### and experimental measurements of the thermal conductivity of snow

### N. Calonne

^{1}

### , F. Flin

^{1}

### , S. Morin

^{1}

### , B. Lesaffre

^{1}

### , S. Rolland du Roscoat

^{2}

### and C. Geindreau

^{2}

### .

1

### Meteo-France - CNRS, CNRM-GAME URA 1357, CEN, Grenoble, France

2

### 3S-R CNRS UMR 5521, Universite Joseph Fourier - Grenoble INP, Grenoble, France

**1 mm**

## T

##

## **k** **F**

### Heat transfer and thermal conductivity

### Fourier’s law:

### Heat transfer:

### - Conduction in ice - Conduction in air

### - Phase change effects

### - Air convection in the pores - Water vapor diffusion

### Effective thermal conductivity: k

_{eff}### - from computations - only conduction

### ** Apparent thermal conductivity: k**

^{*}

### - from measurements - all processes

### - scalar (isotropic medium or 1D) where k is a tensor, in W m

^{-1 }

### K

^{-1}

### Thermal conductivity k

**F** _{z}

_{z}

**Atmosphere, T**

_{1}###

^{z}### T

**Ground, T**

_{2}### Thermal conductivity in snowpack models

### Yen, 1981

### Sturm et al., 1997

### Experimental fits Experimental data

### k

^{*}

### = f ( density )

### Accepted idea:

### large scatter

### Issues and contributions

**Need**

### - Re-assessment of the variables related to conductivity

### - Partitioning between physical phenomena involved in heat transfer

### - Assessment of the experimental methods

**Our study**

### Numerical simulations of conductivity from 3D snow microstructure provided by tomography (e.g. Kaempfer et al. 2005).

### Innovations:

### - conduction in ice and air was simulated - access to the full tensor of k

_{eff}### - on a wide range of seasonal snow types (30 samples)

5

### Examples of snow samples (out of 30)

**Precipitation** **Particles**

**Decomposing** **and **

**Fragmented** **precipitation** **particles**

**Rounded**

**Grains** **Depth**

**Hoar**

**1 mm** **1 mm**

**1 mm**

**Melt** **Forms**

**1 mm** **1 mm**

### Cold-room experiment

**Estimation of conductivity by 3 different methods:**

### - Needle probe - Flux-gradient

### - Computation from 3D images

**Insulating polystyrene plate**
**Copper plate**

**Snow slab** **Thermal regulation of the copper plates by a fluid circulation**

**T = - 1 °C**
**T = - 7 °C**

**|| T|| = 43 K m**^{-1}**1 m**

**14 cm**

### Needle probe method

**Sturm et al. 1997, Morin et al. **

**2010, Riche and Schneebeli 2010.**

### - Monitors the temperature rise during the heating time.

### - In transient mode

### - Includes all thermal processes - Assumes a homogeneous and isotropic medium

### k

^{*}

_{needle}

^{(scalar)}

### Flux-gradient method

### Heat-flux sensor Temperature probe

### - In steady state condition

### - Includes all thermal processes - Provides vertical component of k

^{*}

### k

^{*}

_{z}

### (scalar)

### ΔT

### F Δz

### k

^{z}

^{*} ^{} ^{}

^{z}

**Sturm et al. 1997, Morin et al. **

**2010, Riche and Schneebeli 2010.**

**Schneebeli and Sokratov 2004, **
**Satyawali et al. 2008.**

### Needle probe method

### - Monitors the temperature rise during the heating time.

### - In transient mode

### - Includes all thermal processes - Assumes a homogeneous and isotropic medium

### k

^{*}

_{needle}

^{(scalar)}

**1.6 cm**
**5.35 cm**

### 1. Sampling 4. Machining

### 3. Freezing 2. Impregnation

### Numerical computation - Acquisition of 3D images

### 5. X-ray tomography 6. Image processing

**1 mm**

### - Image size: 2.5 - 9.5 mm

### - Voxel size: 5 - 10 µm

### L

### Snow layer

### Macroscopic scale Microscopic scale

### T

α###

###

###

_{α}**α**

**k**

**F**

### T

###

### **k**

_{eff}**F**

### ice) air, ( *α*

### Condition

### Separation of scale L >> l

### l

### Numerical computation - Homogenisation

### Auriault et al. 2009

**Periodic homogenisation**

**k**

_{eff}### (micros., k

_{air}

### , k

_{ice}

### )

###

###

###

###

###

###

###

###

###

### zz zy

### zx

### yz yy

### yx

### xz xy

### xx

### k k

### k

### k k

### k

### k k

### k

**k**

_{eff}### Solving a specific boundary problem arising from the homogenisation process, on a 3D image, by Geodict software.

### Numerical computation - Homogenisation

### L

### Snow layer

### Macroscopic scale Microscopic scale

### T

α###

###

###

_{α}**α**

**k**

**F**

### T

###

### **k**

_{eff}**F**

### ice) air, ( *α*

### l

### Auriault et al. 2009

**Periodic homogenisation**

**k**

_{eff}### (micros., k

_{air}

### , k

_{ice}

### )

**0,1**
**0,2**
**0,3**
**0,4**
**0,5**
**0,6**
**0,7**

**0** **1** **2** **3** **4** **5** **6**

**Size (mm)**
**k****e****ff**** (W m****-1** ** K****-1** **)**

**kxky**
**kz**

**0**
**0,05**
**0,1**
**0,15**
**0,2**

**0** **1** **2** **3** **4**

**Size (mm)**
**k****ef****f**** (W m****-1** ** K****-1** **)**

**kx**
**ky**
**kz**

### REV

### Representative Elementary Volume (REV) for k

_{eff}### Precipitation Particles Depth Hoar

### REV

### • Smallest volume from which a variable representative of the whole can be estimated

### • Computation of k

_{eff}### on volume at least equal to the REV

### • REV estimated by calculating k

_{eff}### on increasing volumes

### Results: k

_{eff }### versus density

### • Good correlation between k

_{eff}### and

### density.

### • Anisotropy of k

_{eff }### = vertical component horizontal component

### Anisotropy range:

### 0.7 – 1.5

### Results: k

_{eff }### versus density

### • Anisotropy is linked to snow microstructure:

### - horizontal: some natural snow (depending on the type of snowfall, settling…) .

### - vertical: snow exposed to a vertical temperature gradient.

### Results: k

_{eff }### versus density

### Snow

### type Density

(kg m^{-3})

### Vertical component of k

_{eff }^{(W m}

^{-1}

^{K}

^{-1}

^{)}

**with air** **without air**

### + PP 103 **0.06** **0.006**

### RG 256 ^{0.16} ^{0.10}

^{0.16}

^{0.10}

### ^ DH 315 ^{0.25} ^{0.18}

^{0.25}

^{0.18}

### • At –2°C, k

_{air}

### = 0.024 and k

_{ice}

### = 2.107 W m

^{-1 }

### K

^{-1}

### .

### • Even if k

_{air}

### << k

_{ice }

### air plays a vital role in heat conduction

### • Kaempfer et al., 2005: k

_{air }

### neglected underestimation?

### Influence of conduction in air

**- 90%**

**- 40%**

**- 30%**

### Comparison of k estimations

### Method Average value of k (W m

^{-1}

### K

^{-1}

### )

### Vertical component Isotropic assumption

**Flux-gradient** 0.240 -

**Computation normalized** 0.238 0.203

**Needle probe** - 0.158

### 6 estimations of k from our TG experiment

### by 3 differents methods

### on same snow samples

### • k

^{*}

_{needle}

### values are significantly lower than computed values.

### Comparison of k estimations

### • k

^{*}

_{needle}

### values are significantly lower than computed values.

**1) Influence of the temperature?**

### Sturm et al. 1997: measurements at low temperatures.

### k

_{eff}** increases with decreasing temperature.**

**2) Problem with needle probe parameters?**

### Sturm et al.1997: Δt

_{heating}

### =120s et ΔQ such as ΔT = +2-4K.

### OK according to Morin et al. 2010.

**3) Bias linked to microstructural disturbance? **

### Riche and Schneebeli 2010

### Morin et al. 2010: no signifiant effects on k

^{*}

### provided that the first 30s of heating are discarded.

### Need to a specific study on this subject.

### Why does needle-probe underestimate the thermal conductivity compared to other methods?

Riche and Schneebeli 2010

### 3 different estimations:

### **Needle probe - assumes a homogeneous and isotropic medium ** - transient mode (strong local temperature gradients)

### - all thermal processes

### **Flux-gradient - vertical component of k***

### - steady state condition - all thermal processes

### Numerical computation - full tensor of k

_{eff}### - pure conduction

### Differences between these 3 variables may depend on the snow type and on temperature and temperature gradient because of the non- conductive processes.

### Why needle-probe underestimates the thermal

### conductivity compared to other methods?

**Conclusions**

### • k

_{eff}### strongly correlated with snow density (Yen,1981)

### support the use of the k-density relationship in snowpack models

### • Anisotropy of k

_{eff}### linked to snow microstructure (0.7-1.5)

### • Air plays an important role in heat conduction

### • Underestimation of k

^{*}