled) floe size criterion and a concentration criterion for the MIZ may give different predictions for its boundary. It is likely that a combination of the two should be used to model the large scale deformations of the ice. This emphasizes the need for more measurements of floe sizes and large scale deformations in the MIZ to determine how the two criteria are related and how they should be used to precisely define the MIZ. It is also highly likely that floe size and concentration are interlinked variables that will need to be consistently related to one another, once incorporated in a sea ice model. The above results also highlight the importance of obtaining a better understanding of the attenuation process and, in particular the lower-than-observed attenuation of long waves than is predicted by scattering theory. It also shows the urgent need for more measurements of attenuation and of more reliable thickness data.
我们的结果确实表明，(模型化的)冰块大小标准和边缘区的浓度标准可能会对其边界给出不同的预测。很可能应该结合这两者来模拟冰的宏观变形。这强调了在边缘区进行更多冰块大小和宏观变形测量的必要性，以确定这两个标准之间的关系以及如何使用它们来精确定义边缘区。冰块大小和浓度很可能是相互关联的变量，一旦纳入海冰模型，就需要始终保持它们之间的一致关系。上述结果还突显了更好理解衰减过程的重要性，特别是长波的衰减低于观察值，而这一点与散射理论的预测不符。这也显示了对衰减和更可靠的厚度数据进行更多测量的迫切需求。

6. Incorporating the WIM into coupled ice-ocean models
6. 将 WIM 纳入耦合冰-海洋模型

The WIM presented in this two-part paper is designed to be integrated into an ice/ocean model (IOM), such as HYCOM. Specifically, by WIM we mean the numerical scheme N1, as this is more easily generalized to two horizontal dimensions. The A0 and A1 schemes were implemented to provide checks for the N1 scheme in different limiting cases and also to provide fast results in this one-dimensional setting. However, it is much more difficult to generalize the semi-analytical schemes to two dimensions.
本文的 WIM 设计用于集成到冰/海洋模型（IOM）中，例如 HYCOM。具体来说，我们所指的 WIM 是数值方案 N1，因为它更容易推广到两个水平维度。A0 和 A1 方案的实施旨在为 N1 方案在不同极限情况下提供检查，并在这一维度设置中提供快速结果。然而，将半解析方案推广到二维则要困难得多。

We envisage the WIM to be a separate module that is called periodically to update the floe size distribution (FSD). The wave model component of the IOM, e.g., WAM or WAVEWATCH III®, will provide the wave forcing boundary condition required in the WIM. The sea ice model component of the IOM, e.g., CICE, will provide the ice conditions for the WIM. We note that the quantities provided by the wave and sea ice models to the WIM are likely to require interpolation onto the high resolution grid it uses. The FSD computed by the WIM will be an input parameter for a number of other parameterizations in the IOM. For example, it can be used to distinguish between the pack ice and the MIZ, and thus to decide which large-scale rheology should be used to determine the ice deformation. The thermodynamic model of Steele (1992), for example, could also be applied to the FSD to allow for lateral melting (or freezing). We also note that the sub-components of the WIM (cf. Section 4 of Part 1) are independent and so, once the skeleton of the WIM is implemented, they can be easily updated whenever new data are obtained or new theoretical progress is made. The implementation of the WIM inside the TOPAZ operational analysis and forecasting system, which is based on the Hybrid Coordinate Ocean Model (HYCOM) and the sea ice model of Drange and Simonsen (1996), is in progress and will be the focus of a future manuscript. The sea ice model of Drange and Simonsen (1996) is similar to CICE.
我们设想 WIM 是一个独立的模块，定期调用以更新浮冰大小分布（FSD）。IOM 的波浪模型组件，例如 WAM 或 WAVEWATCH III®，将提供 WIM 所需的波浪强迫边界条件。IOM 的海冰模型组件，例如 CICE，将为 WIM 提供冰情条件。我们注意到，波浪和海冰模型提供给 WIM 的量可能需要插值到其使用的高分辨率网格上。WIM 计算的 FSD 将作为 IOM 中多个其他参数化的输入参数。例如，它可以用于区分浮冰和边缘冰区（MIZ），从而决定应使用哪种大尺度流变学来确定冰的变形。例如，Steele（1992）的热力学模型也可以应用于 FSD，以考虑横向融化（或冻结）。我们还注意到，WIM 的子组件（参见第 1 部分第 4 节）是独立的，因此，一旦 WIM 的框架实现，它们可以在获得新数据或取得新理论进展时轻松更新。在基于混合坐标海洋模型（HYCOM）和 Drange 与 Simonsen（1996）海冰模型的 TOPAZ 操作分析与预测系统中，WIM 的实施正在进行中，并将成为未来手稿的重点。Drange 与 Simonsen（1996）的海冰模型类似于 CICE。

7. Summary and overall conclusions
7. 摘要和总体结论

In Part 2 of this two-part series, we have developed the theory of wave-ice interactions, presented in Part 1, into a numerical algorithm that predicts the FSD and wave spectrum in the MIZ, given an incident wave spectrum at the ice edge, and ice thickness and surface concentration profiles. Our investigation focused on the predictions of the FSD.
在本系列的第二部分中，我们将第一部分中提出的波浪与冰相互作用的理论发展为一个数值算法，该算法可以预测在冰缘处给定入射波谱、冰厚度和表面浓度剖面的情况下，MIZ 中的 FSD 和波谱。我们的研究重点是 FSD 的预测。

The numerical WIM was outlined for one-dimensional transects, as a step towards the full two-dimensional model. But the restriction was also imposed to facilitate a thorough sensitivity study, with respect to the key numerical and physical parameters in the WIM. This is especially important because of the high degree of uncertainty in many of these quantities. Sensitivity studies were conducted, in the first instance, using idealized ice thickness and concentration profiles. The most substantive observations follow.
数值 WIM 被概述为一维横截面的步骤，以便为完整的二维模型铺平道路。但也施加了限制，以便对 WIM 中的关键数值和物理参数进行全面的灵敏度研究。这一点尤其重要，因为许多这些量存在很高的不确定性。灵敏度研究首先使用理想化的冰厚度和浓度剖面进行。以下是最实质性的观察结果。

1.
Sufficient convergence of the FSD is given by a spatial resolution of approximately 5 km.
FSD 的充分收敛由大约 5 公里的空间分辨率提供。
2.
The waves can be forced to travel through an arbitrary proportion of the ice they break by adjusting the Courant number in the time stepping component of the numerical algorithm. This serves as a proxy for wave energy lost during ice breakage. When the waves only travel through a small proportion of broken ice, small changes in the exact proportion can lead to large changes in the width of the MIZ. But, this sensitivity quickly reduces, and the MIZ width is unaffected by the exact proportion when the waves travel through more than 30% of the broken ice.
通过调整数值算法时间步进组件中的 Courant 数，可以强制波浪穿过它们破坏的任意比例的冰。这作为波浪在冰破坏过程中损失的能量的代理。当波浪仅穿过一小部分破碎冰时，确切比例的微小变化会导致 MIZ 宽度的巨大变化。但是，这种敏感性很快减弱，当波浪穿过超过 30%的破碎冰时，MIZ 宽度不受确切比例的影响。
3.
Neglecting dispersion of the wave spectrum does not affect the FSD predicted by the WIM. Semi-analytical models, which do not incorporate dispersion, were therefore proposed for the two limiting cases of wave energy loss during ice breakage. The semi-analytical models are numerically efficient and helped to validate the full numerical model. However, it was noted that it will be difficult to generalize these models to two-dimensions.
忽略波谱的色散不会影响 WIM 预测的 FSD。因此，提出了不考虑色散的半解析模型，用于冰破裂过程中波能量损失的两种极限情况。半解析模型在数值上高效，并有助于验证完整的数值模型。然而，注意到将这些模型推广到二维将是困难的。
4.
The FSD is highly sensitive to the values of the damping parameter Γ, the ice thickness, and the breaking strain parameter. This emphasizes the need for more measurements of ice thickness, wave attenuation and breaking strains, as well as in situ observations of ice breaking.
FSD 对阻尼参数Γ、冰厚度和破坏应变参数的值非常敏感。这强调了对冰厚度、波衰减和破坏应变的更多测量的必要性，以及对冰破裂的原位观察。

The WIM was also tested using realistic input parameters that represented the Fram Strait in 2007. The sensitivity of the FSD predicted by the WIM to the ice thickness and breaking strain parameter was lower than in the idealized simulations. However, sensitivity to Γ remained high. It is therefore crucial to resolve the problem of how long waves are attenuated theoretically, and also to conduct more experiments to confirm the observations of Squire and Moore (1980) and to extend them to different ice types.
WIM 还使用了代表 2007 年弗拉姆海峡的现实输入参数进行了测试。WIM 预测的 FSD 对冰厚度和破裂应变参数的敏感性低于理想化模拟。然而，对Γ的敏感性仍然很高。因此，解决长波如何在理论上衰减的问题至关重要，并且还需要进行更多实验以确认 Squire 和 Moore（1980）的观察结果，并将其扩展到不同的冰类型。

To conclude, the MIZ widths obtained from the realistic simulations, i.e., the distance of broken ice in the model, were compared to MIZ widths determined from contemporaneous AMSR-E (University of Bremen) concentration data. In winter months, when waves are at their strongest, the MIZ widths predicted by the WIM were roughly half those predicted by the concentration criterion. In the summer, the model results and the concentration results give quite different boundaries to the MIZ. Probably this is partly due to smaller waves and thicker ice in this time period, partly to neglected effects like ice advection and thermodynamic effects, and partly due to the two different definitions of the MIZ. This highlights the need for more measurements of the FSD, and also more research on how to define the MIZ (in our case, for the purpose of determining which large-scale ice rheology to use) more precisely. The model proposed in this two-part series has been motivated by the many observations that suggest a primary role for ocean waves in shaping the morphology of ice fields. Waves habitually limit the size of the constituent ice floes throughout the MIZ, by fracturing those floes that are too large to exist as the waves permeate further into the ice pack. Attenuation, arising due to scattering and supplementary inelastic processes such as turbulence, bending hysteresis and interfloe collisions and rafting, also occurs causing a gradual reduction of the wave energy envelope with distance from the ice edge that, cæteris paribus, results in a gradual increase in floe size with penetration. The FSD is therefore continuously modified by pervasive incident ocean wave trains that, according to their period, may either travel long fetches from distant storms or else be more locally generated. They are then preferentially filtered by the sea ice in a manner that favors the survival of longer wavelengths.
总之，从现实模拟中获得的 MIZ 宽度，即模型中破碎冰的距离，与同时期 AMSR-E（不来梅大学）浓度数据确定的 MIZ 宽度进行了比较。在冬季，当波浪最强时，WIM 预测的 MIZ 宽度大约是浓度标准预测值的一半。在夏季，模型结果和浓度结果给出了截然不同的 MIZ 边界。这可能部分是由于这一时期波浪较小和冰层较厚，部分是由于忽略了冰的平流和热力学效应等因素，部分是由于对 MIZ 的两种不同定义。这突显了对 FSD 进行更多测量的必要性，以及对如何更精确地定义 MIZ（在我们的案例中，目的是确定使用哪种大规模冰流变学）进行更多研究的需求。本系列的模型提案是受到许多观察结果的启发，这些结果表明海洋波浪在塑造冰场形态方面起着主要作用。波浪习惯性地限制了边缘冰区内组成冰块的大小，通过破碎那些过大而无法在波浪进一步渗透冰层时存在的冰块。由于散射和附加的非弹性过程（如湍流、弯曲滞后以及冰块间的碰撞和漂浮），衰减现象也会发生，导致波能包络随着距离冰缘的增加而逐渐减小，这在其他条件相同的情况下，导致冰块大小随着渗透的增加而逐渐增大。因此，浮冰大小分布（FSD）不断受到普遍入射海洋波列的影响，根据其周期，这些波列可能来自远处风暴的长距离传播，或者是更局部生成的。然后，它们会被海冰以一种有利于较长波长存活的方式进行优先过滤。

Given these several influential factors relating to the composition of MIZs, it is perhaps surprising to the reader that wave–ice interactions have not been included in ice/ocean models hitherto. While it has been discussed in the past, the complexity of doing this has proved insuperable until now. We have provided a potential way to do it, and have given some first predictions of how floe sizes and MIZ width are manipulated by waves in a one-dimensional spectral setting. Most importantly, we have established the machinery to deal with the next stage of development, which is to incorporate two-dimensional interactions arising from a directional sea comprising energy at a comb of different frequencies distributed angularly.
考虑到与混合冰区（MIZ）组成相关的几个影响因素，读者可能会感到惊讶的是，波浪与冰的相互作用迄今为止并未被纳入冰/海洋模型中。尽管过去曾对此进行讨论，但实现这一点的复杂性直到现在仍然难以克服。我们提供了一种潜在的方法，并在一维谱设置中给出了一些关于波浪如何影响浮冰大小和混合冰区宽度的初步预测。最重要的是，我们建立了处理下一阶段发展的机制，即将来自方向性海洋的二维相互作用纳入考虑，这种海洋包含以不同频率分布的能量。

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。

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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.
Nonlinear simulation of wave group attenuation due to scattering in broken floe fields
破碎冰块场中波群衰减的非线性模拟由于散射
2023, Ocean Modelling 2023 年，海洋建模
Direct phase-resolved simulations are performed to investigate the propagation and scattering of nonlinear ocean waves in fragmented sea ice. The numerical model solves the full time-dependent equations for nonlinear potential flow coupled with a nonlinear thin-plate representation of the ice cover, and it neglects dissipative processes. The two-dimensional setting with incident wave groups on deep water is considered, in view of applications to wave attenuation along transects of the marginal ice zone. A spatially-varying weight is assigned to the surface pressure so that irregular distributions of ice floes can be directly specified in the physical domain. For various wave regimes and floe configurations, a local wave spectrum across the ice field is computed and then least-squares fitted to extract a spatial attenuation rate as a function of wave frequency. A general increase with frequency is found, consistent with typical predictions from linear theory. However, a non-monotonic behavior around a certain frequency is also observed, especially in cases with a highly fragmented ice cover. The key role played by nonlinear interactions in this phenomenon is highlighted.
Assessment of a porous viscoelastic model for wave attenuation in ice-covered seas
冰盖海域波衰减的多孔粘弹性模型评估
2022, Applied Ocean Research
2022 年，应用海洋研究
Chen et al. (2019) recently proposed a two-dimensional continuum model for linear gravity waves propagating in ice-covered seas. It is based on a two-layer formulation where the ice cover is viewed as a porous viscoelastic medium. In the present paper, extensive tests against both laboratory experiments and field observations are performed to assess this model’s ability at describing wave attenuation in various types of sea ice. The theoretical predictions are fitted to data on attenuation rate via error minimization and numerical solution of the corresponding dispersion relation. Detailed comparison with other existing viscoelastic theories is also presented. Estimates for effective rheological parameters such as shear modulus and kinematic viscosity are obtained from the fits and are found to vary significantly among the models. For this poroelastic system, the range of estimated values turns out to be relatively narrow in orders of magnitude over all the cases considered. Against field measurements from the Arctic Ocean, this model is able to reasonably reproduce the roll-over of attenuation rate as a function of frequency. Given the rather large number of physical parameters in such a formulation, a sensitivity analysis is also conducted to gauge the relevance of a representative set of them to the attenuation process.
ON TRANSITIONS IN WATER WAVE PROPAGATION THROUGH CONSOLIDATED TO BROKEN SEA ICE COVERS
关于水波在固结到破碎海冰覆盖层传播中的过渡
2023, arXiv 2023，arXiv
Summer sea ice floe perimeter density in the Arctic: High-resolution optical satellite imagery and model evaluation
北极夏季海冰浮冰周长密度：高分辨率光学卫星影像与模型评估
2023, Cryosphere 2023，冰冻圈
Aerial observations of sea ice breakup by ship waves
船舶波浪对海冰破裂的空中观察
2023, Cryosphere 2023，冰冻圈

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