## Abstract

A model of the thermo-elastic behaviour of saline ice is formulated, and model solutions describing thermo-elastic waves (TEW) propagating into a half-space of the ice are investigated. The model is based on a proposal that saline ice is a matrix, which encompasses both closed brine pockets and permeable channels filled with brine. Experiments on the thermal expansion of saline ice samples, and on TEW in saline ice, have been performed in the cold laboratories of the University Centre in Svalbard and in University College London. The experimental data are compared with theoretical conclusions. The experimental data support our hypothesis that the brine in saline ice is divided between closed pockets and open, permeable channels.

This article is part of the themed issue ‘Microdynamics of ice’.

## 1. Introduction

Sea ice expands and contracts when its temperature changes. These volume changes can cause structural loads, and can induce thermal cracking, which makes the sea ice more fragile [1], and so understanding this behaviour can be important to a range of engineering and geophysical problems. Saline ice is a composite material—a solid ice matrix containing liquid and gas inclusions—and so understanding its thermal properties requires an understanding of the behaviour of the ice, brine and air, and their interactions. The ice matrix itself is made up of pure ice grains (columns or platelets) and pockets and channels that contain brine and air. The ice is permeable, and the brine is able to migrate through the matrix under the influence of pressure gradients [2]. The permeability may be affected by thermal changes in the ice, as brine pockets separate and merge during freezing and melting cycles. Variations in permeability mean that sea ice may have material memory of thermal expansion and contraction, i.e. the current state may depend on previous conditions as well as current conditions.

Sea ice also exhibits unusual thermal expansion due to phase changes that occur in the matrix. Both fresh ice and brine exhibit typical thermal behaviour: they expand when heated and contract when cooled (i.e. their density decreases with increasing temperature). However, when brine freezes to form ice, this leads to a density reduction, because ice is less dense than water. When cooling leads to the freezing of brine, therefore, this can lead to a net density reduction: cooling in saline ice can cause expansion of the ice. The inverse can also occur, when warming of the saline ice causes brine to replace ice, leading to a reduction in volume. This behaviour was first discussed by Pettersson [3] and later studied experimentally and modelled by Malmgren [4].

Malmgren assumed that all brine remained confined within the ice sample [4]. This is equivalent, within his formulation, to saying that the ice is impermeable. In his experiments, he immersed a sea ice sample in fluid and measured changes in fluid volume with temperature. He then used these total fluid volume changes to calculate thermal expansion coefficients. This experimental set-up makes no allowance for the possibility that brine and air may be expelled into the confining fluid, and the confining fluid may be impelled into the ice [1]. In Malmgren's model, the volumetric effects of phase changes are entirely accommodated within the ice matrix, and therefore saline ice shows the atypical thermal properties discussed in the previous paragraph. *In situ* observations [5] show that the average salinity of ice sheets remains virtually constant for the period of observations despite the prevalence of brine drainage networks and brine corrosion tubes that develop when the sheet warms.

Cox [6] revisited Malmgren's assumptions, and proposed that ice and brine are in fact free to expand and contract independently (this is equivalent to assuming that the ice is infinitely permeable). In Cox's formulation, phase changes are accommodated by brine drainage (or by brine being impelled into the ice) and so have no effect on the thermal expansion coefficient. He therefore proposes that the volumetric thermal expansion of saline ice is the same as that of fresh ice: around 1.5 × 10^{−4 }K^{−1}. Cox's model and Malmgren's model therefore represent two ends of a spectrum that depends on sea ice permeability.

Later experiments give further insight. Johnson & Metzner [1] conducted a series of experiments on the linear thermal expansion of cylindrical ice samples taken from a sheet of first-year congelation sea ice in Harrison Bay, Alaska. They used a Michelson interferometer type laser dilatometer, along with a temperature control unit and computer-controlled data acquisition. They worked on two sets of samples, both with 38 mm diameter: (i) 2 ppt saline ice samples, 71.25 mm long, and (ii) 4 ppt saline ice samples, 69.33 mm long. To measure the samples with the dilatometer, they froze optically flat mirrors to microtomed sample surfaces. Their displacement resolution was 316.4 nm, which corresponds to a strain resolution of 5 × 10^{−5}. They measured the thermal expansion coefficient to be very similar to the fresh ice thermal expansion coefficient of 1.5 × 10^{−4} K^{−1}. They concluded that their results supported the analysis of Cox [6].

Further systematic investigations were performed in the cold laboratories of the University Centre in Svalbard, Norway, and University College London, UK, from 2012 to 2015 [7–9]. These experiments used strain and temperature sensors based on fibre optics with fibre Bragg gratings (FBG). FBG reflect light of a certain wavelength, depending on the extension of the grating. By measuring the wavelength of the reflected light, experimenters can determine the strain in the grating, and ultimately the temperature and strain in the ice sample. Thermal strains were measured with a sampling rate of 1 Hz. The FBG system was designed by Advance Optic Solutions GmbH (Germany). In these experiments, the volumetric coefficient of thermal expansion was found to be close to 1.5 × 10^{−4} K^{−1} for fresh ice and ice with 2 ppt salt. However, with 6 ppt or higher salinity, negative values of thermal expansion were observed in several experiments.

The experiments show that the mean thermal expansion coefficient depends on temperature and salinity. They also show that thermal expansion coefficients vary around a mean, for a given temperature and salinity. The thermal expansion is observed to be a non-quasi-static process, where we assume that individual small deformations occur over time within the ice matrix, and these lead to changes in the thermal expansion over time. This suggests that modelling of thermal expansion should take into account temperature diffusion and liquid brine migration through the sample. A thermo-mechanical model of saline ice, accounting for migration of liquid brine through ice of finite, non-zero permeability, was proposed by Marchenko & Lishman [10]. The paper gave a theoretical model of the propagation of thermo-elastic waves (TEW), caused by temperature fluctuations at the ice surface, through an infinite half-space of saline ice. One conclusion was that temperature fluctuations and ice deformation are concentrated in a layer around 1 cm from the surface when the period of the temperature fluctuations is around 10 min.

In the first part of this paper, we describe a thermo-mechanical model of saline ice incorporating both (i) closed pockets of liquid brine and (ii) permeable channels of liquid brine. We derive model solutions that describe the evolution of TEW in this ice. In the second part of the paper, we describe experiments on the thermal expansion of saline ice samples, and on the propagation of TEW through these samples. In the conclusions, we compare theoretical predictions to experimental results.

## 2. Model equations

### (a) A model of sea ice brine distribution

In this paper, we assume that the brine contained in sea ice can be characterized in two ways: some fraction of the brine is contained in closed pockets, entirely surrounded by ice, while the remainder of the brine is in permeable channels. We assume that the amount of precipitated salt is very small and can be ignored. For sea ice, this is true when the ice temperature is higher than −8.2°C [11]. We can then divide the sea ice matrix into two components:

—

*ice with closed brine pockets*(abbreviated to ICP); this fraction of the sea ice incorporates both the pure ice, and the brine held in impermeable pockets;—

*liquid brine in permeable channels*(abbreviated to BPC).

This division is illustrated schematically in figure 1. This view allows us to describe observed Malmgren-like behaviour while simultaneously understanding sea ice as a permeable material.

In this section, we explore the properties of this ice: how the mass and salt are distributed in the ice; how brine flows through the permeable channels; and how heat flows through the bulk ice. We then use this understanding to develop a model of TEW in sea ice: as the temperature at the surface of an ice sheet changes, how does the heat propagate through the ice, and how does the ice then deform? This model can then be directly compared with experimental results in the next section.

The proposed structural model of saline ice is similar to models describing the flow of partially molten rock through the Earth's convecting mantle, assuming viscous deformations of the permeable matrix (see e.g. [12]). The details of the assumptions make a difference, with traditional models describing the flow of a low-viscosity fluid through a solid matrix (see e.g. [13]). In the case of saline ice, the permeable matrix, the ICP, has strain-rate-dependent rheology (see e.g. [14]). Within our approach, the mass balance equations include a source term equal to the rate of mass transfer from matrix to melt (melting rate), and the permeability of the matrix changes with porosity depending on thermo-mechanical processes.

### (b) Mass balance

We begin by defining the volumetric concentrations (or volume fractions) of each component in a unit volume of sea ice. The volumetric concentration of BPC (*ν*_{b}) in the volume of saline ice *V*_{si} shown in figure 1*a* is equal to *V*_{bpc}/*V*_{si}, where *V*_{bpc} is the BPC volume. The volumetric concentration of ICP (*ν*_{icp}) is equal to the ratio (*V*_{si }− *V*_{bpc})/*V*_{si}, where *V*_{si }− *V*_{bpc} is the volume of ICP in the saline ice volume *V*_{si}, and therefore
2.1
This is the same as saying that the entire volume of sea ice is composed of either ICP or BPC. The ratio of ICP to BPC may change with temperature.

ICP is made up of ice and brine only: we denote the volume fractions of ice and brine in ICP as *ν*_{i} and *ν*_{bi}, respectively (figure 1*b*), such that
2.2
ICP is not permeable by brine. Temperature changes will affect the ratio of *ν*_{i} and *ν*_{bi}. The quantity *ν*_{bi}—the brine fraction in ICP—will never fall to zero because the increasing brine salinity will eventually prevent freezing. A decrease in *ν*_{b} may lead to a transformation from BPC to a chain of closed pockets if the temperature is sufficiently low.

The sea ice density can be determined from the densities and volume fractions of its constituent parts: 2.3

### (c) Salt balance

The salinity of sea ice can be determined using the formula
2.4
where *σ* represents salinity (this equation states that the mass of salt per unit volume of sea ice is equal to the mass of salt in ICP in that unit volume plus the mass of salt in BPC in that volume).

We assume that the liquid brine and pure ice are in thermodynamic equilibrium, and therefore the salinities (and hence densities) of the brine in closed pockets and of brine in permeable channels are the same. Therefore, the ICP salinity can be determined as follows: 2.5 Schwerdtfeger [11] gives the density of ice containing brine as eq. (2.3.1). Adapting this for ICP, we find 2.6

Pure ice density depends on the temperature, and is given by the formula [15]
2.7
The volumetric coefficient of thermal expansion *α*_{i} = 1.53 × 10^{−4} K^{−1}. The density of brine (and hence of BPC) is also given by Schwerdtfeger [11]:
2.8
where *ρ*_{w} is the density of pure water and *S* is the fractional salt content of the brine,
2.9
(The fractional salt content is related to the salinity by *σ* = *S*/(1 + *S*), and hence *σ*_{b} can also be calculated as a function of temperature.)

To recap: given the temperature, we can find from formulae (2.8) and (2.9) the density and salinity of the brine everywhere in the ice, as well as the pure ice density from formula (2.7). If we then knew the ICP salinity, we could calculate the density of ICP from (2.6) and the volume fraction of brine in ICP from (2.5). This would allow us to calculate the ice density and the volumetric concentration of ICP from (2.3) and (2.4): 2.10 Overall, then, in our formulation, the bulk temperature, and ice and ICP salinities are sufficient to define all densities and volume fractions.

### (d) Thermal expansion of ice with closed brine pockets

A change of ICP mass may occur due to brine transfer between ICP and BPC, and can be related to a change of the ICP salinity by the formula
2.11
where *m*_{icp} = *ρ*_{icp}*V*_{icp} is the mass of an ICP particle with volume *V*_{icp}. In the case when mass change is caused by temperature change, from (2.11) it follows that
2.12

The volumetric coefficient of thermal expansion of ICP is determined by the formula 2.13 From (2.12) and (2.13), we find the standard formula for the calculation of the volumetric coefficient of thermal expansion of ICP: 2.14

### (e) System dynamics

The mass balance of the BPC is formulated as
2.15
where *t* is the time, *x*, *y* and *z* are the spatial coordinates, **v**_{b} is the BPC velocity relative to the ICP and **v**_{icp} is the ICP velocity in an absolute frame of reference. The term *ρ*_{b}∂*ν*_{b}/∂*t* is the source or sink of the brine due to the ice melting, brine refreezing and brine exchange with the ICP.

The momentum balance, and hence velocity, of BPC is given by Darcy's law:
2.16
where *k*_{si,b} is the permeability of sea ice by brine, *μ*_{b} = 1.8 × 10^{−3} Pa s is the dynamic viscosity of water, and *p* is the pressure in the brine channels. The permeability of saline ice by brine is determined by the formula
2.17
where the coefficient *k*_{si,b0} = 10^{−13} m^{2} characterizes ice permeability at low values of the liquid brine content. Formula (2.17) approximates the data of the laboratory and numerical simulations shown in fig. 2 in [16].

We assume that ice deformations are sufficiently slow and inertial effects can be ignored. The equations of static equilibrium based on Hooke's law can then be used, assuming an elastic model for the ice (i.e. for small displacements and short-term effects) [17]
2.18
2.19
where *p*_{icp} is the ICP pressure, **σ**_{d} and **ϵ**_{d} are the deviators of stresses and strains, *ϵ*_{icp} is the volumetric strain, *G* = 3.5 GPa and *K *= 8.9 GPa are the shear modulus and the bulk modulus of ice [18], *α*_{icp} is the volumetric coefficient of thermal expansion of the ICP, and *T*_{0} is a reference temperature. Equation (2.18) describes a static momentum balance in ICP, and formulae (2.19) describe Hooke's law for thermo-elastic materials. Formulae (2.19) would need to be modified to describe ice creep and delayed elasticity if long-term processes are considered. The strain-rate deviator is related to the ICP velocity by standard formulae specifying strain rates in continuum mechanics (see e.g. [17]).

The energy balance is given by the heat transfer equation [19,20]
2.20
where the material derivative is expressed by the formula d/d*t* = *∂*/*∂t* + ν_{b}**v**_{b} · ∇, and the coefficients of specific heat capacity and thermal conductivity are calculated by the formulae
2.21
Here, *c*_{i} and *c*_{b} are the specific heat capacities, and λ_{i} and λ_{b} are the thermal conductivities of ice and brine, *L*_{i} = 333.4 kJ kg^{−1} is the latent heat of ice, and **e**_{b} represents the strain rate of the brine. The second term on the left-hand side of equation (2.20) describes the latent heat rate due to the brine freezing or ice melting. The last two terms on the right-hand side of equation (2.20) are related to the heat produced due to the thermal expansion of ice and due to viscous energy dissipation in the brine.

Further, it is assumed that the specific heat capacity and thermal conductivity of brine are equal to the specific heat capacity and thermal conductivity of water. The specific heat capacities and thermal conductivities of ice and water are given by the formulae [11]
2.22
The time-varying salt balance in the ice is written in the form
2.23
The term *ρ*_{b}*σ*_{b}*ν*_{b} is equal to the mass of salt in BPC per unit volume of sea ice, and the term *ρ*_{b}*σ*_{b}*ν*_{b}**v**_{b} is equal to the salt flux.

The three scalar equations describing balance of mass (2.15), energy (2.20) and salt (2.23) and the two vector equations describing momentum balance in PBC (2.16) and ICP (2.18) form a system for the calculation of the three scalar variables (*p*, *T* and *σ*_{si}) and two velocity vectors (**v**_{b} and **v**_{icp}). The model is not complete since equations describing brine exchange between ICP and BPC are not formulated. We assume that the brine exchange can be described by an empirical dependence:
2.24
Further, we investigate this dependence using data from laboratory experiments on the thermal expansion of saline ice and on measurement of TEW in saline ice.

### (f) Thermo-elastic waves in saline ice

In experiments described later in this paper, we measure the thermal expansion of sea ice as temperature changes propagate through the ice. In order to describe these experiments, we develop our model of saline ice to describe TEW propagating into an ice half-space.

If the temperature of the upper surface of an ice sheet changes, then these temperature changes propagate down into the ice sheet, over time and with damping. Stresses due to these temperature changes also propagate, and, in the case of saline ice, may cause brine migration. This complicated interplay is modelled in this section of the paper. It is assumed that the steady state of a saline-ice-filled half-space *x *> 0 is described by the following characteristics:
2.25
We are then interested in the fluctuations around this steady state. Let us consider deformations of a saline ice half-space (*x* > 0) initiated by small temperature variations around *T* = *T*_{0} at *x* = 0:
2.26
where *A _{T}* is the amplitude and

*ω*is the frequency of the temperature wave at the ice surface.

We investigate the evolution of saline ice characteristics caused by periodic variations of the temperature at the surface described by formulae (2.26) and assume that the saline ice characteristics depend on the time *t* and spatial variable *x,* and are represented as follows:
2.27
Each component of vector **X** represents some characteristic of the saline ice, and each component of vector *δ***X** represents a perturbation of the ice characteristic near its value at the steady state specified by vector **X**_{0}, according to the formulae
2.28
where *v*_{ix} and *v*_{b} are the velocities of ICP and BPC in the *x* direction. Components of the vector **X**_{0} are background characteristics of ice. In this paper, we use the subscript ‘0’ to denote variables calculated using the background characteristics of ice.

A set of equations used for the investigation of the temporal and spatial evolution of *δ***X** is specified further in the case when all variables depend only on the time *t* and spatial coordinate *x*. In this case, non-zero components of the ICP strain rate and vertical displacement are determined by the formulae
2.29
From Darcy's law (2.16) it follows that
2.30

Substitution of (2.30) into equations (2.15) and (2.23)—the mass balance of BPC and the salt balance—and linearization leads to the equations
2.31
and
2.32
where the ice density perturbation *δρ*_{si} is, according to the first part of formula (2.10), a linear function of *δT*, *δσ*_{si} and *δσ*_{icp}:
2.33
The heat transfer equation (2.20) is written after linearization as
2.34
where according to formulae (2.1), (2.5), (2.6) and (2.10) the perturbations *δν*_{b} and *δν*_{bi} are linear functions of *δT*, *δσ*_{si} and *δσ*_{icp}:
2.35

Equations (2.18) and (2.19) are differentiated with respect to time and reduced to one equation: 2.36 The perturbation of the ICP salinity is described according to equations (2.24) as follows: 2.37

The six equations (2.30)–(2.32), (2.34), (2.36) and (2.37) form a closed system describing the evolution of the vector *δ***X** specified by the second formula (2.28). We consider the solution in the form
2.38
where the products of the temperature amplitude *A _{T}* on the coefficients

*a*,

_{p}*a*

_{σ}_{si},

*a*

_{σ}_{icp},

*a*

_{v}_{ix}and

*a*

_{v}_{b}are the perturbation amplitudes of the BPS pressure, ice salinity, ICP salinity, ICP velocity and BPC velocity. Substitution of (2.37) in equations (2.31)–(2.36) leads to a system of five linear algebraic equations with respect to five components of the vector

**A**. This homogeneous system has a non-trivial solution when the determinant is equal to zero. The last condition is called the dispersion equation of TEW: 2.39 where

*X*is given in formulae (5.1) in the Additional information section (§5). The dispersion equation is a rule for the calculation of the wavenumber

_{T}*k*corresponding to a given wave frequency

*ω*. If

*ω*and

*k*satisfy the dispersion equation, then all numbers

*a*,

_{p}*a*

_{σs}_{i},

*a*

_{v}_{ix}and

*a*

_{v}_{b}are calculated from equations (2.30)–(2.32), (2.34) and (2.36), while the amplitude

*A*remains an arbitrary quantity. From equation (2.39), it follows that the wavenumber is a complex quantity, i.e. it is the sum of a real part describing periodic oscillations of ice characteristics in the direction of wave propagation and an imaginary part describing the damping of wave amplitude. Similar dispersion equations describe pure thermal waves caused by temperature oscillations at the surface of materials, and Stokes waves in viscous fluids caused by the oscillations of shear displacements at the surface (see e.g. [20]).

_{T}Numerical estimates using formulae (5.1) show very little dependence of *X _{T}* on ∂σ

_{icp}/∂

*T*, when σ

_{si}< 10 ppt,

*σ*

_{icp}<

*σ*

_{si}, , and ∂

*σ*

_{icp}/∂

*T*is approximated by the linear dependence ∂

*σ*

_{icp}/∂

*T*≈ −

*σ*

_{icp}/6 constructed according to the temperature range. Therefore, we neglect all terms proportional to ∂

*σ*

_{icp}/∂

*T*when processing experimental data. In our experiments (described below), we did not observe significant changes of the ice salinity. Therefore, in the analysis of the experimental results, we also neglect all terms proportional to ∂σ

_{si}/∂

*T*.

The amplitude of vertical displacement of the ice surface is then equal to
2.40
The amplitudes *A _{u}* and

*A*can be measured in experiments. Their absolute ratio |

_{T}*A*/

_{u}*A*| (hereafter named the DT ratio, or displacement–temperature ratio) depends on the ice characteristics and wave frequency. The model given can be adapted to describe experimental data by varying the background characteristics

_{T}*ν*

_{b0},

*ν*

_{bi0}or

*σ*

_{icp}. From equations (2.31) and (2.32), it follows that only pore pressure amplitude

*a*depends on the ice permeability

_{p}*k*

_{si,b0}. According to formula (2.40), the amplitude of vertical displacement of the ice surface

*A*is proportional to the amplitude of surface temperature

_{u}*A*, the square root of the period of the temperature oscillations √(2

_{T}*π*/

*ω*) and other characteristics of saline ice excluding the ice permeability. Experimental data can therefore be used to test the validity of the structural model above, using equation (2.40), and also to infer parameters that are not well known, for example the ICP salinity

*σ*

_{icp}.

## 3. Experimental investigations

### (a) Thermal expansion of unconfined saline ice

Experiments on thermal expansion of unconfined samples of fresh and saline ice were performed in the cold laboratory of the University Centre in Svalbard (UNIS) [7,8]. The ice samples had a rectangular shape, with the long dimension around 20 cm and the shorter dimensions 5–10 cm. The optical fibres with FBG strain sensors had an integrative working length of 20 cm that was defined by two brass anchor bolts with nuts and washers. The optical fibres were placed into 2 mm-wide cuts that were sawn in the ice samples, and were fastened at the edges of the ice samples with nuts and washers (figure 2*a*). The ice block's thermal expansion or shrinkage is therefore transferred to the optical fibre with the FBG inside (this fibre is prestrained to around 0.3% by adjusting the nuts accordingly). The FBG fibre was not frozen into place, because this would allow localized shear around the FBG itself to distort the measurements. The FBG temperature sensor (thermistor string) was encapsulated in a 1 mm stainless steel capillary tube and inserted into a drilled hole in the ice sample, next to the FBG strain sensor.

The equipment used in our measurement set-up includes a broadband LED light source with a central wavelength of 1550 nm and a bandwidth (FWHM) or approximately 90 nm (AOS GmbH, Germany), an NIR spectrometer ‘I-MON 512E-USB 2.0 interrogation monitor’ with a detectable spectral width of 1510–1595 nm (Ibsen Photonics S/A, Denmark), a PC, and two FBG sensors, one for strain measurement and one for temperature measurement (both AOS GmbH, Germany). The Ibsen monitor is distributed with operating software, including a LabView source code. The experimental schematic is shown in figure 3*a*. All electronic devices—the LED source, the spectrometer and the PC—were installed outside the cold laboratory. Regular optical single-mode cables were used to connect the equipment to the FBG sensors embedded into the ice sample in the cold laboratory.

The ice sample was covered by a plastic housing to avoid evaporation (figure 2*b*). The temperature in the cold laboratory (2.5 × 2.5 × 1.8 m) was changed in increments of 2°C from −20°C to a nominal 0°C, programmed and displayed on the laboratory's control system. Measurements of actual temperature in the cold laboratory show temperature oscillations with amplitude about 1.8°C across a range of mean temperature. The period of this temperature oscillation (when the room temperature was programmed to be constant) was 12 min. During the experiment, we realized that the actual air temperature close to the ice samples was lower than the displayed temperature by several degrees. Temperature and extension are measured at 1 sample s^{−1}.

A representative time of temperature equilibration over a sample is estimated by the formula *t** = *ρ*_{si}*c*_{si}*l*_{0}^{2}/λ_{si}, where *ρ*_{si} = 920 kg m^{−3} is a typical value of sea ice density, λ_{si} = 2 W (m K)^{−1} is a typical value of the thermal conductivity of sea ice, *l*_{0} = 0.1 m is a representative length of the sample, and *c*_{si} is the specific heat capacity of saline ice. The time taken for temperature to equilibrate in fresh ice samples is estimated at 2.5 hours when *c*_{si} is equal to the specific heat capacity of fresh ice: 2 kJ (kg K)^{−1}. Because of the phase changes around brine pockets, the specific heat capacity of saline ice can be much greater than the specific heat capacity of fresh ice [11]. Therefore, time intervals for the temperature increments were set to 2–3 h or greater.

A mixture of sea water, pumped from a nearby fjord, and fresh water was used to prepare saline ice samples. The salt water was added to a rectangular vessel and allowed to freeze layer by layer. The thickness of each layer was about 1 cm. Fresh ice samples were frozen by the same method from fresh water. The laboratory-grown saline ice had a columnar structure in each horizontal layer with clear boundaries between the layers (figure 3*b*). The salinity of the ice samples was measured before and after the experiments with a Mettler Toledo Seven Pro conductivity meter SG7, with resolution 0.01 ppt. Multiple ‘dummy’ samples were kept under the same conditions as the strain-gauged sample: these samples were then used to measure changes in salinity and density during the experiments. To measure ice density, we measure sample weight and sample volume. The volume measurement was performed by submerging the ice sample in a graded tube filled with oil with a density less than the ice density. The sample volume is taken as equivalent to the volume of oil displaced. Table 1 shows the salinity and density variations, measured twice per day, when the room temperature changed between −5°C and −15°C with one closed cycle per day. One can see that the laboratory ice salinity remained similar over the 4-day experiment. The mean value of the ice density is 904 kg m^{−3}. The gas content, estimated with air temperature −8°C and ice salinity 6 ppt, is about 2.4% [14].

From the experimental data, we calculate the effective coefficient of linear thermal expansion (ECTE) with the formula
3.1
where Δ*L*/*L* is the lengthening or shortening of ice samples measured with the FBG strain sensor when the ice temperature changes from *T* to . The ECTE describes measured properties of ice samples.

The coefficient of linear thermal expansion (CTE) is a material property, while ECTE is a characteristic of thermal expansion measured in the experiment. We consider saline ice as an isotropic material and assume that the CTE is one-third of the coefficient of volumetric thermal expansion. The linear coefficient of thermal expansion of fresh ice (CTEFI) is *α _{l}*

_{i}=

*α*

_{i}/3 = 5.1 × 10

^{−5}K

^{−1}. Results of the tests with fresh ice samples are shown in figure 4. Figure 4

*a*shows temperature variations in the ice samples as measured with the FBG thermistor string. The inset graph in figure 4

*a*shows temporal variations induced by the cooling cycle of the fans. The data are averaged over 15 min intervals to minimize the influence of the fan cycle. Calculated values for the ECTE for fresh ice are shown in figure 4

*b*, and these are close to the CTEFI, shown by the dashed line.

Figure 5 shows the results of a test with ice sample salinity 6 ppt. Thick black lines in figure 5*a* correspond to the values of the ice sample extension and temperature measured in the experiment. Thin grey lines in figure 5*a* show polynomial interpolations of the experimental data. Thick lines in figure 5*b* show ECTE versus the temperature calculated using the polynomial interpolation of the experimental data. Thin lines in figure 5*b* show theoretical dependences of CTE on temperature, calculated with adjusted values of the ice salinity *σ*_{icp} calculated from formula (2.14) using formulae (2.6)–(2.10). The test shows thermal expansion similar to that described by Malmgren [4], with a negative ECTE, when the ice temperature is higher than −8°C. Formula (2.11) gives a good approximation of the experimental results with *σ*_{icp} = 3.7 ppt when the ice temperature is less than −14°C, and with *σ*_{icp} = 1.2 ppt when the ice temperature is higher than −8°C. We explain the observed decrease of the ECTE in warm ice as due to the variation in the amount of brine trapped in closed pockets. This amount is estimated to be about 60% of total brine content when the ice temperature is below −14°C, and about 20% of total brine content when the ice temperature is above −8°C.

### (b) Thermal expansion of constrained saline ice

A series of experiments were undertaken in the cold rooms of University College London [9]. The FBG strain sensor was used to measure the linear strain in ice samples of cylindrical shape constrained between two steel pipes. The initial intent of this experiment was to investigate slip between ice and steel during thermal expansion and contraction. The ice was formed between two steel pipes: an outer pipe with inside diameter 11 cm, and an inner pipe with outside diameter 2 cm (figure 6). Fresh ice was made from London tap water; saline ice from the same, with 8 ppt NaCl added. The samples were formed in layers, so that up to 1 cm depth of water was added and allowed to freeze before the next layer was formed. No evidence was seen of supercooled water or large bubbles within these layers. The outer pipe has length 180 mm, and the samples are milled 5 mm from either end of the pipe, such that the initial ice sample has two flat parallel ends 190 mm apart. On these flat ends, we place an aluminium spacer of width 4 mm which supports the FBG strain sensor such that expansion in the ice in the along-pipe direction stretches the sensor.

Temperatures are measured using the FBG 12-thermistor string inserted through a drilled hole in the ice (or ice and pipe), such that temperatures are recorded through the sample in the horizontal direction. The temperature used for calibration of the strain sensor is that closest to the sensor. For calculation of thermal expansion, we use the average of thermistors located inside the ice sample, which gives a mean value through the ice, but eliminates the thermistors in the air, which respond much faster to temperature changes in the room. Temperature and extension are measured at 1 sample s^{−1}. The entire apparatus is shown in a photograph and schematic in figure 6.

Figure 7*a* shows strain, as a function of temperature, measured at two different stages of a single experiment. Figure 7*b* shows this strain–temperature relationship expressed as ECTE. Measurements of thermal expansion are taken during periods when the temperature variation in the ice is greater than 0.1°C s^{−1} (i.e. only when the temperature gradient is sufficiently large). Two clusters of results are visible in figure 7, and marked as 1 and 2. This split set of results is derived from one single experiment. The two clusters represent two different stages of the experiment, around 24 h apart, and separated by cooling the ice to −16°C: the more negative values of ECTE (i.e. cluster 1) come from the first stage of the experiment, and the values close to zero (cluster 2) from a later stage of the experiment. In both clusters, the ice was warming as measurements were taken. In cluster 1, the ice clearly contracts with increasing temperature, whereas in cluster 2 the ice expands with increasing temperature up to around −7°C, then contracts. It is surprising that the same ice, during the same experiment, shows such variation in ECTE against temperature. We note that the constrainment of the ice means the ECTE could take values between the volumetric and linear coefficients of thermal expansion, but this switching between the volumetric and linear expansion does not explain the two regions of ECTE in the temperature range from −10°C to −6°C. It is possible that the constraining pipes cause some brine regions to switch over time between BPC and ICP, but this effect needs further investigation.

### (c) Experiment on thermo-elastic waves

Experiments to measure TEW were undertaken in the UNIS cold laboratory in 2015. Saline ice samples were prepared from a mixture of seawater and fresh water. Cylindrical samples with inner vertical hole were frozen by adding saline water, layer by layer, into a cylindrical plastic vessel with a fixed coaxial vertical pipe inside. The FBG was fixed by steel plates and bolts on the top and the bottom of the ice sensor, as shown in figure 8. The FBG thermistor (FBGT) was placed near the FBG strain sensor. The ice sample was insulated from the environment by foam plastic (figure 8*a*).

The upper surface of the sample was subjected to periodic variations of air temperature, caused by the cycle of the fans in the cold laboratory. The period of the air temperature variations was 12 min. The inverse wavenumber *k*^{−1}, calculated from formula (2.39) and with a wave period of 12 min, is around 1 cm. This quantity characterizes the depth of penetration of the air temperature fluctuations into the ice. The FBG strain sensor recorded the vertical deformation of the ice sample created by the temperature oscillations with sampling interval 1 s. The air temperature was measured less than 1 cm from the ice surface with a Testo 176T4 rod-probe, and on the ice surface with a Testo 176T4 wire-probe (figure 9).

The mean air temperature was kept the same for 5–6 h to get stable ice temperatures several centimetres below the ice surface, and stable amplitude of TEW at constant mean air temperature. The amplitude of the temperature oscillations measured in the air was 1.8°C, across a range of mean temperatures. Figure 10 shows an example of the temperature and strain records. The general trend of the strain record, and the temperature-induced oscillations of strain, are clearly visible. The amplitude of oscillations around 1 µstrain (1 micro-strain unit, 10^{−6}) corresponds to the resolution of the FBG strain sensors. A moving average was applied to extract the trend of the strain. The difference between these averaged data and the actual data gives a periodic signal with amplitude proportional to the amplitude *A _{u}* of vertical displacements of the ice surface (i.e. the TEW amplitude).

The amplitudes of the temperature oscillations measured with the wire-probe on the ice surface were smaller than the amplitudes measured with the rod-probe in air, and were highly dependent on the nature of the freezing between probe and ice surface. The actual amplitudes of the ice surface temperature are assumed to be between the wire-probe and rod-probe measurements. Therefore, we include a calibration coefficient to compare the shape of the experimental and theoretical dependences of the DT ratio on the mean surface temperature. The theoretical dependence of the DT ratio on the mean surface temperature was calculated using formula (2.40).

These theoretical dependences of the DT ratio on the mean ice temperature are shown in figure 11*a* when closed brine pockets are absent (*σ*_{icp} *=* 0). The values of ice salinity *σ*_{si} are shown near the lines. Figure 11*b* shows the theoretical dependences of the DT ratio on the mean ice temperature when *σ*_{si} = 5 ppt. The values of ICP salinity *σ*_{icp} are shown near the lines. One can see that the DT ratios shown in figure 11*a* are much smaller than those shown in figure 11*b*. The DT ratios decrease when the mean temperature increases in figure 11*a*, and the DT ratios increase when the mean temperature increases in figure 11*b*. Examples of the DT ratios (as a function of mean temperature) reconstructed from experiments are shown in figure 12*a* for *σ*_{si} *=* 9 ppt by black dots and for *σ*_{si} *=* 5 ppt by grey dots. These results show increasing DT ratios when the mean temperature increases. This type of behaviour is similar to that shown in figure 11*b*. The values of the DT ratios in figure 12 are smaller than the values shown in figure 11*b* because only a fraction of the liquid brine is trapped in closed brine pockets of saline ice in the laboratory experiment. The solid lines in figure 12*a* show a (second-order) polynomial fit of the DT ratios. Figure 12*b* shows the dependences of the ICP salinity on the temperature found from formula (2.40) after the substitution of the polynomial fits of the DT ratios shown in figure 12*a*. Both of the dependences demonstrate a decrease of the ICP salinity with increasing temperature.

## 4. Conclusion and discussion

A thermodynamic model of saline ice, accounting for phase changes and liquid brine migration in the pore space, has been formulated, and basic equations derived. It is assumed that sea ice is composed of two intermingled parts: ice with closed brine pockets (ICP) and liquid brine in permeable channels (BPC). The liquid brine content of the ICP is characterized by the ICE salinity *σ*_{icp}, which is lower than the total ice salinity *σ*_{si}. The brine migration through the permeable channels is described by Darcy's law in the model. The ICP salinity *σ*_{icp} can be determined from the temperature *T* and the total ice salinity *σ*_{si} through an empirical equation. It has been shown that the coefficient of thermal expansion of sea ice, as given by the model, is similar to the coefficient of thermal expansion derived by Malmgren [4] when the sea ice salinity is equal to the ICP salinity.

We have conducted theoretical investigations of the thermal deformations of a sea ice half-space caused by periodic changes of the surface temperature. These temperature oscillations penetrate into the ice and cause ice deformations. These are TEW. We analysed the dependence of the amplitude of vertical displacements of the ice surface (i.e. TEW amplitude) on the mean ice temperature when the amplitude of the surface temperature oscillations is fixed. It was shown that the TEW amplitude is expected to decrease with increasing mean ice temperature when all the liquid brine is located in permeable brine channels, and *σ*_{icp} = 0. The TEW amplitude increases with increasing mean ice temperature when the ICP salinity is greater than zero (*σ*_{icp} > 0).

We investigated the dependence of the ECTE on temperature, in laboratory experiments. Values of ECTE around 5 × 10^{−5} K^{−1} were found for fresh ice, and these are close to the widely accepted value. Negative values of ECTE were found in samples with salinity of 6 ppt and higher. Atypical thermal behaviour—contraction with increasing temperature—was observed across several experiments at temperatures higher than −8°C.

The absolute values of the ECTE were lower than those given by Malmgren [4] but can be modelled by assuming that only a part of the liquid brine exists in closed brine pockets. The experimental dependence of ECTE on temperature was compared with theoretical predictions, and a reasonable match was achieved by tuning the ICP salinity *σ*_{icp}. We find that the ICP salinity (which tells us, effectively, how much brine is in closed pockets and how much in open channels) should be lower than the total ice salinity. The amount of salt trapped in closed pockets decreases with temperature increase. In our experiments, it is about 60% of total ice salinity when the ice temperature was below −14°C and drops below 20% of total ice salinity when the ice temperature was above −8°C.

In the experiment with a constrained ice sample of 8 ppt salinity, we discovered that the ECTE varies within the same experiment, so that two distinct clusters of strain (and hence trends of ECTE) are visible. Such behaviour could be explained by temporal locking and unlocking of BPC, perhaps due to the influence of the constraining pipes. Therefore, we think that thermal expansion of saline ice is a highly dynamic process, accompanied by temporal variations of the fraction of closed brine pockets, depending on spatial sizes and confinement of the ice. Numerical simulations of the experiment [9] show that the maximum value of thermally induced axial displacement of the confined ice sample, calculated with an elastic model, is similar to the axial displacement of an unconfined sample. If creep is incorporated, the maximum displacement is modelled to increase around 20% (assuming no slip between ice and steel). Frictional sliding between ice and steel may influence the measured thermal expansion in our experiments. This effect should be further studied.

Theoretically derived TEW have been validated in laboratory experiments with saline ice samples. In the experiments, we registered periodic elongation and shortening of ice samples, caused by periodic cycles of the air temperature around some mean value. The amplitudes of these TEW increase with increasing room temperature. Within our model, this occurs only when the ice includes some amount of brine in closed pockets (*σ*_{icp} > 0).

We have managed to reproduce the main trends of the ECTE dependence on temperature and the dependence of TEW amplitudes on temperature by adjusting ICP salinity *σ*_{icp}. Phase changes occur at the boundaries of the closed brine pockets, and these phase changes are accompanied by changes of local pressure and deformations due to the 10% difference between ice and brine densities. These stresses are not accounted for in the model. The gas content of saline ice [21,22] may play a role in damping this effect, and the presence of gas content could be incorporated into later models.

Measurements of strain and temperature were performed with fibre Bragg grating sensors, which can be recommended as a productive tool for laboratory measurements of the thermo-mechanical properties of saline ice. We were able to perform experiments with samples of various sizes, geometries and configurations; sampling at 1 Hz; and with high accuracy and resolution across a temperature range from 0°C to −20°C. The sensors, and their associated hardware and software, were stable and robust. The main experimental complication was in developing techniques to mount the FBG sensors onto the ice samples.

The results in this paper suggest that some brine within sea ice may be contained in closed pockets, and unable to permeate through the ice under typical pressure differences. This in turn may influence the thermal expansion of sea ice. This work may be of interest to those modelling sea ice thickness and salinity distribution on both local scales and large scales [23,24].

## 5. Additional information

### (a) Dispersion equation

5.1

### (b) Instrumentation

Experiments on thermal expansion and measurements of TEW in saline ice were performed with fibre Bragg grating (FBG) sensors manufactured by AOS GmbH (Dresden, Germany). A fibre Bragg grating is a periodic index change in the refractive index *n* along an optical silica fibre's core, formed by an interference pattern of two UV laser beams that the fibre is exposed to. With a period of less than 10^{−6 }mm, areas of higher *n* alternate with areas of lower *n* while Δ*n* is relatively small (Δ*n* < 0.001 *at n* = 1.45). Owing to the index change, each ‘grating line’ reflects a very small portion of the light wave propagating along the fibre, back to the light source. Although a single reflected portion is negligible compared with the transmitted power, the effect becomes noticeable because the number of ‘grating lines’ in a conventional FBG is about 4000 mm^{−1}, and a typical FBG with 10 mm length consists of 40 000 reflections. If the light's wavelength matches the condition
5.2
where *n*_{eff} is the fibre's effective index, λ_{Bragg} is the light's wavelength and *Λ* is the index change's period, then all reflected light wave portions are propagating ‘in-phase’ and interfere constructively. It is easy to see that λ_{Bragg} will change when the fibre is strained or compressed, whereas the effective refractive index is a material property and thus *n*_{eff} is sensitive to changes in temperature. This sensitivity of the peak wavelength with respect to thermal and mechanical loads allows the usage of FBG as strain and temperature sensors [25,26].

In our experiment, FBG sensors are practicable for measuring the thermal expansion of large samples because, in contrast with standard dilatometers, the fibre can be embedded directly inside the ice sample. An FBG thermistor string and strain sensor are shown in figure 13. Typical strain resolution for FBG systems is 10^{−6} (1 µstrain) or better, and the accuracy is typically 5 × 10^{−6} (5 µstrain). These characteristics are comparable with Michelson interferometer laser dilatometers as used by Johnson & Metzner [1] and discussed above.

The variation (Δλ_{Bragg}) of the peak wavelength caused by the extension (Δ*L*/*L*) and the change of the temperature (Δ*T*) of the sensor is described by the equation
5.3
where the gauge factor GF = 0.719 and a linear temperature coefficient TK = 5.5 × 10^{−6} are the constants obtained from a calibration cycle for our FBG sensors in a standard single-mode fibre, within a temperature range from −20°C to 0°C.

The variation of the peak wavelength Δλ is measured with a spectrometer that receives the reflected signal from the FBG sensor. For the calculation of the strain (Δ*L*/*L* according to formula (2.40)), it is necessary to measure the temperature change (Δ*T*) at the strain sensor's position in order to compensate TK. The temperature measurements can easily be performed with another FBG sensor protected from mechanical deformation, or alternatively with a thermometer. In our experiments, we used FBG strain sensors with a reference peak wavelength in the vicinity of around 1534 nm, and FBG temperature sensors with a reference peak wavelength in the vicinity of around 1565 nm. The strain and temperature sensors were cascaded in one optical fibre. The FBG measurement system's nominal resolution and accuracy in our experiment was 0.08°C and 0.4°C, respectively.

## Authors' contributions

A.M. helped in the formulation of the model, experiments on thermal expansion, experiments on TEW, data processing, drafting the article, and final approval of the version to be published. B.L. helped in the critical analysis of the model equations, experiments on thermal expansion, data processing, drafting the article, revising the article critically for important intellectual content and final approval of the version to be published.

## Competing interests

We have no competing interests.

## Funding

Budget funding of the University Centre in Svalbard to A.M. Budget funding of University College London to B.L.

## Acknowledgements

The authors wish to acknowledge the support of the Research Council of Norway through the SFI SAMCoT and Petromaks2 project ‘Experiments on waves in oil and ice’.

## Footnotes

One contribution of 11 to a theme issue ‘Microdynamics of ice’.

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3580721.

- Accepted June 2, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.