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Volume 65, Issue 2 p. 247-259
Article
Free Access

Under-ice salinity transport in low-salinity waterbodies

Jason Olsthoorn

Corresponding Author

Jason Olsthoorn

Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia, Canada

Correspondence: [email protected]Search for more papers by this author
Cynthia E. Bluteau

Cynthia E. Bluteau

Institut des sciences de la mer, Université du Quebec à Rimouski, Rimouski, Quebec, Canada

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Gregory A. Lawrence

Gregory A. Lawrence

Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia, Canada

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First published: 14 August 2019
Citations: 5
Associate editor: Leon Boegman

Abstract

In cold and temperate climates, ice typically covers the surface of waterbodies during winter. Many of these systems are also weakly saline where, unlike seawater, the temperature of maximum density, urn:x-wiley:00243590:media:lno11295:lno11295-math-0001, is higher than its freezing temperature, urn:x-wiley:00243590:media:lno11295:lno11295-math-0002. This feature of the equation of state results in a stable temperature stratification when surface waters cool below urn:x-wiley:00243590:media:lno11295:lno11295-math-0003. Conversely, salts excluded from the growing ice can destabilize the underlying water. Previous laboratory and field experiments demonstrated that excluded salts generate localized overturning and downward transport of salt, despite the persistence of a stable temperature gradient. Those experiments were not able to determine the processes responsible for this transport. Here, we use direct numerical simulations to visualize and characterize the plumes generated when ice excludes salt into a stable temperature gradient. We restrict our analysis to times much earlier than the diffusion timescale of temperature over the domain. We define a mass flux parameter that considers the strength of the reverse-temperature stratification relative to the rate of salt exclusion. We identify two types of plumes whose characteristics depend mainly on : double-diffusive salt-fingering plumes and convective plumes. The former encourages transport of salt to the bottom without significantly mixing the temperature stratification, while the latter tends to mix the water column. We apply a scaled mass flux parameter to published laboratory and field observations in low-salinity systems. These limited observations compare favorably with our numerical analysis.

The impact of salt exclusion during ice formation on circulation is well understood for oceans (Carmack 1990), and for aquatic systems with salinities comparable or higher than seawater (Gibson 1999; Dugan and Lamoureux 2011). In these saline systems (i.e., urn:x-wiley:00243590:media:lno11295:lno11295-math-1001 ≳ 24 g L−1), the freezing temperature (urn:x-wiley:00243590:media:lno11295:lno11295-math-0004) is higher than the temperature of maximum density (urn:x-wiley:00243590:media:lno11295:lno11295-math-0005), so both surface heat loss and excluded salts will increase the water density beneath the ice. This density increase typically results in an isothermal and isohaline convection layer that progressively deepens as ice continues to form (e.g., Gibson 1999; Willemse et al. 2004). When ice first forms near shallow boundaries, density currents may also form, which transport near-surface waters to the bottom (e.g., Miller and Aiken 1996)—promoting the formation of stagnant bottom waters (e.g., Ferris et al. 1991). The formation of dense (saline) bottom waters can inhibit oxygen replenishment at depth generating hypoxic, and even anoxic conditions. In some instances, such as for treating mine pit lakes, it can even be desirable to isolate bottom waters from the surface (Pieters and Lawrence 2009). Therefore, understanding the conditions that can lead to these stagnant waters in low-salinity systems is both ecologically and industrially relevant.

Low-salinity systems (urn:x-wiley:00243590:media:lno11295:lno11295-math-1002 ≲ 24 g L−1), where urn:x-wiley:00243590:media:lno11295:lno11295-math-0006, are distinctively different from seawater since water becomes more buoyant as it cools below urn:x-wiley:00243590:media:lno11295:lno11295-math-0007. This reverse temperature stratification partially balances the destabilizing effect of the excluded salts. With increasing salinity, urn:x-wiley:00243590:media:lno11295:lno11295-math-0008 approaches urn:x-wiley:00243590:media:lno11295:lno11295-math-0009, weakening the maximum stability that can be provided by a reverse-temperature gradient beneath the ice. In these aquatic systems, the impact of salt exclusion on transport is still debated. This debate is partly a result of the small amount of excluded salts (Ouellet and Pagé 1987), but also because other processes can cause circulation underneath the ice, such as radiative forcing (e.g., Kirillin and Terzhevik 2011; Kirillin et al. 2012). Some measurements in low-salinity lakes (urn:x-wiley:00243590:media:lno11295:lno11295-math-0010) have shown that localized overturning of 1–2 m thick can occur at discrete depths in the water column (Pieters and Lawrence 2009). These overturning events occurred regularly while the water column remained temperature-stratified throughout winter. Because of the nature of field observations, Pieters and Lawrence (2009) could not determine the processes responsible for redistributing excluded salts in the water column.

Motivated by the above field observations, Bluteau et al. (2017) conducted laboratory experiments over a range of initial salinities (urn:x-wiley:00243590:media:lno11295:lno11295-math-0011) to isolate the influence of salt-exclusion on under-ice circulation. In their experiments, excluded salts generated cascades of localized overturning, while the water column remained reverse-temperature stratified. At the highest salinity (urn:x-wiley:00243590:media:lno11295:lno11295-math-0012), the temperature gradients were regularly erased because the reverse-temperature gradient was insufficient to support the higher amounts of salt excluded by the ice. An essential feature of their experiments was the downward transport of salt (Bluteau et al. 2017). The mechanism responsible for this transport could not be identified from their experimental setup. They hypothesized, like Stewart and Platford (1986) who took measurements in low-salinity lakes, that salt-fingering plumes could be responsible for this transport. This double-diffusive process is theoretically possible when two constituents, with differing molecular diffusivities, stratify the water column, provided the fastest diffusing species (heat) stabilizes the water column, while the slowest diffusing species (salt) destabilizes it (for further details, see Ruddick and Gargett 2003). The problem studied herein represents the cold-climate variant of the salt-fingers produced in tropical waters, when evaporation generates an unstable salinity stratification in an otherwise temperature stable water column. We clarify that this process is different from double-diffusive convection as discussed in Bouffard and Wüest (2019), where salinity is the stabilizing component and temperature is destabilizing.

In this article, we present direct numerical simulations (DNS) used to determine whether double-diffusive salt-fingers could be responsible for transporting salt, and if the characteristics of these plumes varied with the imposed salt exclusion rate and/or with the strength provided by the stable reverse-temperature stratification. We present the model's framework (“Methods” section), which was set up for similar initial salinities, but a wider range of salt exclusion rates than investigated by Bluteau et al. (2017). We then describe the resulting scalar dynamics before comparing our results to the limited field measurements on under-ice circulation in low-salinity waterbodies. We will limit our analysis to early times compared with the diffusive timescale of heat over the domain depth.

Methods

To identify the small-scale processes responsible for transporting excluded salts through a reverse-temperature gradient, we performed two-dimensional DNS based on the physical experiments described in Bluteau et al. (2017). In their experiments, an insulated container measuring 0.31 m (length) × 0.19 m (width) × 0.30 m (depth) was filled with a potassium chloride solution at room temperature to a depth of 0.245 m. The insulated container, with an exposed top, was then placed into a freezer where the air temperature was urn:x-wiley:00243590:media:lno11295:lno11295-math-0013. The initial water temperature (urn:x-wiley:00243590:media:lno11295:lno11295-math-1003) was above its temperature of maximum density (urn:x-wiley:00243590:media:lno11295:lno11295-math-0014) such that heat lost at the surface destabilized the water column. This heat loss resulted in convection and a near-uniform fluid temperature throughout the entire water column. The water temperature eventually cooled to urn:x-wiley:00243590:media:lno11295:lno11295-math-0015, at which point continued heat loss at the surface created a stable reverse-temperature stratification. The surface then rapidly cooled to its freezing temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0016, initiating ice formation, and the downward transport of excluded salts throughout the underlying water (Bluteau et al. 2017).

Brine rejection at the ice-water interface

We use the results of Bluteau et al. (2017) and their analysis of the ice formation rates to set our surface boundary conditions. They demonstrated, by comparing their ice thicknesses to the theoretical growth curves for thin ice (Ashton 1989), that surface ice grew nearly linearly with time, consistent with field observations in the early stages of ice formation. The authors also showed that the ice excluded approximately 95% of the salt, consistent with observations in low-salinity lakes (Belzile et al. 2002; Pieters and Lawrence 2009). Combining their fraction of excluded salts f with the linear ice growth model of Ashton (1989), Bluteau et al. (2017) suggested the following salt exclusion rate model during the early stages of ice formation:
urn:x-wiley:00243590:media:lno11295:lno11295-math-0017(1)
Here, L is the latent heat of fusion for water (3.3 × 105 J kg−1), urn:x-wiley:00243590:media:lno11295:lno11295-math-0018 is the density of ice (917 kg m−3), and Hia is the heat transfer coefficient between the ice and overlying air, which depends on the prevailing wind conditions (Ashton 1989). The value for still air ( Hia = 10 W m−2 ° C−1) agreed well with the experimental results of Bluteau et al. (2017). We estimate the freezing temperature (urn:x-wiley:00243590:media:lno11295:lno11295-math-0019) as that of diluted seawater
urn:x-wiley:00243590:media:lno11295:lno11295-math-0020(2)
given by Chen and Millero (1986). Equation 2 differs slightly from the one presented by Bluteau et al. (2017), since they used potassium chloride solutions for their experiments.
Bluteau et al. (2017)'s model (see Eq. 1) assumes that during the early stages of ice formation, the salt exclusion rate depends primarily on the initial salinity of the water (urn:x-wiley:00243590:media:lno11295:lno11295-math-0021) and the overlying air temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0022. In their laboratory experiments, urn:x-wiley:00243590:media:lno11295:lno11295-math-0023 was kept approximately constant, implying that the salt exclusion rate Fi was also approximately constant for a given initial salinity urn:x-wiley:00243590:media:lno11295:lno11295-math-0024. An increase in urn:x-wiley:00243590:media:lno11295:lno11295-math-0025 will result in an increase in Fi, while the maximum reverse temperature gradient weakens as the temperature of maximum density (urn:x-wiley:00243590:media:lno11295:lno11295-math-0026) for diluted seawater (Chen and Millero 1986),
urn:x-wiley:00243590:media:lno11295:lno11295-math-0027(3)
approaches its freezing temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0028 (Eq. 2). By varying urn:x-wiley:00243590:media:lno11295:lno11295-math-0029, Bluteau et al. (2017) simultaneously increased the salt exclusion rate, while decreasing the strength of the reverse temperature stratification. The present numerical study will control these two effects independently to highlight their role in transporting scalars.

Numerical framework

A sketch of the idealized brine rejection model proposed by Bluteau et al. (2017) is provided in Fig. 1a. As the ice thickness h grows with time t, brine rejection results in a salt flux into the liquid water underneath the ice, which then creates a saline boundary layer of thickness δ. The ice thickening depends on the atmospheric temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0030, while the initial salinity urn:x-wiley:00243590:media:lno11295:lno11295-math-0031 determines both the freezing temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0032 and urn:x-wiley:00243590:media:lno11295:lno11295-math-0033. To quantify the mechanism transporting the excluded salts throughout the underlying water column, we rely on two-dimensional DNS. Two-dimensional simulations will create larger length scales and tend to under-predict the vertical temperature and salinity fluxes when compared to three-dimensional simulations (Traxler et al. 2011). Here, we will restrict ourselves to two-dimensional DNS and will analyze three-dimensional simulations in future work.

Details are in the caption following the image
Diagram of the (a) temperature and salinity profiles and the ice-water boundary conditions. A sketch (b) of the numerical domain in which the dynamics of the liquid water under the ice are modeled. The numerical domain is periodic in the horizontal direction. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 1b is a schematic of the numerical implementation of the ice exclusion problem depicted in Fig. 1a. The numerical domain was square with depth denoted as H. As discussed above, the temperature (urn:x-wiley:00243590:media:lno11295:lno11295-math-1004) and salinity (urn:x-wiley:00243590:media:lno11295:lno11295-math-1005) fields are forced by boundary conditions set by the ice-water interface. That is, the temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-1006 beneath the ice was set to urn:x-wiley:00243590:media:lno11295:lno11295-math-0034, while we prescribed a constant salt flux at the surface. We ignore radiative forcing, which can enhance mixing beneath clear ice, to focus on the convection induced by excluded salts.

For our simulations, we used a simplified equation of state for brackish water. The density (urn:x-wiley:00243590:media:lno11295:lno11295-math-1007) of water between urn:x-wiley:00243590:media:lno11295:lno11295-math-0035 and urn:x-wiley:00243590:media:lno11295:lno11295-math-0036 depends (nearly) quadratically on temperature and linearly with salinity. In shallow waters, we can neglect pressure effects on the equation of state, and so we write
urn:x-wiley:00243590:media:lno11295:lno11295-math-0037(4)

Here, urn:x-wiley:00243590:media:lno11295:lno11295-math-0038, and urn:x-wiley:00243590:media:lno11295:lno11295-math-0039 is the fit coefficient that plays the role of the thermal-expansion coefficient and is dependent upon the reference salinity urn:x-wiley:00243590:media:lno11295:lno11295-math-0040. The haline contraction coefficient β = 8 × 10−4(g kg−1)−1. This equation of state, similar in form to that of Pieters and Lawrence (2009), is a simplified version of that given by Chen and Millero (1986). For freshwater, the maximum error of Eq. 4 is under 2% of the maximum density variation between Tf and urn:x-wiley:00243590:media:lno11295:lno11295-math-0041. Equation 4 greatly simplifies the analysis compared to the more complete equation of state of Chen and Millero (1986).

The numerical simulations were performed in nondimensional coordinates to reduce the number of free parameters. As a spatial scale, we used the vertical depth of the domain H, while a natural choice of time-scale was the time heat takes to diffuse over this distance H via molecular diffusion κ T. These choices of characteristic scales define the following nondimensional parameters for time (t), position vector (x), fluid velocity (u), and pressure (P):
urn:x-wiley:00243590:media:lno11295:lno11295-math-0042(5)
Here, bold-face variables denote vector quantities. We further nondimensionalize the temperature field using the maximum possible temperature difference for the reverse-temperature system urn:x-wiley:00243590:media:lno11295:lno11295-math-0043. Similarly, we nondimensionalize salinity by the amount of salt excluded over the diffusive time-scale (t κ). As such,
urn:x-wiley:00243590:media:lno11295:lno11295-math-0044(6)

We fix the urn:x-wiley:00243590:media:lno11295:lno11295-math-0045 and urn:x-wiley:00243590:media:lno11295:lno11295-math-0046 according to urn:x-wiley:00243590:media:lno11295:lno11295-math-0047, therefore ignoring small changes in salinity during the simulations. With the above definitions, T varies from −1 to 0, which corresponds to a temperature range of urn:x-wiley:00243590:media:lno11295:lno11295-math-0048 to urn:x-wiley:00243590:media:lno11295:lno11295-math-0049, respectively. The initial salinity was initially assumed to be well mixed (urn:x-wiley:00243590:media:lno11295:lno11295-math-0050) and set S = 0. The salinity increased over the simulation, such that its spatially averaged value <S>A would be 1 at t = 1. In this article, we focus on the initial dynamics and restrict our analysis to t ≪ 1.

Four dimensionless parameters characterize the numerical simulations:
urn:x-wiley:00243590:media:lno11295:lno11295-math-0051(7)
where g is the gravitational acceleration, ν is the molecular viscosity and κ S is the diffusivity of salt. Here, urn:x-wiley:00243590:media:lno11295:lno11295-math-0052 represents the maximum density difference due to the temperature for a reverse-temperature system. The Rayleigh number (Ra) characterizes the strength of the temperature stratification. As temperature is stabilizing, Ra quantifies how quickly the reverse temperature stratification responds to perturbations relative to a diffusive response. The Prandtl number (Pr) is the ratio of viscosity to molecular heat diffusion, while τ represents the ratio between heat and salt diffusion. The physical values of the fluid viscosity and constituent diffusivities prescribe Pr and τ (we select Pr = 7 and τ = 100). As water approaches its freezing temperature, the Pr will also increase to ≈13 (see Lide 2006, p. 136 for details of the fluid properties with temperature). Our study investigates the potential types of transport regimes, and so accounting for this increase Pr is beyond our article's scope. Last, the mass ratio ( ) characterizes the density contribution of the temperature stratification (urn:x-wiley:00243590:media:lno11295:lno11295-math-0053) relative to the increase in mean salt concentration over one nondimensional time unit, i.e., over the diffusive timescale for temperature urn:x-wiley:00243590:media:lno11295:lno11295-math-0054. There are thus only two free dimensionless quantities that control the dynamics of this system: Ra and . To quantify the effect of these two parameters on the transport of salt through the reverse-temperature stratification, our numerical experiments encompass a range of salinity, air temperatures, and hence salt-exclusion rates (Fig. 2).
Details are in the caption following the image
Variation of (a) ℜ and (b) the imposed salt fluxes Fi as a function of air temperature Ta and initial salinity So with the simulated conditions denoted by the gray circles. The black line in (a) represents ℜ = 0.25, the approximate transition between the salt-fingering plumes and convective plumes, while the variation of Ra is shown in (b) via the secondary blue y-axis. [Color figure can be viewed at wileyonlinelibrary.com]
The fluid equations of motion are the Navier–Stokes equations under the Boussinesq approximation. With our nondimensionalization, these equations are written
urn:x-wiley:00243590:media:lno11295:lno11295-math-0055(8)
urn:x-wiley:00243590:media:lno11295:lno11295-math-0056(9)
urn:x-wiley:00243590:media:lno11295:lno11295-math-0057(10)
Over the limited temperature range considered in this article (−1 ≤ T ≤ 0), we use the simplified equation of state (Eq. 4) in nondimensional form
urn:x-wiley:00243590:media:lno11295:lno11295-math-0058(11)
Here, we define the density perturbation as urn:x-wiley:00243590:media:lno11295:lno11295-math-0059. The boundary conditions at the surface (z = 0.5) and the bottom (z = −0.5) simplify to
urn:x-wiley:00243590:media:lno11295:lno11295-math-0060(12)

We numerically implement the model equations using Dedalus, a pseudospectral partial differential equation solver (Burns et al. 2016). We use fast Fourier transforms in the horizontal to impose the periodic boundary conditions, and a Chebyshev grid in the vertical, which clusters grid points at the boundaries (see Fig. 1b), to resolve the thin surface boundary layer generated by excluded salts. We use periodic lateral boundary conditions instead of rigid walls since our study focuses on the growth and evolution of the saline plumes rather than their interaction with the sidewalls. The width of the numerical domain is much larger than any individual saline plume. The numerical scheme implements a fourth-order Runge–Kutta (explicit) time-stepping scheme. We performed all the simulations with Nx × Nz = 1024 × 512 grid points. Due to the reverse-energy cascade in two-dimensional simulations, the smallest scales in the flow will be comparable to the thickness of the top salinity boundary layer. With the grid clustering near the boundary, we have more than 20 grid points resolving the mean salinity boundary layer. We performed grid resolution studies, which demonstrated that the simulations adequately resolved the smallest scales for our problem.

In our numerical setup, the fluid volume was specified to be constant, and so the ice-water interface was always located at z = 0.5. That is, the total height of the domain, H, is constant. Two physical processes would decrease the total volume of the liquid water in the system. First, water becomes denser as it cools, which due to the conservation of mass, results in a smaller fluid volume. For our simulations, water can be cooled from the urn:x-wiley:00243590:media:lno11295:lno11295-math-0061 to urn:x-wiley:00243590:media:lno11295:lno11295-math-0062, leading to a negligible depth change of approximately 0.01%. Second, as liquid water changes to ice, the ice-water interface moves downward (Fig. 1a). For the coldest air temperatures simulated (−28°C), about 7% of the initial water volume is expected to convert into ice at t = 0.125. For milder air temperatures (−5°C), about 1% of the water volume would be converted into ice. We, therefore, restrict the simulations to t ≤ 0.125 (urn:x-wiley:00243590:media:lno11295:lno11295-math-1008 ≤ 2.4 h) to neglect changes in the fluid volume.

Parameter space evaluation

As detailed above, the current problem depends on two free dimensionless parameters: the Rayleigh number (Ra) and the mass flux ratio ( ). We performed a series of numerical simulations designed to cover a wide range of by controlling the destabilizing salt flux Fi independently of the stabilizing temperature stratification urn:x-wiley:00243590:media:lno11295:lno11295-math-0063. In total, 13 cases were simulated at two different Ra by keeping H = 0.1 m constant, while allowing urn:x-wiley:00243590:media:lno11295:lno11295-math-0064 and urn:x-wiley:00243590:media:lno11295:lno11295-math-0065 to vary (Table 1 and Fig. 2). For the first six simulations, Ra = 8.40 × 106 given by setting the initial salinity to urn:x-wiley:00243590:media:lno11295:lno11295-math-0066. For these runs, was varied between 0.25 and 5, which is equivalent to increasing the air temperatures from −28°C to −1.5°C—mimicking the impact of decreasing the salt flux Fi entering the surface by an order of magnitude. Runs 7–13, simulating a lower Ra = 3.97 × 106, are equivalent to a higher salinity of urn:x-wiley:00243590:media:lno11295:lno11295-math-0067. Thus, lower could be modeled over the range of air temperatures encountered in natural systems (Fig. 2). Our simulations covered a wider range of air temperatures and salt exclusion rates, and therefore a wider range of , than the laboratory experiments of Bluteau et al. (2017). We have considered a limited range of Ra since it is weakly affected by urn:x-wiley:00243590:media:lno11295:lno11295-math-0068.

Table 1. Details of the numerical experiments, each performed with H = 0.1 m, ν = 10−6 m2 s−1, and 1024 × 512 grid points in the horizontal (x) and vertical (z) planes. Here, Pr = 7 and τ = 100. We used (Eq. 1) to estimate the imposed salt fluxes Fi from the air temperatures Ta associated with the simulated . We also included the computed δ based upon the boundary layer thickness from the simulations.
Run # urn:x-wiley:00243590:media:lno11295:lno11295-math-0069 (g kg−1) Ta (°C) F i × 10−7 (kg m−2 s−1) Ra × 106 δ
1 1 −28 8.7 8.40 0.25 0.22
2 1 −14 4.3 8.40 0.5 0.39
3 1 −5.7 1.7 8.40 1.25 0.80
4 1 −2.9 0.87 8.40 2.5 1.4
5 1 −1.9 0.58 8.40 3.75 2.0
6 1 −1.5 0.44 8.40 5 2.5
7 8 −17 41 3.97 0.025 0.03
8 8 −8.8 21 3.97 0.05 0.05
9 8 −3.8 83 3.97 0.125 0.11
10 8 −2.1 4.1 3.97 0.25 0.19
11 8 −1.3 2.1 3.97 0.5 0.32
12 8 −0.8 0.83 3.97 1.25 0.66
13 8 −0.6 0.41 3.97 2.5 1.2

Results

Plume characteristics as a function of and Ra

To illustrate the scalar transport generated by excluded salts, we present temperature (T), salinity (S), and density (ρ) snapshots for Ra = 8.40 × 106 and = 2.5 (Fig. 3), along with their horizontal average, denoted as <·> H = ∫·dx. Snapshots are presented at t = {0.01, 0.02, 0.03, 0.04}. At the start of each simulation, the salt accumulated in a thin layer at the top of the domain (Fig. 3e). This layer was statically unstable because the destabilizing salt gradient overwhelmed the stable reverse-temperature stratification (Fig. 3m). The relatively cold and saline layer continued to grow until it reached a critical thickness at which point an instability developed (Fig. 3f), generating saline plumes that descended through the reverse-temperature stratification (Fig. 3g). Shortly after the onset of the plumes' descent, the thickness of the unstable surface layer quickly decreased (Fig. 3m–p) and remained approximately constant throughout the remainder of the simulation. Distinct plumes, which often merged (e.g., Fig. 3g,h), continued to emanate from the relatively saline and cold surface layer. In the temperature snapshots, the plumes were less evident, although the relatively cold plumes displaced the isotherms (see Fig. 3c,d). The saline plumes perturbed the diffusely evolving temperature field, while transporting relatively saline waters to the bottom (Fig. 3g,h,k,l).

Details are in the caption following the image
Snapshots of the nondimensional temperature T (ad), salinity S (eh), and density ρ (il), along with a profile of their horizontal averages (mp) at times t = 0.01 (a, e, i, m), t = 0.02 (b, f, j, n), t = 0.03 (c, g, k, o), and t = 0.04 (d, h, l, p) for run 4 with Ra = 8.40 × 106 and = 2.5. [Color figure can be viewed at wileyonlinelibrary.com]

During their descent, the plumes were always denser than the ambient fluid and their characteristics varied with . Figure 4 is a series of snapshots at the same times, t, as Fig. 3 but for = 0.125 and Ra = 3.97 × 106. For this lower , salinity dominated the plume dynamics such that the stabilizing effect of the temperature field was weak. As such, the instability in the upper boundary layer developed faster, and the plumes were already descending at t = 0.01 (Fig. 4e). Similarly, the temperature of the plumes did not significantly modify their density such that when they emanated from the saline surface layer, the plumes were denser than those created at higher = 2.5 (Fig. 3). Furthermore, each plume was thinner and descended more rapidly through the reverse-temperature stratification, resulting in isotherms displacing more vigorously (Fig. 4c,d). That is, with decreasing , the plumes' horizontal length scale decreased (Fig. 3g vs. Fig. 4g), while the vertical temperature flux increased.

Details are in the caption following the image
Similar to Fig. 3 but for run 9 with Ra = 3.97 × 106 and = 0.125. [Color figure can be viewed at wileyonlinelibrary.com]

For = 0.125, salinity dominated the plumes' buoyancy and they accelerated until reaching the bottom of the domain. Conversely, for = 2.5, the temperature stratification was significant such that the plumes were marginally denser than the surrounding fluid (Fig. 3g) and became noticeably more dense as they descended through the reverse-temperature stratification (Fig. 3k,l). The plumes essentially gained heat faster than they lost salt becoming dominated by double-diffusive processes with increasing . Thus, we have identified two fluid regimes for the plumes that are parameterized by : a double-diffusive regime where the salt plumes do not significantly modify the temperature stratification, and a convective regime where the salt-exclusion vigorously mixes the temperature field. The distinction between the two parameter regimes is further illuminated in the Supporting Information Videos S1, S2 that correspond to the parameters of Figs. 3, 4.

Unlike the mass flux ratio , the plumes' characteristics depended only weakly on Ra. Figure 5 contains the horizontally averaged T, S, and ρ depth-time contours for four different runs. We present two runs with the same = 0.25 but different Ra = 3.97 × 106 (Fig. 5d–f) and Ra = 8.40 × 106 (Fig. 5g–i). From these two examples, Ra only weakly modifies the horizontally averaged fields, except near the surface before plumes start descending (Fig. 5e vs. h, t < 0.01). At lower Ra, the unstable surface layer became thicker and took longer before generating plumes (Fig. 5e). Figure 5 also presents the results for the low = 0.125 and high = 2.5 examples presented above, highlighting that the plumes depended more on than on Ra. For the low example, convective-type plumes were generated that weakened the background temperature gradient (Fig. 5a). Conversely, for high , the “double-diffusive” plumes only weakly perturbed the background temperature stratification (Fig. 5j). This persistent temperature stratification, in turn, enables more salt to accumulate in the upper third of the domain, giving more structure to the salinity profiles (Fig. 5b,e,h,k). We have considered a limited range of Ra due to its weak dependence on S0.

Details are in the caption following the image
Depth-time contours of the horizontally averaged (a, d, g, j) temperature <T> H, (b, e, h, k) salinity <S> H, and (c, f, i, l) density <ρ> H for various simulations. Results are presented for (ac) Ra = 3.97 × 106 and = 0.125; (df) Ra = 3.97 × 106 and = 0.25; (gi) Ra = 8.40 × 106 and = 0.25; and (jl) Ra = 8.40 × 106 and = 2.5. Vertical dashed lines represent the time of the snapshots presented in Figs. 3, 4. The two center columns show simulations at the same but different Ra demonstrating that the flow dynamics are dominated by . [Color figure can be viewed at wileyonlinelibrary.com]

Temperature time series

In Fig. 6, we present temperature time series at four depths for various simulated at Ra = 3.97 × 106 to provide later, in our discussion, a more direct comparison with the types of measurements collected in previous laboratory and field measurements. This figure additionally includes the time series predicted for a purely diffusive temperature profile, where the salt exclusion rate is zero (  → ∞). Without an imposed salt flux, the temperatures at each depth decrease monotonically and smoothly with time via molecular diffusion (Fig. 6a). As  → 0, the saline plumes increasingly perturb the diffusive temperature profile creating oscillations in the temperature time series (Fig. 6b–e). These oscillations were more pronounced near the surface (e.g., z = 0.375 in Fig. 6c) and occurred more frequently with decreasing . In the double-diffusive regime (  ≳ 0.25), the temperature time series resemble the diffusive case superimposed with small-amplitude oscillations (Fig. 6b,c), while the time series for  ≲ 0.25 were more consistent with a convectively mixed water column (Fig. 6d,e). In the convective-regime, the imposed salt flux disrupted the temperature gradients, resulting in larger vertical heat fluxes.

Details are in the caption following the image
Temperature time-series at four vertical positions (z = {−0.375, −0.125, 0.125, 0.375}) for Ra = 3.97 × 106 but differing : (a)  → ∞, (b) = 2.5, (c) = 0.5, (d) = 0.125, and (e) = 0.05. The horizontal position of the time series was arbitrarily chosen to be x = −0.5. [Color figure can be viewed at wileyonlinelibrary.com]

Heat transport

Initially, brine rejection created a laminar, saline, surface boundary layer, which grew in time. Associated with that boundary layer was an increase in the potential energy of the system. Once the boundary layer became dynamically unstable, the potential energy of the boundary layer was released, and the kinetic energy peaked. Figure 7a is a plot of the area-averaged (<·> A = ∫·dA) kinetic energy (per unit density) as a function of time. After the initial peak, the kinetic energy decreased to a relatively constant production rate in balance with the potential energy generated by the brine rejection—implying the surface salinity flux balanced the salt transported by the descending plumes. The increase in fluid transport (i.e., kinetic energy) also increased the temperature perturbation flux (Fig. 7b), which we define as
urn:x-wiley:00243590:media:lno11295:lno11295-math-0070(13)
Details are in the caption following the image
(a) Area-averaged kinetic energy (per unit density) as a function of time. The inset demonstrates that the flow was dynamically stable until the surface boundary layer reached a critical thickness. (b) Area-averaged temperature perturbation flux throughout the domain, which were normalized in (c) with the surface heat flux. These normalized fluxes were temporally averaged in (d) over the period after the boundary layer initially became unstable (see horizontal lines in c) to highlight their dependency on . The solid line in (d) corresponds to urn:x-wiley:00243590:media:lno11295:lno11295-math-0071. Only runs 1–6 for Ra = 8.40 × 106 are presented in (ac) to avoid data overlap, while (d) presents all runs. [Color figure can be viewed at wileyonlinelibrary.com]
This quantity, FT, denotes the nondiffusive vertical heat transport within the water column. In the simulations, the vertical heat transport balanced the heat flux across the surface boundary layer. Therefore, the area-averaged FT, normalized by the surface heat flux, was relatively constant over time (Fig. 7c). To illustrate how this quantity varied with , we time-averaged it over the period when the surface boundary layer generated plumes (Fig. 7d). For  > 0.25, the normalized heat flux decreased with , roughly as
urn:x-wiley:00243590:media:lno11295:lno11295-math-0072(14)
while for  ≲ 0.25, the normalized heat flux approached a constant value (Fig. 7d). These observed scalings were not significantly affected by the simulated Ra, but were distinctly different between the convective (  ≲ 0.25) and double-diffusive (  ≳ 0.25) regimes.

Saline boundary layer thickness

The mass flux ratio ( ) is a bulk density ratio between the strength of the reverse-temperature stratification and the mean increase in salinity over the diffusive timescale. To define the mean increase in salinity, we averaged the salinity flux from the boundary over the domain depth H. An alternative to the total depth for computing the mean salinity increase would be to average over the thickness of saline boundary layer δ (Fig. 1a). Obtaining an a priori estimate of δ is challenging due to the multicomponent (T and S) nature of the flow. We can, nonetheless, empirically investigate δ from the simulation results.

Similar to Howard (1966), we investigate the mean turbulent boundary thickness over a range of nondimensional parameters. We define the boundary layer thickness as the distance from the boundary to where the salinity gradient diminishes to 10% of its boundary value:
urn:x-wiley:00243590:media:lno11295:lno11295-math-0073(15)

This prescribed cutoff criterion provides a consistent method of determining δ across the different numerical simulations. As discussed above, the saline boundary layer initially grew diffusively until it reached some critical thickness, at which point it became unstable. The subsequent instability resulted in a contraction of the boundary layer, where the forming saline plumes at the base of the layer transported saline fluid into the interior of the domain (see Fig. 8a). The boundary layer thickness (δ) for each simulation was computed and determined to be approximately constant with time after the flow became unstable. Figure 8b is a plot of the time-averaged turbulent boundary-layer thickness urn:x-wiley:00243590:media:lno11295:lno11295-math-0074 as a function of . We observe that urn:x-wiley:00243590:media:lno11295:lno11295-math-0075 depended weakly on ( δ~0.17), increasing by less than a factor of three over the two orders of magnitude that we varied . Similarly, there is a weak dependence of urn:x-wiley:00243590:media:lno11295:lno11295-math-0076 on the Rayleigh number.

Details are in the caption following the image
(a) Boundary layer thickness for three different simulations at Ra = 3.97 × 106. (b) Computed mean turbulent salinity boundary layer thickness (urn:x-wiley:00243590:media:lno11295:lno11295-math-0077) as a function of . The dashed line in (b) represents δ = 4l. [Color figure can be viewed at wileyonlinelibrary.com]
The salinity gradient at the ice-water interface is fixed, enabling us to define a compensation depth (~l); the depth at which the salinity gradient would balance urn:x-wiley:00243590:media:lno11295:lno11295-math-0078. That is,
urn:x-wiley:00243590:media:lno11295:lno11295-math-0079(16)
The top saline boundary condition (urn:x-wiley:00243590:media:lno11295:lno11295-math-0080) then enables us to nondimensionalize the compensation depth for comparison with the simulated results. Here,
urn:x-wiley:00243590:media:lno11295:lno11295-math-0081(17)

This definition provides a physical interpretation for the critical presented in Fig. 7d. When the boundary layer thickness (δ) is less than this compensation length (δ ≲ 4l), the salinity of the forming plumes is insufficient to overcome the strength of the stable temperature stratification (urn:x-wiley:00243590:media:lno11295:lno11295-math-0082). Therefore, the dynamics of these plumes are controlled by double-diffusive processes. Conversely, if the boundary layer thickness is greater than this compensation length (δ ≳ 4l), the saline plumes are such that urn:x-wiley:00243590:media:lno11295:lno11295-math-0083 and convective plumes are generated. We include this critical value of δ = 4l as a dashed line in Fig. 8b.

The model parameters for the numerical simulations were computed for a fixed value of H, the maximum depth of the domain. However, as discussed above, the critical parameter that determines the plume regime (double-diffusive or convective) is the ratio urn:x-wiley:00243590:media:lno11295:lno11295-math-0084. For comparisons with field measurements, we define a rescaled , denoted δ
urn:x-wiley:00243590:media:lno11295:lno11295-math-0085(18)
which is defined in terms of δ. In the numerical simulations, τ = 100 and δ ≈ 0.01 such that  ≈  δ. We thus expect a change in plume dynamics, from convective to double-diffusive plume regimes, for δ ≈ 0.25. We have included the computed value of δ for each simulation in Table 1.

Discussion and conclusions

We demonstrated that when growing ice excludes salt into a reverse-temperature system, the scalar evolution primarily depends on two free parameters: the Rayleigh number (Ra) and the mass flux ratio ( ). In our simulations, the imposed salt flux was initially constrained to a boundary layer at the surface, which slowly grew with time. Once this layer reached a critical thickness, the system became dynamically unstable and plumes formed, which descended through the water column. The structure of the relatively cold and saline plumes depended primarily on the mass flux ratio . This bulk quantity, , characterizes the relative density contribution of the reverse-temperature stratification to the mean increase in salinity caused by excluding salts over the diffusive timescale. For large , double-diffusive processes affected the plumes since they became denser by gaining heat faster than losing salt during their descent. These double diffusive plumes mixed the reverse-temperature stratification significantly less than the convective plumes at low . With decreasing , the system evolved more energetically, while the water column lost significantly more heat. Overall, lower salinity environments can support stronger reverse-temperature gradients, requiring a higher salt exclusion rate (i.e., faster-growing ice) for convective plumes to be generated.

The critical that separates the convective and double-diffusive plume regimes depends on the boundary layer thickness δ. Thus, we define a scaled δ (Eq. 18), where  ≈  δ in the numerical simulations. We predict the likelihood of forming convective rather than double-diffusive plumes by estimating δ for three distinct studies: the laboratory measurements of Bluteau et al. (2017), the field measurements from Tailings Lake (Pieters and Lawrence 2009), and those from Base Mine Lake (Lawrence et al. 2016). To compare the laboratory and field studies with our simulations, we evaluated δ in Eq. 18 by setting δ = 1 mm, in agreement with the numerical simulations (Fig. 8). We considered brine rejection rates that are similar to field conditions, and therefore we expect that δ will have a comparable value to that determined here. However, complex field conditions, such as cross-flow, may modify this estimate. We further assumed a steady salinity flux from the ice in these studies, despite the prevailing meteorological conditions and change in ice growth rates, which will change the salt fluxes overtime (see Ashton 1989).

The laboratory experiments of Bluteau et al. (2017) covered a range of salt exclusion rates and urn:x-wiley:00243590:media:lno11295:lno11295-math-0086, yielding δ between 0.05 and 0.52. The temperature time series from their low-salinity experiment (urn:x-wiley:00243590:media:lno11295:lno11295-math-0087 and δ ≈ 0.52), resemble those obtained from our simulations within the double-diffusive regime (Fig. 6c). The laboratory temperature gradients in the lower three-quarters of the tank appeared weaker—likely because of heat loss through the tank's sidewalls (Bluteau 2006). For their most saline experiments, we estimate a value of δ ≈ 0.045, which is within the convective-regime. Hence, the laboratory time series displayed features consistent with the simulations conducted at an equivalent δ (e.g., Fig. 6e). In fact, fig. 2 of Bluteau et al. (2017) agrees remarkably well with Fig. 6. This comparison supports the notion that the flow regime, and the associated vertical heat transport, can be categorized according to δ.

Previous field studies of saline waterbodies motivated both our current discussion and the laboratory work of Bluteau et al. (2017). In particular, Pieters and Lawrence (2009) observed localized overturning at discrete depths in Tailings Lake, despite the presence of a reverse-temperature stratification that progressively weakened throughout winter. In our current framework, we can estimate δ for this lake (Table 2). The estimated δ ≫ 0.25 suggests that plumes—dominated by double-diffusive processes—transported salt throughout the water column. Measured temperature time series during the winter in another lake, Base Mine Lake, also displayed perturbations despite the presence of stable reverse-temperature gradients (Lawrence et al. 2016). Similar to Tailings Lake, its relatively low salinity and salt exclusion rates yield a high δ ≈ 1 (Table 2). In higher salinity waterbodies, we would expect lower δ because of the (likely) higher salt exclusion rates Fi and the weakened reverse-temperature stratification urn:x-wiley:00243590:media:lno11295:lno11295-math-0088 as the urn:x-wiley:00243590:media:lno11295:lno11295-math-0089 approaches Tf (Fig. 2). At these lower δ, the formation of convective plumes is more likely, promoting the creation of well-mixed convective layers. Gibson (1999) documented, in many aquatic systems of Antarctica, convective layers that progressively deepened as ice formed. Of the 21 lakes surveyed, about eight were less saline than seawater near the surface with the lowest values of about 4 g L−1 recorded in Ephyra Lake. These lakes are ice-free for a few months at most, with ice formation beginning in February or March, i.e., late summer (Gibson 1999). The total duration of ice growth for Ephyra lake is unknown, but given the 2-m thick ice cover present in December (late spring), we expect F i ≳ 3 × 10−7 kg m−2 s−1, and so δ ≲ 0.25 for both Ephyra and the other lakes. These field studies suggest that δ can be used as a bulk quantity to predict whether salt exclusion will promote convection or the transport of salt through double-diffusive processes.

Table 2. Brine rejection parameters associated with previous laboratory and field experiments. Here, Fi is the mean brine rejection rate as estimated by Bluteau et al. (2017) and T i is an estimate of the depth-averaged water temperature prior to ice-on. For the laboratory experiments, urn:x-wiley:00243590:media:lno11295:lno11295-math-0090. As no measurements of the boundary layer thicknesses exist for the field and laboratory experiments, we set urn:x-wiley:00243590:media:lno11295:lno11295-math-0091, in agreement with our simulations.
Mean depth (m) Summer salinity (urn:x-wiley:00243590:media:lno11295:lno11295-math-0092) F i (×10−7 kg m−2 s−1) T i (°C) δ
Laboratory 0.245 1 4.3 3.8 0.52
Laboratory 0.245 8 34 2.7 0.045
Tailings Lake 7 0.9 0.7 1.5–2 2.1–2.5
Base Mine Lake 9 2.0 1.1 1–1.5 0.95–1.2

In field settings, several other factors may influence the transport of excluded salts. Solar radiative heating through the ice would reduce the strength of the reverse-temperature stratification beneath the ice (see Kirillin et al. 2012), effectively reducing δ and favoring the creation of convective plumes. The presence of a background flow, such as those induced by internal waves underneath the ice, may also mix and homogenize the plumes with the background fluid, potentially disrupting their ability to descend through the water column. Further work is thus required to develop theoretical predictions, applicable under field conditions, that can quantify transport caused by salt exclusion during ice formation in reverse-temperature systems. In particular, the numerical simulations considered here did not the effect of change domain size on the heat and saline fluxes. Future work will quantify the effect of domain size and, thus, a much wider range of Raleigh number. We are in the process of constructing laboratory experiments capable of visualizing the flow field induced by the downward-propagating brackish plumes.

We defined a simple nondimensional parameter δ to differentiate the plumes resulting from brine rejection into two possible regimes: convective or double-diffusive. The latter encourages the formation of a more stable water column, while the former tends to mix the water column. Our article thus presents a simple framework to categorize the impact of brine rejection during ice formation on the stability of low-salinity aquatic systems.

Acknowledgments

CEB thanks the Natural Sciences and Engineering Research Council of Canada for funding a postdoctoral fellowship to carry out the work. JO was funded under a Canada Collaborative Research and Development Grant from Syncrude Canada Limited and the Natural Sciences and Engineering Research Council of Canada. The authors also thank Jeff Carpenter for helpful discussions on the boundary conditions during the early stages of this project. We thank two anonymous referees who provided helpful comments when reviewing this article.

    Conflict of Interest

    None declared.