# Modeling Dry-Snow Densification without Abrupt Transition

## Abstract

**:**

## 1. Introduction

#### 1.1. Time-Varying Conditions

#### 1.2. Stage 1 and Stage 2 Densification

## 2. Transition Model

#### 2.1. Calibration and Validation

#### 2.2. Strain-Rate Profiles

#### 2.3. Density Profiles

## 3. Discussion

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\overline{a}$ | mean annual accumulation rate, m w.e. a${}^{-1}$ or kg m${}^{-2}$ a${}^{-1}$ |

$\mathrm{A}$ | constant in activation equation, a${}^{2}$ |

$\mathrm{B}$ | constant in activation equation, kg m${}^{-3}$ |

c | density-corrected volumetric strain rate, a${}^{-1}$ |

$\mathrm{C}$ | constant in activation equation, kg m${}^{-3}$ |

$\mathrm{D}$ | constant in activation equation, a${}^{-1}$ |

E | activation energy, J mol${}^{-1}$ |

${f}_{0}$ | parameter in Simonsen model |

${f}_{1}$ | parameter in Simonsen model |

F | average density-corrected volumetric strain rate, a${}^{-1}$ |

I | integral |

k | local vertical densification rate, m w.e.${}^{-1}$ |

${k}^{*}$ | global vertical densification rate, m w.e.${}^{-1}$ |

$\mathrm{M}$ | constant in transition model equation |

q | water equivalent height, m w.e. |

Q | mass of section of profile, m w.e. |

$\mathrm{R}$ | gas constant, 8.314 J mol${}^{-1}$ K${}^{-1}$ |

t | time, a |

T | temperature, K |

${T}_{m}$ | mean annual temperature, K |

w | vertical velocity, m a${}^{-1}$ |

X | scaled density, kg m${}^{-3}$ |

z | vertical co-ordinate, m |

Z | length of section of profile, m |

$\Delta t$ | time between measurements at a given site, a |

$\dot{\epsilon}$ | volumetric strain rate, a${}^{-1}$ |

${\dot{\epsilon}}_{H}$ | horizontal velocity divergence, a${}^{-1}$ |

${\dot{\epsilon}}_{zz}$ | vertical strain rate, a${}^{-1}$ |

$\rho $ | density, kg m${}^{-3}$ |

${\rho}_{i}$ | density of ice, 917 kg m${}^{-3}$ |

${\rho}_{T}$ | transition density, kg m${}^{-3}$ |

${\rho}_{0}$ | vertically-smoothed density, kg m${}^{-3}$ |

$\sigma $ | stress, Pa |

$\tau $ | time since deposition of snow, a |

$\Psi $ | cost function |

## Appendix A. Analytical Solution for Depth as a Function of Density

## Appendix B. Analytical Solution for Depth-Integrated Porosity as a Function of Density

## References

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**Figure 1.**Vertical densification rates for (

**a**) Stage 1 and (

**b**) Stage 2 densification. Black lines show the Herron and Langway values (solid lines) with their uncertainties (dashed lines). Colored lines show the Ligtenberg values for mean annual temperatures of 230 (blue), 240 (cyan), 250 (green), 260 (magenta), and 270 K (red), with solid lines for Antarctic sites and dashed lines for Greenland sites.

**Figure 2.**Mean annual temperature and accumulation at sites for which strain rate data are available (in Greenland [16] and Antarctica [8]), where high-resolution (gamma-ray) core-density profiles are available [7], and where the cores used to calibrate the Herron and Langway densificaton equation were collected [6].

**Figure 3.**Density and volumetric strain-rate profiles from Site 21 along the iSTAR traverse. (

**a**) Density profiles from the austral summers of 2013/14 (red curve) and 2014/15 (green curve), and a fitted polynomial curve showing ${\rho}_{0}$ for 2013/14. (

**b**) Volumetric strain rate at intervals of 1 cm w.e. (red curve) and smoothed over ≈3 cm w.e. (blue curve).

**Figure 4.**Measured values of F as a function of density for iSTAR Site 21 (red dots), values of c for Stage 1 and Stage 2 from the Herron and Langway model (cyan lines) and c as a function of density from the transition model with ${\rho}_{T}$ = 590 kg m${}^{-3}$, M = 2.8 (black line).

**Figure 5.**Measured values of F as a function of density for iSTAR Site 21 (red dots), values of c for Stage 1 and Stage 2 from the Herron and Langway model (cyan lines), and c as a function of density from the transition model with ${\rho}_{T}$ = 580 kg m${}^{-3}$, M = 7 (black line).

**Figure 6.**Profiles of ln($\rho /({\rho}_{i}-\rho ))$ for iSTAR Site 21. Measured values (green), modelled values using Herron and Langway Stage 1 (red) and Stage 2 (blue) densification rates, and modelled values using the transition model with ${\rho}_{T}$ = 580 kg m${}^{-3}$, M = 7 (black line).

**Figure 7.**Profiles of ln($\rho /({\rho}_{i}-\rho ))$ for iSTAR Site 4. Measured values from ice cores (magenta dots) and neutron-probe profiles (green), modelled values using Herron and Langway Stage 1 (red) and Stage 2 (blue) densification rates and modelled values using the transition model with ${\rho}_{T}$ = 580 kg m${}^{-3}$, M = 7 (black line).

**Figure 8.**Profiles of ln($\rho /({\rho}_{i}-\rho ))$ for ice core B39. Measured values (magenta line), modelled values using Herron and Langway Stage 1 (red) and Stage 2 (blue) densification rates, and modelled values using the transition model with ${\rho}_{T}$ = 580 kg m${}^{-3}$, M = 7 (black line).

**Figure 9.**Profiles of ln($\rho /({\rho}_{i}-\rho ))$ for ice core B26. Measured values (magenta line), modelled values using Herron and Langway Stage 1 (red) and Stage 2 (blue) densification rates, and modelled values using the transition model with ${\rho}_{T}$ = 530 kg m${}^{-3}$, M = 7 (black line).

**Figure 10.**Variation of steady-state BCO depth with $\overline{a}$ using the Herron and Langway (brown line), Ligtenberg (green line), Simonsen (pink line), and transition (blue line) models. Simonsen and Ligtenberg results are taken from [1].

**Figure 11.**Variation of steady-state DIP with $\overline{a}$ using the Herron and Langway (brown line), Ligtenberg (green line), Simonsen (pink line) and transition (blue line) models. Simonsen and Ligtenberg results are taken from [1].

**Table 1.**Cost functions over the range of 500–600 kg m${}^{-3}$ for the Herron and Langway, Ligtenberg, and transition models for sites in the Pine Island Glacier basin.

Site | $\overline{\mathit{a}}$ | Herron and Langway | Ligtenberg | Transition |
---|---|---|---|---|

m w.e. a${}^{-1}$ | $\mathbf{\Psi}$ | $\mathbf{\Psi}$ | $\mathbf{\Psi}$ | |

1 | 0.35 | 0.202 | 0.185 | 0.097 |

2 | 0.34 | 0.186 | 0.148 | 0.036 |

3 | 0.43 | 0.154 | 0.098 | 0.053 |

4 | 0.58 | 0.212 | 0.162 | 0.037 |

5 | 0.45 | 0.237 | 0.217 | 0.094 |

6 | 0.45 | 0.125 | 0.076 | 0.072 |

7 | 0.33 | 0.275 | 0.250 | 0.204 |

8 | 0.32 | 0.169 | 0.133 | 0.094 |

9 | 0.37 | 0.149 | 0.125 | 0.064 |

10 | 0.23 | 0.167 | 0.083 | 0.128 |

11 | 0.23 | 0.135 | 0.060 | 0.086 |

12 | 0.28 | 0.209 | 0.155 | 0.134 |

13 | 0.43 | 0.164 | 0.129 | 0.030 |

14 | 0.47 | 0.186 | 0.162 | 0.035 |

15 | 0.80 | 0.202 | 0.284 | 0.105 |

16 | 0.51 | 0.143 | 0.110 | 0.045 |

17 | 0.52 | 0.192 | 0.140 | 0.013 |

18 | 0.69 | 0.164 | 0.163 | 0.132 |

19 | 0.69 | 0.258 | 0.219 | 0.019 |

20 | 0.64 | 0.195 | 0.156 | 0.042 |

21 | 0.75 | 0.214 | 0.181 | 0.042 |

22 | 0.78 | 0.198 | 0.170 | 0.041 |

**Table 2.**Cost functions over the range 500–800 kg m${}^{-3}$ for the Herron and Langway, Ligtenberg, and transition models for sites in the Pine Island Glacier basin.

Site | $\overline{\mathit{a}}$ | Herron and Langway | Ligtenberg | Transition |
---|---|---|---|---|

m w.e. a${}^{-1}$ | $\mathbf{\Psi}$ | $\mathbf{\Psi}$ | $\mathbf{\Psi}$ | |

1 | 0.35 | 0.264 | 0.145 | 0.097 |

4 | 0.58 | 0.228 | 0.092 | 0.088 |

6 | 0.45 | 0.205 | 0.106 | 0.093 |

7 | 0.33 | 0.291 | 0.287 | 0.270 |

8 | 0.32 | 0.182 | 0.114 | 0.093 |

10 | 0.23 | 0.168 | 0.116 | 0.147 |

15 | 0.80 | 0.293 | 0.186 | 0.155 |

18 | 0.69 | 0.222 | 0.137 | 0.103 |

20 | 0.64 | 0.297 | 0.184 | 0.072 |

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**MDPI and ACS Style**

Morris, E.
Modeling Dry-Snow Densification without Abrupt Transition. *Geosciences* **2018**, *8*, 464.
https://doi.org/10.3390/geosciences8120464

**AMA Style**

Morris E.
Modeling Dry-Snow Densification without Abrupt Transition. *Geosciences*. 2018; 8(12):464.
https://doi.org/10.3390/geosciences8120464

**Chicago/Turabian Style**

Morris, Elizabeth.
2018. "Modeling Dry-Snow Densification without Abrupt Transition" *Geosciences* 8, no. 12: 464.
https://doi.org/10.3390/geosciences8120464