ISSUE 17

Stratification in winter lakes: The effect of cabbeling

Contact

Marek
Marek Stastna
Department of Applied Mathematics

Introduction

In late winter, lakes in temperate climates often remain ice covered while warmer water from adjacent rivers flows into water beneath the ice. As freshwater has a temperature of maximum density (T_md) around four degrees, river inflows can initiate a mixing process during which two masses of freshwater mix to form a denser mass. This denser mass in turn leads to more mixing as the mass sinks in what is called the “cabbeling”. Cabbeling, which comes from a German origin hence the unusual spelling, has been observed in several Canadian lakes (e.g. Kamloops Lake, Great Bear Lake) but comparatively little work has been done to study the detailed fluid dynamics of cabbeling at a small scale.

This paper studies the transition to, and evolution of, turbulence in the cabbeling regime using high resolution numerical simulations. Unlike previous work, the study allows the denser water mass developed via cabbeling to extend downwards, as it would in the near-shore region of a lake. A key emergent feature that is not observed in water outside of the cabbeling regime is documented and the roles of dimensionless parameters (e.g the Richardson number) in the evolution process are discussed.

Methodology

Figure 1 shows the conceptual model and simulation setup for the study. The open-source, pseudospectral SPINS model developed at the University of Waterloo was used to solve the incompressible Navier Stokes equations under the Boussinesq approximation. The cold-water regime exhibits were established by choosing a quadratic equation of state. Based on past work, it was expected that both convective and shear instabilities would form in different planes, with the latter aligned with the background shear. Unlike many lake-scale models, a direct numerical simulation approach was used which did not require turbulence parameterization as molecular values of viscosity and diffusivity were used.

Model Setup

Figure 1: Model Setup
Panel (a) shows a typical setup of a late winter river plume beneath ice. The warm river water (temperature greater than T_md) enters the cool ambient (temperatures less than T_md) and mixing induces cabbeling and turbulence below the plume. The hypothesized interface between the two masses of water is plotted by the dashed line. Panel (b) shows the initial temperature profile for the simulations; a layer of warm water atop a layer of cool water, where the profile crosses T_md. ΔT₁ and ΔT₂ are the temperature differences of each layer about T_md, while the definition for Δb is shown below. Panel (c) highlights the velocity profile. The upper layer has a thickness D with a velocity difference ΔU across the interface of thickness δ. The velocity and temperature transitions are coincident, and gravity g points downward.

While the primary result may be understood in dimensional form, the standard approach to non-dimensionalize the model equations was adopted. The upper layer depth D was selected to scale the spatial coordinates, the velocity difference to scale the velocity field, the layer-wise temperature difference to scale the temperature field and the advective timescale to scale time, resulting in the following set of non-dimensional numbers:

(1)

Re is the Reynolds number of the flow which measures the relative magnitude of fluid inertia to momentum diffusion. Pr is the Prandtl number representing the ratio of kinematic viscosity to thermal diffusion. The values for this parameter were chosen such that the numerical simulations highlight system evolution in the “sub-critical” regime where the entrainment rate is hypothesized to be constant and the “super-critical regime” where the entrainment rate is said to vary with Ri_b. Θ is a parameter governing the degree of nonlinearity of the nonlinear equation of state.

The problem was simplified by assuming that once the instabilities have developed, the interface thickness will no longer play a role in the dynamics. Allowing for time dependence, the problem was simplified to

(2)

where Π is any quantity that satisfies the above assumptions about the flow.

Outcomes

Figure 2 shows the mechanism by which layers initially mix in a river plume setting. It is clear that the process is both multi-scale and complex. Once instabilities collapse, dense water is formed, which in turn triggers further large-scale convection and the entrainment of upper layer water (Figure 2c). A large-scale plume structure penetrating into the lower layer of the system, with return flows on the flanks of this instability, can be seen. Once the cabbeling instability has reached a mature state, the upper layer remains quiescent while the lower layer continues to undergo convection (Figure 2d). Emergent stratification effectively seals the upper layer from the lower layer with the lower layer weakly unstably stratified while the upper layer is completely quiescent (Figure 2e). This is the primary novelty of this work and has the largest potential for impact outside of the fluid mechanics community, since reduced mixing in the upper region would decrease scouring of overlying ice, while at the same time detritus and biota that cannot swim would sink out of the near surface layer due to the absence of turbulence.

The remaining panels of Figure 2 along with Figure 3 provide further detail. The act of cabbeling worked to asymmetrically diffuse momentum downward with the resultant mixing making the bottom layer much denser over time (Figure 2g). While the cabbeling instability had a negligible influence very early in the evolution as instabilities develop, it had a large influence on the level of turbulence below the interface after initial mixing had taken place (Figure 2h).

Figure 2: Simulation Results of Representative Case
Panels (a) and (b) show streamwise and spanwise cross-sections of the temperature field at an early time. Panels (c) and (d) show streamwise cross-sections at mid and late times. Panel (e) shows filled isotherms highlighting the horizontally averaged temperature field. The black contour approximately represents the T_md isotherm. The white vertical dotted lines indicate the times of the snapshots in panels (a)–(d). Panels (f), (g), and (h) show vertical profiles of the horizontal velocity, density, and 〈w²〉/ΔU².

To investigate the behavior of the interface with time, an isotherm was chosen to serve as a proxy for the interface location (Figure 2e) with the normalized height of the isotherm defined as d/D. Figure 3a shows d/D plotted against τ, and shows that the curves for cases A-D briefly collapsed early in their evolution before separating at different times while case E increased at a slightly different rate suggesting that early in the evolution the system enters a time-dependent self-similar regime. In a riverine setting, this could translate to river water being progressively entrained with horizontal distance and then sinking due cabbeling. Thus, river water may progressively be mixed resulting in a thinner river plume further out into the lake. Figure 3b plots τ₀ as a function of time and shows that after a brief period during which the instabilities are growing, τ₀ is approximately constant in time and occurs during the self-similar entraining regime before the scaling changes and re-stratification is established. Thus, during the entrainment regime we can infer that the destabilizing buoyancy flux is relatively insensitive to Ri_b and Re when the entrainment coefficient, E, is relatively constant. This formula is useful to estimate the maximum run-out distance of a river plume given a relationship between E and Ri_b.

Figure 3: Panel (a) shows the normalized approximate height of the interface for the cases. Panel (b) shows τ₀ for each case. Note the during the regime where τ₀ is constant, all cases with Ri_b < 1.5 collapse (cases A-D), whereas case E deviates, probably due to the large value Ri_b.

Conclusions

Using high resolution numerical simulations, the paper proposes a simple model for the vertical fluid flux during the mixing regime and discusses a potential mechanism responsible for the emergent stratification in late winter lakes. The study demonstrates a parallel shear flow where the bottom layer is sufficiently deep and cabbeling instability efficiently entrains upper layer fluid. As the new dense fluid sinks and is replaced by bottom layer fluid, a quasi-steady state forms and the interface between the two layers rises at a rate proportional to time. The process continues until an emergent stratification induced by the nonlinear equation of state strengthened the stratification once again, effectively cutting off the active lower layer. This phenomenon is, to the best of the authors’ knowledge, unique to the cold-water regime.

Study results suggested that during late-winter/early spring river discharge, the cabbeling mechanism is a key process by which mass and heat are redistributed in the water column. Cabbeling dominates turbulence production in the lower layer, while at the same time facilitating a mechanism that limits mixing into the near-surface layer. This has implications for ice scour and the transport of biogeochemical constituents from the near surface region. Lake scale models do not reveal information about flows such as this because common approximations like the hydrostatic approximation do not apply, or commonly used turbulence parametrizations (e.g. the KPP scheme) do not apply in the cold water regime.

Understanding these kinds of flows is vital for understanding the horizontal and vertical heat transport during the climate-change-impacted spring shoulder season in northern lakes. This has applications for the timing of ice-off, and for understanding how winter time dynamics affects biogeochemical dynamics, including potential algal blooms, during the subsequent summer season. Furthermore, parametrizations built on the basis of direct numerical simulations have the potential to fundamentally change how the modeling of transport of river inflows (e.g. of pollutants) over the winter and spring in lakes is carried out. This is vital for accurate modeling of northern lake health during the subsequent summer season over a region that is one of the most climate-change-impacted on the planet.