Morphodynamic modeling of the basal boundary of ice cover on brackish lakes


  • Luca Solari,

    Corresponding author
    1. Department of Civil and Environmental Engineering, University of Florence, Florence, Italy
    • Corresponding author: L. Solari, Department of Civil and Environmental Engineering, University of Florence, via S. Marta 3, 50139 Florence, Italy. (

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  • Gary Parker

    1. Department of Civil and Environmental Engineering and Department of Geology, Hydrosystems Laboratory, University of Illinois, Urbana, Illinois, USA
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[1] The analysis considers a cold-region brackish lake that develops an ice cover on its surface in winter. As the ice cover thickens due to freezing, black ice grows on the bottom side. The freezing process excludes the salt from the ice, resulting in an increase in dissolved salt concentration in the lake water just below the bottom of the ice. The density profile in this type of lake is then governed by two opposing factors: the water temperature distribution generally produces a stable stratification, whereas the excess salinity produced by exclusion tends to increase the density of the upper layers of the lake, resulting in an unstable contribution to stratification. Competition between these two factors can lead to unstable density gradients, so an initially motionless lake with a slowly thickening plane ice cover develops a convective flow field. This flow field can then feed back into the evolution of the morphology of the ice cover itself, resulting in a morphodynamic interaction between water and ice. This paper aims at investigating this morphodynamic instability. A mathematical model for the stability of a lake covered with an initially plane ice cover growing in time is proposed here. The results reveal threshold values of the main dimensionless parameters, expressed in the form of Rayleigh numbers, and in particular define a ratio between buoyancy effects and diffusivity that governs neutral stability conditions. When the system is unstable, i.e., for Rayleigh numbers above ~103 depending on the values of the input parameters, the analysis predicts the growth of convective flow circulation cells responsible for the morphodynamic evolution of the ice-water interface.

1 Introduction

[2] Undulating morphologies such as ripples are commonly seen under ice cover in rivers and lakes, and in glacial meltwater streams [e.g., Carey, 1966; Ashton and Kennedy, 1972; Parker, 1975; Fuhs et al., 1980]. Stability analysis suggests that a plane ice surface in contact with water at a temperature slightly above freezing can become unstable, with the appearance of ripples [e.g., Thorsness and Hanratty, 1979; see also Hanratty, 1981; Epstein and Cheung, 1983]. Experimental findings suggest that ripples grow in amplitude until an equilibrium configuration is attained and the Reynolds number, defined using the ripple wavelength as the characteristic length, approaches a constant value of about 6.2 × 104 [Fuhs et al., 1980].

[3] Studies of freshwater lakes [e.g., Fang et al., 1996; Stefan et al., 1998] suggest that in lakes that have an ice cover, temperature and therefore density are stably stratified, so convective mixing due to unfavorable density stratification and wind forcing are absent. Under these conditions and in the absence of convective flow driven by, e.g., river input, no driving mechanism exists for the self-generation of significant convective motion. Even though the ice-water interface might lower as winter progresses, an initially plane interface can be expected to remain so. In lakes, the so called “black ice” grows from the bottom of the lake ice cover, resulting in a transparent sheet composed of columnar crystals elongated in the vertical direction [Bengtsson, 1986]. Note that fresh water exhibits a density increase with temperature until reaching a maximum at 4°C. Therefore, the thermally driven stratification under an ice cover has a behavior that is opposite of that typically occurring in a lake during, for example, the summer period.

[4] In the case of brackish lakes, however, gradual freezing at the interface causes the exclusion of salt from the black ice. As the interface advances downward, a net downward flux of dissolved salt is generated in the water at the interface. Both salt exclusion, from the ice as the ice cover thickens, and subsequent freshwater ice melt can modify the seasonal structure of a brackish lake.

[5] Pieters and Lawrence [2009] documented the effect of salt exclusion on seasonal circulation in Tailings Lake, a brackish lake with an average salinity of 0.96 g/l located in the Northwest Territories, Canada. They collected under-ice profiles of temperature and conductivity during winter. Salinity was computed from the conductivity measurements. These measurements showed that the water column remained temperature stratified, increasing linearly from about 0°C below the ice to about 2°C at the bottom of the lake, while salinity appeared to be uniform over the entire water column. These observations suggest that for Tailings Lake, the formation of significant black ice, as well as the resulting flux of salt beneath the ice, was sufficient to break down the stable temperature stratification to the degree necessary to fully redistribute this salt throughout the water column. Estimates of the amount of excluded salt needed to break down temperature-mediated stable stratification due to temperature under the ice show that excluded salt can cause under-ice mixing even in relatively fresh lakes. In particular, for Tailings Lake in 2005, with an increase of water temperature of 1.5°C and average salinity of 0.96 g/l, the black ice thickness required to overcome the temperature stable stratification appears to be only 0.04 m [Pieters and Lawrence, 2009, Figure 1].

[6] These observations suggest that, in the case of brackish lakes, a heretofore unrecognized driving mechanism may be available for the formation of rhythmic waveforms at the bottom of the ice cover. This mechanism is salt exclusion, which in the absence of temperature-mediated density differences, generates a sustained unstable stratification in the water column, with density decreasing from the top to the bottom of the lake. The inherent instability of such stratification might be expected to give rise to some spatially periodic flow pattern analogous to Rayleigh-Benard convection. This possibility is pursued here. The difference between the hydrodynamic instability of Rayleigh-Benard convection and the convection pattern explored here is that whereas the former is a purely hydrodynamic instability, the latter is morphodynamic, with the interaction between a rhythmically deforming ice-water interface and the water convection being an essential element of the problem.

[7] In the literature (see for instance the review by Turner [1974]), hydrodynamic double-diffusive phenomena producing an unstable density stratification in a fluid have been investigated in many fields, ranging from the ocean, with studies on thermohaline circulation, to lakes, with the stratification due to intrusion of sea water, and to the case of liquid metals as they solidify. Morphodynamic instability has been studied extensively in the case of alluvial rivers [e.g., Seminara, 1998] but has received relatively little attention in the case of water-ice interfaces. Exceptions include the cases of supraglacial meltwater meandering streams [e.g., Parker, 1975], ripples on the underside of ice covers [Thorsness and Hanratty, 1979], and ripples over an inclined melting-freezing interface bounding a free-surface laminar flow [Camporeale and Ridoldi, 2012]. Interestingly, in the case of freezing of sea water, it has been shown that the morphodynamic instability gives rise, from a microscopic point of view, to a porous mushy layer, which, from a macroscopic, fluid mechanical point of view, can be treated as a porous medium (see for instance the examples reported by Worster [2000]). As an example of such instabilities, a photo of ice undulations observed on the bottom of the ice cover in the St. Croix River near Danbury, WI, USA is shown in Figure 1.

Figure 1.

Photo of the waveforms observed on the bottom of the ice cover in the St. Croix River near Danbury (WI, USA) on 18 February 1965. Wavelengths and amplitudes are in the range of 0.15–0.21 m and 0.02–0.04 m, respectively. The photo was taken by George Ashton, courtesy of Robert Ettema.

[8] The essential difference between hydrodynamic and morphodynamic instability pertains to (a) the time terms in the equations of fluid motion and (b) one or more equations describing the slow evolution of a deformable boundary (e.g., the Exner equation of sediment conservation in the case of alluvial rivers and the Stefan condition in the case of water-ice interfaces). In the case of a purely hydrodynamic instability, element (b) is absent from the problem. In the case of a morphodynamic instability, element (b) plays an essential role, but the slowness of the boundary evolution allows the time terms in the hydrodynamic equations to be neglected. This slowness is expressed in terms of a characteristic time scale of change of the flow that is much shorter than that of the boundary evolution [de Vries, 1965; Parker, 2004].

[9] In the present research, we consider the case of an ice-water interface that moves slowly downward due to freezing in a brackish lake. The freezing excludes salt from the ice, thus forcing a downward flux of dissolved salt in the water at the interface. We study the conditions for double-diffusive morphodynamic instability of the underside of this ice cover and the consequent evolution of flow circulation and morphology of the ice-water interface associated with the diffusion and convection of both heat and salt. The model developed herein provides an interpretation of the observations collected by Pieters and Lawrence [2009] in Tailings Lake; however, since data about the morphology of the ice cover bottom are not available, the predicted undulations cannot be directly compared with data.

[10] In the classical Rayleigh-Benard problem (see for instance the review by Pellew and Southwell [1940]), a layer of fluid heated from below can become unstable so as to give rise to convective motion in the form of periodic cells. In the present problem, the flux of salt from the top can reverse the stable stratification due to the temperature until the fluid is rendered top heavy and thus unstably stratified. The difference in density so induced will tend in time to be counteracted by diffusion of heat; convective motion of the fluid will not result if this diffusivity is sufficiently high. In the problem considered here, however, diffusion is counteracted by the continuous flux of salt from the top of the lake, which continuously forces unstable stratification.

[11] We tackle this problem by considering the stability of a laminar flow subject to stratification effects due to both temperature and salinity. The solution is found by means of linear perturbation analysis. Results allow estimation of conditions under which the system is unstable, so giving rise not only to convective cells, but also concomitant waveforms on the underside of the ice cover. In analogy with the Rayleigh-Benard problem, neutral stability conditions are expressed in terms of Rayleigh-type dimensionless parameters expressing (a) the ratio between buoyancy due to salinity and diffusivity, and (b) the ratio between buoyancy due to temperature and diffusivity.

2 Formulation of the Problem

[12] This analysis considers a simplified lake with a two-dimensional geometry, limited at the bottom by a flat bed and at the top by an ice cover of thickness Li* (Figure 2). As in the case of the Rayleigh-Benard problem, generalization to the three-dimensional case is straightforward. The top surface of the ice layer is flat and horizontal. The lake has an unperturbed water depth inline image, an average salinity inline image, and a temperature difference inline image between the bed and the water-ice interface. The local temperature in the water is inline image, while that in the ice cover is inline image. In the unperturbed state, the water-ice interface is slowly moving downward as it freezes, excluding salt as it does so.

Figure 2.

Sketch and notations. Li* is the ice cover thickness, S* is the salinity concentration, Y0* is the lake flow depth, H* is the elevation of the water-ice interface, η* is the elevation of the bed of the lake, Tb* is the temperature at the bed, Ta* is the temperature of atmosphere, and (x*, z*) denote the reference system. A qualitative plot of water temperature Tw* and salinity S* distribution is also reported.

[13] To render the problem tractable from the point of linear stability analysis, the unperturbed water depth is held constant as the freezing progresses, implying a supply of water from the bottom at the rate necessary to satisfy this condition. This rate is held fixed, even when the system is perturbed. In this way, the system is steady in a frame moving with the prescribed solidification rate. This setup appears to be in analogy with similar problems involving a unidirectional solidification in a mushy layer [Worster, 1991].

[14] The system is referred to Cartesian coordinates with horizontal axis x* and upward vertical axis z*. Here a superscript asterisk will denote a dimensioned quantity, subsequently made dimensionless and denoted without the superscript asterisk. The following dimensionless quantities are defined:

display math(1a)
display math(1b)
display math(1c)
display math(1d)
display math(1e)
display math(1f)
display math(1g)
display math(1h)

where H and η denote the elevation of the water-ice interface and of the bed of the lake, respectively; (u, w) is the local velocity vector (u longitudinal, w vertical); κs is the molecular diffusivity of salt in water; ρr is a reference value for the water density; p is pressure; and t is time.

[15] The dimensioned diffusive heat flux vectors in water and ice, inline image and inline image, as well as the vertical speed of migration of the water-ice interface, inline image, are made dimensionless as follows:

display math(2a)
display math(2b)
display math(3)

[16] In the above relations, the operator ∇ is dimensionless, and Khw and Khi denote the thermal conductivity of water and ice, respectively, given as follows:

display math(4a)
display math(4b)

where cpw and cpi denote the specific heat capacities of water and ice, respectively; ρi is the density of ice, while κhw and κhi denote the diffusivities of heat in water and ice (see Table 1 for the adopted values).

Table 1. Symbols and Values Adopted for the Various Coefficients
Thermal coefficient of expansionα−5.0 × 10−51/K
Salinity coefficient of expansionβ8.05 × 10−4l/g
Latent specific heat of freezingah334 × 103J/kg
Kinematic viscosityν1.75 × 10−6m2/s
Salt diffusivityκs1.35 × 10−9m2/s
Thermal diffusivity of waterκhw1.4 × 10−7m2/s
Thermal diffusivity of iceκhi1.18 × 10−6m2/s
Ice densityρi918kg/m3
Water densityρr1000kg/m3
Water specific heat capacitycpw4.2 × 103J/kg K
Ice specific heat capacitycpi2.05 × 103J/kg K
Thermal conductivity of waterKhw0.59W/m K
Thermal conductivity of iceKhi2.22W/m K
ksDf7.71 × 10−4 
ks /khwDw9.64 × 10−3 
ks /khiDi1.14 × 10−3 

[17] It is assumed that the relevant equation of state for brackish water can be linearized, so the fractional excess density of the water, f, about a reference value ρr at the freezing point can be reduced to the form

display math(5)

where α is the coefficient of thermal expansion and β is the corresponding coefficient associated with salinity, both approximated here as constants.

[18] We now define five key dimensionless parameters. Let ν denote the kinematic viscosity of water (i.e., the kinematic diffusivity of momentum in water). The diffusivity ratios Df, Dw, and Di are defined below:

display math(6a)
display math(6b)
display math(6c)

[19] These ratios quantify the difference related to the molecular diffusivities of the different quantities involved. In particular, it is of interest to note the 2 orders of magnitude difference between salt and heat diffusivities; this large differential diffusion produces the water density differences required to drive the motion.

[20] In addition, two Rayleigh numbers Raα and Raβ can be defined as follows:

display math(7a)
display math(7b)

[21] Here Raα scales the ratio of the tendency of the buoyancy force due to temperature difference to drive convection to diffusive effects that damp this convection, and Raβ scales the corresponding ratio associated with the buoyancy force due to salinity difference. Rayleigh numbers can be interpreted as the ratio between diffusive (in this case associated with dissolved salt) and convective (as driven by with buoyancy) time scales; when the former is much greater than the latter, instabilities are expected to occur.

[22] Using the above notation and assumptions, the Navier-Stokes equations for the hydrodynamics can be written in the following dimensionless form:

display math(8)
display math(9)
display math(10)

[23] Note that changes in density across the flow depth are assumed to be small enough for the well-known Boussinesq approximation to hold (see, for instance, Defina et al. [1999]).

[24] Importantly, the flow is here considered to be in the laminar regime. In the Rayleigh-Benard problem, according to Ahlers and Behringer [1978], onset of turbulence, in the sense that the fluid has a nonperiodic time dependence, is expected to occur when the Rayleigh number exceeds some threshold depending on the geometry (aspect ratio) of the system; in particular, they show that in the case that approximates most closely the laterally infinite system (2-D configuration), nonperiodic fluid flow occurs even for a low Rayleigh number, i.e., just above the critical value for instability. However, such experimental evidence cannot be directly applied to our problem because in our case, instability is related to the exclusion of salt from ice, whereas temperature has a stabilizing effect. Field and laboratory measurements of the flow field would be needed to ascertain this point.

[25] The governing equations for the temperature distributions in the water and the ice cover above it take the following respective forms:

display math(11)
display math(12)

[26] Conservation of salt is governed by following equation:

display math(13)

[27] According to the Stefan relation, the speed of migration of the water-ice boundary is governed by the difference between the heat fluxes of ice inline image and of water inline image across the interface:

display math(14)

where inline image is the unit vector normal to the interface (positive from water to ice), and ah is the coefficient of latent heat of freezing.

[28] The dimensionless normal speed of the boundary can be related to the vertical speed as follows:

display math(15)

where θ is the local angle of the interface to the horizontal (Figure 3). Furthermore, the unit vector normal to the interface is given as

display math(16)
Figure 3.

Sketch of details of the water-ice interface. t is time, H is the elevation of the water-ice interface, inline image is the unit vector normal to the interface (positive from water to ice), and θ is the local angle of the interface to the horizontal.

[29] Combining (14)(16) with the definitions (2a) and (2b), the dimensionless Stefan relation takes the form

display math(17)

having introduced the Stefan number

display math(18)

[30] This number expresses the ratio of latent heat of freezing to the heat required to cool down the water layer.

[31] The differential problem (8)(10) for the hydrodynamics is subject to the following boundary conditions at the bed and at the water-ice interface:

display math(19a)
display math(19b)

[32] The boundary conditions to be applied at the porous ice-water interface (see, for example, Glicksman et al. [1986]) arise from (a) the no-slip condition for the tangential velocity component and (b) from the conservation of mass during the freezing for the vertical component. In the present work, due to the small solidification speed (see also later) and neglecting the density difference between ice and water, the vertical velocity component is also constrained to vanish.

[33] Regarding temperature distributions in the water and the ice, the following boundary conditions are imposed. At the top of the ice cover (z = H + Li), the atmosphere has a constant, below-freezing temperature (−Ta where Ta is defined to be positive). At the ice-water interface (z = H), water and ice are at the freezing point (here assumed to be 0°C as an appropriate approximation for slightly brackish water). At the bed of the lake (z = η), the water has a temperature Tb above the freezing point but not larger than the temperature producing the density maximum (~4°C for fresh water). In other words,

display math(20a)
display math(20b)
display math(20c)

[34] Note that equation (20b) assumes that the freezing temperature has a negligible dependence on salinity. This assumption relies on the relatively low values of the salinity and the related small changes in the temperature of freezing; for instance, in the case of a salinity of 1 g/l, the temperature of freezing falls to about −0.5°C.

[35] Due to the mechanism of salt exclusion, the downward motion of the water-ice interface generates a downward flux of salt at the top water boundary such that

display math(21)

[36] An additional integral condition to be coupled with the equation governing the salinity distribution stipulates the condition of mass conservation of dissolved salt in the water:

display math(22)

[37] It is convenient to normalize the vertical coordinate with the instantaneous flow depth such that

display math(23)

[38] With this in mind, a new reference system inline image is defined as follows:

display math(24a)
display math(24b)
display math(24c)

[39] Using the chain rule, it is found that

display math(25a)
display math(25b)
display math(25c)


display math(26a)
display math(26b)
display math(26c)

3 Solution Procedure

[40] Based on the data from Pieters and Lawrence [2009], it appears that the water-ice interface in Tailings Lake moves downward at a very slow rate, with a characteristic speed inline image, i.e. on the order of 10−7 m/s (corresponding to the development of a thickness of black ice of 0.80 m in about 4 months). A characteristic response velocity of the flow inline image due to buoyancy effects can be estimated as follows:

display math(27)

where inline image and inline image denote reference values for inline image and S*.

[41] In the present case, it is easily shown that inline image << inline image, thus suggesting that the hydrodynamics is able to adapt instantaneously to the moving water-ice interface. This consideration allows us to adopt a quasi-steady approach, according to which time derivatives are neglected in equations (8)(13) governing the flow, but the time derivative is retained in equation (17) governing the morphodynamics. This formulation represents an adaptation of the quasi-steady approximation of sediment transport morphodynamics to the present problem of ice morphodynamics.

[42] Pieters and Lawrence [2009] indicate that during the summer, water temperature increases from the bottom to the top of the lake, and salinity shows a corresponding decline. This trend is opposite to that seen in winter. This suggests the presence, after the summer season, of a transient time period (fall turnover) when the system gradually adapts to the winter conditions.

[43] The present analysis provides a physical interpretation of the system during winter conditions, when temperature and salinity distributions have reached a configuration in equilibrium with the new boundary conditions imposed by the cold climate. The analysis shows that this equilibrium is unstable by introducing an undulating perturbation at the ice-water interface.

[44] Based on a perturbation expansion strictly valid in the case of small-amplitude waveforms of the ice-water interface, the problem formulated in section 2 is now linearized and solved. An equilibrium base state needs to be defined for this purpose. Were the bottom of the domain to be fixed, the unperturbed depth inline image would gradually decrease in time, owing to thickening of the ice cover. It is here assumed for convenience that (a) the bottom boundary moves down at the same speed as the water-ice interface, but otherwise exhibits no perturbations, and (b) water is introduced at the bottom just so as to balance the mass lost to ice at the top. We expect these boundary conditions to play a negligible effect on model results, as observations in the Tailings Lake suggest that (i) speed of the ice-water interface inline image is very small (on the order of 10−7 m/s) and (ii) the changes in the actual flow depth due to growth of black ice from the bottom of the ice cover are minor, being in the range of 6–13% in the case of black ice growth of 0.80 m over flow depths between 6 and 14 m.

[45] Note that the imposed fluid flux at the bottom boundary is equivalent to a unidirectional solidification setup in analogy with research in other fields related to the solidification of alloys and instabilities of liquid and mushy regions, e.g., as described by Worster [1991].

[46] Under these assumptions and the conservation condition (22) imposed by the assumption of perfectly efficient salt exclusion (21), the equilibrium base state is characterized by a constant flow depth, a salinity profile decreasing toward the bottom, and a vertical temperature profile that exhibits stable stratification (see the sketch in Figure 2).

[47] The following expansions are introduced to perturb the base state:

display math(28a)
display math(28b)

[48] In addition, we introduce the forms

display math(29a)
display math(29b)
display math(29c)

[49] Substituting (28a) and (28b), and (29a)(29c) into (8)(13) and (17), and equating likewise powers of ε, a sequence of problems at the various orders of approximation are obtained.

3.1 Leading Order

[50] In the base state, there is no flow motion, and the flow boundaries translate downward at a constant speed:

display math(30)

[51] All quantities are a function of inline image alone, as in the moving frame, there is no dependence on time.

[52] At the leading order of approximation, O(ε0), the vertical component of the Navier-Stokes equations reads

display math(31)

[53] This equation indicates that pressure deviates from a hydrostatic distribution due to stratification effects associated with both temperature and salinity.

[54] The governing problem for the salinity distribution can be written as follows:

display math(32a)

with the boundary conditions:

display math(32b)
display math(32c)

[55] The solution of the above problem is such that salinity is distributed according to an exponential law, monotonically increasing toward the ice-water interface:

display math(32d)

[56] Regarding the temperature distribution in the ice, the governing problem reads

display math(33a)
display math(33b)
display math(33c)

[57] Solving this problem leads to the following:

display math(33d)

[58] The problem for the temperature distribution in the water reads

display math(34a)
display math(34b)
display math(34c)

[59] Therefore, the solution for the temperature distribution in the water is

display math(34d)

[60] Finally, the Stefan condition takes the form

display math(35)

[61] Therefore,

display math(36)

[62] This equation allows one to estimate the speed at which the ice-water interface moves downward. For instance, considering an example with T*b = 1°C, −T*a = −10°C, and Li = 0.1, equation (36) gives c0* ~ 2 · 10−7 m/s. This estimate appears to be in general agreement with the field observations collected by Pieters and Lawrence [2009].

3.2 First Order

[63] The analysis now proceeds to O(ε1). The stability of the above solution is here investigated with respect to small fluctuations of the ice-water interface.

[64] The governing problem for the hydrodynamics reads

display math(37a)
display math(37b)
display math(37c)

with the boundary conditions

display math(37d)
display math(37e)
display math(37f)
display math(37g)

[65] Since the problems (37a)(37g) are 2-D, equation (37a) implies the existence of a stream function ϕ defined as follows:

display math(38a)
display math(38b)

[66] Introducing (38a) and (38b) into (37a)(37g), after some algebraic manipulation, the following problem for ϕ is obtained:

display math(39a)
display math(39b)
display math(39c)

[67] The problem for the perturbation of the ice temperature distribution reads

display math(40a)
display math(40b)
display math(40c)

[68] Regarding the water temperature distribution, it is found that

display math(41a)
display math(41b)
display math(41c)

[69] Finally, the salinity distribution is governed by the following problem:

display math(42a)
display math(42b)
display math(42c)

[70] The various differential problems can be solved by assuming the following perturbation forms:

display math(43a)
display math(43b)

where q is a dimensionless wave number and σ is a dimensionless amplification factor. If σ is positive for any values of q, then the solutions grow exponentially in time and are unstable.

[71] Substituting the expansions (43a) and (43b) into the governing equations, a series of ordinary differential equations for the unknowns inline image are found.

[72] The differential problem (39a)(39c) is readily solved using the shooting method together with the fourth-order Runge-Kutta scheme. It is found that

display math(44a)

with gi (i = 0,1,2) being the solutions to the following differential system:

display math(44b)
display math(44c)
display math(44d)
display math(44e)
display math(44f)
display math(44g)
display math(44h)

[73] Note that in (44a), inline image is linearly proportional to both Raα and Raβ.

[74] The solution for the perturbation of the ice temperature is readily found as follows:

display math(45a)

with λ1,2 being solutions of the following characteristic equation:

display math(45b)

[75] The solution for the perturbation of the water temperature distribution can be readily found using the shooting method, as follows:

display math(46a)

with fi (i = 0,1,2,3) being the solutions of the following differential system:

display math(46b)
display math(46c)
display math(46d)
display math(46e)
display math(46f)
display math(46g)
display math(46h)
display math(46i)

[76] The solution for the salinity distribution is found using the method of variation of parameters and reads as follows:

display math(47a)

with λ1,2 being solutions of the following characteristic equation:

display math(47b)

[77] The functions u1,2 are defined as follows:

display math(47c)
display math(47d)

where g and W take the following expressions:

display math(48a)
display math(48b)

[78] Finally, a1,2 are the following constants of integration:

display math(49a)
display math(49b)

[79] Finally, in the moving frame, using the Stefan equation, the following equation for the amplification factor is found:

display math(50)

[80] The solution of the above system is iterative; in the first trial, inline image is set to 0, and an expression for inline image and inline image is found; with these findings, equation (44a) is employed to calculate a tentative distribution of the stream function. This process is repeated until convergence.

4 Results

[81] We first show results for the dependence of the amplification factor σ on the two Rayleigh numbers Raα and Raβ. It is useful to recall that according to equations 7a,b), increasing values of Raα correspond to increasing stratification in temperature, which increases stability, and increasing values of Raβ correspond to increasing stratification in dissolved salt, which decreases stability.

[82] Figure 4 shows the amplification factor σ as a function of Raβ for different values of the wave number q, for the specified value Raα = 1 × 105. This example can be interpreted as the case of a lake with a specified temperature distribution but different average salinities. As can be seen from equations (50) and (46a), σ increases linearly with Raβ, but with different slopes depending on the value of q. Physically, this behavior can be interpreted as follows: for a given flow depth and temperature at the bed Tb, the system becomes less and less stable with increasing values of the average salinity. The values of Raβ for which σ = 0 define marginal stability conditions corresponding to neutral stability; above these values, the analysis predicts that the system is unstable (σ > 0). This means that for a given flow depth and temperature distribution, when the average salinity is sufficiently large, the destabilizing contribution associated to salinity becomes greater than the stabilizing contribution due to temperature.

Figure 4.

The dimensionless amplification factor σ as a function of Raβ for different values of dimensionless wave number q in the case of a constant Raα = 105 (Tb* = 1°C, −Ta* = −2°C, Li = 0.5).

[83] The effect of varying water temperature gradient on the stability of the system is illustrated in Figures 5 and 6.

Figure 5.

The dimensionless amplification factor σ as a function of Raβ for two water temperatures at the bed Tb* and constant flow depth Y0*(−Ta* = −2°C, Li = 0.5, q = 10).

Figure 6.

The amplification factor σ as a function of Raβ for two different flow depths and constant water temperature at the bed (Tb* = 1°C, −Ta* = −2°C, Li = 0.5, q = 10).

[84] As one can expect, for a given flow depth, an increase of T*b leads to a greater gradient in water temperature distribution, which has a more pronounced stabilizing effect on the system. This can be seen in Figure 5, where marginal stability conditions are reached for a higher value of Raβ in the case T*b = 2°C (Raα = 2 × 105) than the case T*b = 1°C (Raβ = 1 × 105). This suggests that a higher quantity of salt is needed to destabilize the lake when the temperature gradient is higher. This is confirmed also by results in Figure 6, where base states with constant T*b but with two different flow depths Y*0 are considered. In particular, σ is plotted as a function of Raβ for two cases: one (Raα = 8 × 105) having the flow depth double than the other (Raα = 105), and constant T*b = 1°C. It appears that in the case Raα = 105, σ = 0 when Raβ = 8 × 103, which leads to an average salinityinline image = 5 × 10−3 g/l required to destabilize the system. In the case Raα = 8 × 105, σ = 0 when Raβ = 4 × 104 giving inline image = 3 × 10−3 g/l, i.e., smaller than the previous case. This suggests that a lake with a smaller temperature gradient can reach an unstable condition with a smaller quantity of salt.

[85] Marginal stability curves plotted in terms of Raβ versus q for various values of Raα in the case of constant T*b are shown in Figure 7. The curves attain a minimum value of Raβ for wave numbers q in the range of 14–16. In the context of a linear stability analysis, this minimum value of Raβ corresponds to the weakest salinity for the onset of instability. The minimum values of average salinity required to render the system unstable appear to be relatively low; for instance, in the case Raα = 104, the curve exhibits a minimum for Raβ ~ 3 × 103, corresponding to an average salinity inline image = 1.88 × 10−2 g/l. These findings suggest that, in nature, brackish lakes during winter conditions can easily be subject to unstable conditions due to salt exclusion.

Figure 7.

Marginal stability curves for various values of Raα (Tb* = 1°C, −Ta* = −2°C, Li = 0.5).

[86] Figure 8a shows vertical temperature and salinity profiles for an example base state. Note the linearly decreasing water temperature distribution (see equation (34d) which can be safely approximated to a linear relationship in inline image, as Dwc0 << 0) and the exponential increase in salinity (see equation (32d)) from bottom to top. As can be seen in Figure 7, in the case Raα = 105 and Raβ = 7 × 103, the system is unstable in so far as there is a range of wave numbers having positive values of the amplification factor.

Figure 8.

(a) Example of a base state (Tb* = 1°C, −Ta* = −2°C, Li = 0.5). (b) Example of flow field inline image (Raα = 105, Raβ = 7 × 103, q = 16). (c) Example of perturbation of water temperature distribution field (Raα = 105, Raβ = 7 × 103, q = 16). (d) Example of perturbation of salinity distribution (Raα = 105, Raβ = 7 × 103, q = 16).

[87] In Figure 8b, the flow field for the case of a sample unstable configuration is shown. Results indicate the presence of two cells of circulation (this can be seen more clearly in Figure 9) rotating in opposite directions: a stronger cell confined to a region close to the ice cover and a much weaker cell distributed over the remaining lower portion of the water column. It can be expected that these cells will drive mixing of salinity and temperature in the lake, an effect that should appear in a nonlinear analysis.

Figure 9.

Example of a flow field in an unstable configuration; the undulations of the ice-water interface are also shown (Raα = 105, Raβ = 7 × 103, q = 16, ε = 0.015).

[88] The perturbation of water temperature distribution is shown in Figure 8c, where it is seen that Tw1 follows the distribution of the vertical component of the flow velocity, with positive values close to the ice cover and negative values in the remaining part of the water column.

[89] Finally, the distribution of the perturbation of salinity (Figure 8d) appears to be positive in the top layers of the lake just below the ice cover and negative toward the bed. This behavior reflects the mechanism of salt exclusion from the ice.

[90] In addition to the flow field, Figure 9 also illustrates the undulating morphology of the water-ice interface. Note that in the example shown, the undulations are such that the ice cover is thicker (ice-water interface elevation lower) in regions of downwelling and thinner (interface elevation higher) in regions of upwelling. It therefore appears that in the present morphodynamic stability analysis, the presence of these undulations is inextricably linked to the flow pattern itself.

5 Conclusions

[91] We have investigated a double-diffusive morphodynamic instability in brackish, ice-covered lakes driven by salt exclusion as the ice cover thickens. This instability is reflected not only in the flow, but also in the morphology of the ice-water interface. The analysis indicates that instability of this ice-interface is governed by the competition between two opposite effects: (a) the stabilizing contribution associated with the water temperature distribution which causes density to increase from top to bottom in the water column and (b) the destabilizing effect caused by the exclusion of salt from the ice cover as it thickens, which intensifies the density of the top layer of water.

[92] In analogy with the Rayleigh problem formulated for the stability of a layer of fluid heated from below, the system is found to be unstable when the ratio between buoyancy effect due to salinity and diffusivity effects is sufficiently large for a given temperature distribution. When unstable conditions are reached, a convective motion occurs in the form of rotating cells. These cells drive a mixing of salinity and temperature.

[93] Our results provide a mechanistic explanation of the field observations collected by Pieters and Lawrence [2009], who found out that under-ice profile distributions of salinity are, despite the presence of mechanism of salt exclusion, nearly constant. This suggests the presence of some under-ice motion producing an efficient mixing of dissolved salt. However, field measurements showed that the temperature profiles remained stratified throughout the winter, though at times the top layers became isothermal, indicating localized mixing. According to Pieters and Lawrence [2009], this suggests that the excluded salt drives successive episodes of convective mixing over limited regions of depth. The present analysis predicts a flow field characterized by two counteracting convective cells, a much stronger cell confined to the top layer close to the ice cover and a much weaker and larger cell located in the remaining lower portion of the water column. This predicted pattern indicates that the mixing of salinity and temperature is more efficient in the top layers where, due to mechanism of salt exclusion, unstable density gradients are much stronger.

[94] The under-ice undulations predicted in Figure 8 remain to be observed. The common existence of undulating bedforms at water-ice interfaces is, however, observed in other contexts [e.g., Ashton and Kennedy, 1972; Fuhs et al., 1980; Camporeale and Ridoldi, 2012], suggesting their likely presence in the case considered herein. Future developments in the present line of research will be partly predicated on the experimental or field verification of these undulations.


[95] This work was partially supported by the CNR within the 2009 Short Term Mobility Program and by the University of Florence. The participation of the second author was supported in part by the National Center for Earth-Surface Dynamics, a Science and Technology Center of the U.S. National Science Foundation. We are grateful for helpful comments from three referees and the Editor.