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tmain.tex (20080B) |
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1 %\documentclass[11pt,a4paper]{article} |
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2 \documentclass[11pt]{article} |
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3 |
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4 %\usepackage{a4wide} |
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5 \usepackage[margin=1.4in]{geometry} |
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6 |
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7 \usepackage{graphicx} |
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8 %\usepackage[german, english]{babel} |
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9 %\usepackage{tabularx} |
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10 %\usepackage{cancel} |
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11 %\usepackage{multirow} |
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12 %\usepackage{supertabular} |
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13 %\usepackage{algorithmic} |
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14 %\usepackage{algorithm} |
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15 %\usepackage{amsthm} |
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16 %\usepackage{float} |
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17 %\usepackage{subfig} |
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18 \usepackage{rotating} |
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19 \usepackage{amsmath} |
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20 |
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21 \usepackage[T1]{fontenc} % Font encoding |
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22 \usepackage{charter} % Serif body font |
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23 \usepackage[charter]{mathdesign} % Math font |
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24 \usepackage[scale=0.9]{sourcecodepro} % Monospaced fontenc |
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25 \usepackage[lf]{FiraSans} % Sans-serif font |
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26 \usepackage{textgreek} |
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27 |
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28 \usepackage{listings} |
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29 |
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30 \usepackage{hyperref} |
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31 |
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32 \usepackage{soul} % for st strikethrough command |
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33 |
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34 %\usepackage[round]{natbib} |
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35 \usepackage[natbib=true, style=authoryear, bibstyle=authoryear-comp, |
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36 maxbibnames=10, |
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37 maxcitenames=2, backend=bibtex8]{biblatex} |
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38 \bibliography{/Users/ad/articles/own/BIBnew.bib} |
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39 |
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40 |
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41 \begin{document} |
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42 |
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43 \title{Discrete-element simulation of sea-ice mechanics:\\ |
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44 Parameterization of contact mechanics} |
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45 |
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46 \author{\small Anders Damsgaard} |
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47 \date{\small \today} |
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48 |
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49 \maketitle |
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50 |
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51 \section{Rationale} |
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52 Lagrangian parameterizations of sea ice offer several advantages to Eule… |
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53 continuum methods. The continuum assumption of sea-ice mechanics become… |
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54 increasingly questionable as spatial resolution and cell size approaches… |
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55 size of ice floes. Ice-floe scale discretizations are natural for Lagra… |
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56 models, which additionally offer the convenience of being able to handle |
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57 arbitrary sea-ice concentrations. This is likely to improve model perfo… |
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58 in ice-marginal zones with strong advection. Furthermore, phase transit… |
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59 granular rheology around the jamming limit, such as observed when sea ic… |
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60 through geometric confinements, includes sharp thresholds in effective |
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61 viscosity, which are typically ignored in Eulerian models. |
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62 Granular jamming is a stochastic process dependent on having the right g… |
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63 the right place at the right time, and the jamming likelihood over time … |
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64 described by a probabilistic model |
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65 \citep[e.g.][]{Tang2009}. Difficult to parameterize in continuum formul… |
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66 \citep[e.g.][]{Rallabandi2017}, jamming comes natural to dense granular … |
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67 simulated in a Lagrangian framework, and is a very relevant process comp… |
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68 sea-ice transport through narrow straits \citep[e.g.][]{Kwok2010}. |
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69 However, computational performance is a central concern for coupled |
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70 ocean-atmosphere models, with traditional discrete-element method (DEM) |
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71 formulations being too costly for global-scale models. Here we investig… |
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72 behavior of a Lagrangian sea-ice model and assess how mechanical |
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73 parameterizations of differing complexity affect collective behavior at … |
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74 much larger than the individual ice floes. |
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75 |
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76 |
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77 \section{Contact models} |
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78 The presented experiments compare jamming behavior between two differing |
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79 ice-floe contact models. Common to both models, the resistive force |
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80 $\boldsymbol{f}_\text{n}$ to axial compressive strain between to cylindr… |
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81 floes $i$ and $j$ is provided by (Hookean) linear-elasticity based on th… |
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82 overlap distance $\boldsymbol{\delta}_\text{n}$: |
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83 \begin{equation} |
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84 \boldsymbol{f}_\text{n}^{ij} = |
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85 A^{ij} E^{ij} \boldsymbol{\delta}_\text{n}^{ij} |
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86 \quad \text{when} \quad |
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87 0 > |\boldsymbol{\delta}_\text{n}^{ij}| \equiv |\boldsymbol{x}^i - |
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88 \boldsymbol{x}^j| - (r^i + r^j) |
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89 \label{eq:f_n} |
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90 \end{equation} |
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91 where the contact cross-sectional area $A^{ij} = R^{ij} \min(T^i, T^j)$ … |
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92 determined by the harmonic mean $R^{ij} = 2 r^i r^j/(r^i + r^j)$ of the … |
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93 radii $r^i$ and $r^j$, as well as the smallest of the involved ice-floe |
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94 thickness $T^i$ and $T^j$. The linear elastic force resulting from axia… |
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95 of a distance $|\boldsymbol{\delta}_\text{n}^{ij}|$ of the contact is sc… |
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96 the harmonic mean of Young's modulus $E^{ij}$. The stiffness scaling ba… |
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97 Young's modulus is scale invariant \citep[e.g.][]{Obermayr2013}, and is … |
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98 the principle that the elastic stiffness of the ice itself is constant, |
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99 regardless of ice-floe size. |
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100 |
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101 Nonlinear elasticity models based on Hertzian contact mechanics may |
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102 alternatively be applied to determine the stresses resulting from contac… |
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103 compression \citep[e.g.][]{Herman2013}. However, with nonlinear models … |
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104 numerical stability of the explicit temporal integration scheme will dep… |
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105 the stress and packing state of the granular assemblage, and will under |
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106 localized compressive extremes require very small discretization in time. |
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107 |
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108 As demonstrated later, models based compressive strength alone are not |
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109 sufficient to cause granular jamming. We explore two different addition… |
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110 contact model presented in Eq.~\ref{eq:f_n}. The first approach is typi… |
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111 DEM models and is based on resolving including shear resistance through |
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112 tangential (contact parallel) elasticity and associated Coulomb friction… |
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113 limits. An alternative approach, fundamentally complementary to compres… |
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114 elasticity and shear friction, is tensile strength of ice-floe contacts … |
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115 leads to a cohesive bulk granular rheology. |
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116 |
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117 \subsection{Contact-parallel elasticity with Coulomb friction} |
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118 The contact-parallel (tangential) elasticity is resolved by determining … |
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119 contact travel distance $\boldsymbol{\delta}_\text{t}$ on the contact pl… |
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120 the duration of the contact $t_\text{c}$: |
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121 \begin{equation} |
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122 \boldsymbol{\delta}_\text{t} = \int_0^{t_\text{c}} |
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123 \left[ |
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124 (\boldsymbol{v}^i - \boldsymbol{v}^j) \cdot \hat{\boldsymbol{t}}… |
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125 R^{ij} \left(\omega^i + \omega^j\right) |
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126 \right] |
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127 \label{eq:d_t} |
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128 \end{equation} |
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129 where $\boldsymbol{v}$ and $\omega$ denotes linear and rotational veloci… |
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130 respectively. |
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131 The deformation distance $\boldsymbol{\delta}_\text{t}$ is corrected for… |
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132 rotation over the lifetime of the interaction, and is used to determine … |
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133 contact-tangential elastic force: |
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134 \begin{equation} |
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135 \boldsymbol{f}_\text{t} = \frac{E^{ij} A^{ij}}{R^{ij}} |
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136 \frac{2 (1 - \nu^2)}{(2 - \nu)(1 + \nu)} \boldsymbol{\delta}_\text{t} |
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137 \end{equation} |
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138 with $\nu$ being the dimensionless Poisson's ratio characteristic for th… |
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139 material. However, the frictional force above is restricted to the |
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140 Coulomb-frictional limit, relating the magnitude of the normal force to … |
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141 magnitude of the tangential force: |
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142 \begin{equation} |
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143 |\boldsymbol{f}_\text{t}| \leq \mu |\boldsymbol{f}_\text{n}| |
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144 \label{eq:coulomb_friction} |
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145 \end{equation} |
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146 The Coulomb-frictional coefficient $\mu$ may be parameterized as state |
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147 dependent, for example with different values for sliding and stationary |
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148 contacts. However, for the following a single and constant value is use… |
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149 the frictional coefficient $\mu$. |
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150 |
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151 In the case of slip ($|\boldsymbol{f}_\text{t}| > \mu |
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152 |\boldsymbol{f}_\text{n}|$) the length of the contact-parallel travel pa… |
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153 $\boldsymbol{\delta}_\text{t}$ is reduced to be consistent to be consist… |
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154 the Coulomb limit. This accounts for the tangential contact plasticity … |
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155 irreversible work associated with sliding. |
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156 |
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157 Since the above model of tangential shear resistance is based on deforma… |
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158 distance on the inter-floe contact plane, it requires solving for ice-fl… |
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159 rotational kinematics of each ice floe and a bookkeeping algorithm for s… |
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160 contact histories. |
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161 |
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162 \subsection{Tensile contact strength} |
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163 Cohesion is introduced by parameterizing resistance to extension beyond … |
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164 overlap distance between a pair of ice floes (i.e.\ $\delta_\text{n}^{ij… |
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165 In real settings tensile strength may be provided by refreezing processe… |
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166 ice-floe interface or mechanical ridging. An ideal formulation of bond |
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167 deformation includes resistance to bond tension, shear, twist, and rolli… |
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168 \citep[e.g.][]{Potyondy2004, Obermayr2013, Herman2016}. However, for th… |
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169 we explore the possibility of using bond \emph{tension} alone as a mecha… |
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170 component contributing to bulk granular shear strength. |
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171 |
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172 Tensile strength is parameterized by applying Eq.~\ref{eq:f_n} for the e… |
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173 regime ($\delta_\text{n} > 0$). Eq.~\ref{eq:f_n} is applied until the t… |
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174 stress exceeds the tensile strength $\sigma_\text{c}$ defined for the bo… |
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175 The implementation allows for time-dependent tensile strengthening based… |
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176 contact age, but for the following we parameterize the bonds to obtain f… |
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177 tensile strength as soon as a pair of ice floes first undergo compressio… |
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178 ($\delta_\text{n} < 0$). |
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179 |
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180 |
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181 \subsection{Implementation} |
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182 The intent of this work is to develop and apply an offline sea-ice dynam… |
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183 code in order to explore strengths and limitations of different methods … |
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184 to sea-ice mechanics. Once the appropriate parameterization has been id… |
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185 it will be built in to the GFDL model of coupled ocean/atmosphere dynami… |
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186 |
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187 The presented experiments are performed with the |
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188 \texttt{SeaIce.jl}\footnote{\url{https://github.com/anders-dc/SeaIce.jl}… |
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189 package for offline simulations of sea-ice mechanics, which includes the… |
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190 parameterizations. Furthermore, the model includes the option to includ… |
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191 (Newtonian) contact viscosity in the normal and tangential components, b… |
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192 analysis of these is not included here. Simulation-specific scripts are |
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193 provided in the repository |
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194 \texttt{SeaIce-experiments}\footnote{\url{https://github.com/anders-dc/S… |
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195 |
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196 The offline model implementation identifies ice-floe contacts and placem… |
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197 the ocean/atmosphere grid through a cell-based spatial discretization, w… |
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198 reduces the computational overhead significantly (Fig.~\ref{fig:performa… |
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199 and additionally mirrors the current Lagrangian ice-berg implementation … |
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200 GFDL ocean model. |
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201 |
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202 \begin{figure} |
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203 \begin{center} |
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204 \includegraphics[width=0.7\textwidth]{graphics/profiling-cpu.pdf} |
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205 \end{center} |
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206 \caption{\label{fig:performance} Single-threaded performance of the … |
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207 algorithm with all-to-all contact search, or a contact search using … |
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208 decomposition based on the ocean/atmosphere grid (Profiling script: |
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209 \url{https://github.com/anders-dc/SeaIce.jl/blob/master/test/profili… |
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210 \end{figure} |
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211 |
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212 The experiments rely on pseudo-random number generation for generating i… |
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213 particle size distributions (PSDs). In order to assess the statistical |
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214 probability of granular jamming we seed the system with different values… |
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215 repeat each experiment ten times. |
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216 |
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217 For the presented runs the ice floes are forced with a uniform and const… |
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218 field oriented from north to south, with a simplified channel-like const… |
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219 in the middle. The ocean also flows from north to south but is constrai… |
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220 the basin geometry. The spatial velocity pattern is determined by a str… |
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221 function under mass conservation, meaning that currents increase as |
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222 the channel narrows. Ice floes are initially placed in a pseudo-random |
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223 arrangement north of the channel. All parameters are kept constant betw… |
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224 experiment sets unless explicitly noted. |
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225 |
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226 \section{Results} |
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227 |
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228 \subsection{Experiment set 1} |
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229 |
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230 Increasing frictional coefficients, increasing tensile strength. Consta… |
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231 uniform size distribution. |
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232 Generated by running \texttt{`make`} from |
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233 \url{https://github.com/anders-dc/SeaIce-experiments/tree/master/cohesio… |
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234 Animation with \texttt{seed=1} can be found at: |
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235 \url{https://youtu.be/JmWvoi9Z6Zk} |
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236 |
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237 \begin{sidewaysfigure} |
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238 \begin{center} |
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239 \includegraphics[width=0.99\textheight]% |
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240 {graphics/cohesion-friction-comparison.png} |
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241 \includegraphics[width=0.99\textheight]% |
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242 {{graphics/cohesion-friction-comparison-jam_fraction.png}} |
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243 \end{center} |
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244 \caption{\label{fig:set1}% |
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245 \textbf{Experiment set 1}: |
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246 Comparison between the frictional and cohesive model with different |
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247 Coulomb-frictional coefficient values and tensile strengths. |
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248 First row: Ice fluxes over time with increasing friction ($\mu = [0.… |
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249 0.2, 0.3, 0.4]$). |
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250 Second row: Ice fluxes over time with increasing tensile strength |
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251 ($\sigma_\text{c} = [10, 100, 200, 400, 800]$ kPa). |
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252 Third row: Fraction of runs jammed over time with increasing frictio… |
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253 Fourth row: Fraction of runs jammed over time with increasing tensil… |
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254 strength. |
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255 } |
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256 \end{sidewaysfigure} |
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257 |
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258 While the frictionless runs rarely jam, we observe that increasing eithe… |
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259 friction or cohesion greatly increases the likelihood of jamming |
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260 (Fig.~\ref{fig:set1}). Under cohesive runs with large tensile strengths… |
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261 floes north of the assemblage behave like a single rigid block, while th… |
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262 corresponding frictional runs show more spread in the jam-fraction proba… |
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263 At intermediate to low frictional coefficients ($\mu \leq 0.3$ or |
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264 $\sigma_\text{c} \leq$ 200 kPa) there is good agreement between friction… |
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265 cohesive models, where the runs with $\mu = 0.3$ most closely resemble t… |
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266 jam-fraction distribution of $\sigma_\text{c} = 200$ kPa. We base furth… |
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267 on comparing the differences between these two parameter sets under vary… |
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268 circumstances. |
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269 |
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270 \subsection{Experiment set 2} |
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271 Constant friction coefficient ($\mu = 0.3$) for frictional runs, constan… |
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272 tensile strength ($\sigma_\text{c} = 200$ kPa) for cohesive runs. Varia… |
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273 channel width. Monodisperse ice-floe size distribution. |
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274 Generated by running \texttt{`make`} from |
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275 \url{https://github.com/anders-dc/SeaIce-experiments/tree/master/cohesio… |
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276 Animation with \texttt{seed=1} can be found at: |
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277 \url{https://youtu.be/tPMSxdw1UZ8}. |
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278 |
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279 \begin{sidewaysfigure} |
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280 \begin{center} |
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281 \includegraphics[width=0.99\textheight]% |
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282 {graphics/cohesion-friction-width-monodisperse-comparison.png} |
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283 \includegraphics[width=0.99\textheight]% |
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284 {{graphics/cohesion-friction-width-monodisperse-comparison-jam_f… |
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285 \end{center} |
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286 \caption{\label{fig:set2}% |
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287 \textbf{Experiment set 2}: |
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288 Comparison between the frictional ($\mu = 0.3$ and $\sigma_\text{c}$… |
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289 kPa) and cohesive ($\mu = 0$ and $\sigma_\text{c}$ = 200 kPa) models… |
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290 monodisperse (single) ice-floe size and increasing channel width. |
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291 First row: Ice fluxes over time with friction ($\mu = 0.3$) and incr… |
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292 channel width. |
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293 Second row: Ice fluxes over time with cohesion ($\sigma_\text{c} = 2… |
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294 and increasing channel width. |
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295 Third row: Fraction of frictional runs jammed over time. |
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296 Fourth row: Fraction of cohesive runs jammed over time. |
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297 } |
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298 \end{sidewaysfigure} |
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299 |
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300 The jamming probability of two-dimensional granular assemblages under gr… |
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301 well described in the literature, especially for monodisperse (same-size… |
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302 granular materials \citep[e.g.][]{Beverloo1961, To2001, Tang2009}. |
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303 While the ocean/atmosphere forcing is different from the gravity drag in… |
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304 silo flows in these publications, we observe the same relationship betwe… |
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305 conduit width and grain size in monodisperse granular assemblages as pre… |
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306 reported. The consistency between the frictional and cohesive model |
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307 (Fig.~\ref{fig:set2}) indicates that the simpler cohesive model is perfo… |
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308 well in this setting. |
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309 |
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310 \subsection{Experiment set 3} |
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311 Constant friction coefficient ($\mu = 0.3$) for frictional runs, constan… |
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312 tensile strength ($\sigma_\text{c} = 200$ kPa) for cohesive runs. Varia… |
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313 channel width. Constant and uniformly-distributed ice-floe size distrib… |
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314 Generated by running \texttt{`make`} from |
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315 \url{https://github.com/anders-dc/SeaIce-experiments/tree/master/cohesio… |
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316 |
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317 \begin{sidewaysfigure} |
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318 \begin{center} |
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319 \includegraphics[width=0.99\textheight]% |
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320 {graphics/cohesion-friction-width-comparison.png} |
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321 \includegraphics[width=0.99\textheight]% |
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322 {{graphics/cohesion-friction-width-comparison-jam_fraction.png}} |
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323 \end{center} |
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324 \caption{\label{fig:set3}% |
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325 \textbf{Experiment set 3}: |
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326 Comparison between the frictional ($\mu = 0.3$ and $\sigma_\text{c}$… |
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327 kPa) and cohesive ($\mu = 0$ and $\sigma_\text{c}$ = 200 kPa) models… |
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328 uniform ice-floe size distribution and increasing channel width. |
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329 First row: Ice fluxes over time with friction ($\mu = 0.3$) and incr… |
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330 channel width. |
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331 Second row: Ice fluxes over time with cohesion ($\sigma_\text{c} = 2… |
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332 and increasing channel width. |
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333 Third row: Fraction of frictional runs jammed over time. |
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334 Fourth row: Fraction of cohesive runs jammed over time. |
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335 } |
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336 \end{sidewaysfigure} |
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337 |
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338 While the monodisperse ice-floe size used in Set 2 provides a comparativ… |
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339 to the literature on granular jamming, we have for Set 3 used a wide gra… |
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340 distribution and varied the channel width as before. We observe that th… |
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341 of ice-floe size distribution decreases the likelihood of jamming, and t… |
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342 these runs are less likely to jam than the corresponding monodisperse en… |
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343 runs. Again we see a good correspondence between the frictional and coh… |
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344 model runs (Fig.~\ref{fig:set3}). |
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345 |
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346 \subsection{Experiment set 4} |
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347 Constant friction coefficient ($\mu = 0.3$) for frictional runs, constan… |
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348 tensile strength ($\sigma_\text{c} = 200$ kPa) for cohesive runs. Const… |
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349 channel width. Different widths in power-law distributed ice-floe size |
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350 distribution. |
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351 Generated by running \texttt{`make`} from |
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352 \url{https://github.com/anders-dc/SeaIce-experiments/tree/master/cohesio… |
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353 Animation with \texttt{seed=1} can be found at: |
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354 \url{https://youtu.be/I9P8XsA1S0I} |
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355 |
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356 \begin{sidewaysfigure} |
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357 \begin{center} |
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358 \includegraphics[width=0.99\textheight]% |
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359 {graphics/cohesion-friction-psd-comparison.png} |
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360 \includegraphics[width=0.99\textheight]% |
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361 {{graphics/cohesion-friction-psd-comparison-jam_fraction.png}} |
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362 \end{center} |
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363 \caption{\label{fig:set4}% |
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364 \textbf{Experiment set 4}: |
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365 Comparison between the frictional ($\mu = 0.3$ and $\sigma_\text{c}$… |
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366 kPa) and cohesive ($\mu = 0$ and $\sigma_\text{c}$ = 200 kPa) models… |
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367 increasingly wide power-law ice-floe size distribution and constant … |
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368 width. |
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369 First row: Ice fluxes over time with friction ($\mu = 0.3$) and incr… |
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370 ice-floe size span. |
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371 Second row: Ice fluxes over time with cohesion ($\sigma_\text{c} = 2… |
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372 and increasing ice-floe size span. |
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373 Third row: Fraction of frictional runs jammed over time. |
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374 Fourth row: Fraction of cohesive runs jammed over time. |
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375 } |
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376 \end{sidewaysfigure} |
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377 |
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378 Ice-floe distributions in natural sea-ice settings have been observed to… |
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379 a power-law probability distribution with an exponent of $\alpha \approx… |
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380 \citep[e.g.][]{Steer2008, Herman2013}. With constant channel size and a… |
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381 increasing width of the ice-floe size distribution, we observe that the |
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382 probability of jamming decreases, and observe good agreement between the |
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383 frictional and cohesive model runs. |
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384 |
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385 \section{Summary} |
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386 We construct a flexible discrete-element framework for simulating Lagran… |
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387 sea-ice dynamics at the ice-floe scale, forced by ocean and atmosphere v… |
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388 fields. While frictionless contact models based on tensile stiffness al… |
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389 very unlikely to jam, we describe two different approaches based on fric… |
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390 tensile strength, both resulting in increased bulk shear strength of the |
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391 granular assemblage. We demonstrate that the discrete-element approach … |
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392 to undergo dynamic jamming when forced through an idealized confinement,… |
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393 the probability of jamming is determined by the channel width, ice-floe … |
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394 and ice-floe size variability. The frictionless but cohesive contact mo… |
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395 with certain tensile strength values display jamming behavior which on t… |
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396 scale is highly similar to a model with contact friction and ice-floe ro… |
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397 The frictionless and cohesive model is likely an ideal candidate for cou… |
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398 computations due to its reduced algorithmic complexity. |
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399 |
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400 \printbibliography{} |
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401 |
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402 \end{document} |
