2 Laboratory measurements: methodology

Two sources of laboratory data are used in all of the following: the data obtained in the present study and the data reported by Newyear & Martin (Reference Newyear and Martin1997), which are available from tables 1 and 2 of their article, comprising two experiments in the same test facilities. They report experimental details in the corresponding article, and the data were used in one later study (Newyear & Martin Reference Newyear and Martin1999). Newyear & Martin (Reference Newyear and Martin1997) used a flap-type paddle (i.e. a plate rotating around its lower edge) to generate waves in a wave tank 3.5 m long, 1 m wide and 1 m deep, which was filled with a depth $H=50$  cm of water. The wave tank was located in a cold room, which allowed the authors to create slush ice in a controlled environment. The slush-ice thicknesses reported by the authors were 11.3 and 14.6 cm for experiments 1 and 2, respectively. The frequency range investigated by Newyear & Martin (Reference Newyear and Martin1997) is relatively small, between 1.0 and 1.6 Hz, and the lower end of the frequency range does not strictly fulfil the deep water waves condition, which adds complexity to the situation in addition to the effect of grease ice.

The data collected in the present study were obtained in a cold room facility located at the University Centre in Svalbard (UNIS). All the details of the actuation and logging system are released as open source material, see appendix A for more details and a link to the description of the system and code. A custom-designed flat-blade paddle (i.e. the whole plate moves back and forth while staying vertical) was used to generate waves over a frequency range between 1.5 and 2.7 Hz, with an increment of 0.1 Hz. This is a wider frequency range than investigated previously by Newyear & Martin (Reference Newyear and Martin1997). Measurements were performed for six different wave–paddle displacement amplitudes at each of those frequencies, for a total of 78 measurements. The wave tank is built in transparent acrylic with the dimensions 3.5 m long, 0.3 m wide and 0.5 m high, and the water depth is 25 cm. A damping beach is located at the opposite end of the wave tank to the paddle.

During the experiments, the effect of the grease ice alone absorbs most of the wave energy. As a consequence, very little water motion is still present at the position of the beginning of the damping beach, and the water at the end of the damping beach is essentially still. Consequently, we do not fear that energy may be reflected at the end of the wave tank. Removable insulation foam is placed under and on the sides of the wave tank in order to avoid ice formation on the wave tank walls during grease-ice growth, and can be taken away for gaining optical access when PIV measurements are performed. The slush ice is generated once at the beginning of the series of measurements by setting the temperature of the cold room to $-6\,^{\circ }\text{C}$ , and turning the paddle on so that waves of frequency 2 Hz are generated with a large paddle motion amplitude. As a consequence, breaking waves are generated that introduce turbulence in the whole wave tank, creating the right conditions for grease-ice formation. In addition, small ice crystals generated in a slush-ice machine are added in the water to speed up the process. As a result, the grease-ice layer obtained is approximately 4 cm thick when the water is at rest and the ice spreads evenly in the whole wave tank. The methodology used for generating the grease ice is similar to the one reported by Newyear & Martin (Reference Newyear and Martin1997). Following ice formation, the room temperature was set to 0 degrees so that no significant ice melting or freezing took place.

A total of six ultrasonic gauges (model U-gauge S18U, from Banner Engineering) are used to measure wave elevation. The gauges are located 50, 66.6, 88, 110.5, 133.9 and 179.9 cm away from the wave maker. The distance between consecutive gauges is deliberately varied, as this helps detecting possible aliasing effects when performing cross-correlation analysis of the wave elevation signal. The gauges have a resolution of 0.5 mm, and perform measurements at a frequency of 100 Hz. The noise level of gauges 4 and 6 is much higher than for the other gauges, possibly because of ice formation on the gauges, and therefore we only use the signals from gauges 1, 2, 3 and 5.

The data from the ultrasonic gauges are used in two ways. For each gauge, the wave amplitude is computed by integrating the Fourier spectrum of wave elevation around the peak frequency. The Fourier spectrum is computed from time series of duration 300 s using a 50 % overlap on 40 s time windows, and the width of the integration domain around the peak frequency is 1.0 Hz. For each run, the wave amplitudes obtained from the different gauges are then fitted to a decreasing exponential in order to extract the coefficient of wave damping, using:

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}a}{\unicode[STIX]{x2202}x}=-\unicode[STIX]{x1D6FC}a,\end{eqnarray}$$

where $a$ is the wave amplitude in the convention that $\unicode[STIX]{x1D702}(t)=a\cos (\unicode[STIX]{x1D714}t)$ is the wave elevation at a fixed point in space, with $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}f$ the wave angular frequency in $\text{rad}~\text{s}^{-1}$ , $f$ the wave frequency, $x$ the distance to the wave maker and $\unicode[STIX]{x1D6FC}$ the spatial decay coefficient that describes wave attenuation. This processing method is similar to what was presented by Sutherland et al. (Reference Sutherland, Halsne, Rabault and Jensen2017).

The spatial damping coefficient obtained experimentally, $\unicode[STIX]{x1D6FC}$ , is compared with the result obtained analytically in (4.15) of Weber (Reference Weber1987), the no-stress boundary condition of Lamb (Reference Lamb1932) and Newyear & Martin (Reference Newyear and Martin1997) and both (10) and (13) of Sutherland et al. (Reference Sutherland, Rabault, Christensen and Jensen2018). More explicitly, equation (4.15) of Weber (Reference Weber1987) corresponds to the wave dissipation due to the presence of an inextensible surface cover (Lamb Reference Lamb1932) and can be written as:

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{in}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FE}k/c_{g},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ is the kinematic eddy viscosity, which can be taken to be higher than the viscosity of water to account for eddies and, in some measure, grease-ice properties, $\unicode[STIX]{x1D6FE}=\sqrt{\unicode[STIX]{x1D714}/2\unicode[STIX]{x1D708}}$ is the inverse thickness of the surface boundary layer, $k$ is the wavenumber and $c_{g}$ is the group velocity.

The damping obtained in the case of a no-stress boundary condition (Lamb Reference Lamb1932; Newyear & Martin Reference Newyear and Martin1997) can be written as:

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{ns}=2\unicode[STIX]{x1D708}k^{2}/c_{g},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ is again a fitting parameter used to model grease ice properties.

Finally, equation (10) of Sutherland et al. (Reference Sutherland, Rabault, Christensen and Jensen2018), can be written as:

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D716}}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D716}hk^{2},\end{eqnarray}$$

where $0<\unicode[STIX]{x1D716}<1$ is the fractional ice thickness, and $h$ is the total ice thickness. A fractional ice thickness $\unicode[STIX]{x1D716}=0.7$ was previously found to model well a wide range of experimental datasets (Sutherland et al. Reference Sutherland, Rabault, Christensen and Jensen2018). Equation (13) of Sutherland et al. (Reference Sutherland, Rabault, Christensen and Jensen2018) can be written as:

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{t}=2h^{2}k^{3}.\end{eqnarray}$$

The difference between (2.4) and (2.5), which correspond respectively with (10) and (13) of the model by Sutherland et al. (Reference Sutherland, Rabault, Christensen and Jensen2018), is that, in the former case, the grease ice is considered thick enough so that the top of the ice is still.

Since the wave tank is of finite width and depth, damping is introduced by boundary layers developing on the side and bottom walls. Those sources of damping must be assessed to confirm that the main damping effect observed arises from the grease ice. The damping introduced by the laminar boundary layers created by the wave motion can be written as (Van Dorn Reference Van Dorn1966; Sutherland et al. Reference Sutherland, Halsne, Rabault and Jensen2017):

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{bs}=\unicode[STIX]{x1D708}_{0}\unicode[STIX]{x1D6FE}_{0}k\left.\left(\frac{1}{\sinh (2kH)}+\frac{1}{kB}\right)\right/c_{g},\end{eqnarray}$$

where $\unicode[STIX]{x1D708}_{0}$ is the kinematic viscosity of water, $\unicode[STIX]{x1D6FE}_{0}=\sqrt{\unicode[STIX]{x1D714}/2\unicode[STIX]{x1D708}_{0}}$ is the inverse boundary thickness and $B$ is the width of the wave tank. The damping coefficient $\unicode[STIX]{x1D6FC}_{bs}$ is found to be two orders of magnitude smaller than the damping measured experimentally, and therefore is neglected in all of the following.

The wavenumber is obtained from performing a cross-correlation analysis of adjacent ultrasonic gauges. For this, the phase difference between waves recorded by gauges $m$ and  $n$ , $\unicode[STIX]{x1D719}_{mn}$ , is obtained from the co-spectral density between adjacent sensors, $S_{mn}$ , as:

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{mn}=\tan ^{-1}\left(\frac{\text{Im}[S_{mn}(f)]}{\text{Re}[S_{mn}(f)]}\right),\end{eqnarray}$$

where Im and Re indicate the imaginary and real part, respectively. The co-spectral density is computed using windows of length 41 s, with a 50 % overlap. The equation describing the propagation of waves phase can be written as:

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{mn}=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{mn},\end{eqnarray}$$

where $\boldsymbol{k}$ is the wave vector and $\boldsymbol{x}_{mn}$ the distance between gauges $m$ and  $n$ . The wave vector corresponding to the frequency generated by the paddle can therefore be obtained from (2.8) since the distance between the gauges is known and the phase shift between adjacent gauges can be computed using (2.7). This methodology is similar to what was presented by Sutherland & Rabault (Reference Sutherland and Rabault2016) and by Marchenko et al. (Reference Marchenko, Rabault, Sutherland, Collins, Wadhams and Chumakov2017) in the case of field data. Results are compared with both the finite depth, open water dispersion relation and the mass-loading effect. The open water dispersion relation is written as:

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}=gk\tanh (kH),\end{eqnarray}$$

with $H$ water depth. The dispersion relation including mass loading is taken from Newyear & Martin (Reference Newyear and Martin1997):

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}=\frac{gk\unicode[STIX]{x1D70C}_{water}\tanh (kH)}{\unicode[STIX]{x1D70C}_{water}+c\unicode[STIX]{x1D70C}_{ice}hk\tanh (kH)},\end{eqnarray}$$

with $c\approx 0.5$ the volumetric fraction of ice.

When analysing laboratory data, we consider that dissipation is produced by the viscous effects introduced in water and/or the ice, as calculated by Weber (Reference Weber1987), Lamb (Reference Lamb1932) or Sutherland et al. (Reference Sutherland, Rabault, Christensen and Jensen2018), while modification of the wavenumber is mostly due to the mass loading of the grease-ice layer, and that as a first approximation both effects can be accounted for separately.

When performing PIV, a white colour linear LED array is used for illumination. Spherical polyamide seeding particles (PSP) of diameter $50~\unicode[STIX]{x03BC}\text{m}$ are used for seeding. A single Falcon2 4M camera is used for taking pictures at a frame rate of 75 Hz. This PIV set-up is similar to that used previously by authors of this paper (Rabault et al. Reference Rabault, Halsne, Sutherland and Jensen2016). The measurement plane is located in the middle of the wave tank, along the main direction of the wave tank and wave propagation, and a mirror inclined at $45^{\circ }$ is used for finely adjusting the position of the light sheet. Filtering is used to detect both the free water surface and the area that corresponds to grease ice, as shown in figure 1. Consecutive frames are used for performing PIV using the Digiflow software (Dalziel Reference Dalziel2012).

Figure 1.

Picture extracted from case 2, which corresponds to an initially discontinuous ice cover. (a) Raw picture before filtering. Both the water surface and the region occupied by grease ice are clearly visible. (b) Picture after filtering.

The velocity fields obtained from PIV are processed using proper orthogonal decomposition (POD). POD is based on the singular value decomposition (or an eigenvalue analysis of the positive semi-definite matrix $\unicode[STIX]{x1D653}^{T}\unicode[STIX]{x1D653}$ , where $\unicode[STIX]{x1D653}$ is the snapshot matrix, see below), and is known by different names depending on the field of study, such as ‘principal component analysis’ (from statistics) and ‘empirical decomposition functions’ (from meteorology). In the present case, the POD is computed in Matlab from the singular value decomposition (SVD) of the snapshots matrix (for detailed explanations about POD and the snapshot method, see for example Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993, Kerschen et al. Reference Kerschen, Golinval, Vakakis and Bergman2005). The snapshot matrix, $\unicode[STIX]{x1D653}$ , is constructed as:

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D653}=\left[\begin{array}{@{}ccc@{}}u_{1}^{1} & \ldots & u_{n}^{1}\\ \vdots & \ldots & \vdots \\ u_{1}^{k} & \ldots & u_{n}^{k}\end{array}\right],\end{eqnarray}$$

where $u_{i}^{j}=u(x_{j},t_{i})$ , with $x_{j}$ the position of the point considered, and $t_{i}$ the time of the snapshot. Both the $X$ and $Z$ components of velocity are stored in  $u$ , by letting $u_{i}^{1..k/2}$ represent the $X$ component and $u_{i}^{k/2+1..k}$ represent the $Z$ component, respectively. As a consequence, each column of $\unicode[STIX]{x1D653}$ contains the two-dimensional (2-D) two component velocity field at a given time reshaped into a 1-D vector. The SVD decomposition is then computed as:

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D653}=\unicode[STIX]{x1D650}\unicode[STIX]{x1D64E}\unicode[STIX]{x1D651}^{\ast },\end{eqnarray}$$

where $\unicode[STIX]{x1D650}$ and $\unicode[STIX]{x1D651}$ are unitary matrices. The diagonal coefficients of $\unicode[STIX]{x1D64E}^{2}$ give the energy of each mode, while $\unicode[STIX]{x1D650}$ and $\unicode[STIX]{x1D651}$ contain the POD modes and POD mode coefficients. The POD modes together with the POD mode coefficients contain the full information about the velocity field at all times. In particular, the description of the velocity field obtained with POD is interesting as it optimizes the energy content in the modes of lower index, therefore effectively extracting an ordered list of the most energetic coherent structures of the flow. Therefore, the POD can be applied to produce a condensed overview of the main structures present in the data of a dataset too large to be analysed by simple visualization techniques.

Figure 2.

Wavenumber reported by Newyear & Martin (Reference Newyear and Martin1997) (noted NM97 in the legend), and obtained in the present study using (2.8) for three different pairs of gauges (1 and 2, 1 and 3 and 2 and 3 are denoted as $k_{12}$ , $k_{13}$ and $k_{23}$ , respectively). The black line indicates the open water dispersion relation. Dotted lines indicate the mass-loading dispersion relation, for different ice thicknesses. Large black dots (▪) denote $k=(ch)^{-1}$ , in the case of each ice thickness. The mean values and 95 % confidence intervals are computed for each frequency based on the experiments performed at different wave amplitudes.

4 Discussion of effective viscosity and scaling of experiments

The use of an effective water viscosity much higher than the kinematic viscosity of water in order to reproduce observations of wave damping is often discussed in the literature. While most authors justify it from considerations about water turbulence and the presence of eddies (De Carolis & Desiderio Reference De Carolis and Desiderio2002), the origin of those eddy structures, especially under grease-pancake ice, where breaking waves are quickly attenuated, is problematic. The eddy structures could arise from wave-induced turbulence, but this topic is after many years still the object of much debate (Babanin Reference Babanin2006; Babanin & Haus Reference Babanin and Haus2009; Beyá, Peirson & Banner Reference Beyá, Peirson and Banner2012). At least two other mechanisms could participate in the generation of eddy structures, influencing wave damping by ice. Ice drifting relative to the underlying water, for example under the influence of wind or waves, could create shear on large temporal and spatial scales and be the origin of eddy structures. Collisions between pancakes, small broken ice floes or small packs of grease ice could also inject energy into the superficial water layer. We document for the first time the existence of such eddy structures in a laboratory experiment, when the waves are started from a situation where some packs of grease ice are separated by open water regions. Unsurprisingly, collision between packs of grease ice leads to the creation of a jet when water is forced out of the gap, and a strong vortex is created which decays slowly with time. Such collision mechanisms could be important in the field for injecting energy in the water under the ice, and therefore increasing the level of eddy viscosity and viscous dissipation.

However, more work is necessary to confirm this mechanism. Indeed, scaling between real sea conditions and wave tank experiments is problematic, and the generality of the experiment presented here would need to be further validated in experiments of larger scale.

More specifically, the very first imperative of laboratory experiments on waves is to be in the right water depth regime. Due to the very limited depth of most wave tanks, this puts a sharp requirement on the minimum frequency that can be used for tests in the laboratory. We consider as a first approximation in all of the discussion that follows that the deep water dispersion relation is enforced, so that $\unicode[STIX]{x1D714}^{2}=g\boldsymbol{k}$ , and that deep water waves are the most common situation for the ocean, which implies that $\boldsymbol{k}H>1$ , i.e. $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{min}=\sqrt{g/H}$ . In addition, the steepness of the waves must be kept moderate to limit nonlinear effects, typically $\unicode[STIX]{x1D716}=\boldsymbol{k}a\approx 0.1$ , i.e. $a<0.1g/\unicode[STIX]{x1D714}^{2}$ , where $\boldsymbol{k}$ is the wave vector and $a$ the wave amplitude. If the amplitude-based Reynolds number defined in Babanin (Reference Babanin2006), $Re_{a}=a^{2}\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708}_{water}$ , is the right non-dimensional parameter for describing turbulence under waves and large eddy structures under the ice, then in a laboratory experiment, with a wave tank of depth  $H$ :

(4.1) $$\begin{eqnarray}Re_{a}=\frac{a^{2}\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D708}_{water}}<\frac{10^{-2}g^{2}}{\unicode[STIX]{x1D714}^{3}\unicode[STIX]{x1D708}_{water}}<\frac{10^{-2}g^{1/2}H^{3/2}}{\unicode[STIX]{x1D708}_{water}}.\end{eqnarray}$$

This implies that, even in the case of a wave tank several metres deep, the maximum Reynolds number that can be achieved in the laboratory for deep water waves of reasonable steepness is much lower than any typical swell on deep water. More specifically, the maximum Reynolds number obtained in a wave tank scales as the water depth to the power $3/2$ . As a consequence, studying phenomena related to water wave turbulence and wave-induced eddy viscosity is challenging in the laboratory, as it is impossible to investigate whether a higher Reynolds number could lead to different physics. However, while this scaling issue is certainly problematic in the study of, for example, wave turbulence, one could expect that it is less of a problem in the present case. Indeed, the dynamics presented here is not relying on the existence of a general instability in the flow, but rather on largely inviscid dynamics, with the effect of viscosity being limited to thin boundary layers that are difficult to observe as they are predicted to be of the order of $\unicode[STIX]{x1D6FF}=\sqrt{2\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}}\approx 5\times 10^{-4}$  m, using an approximate value for water viscosity $\unicode[STIX]{x1D708}=1.7\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and a frequency $f=\unicode[STIX]{x1D714}/(2\unicode[STIX]{x03C0})=2.0$  Hz. For example, the eddy we observed is due to the existence of a water jet created by the reduction of the free volume between the grease-ice packs, rather than (directly) viscous effects. In addition, the dynamics analysed presently is relevant for the grease-ice part of the marginal ice zone, where low frequency waves propagate mostly unaffected and high frequency waves, such as those investigated here, are the ones interacting most with the ice.