Three-dimensional modeling of diffusion-gravity flows in ice-covered lakes

Ramón C.L., Ulloa H.N., Doda T., Winters K.B., Bouffard D.

Abstract. In late winter, solar radiation is the main driver of water motion in ice-covered lakes. The resulting circulation and mixing determine the spatial distribution of heat within the lake and affect the heat budget of the ice cover. Although under-ice lake warming is often modeled as a one-dimensional (1D) vertical process, lake bathymetry induces a relative excess heating of shallow waters, creating horizontal density gradients. This study shows that the dynamic response to these gradients depends sensitively on lake size and latitude – Earth's rotation – and is controlled by the Rossby number. In the ageostrophic limit, horizontal density gradients drive cross-shore circulation that transports excess heat to the lake interior, accelerating the under-ice warming there. In the geostrophic regime, the circulation of the near- and off-shore waters decouples, and excess heat is retained in the shallows. The flow regime controls the fate of this excess heat and its contribution to water-induced ice melt.

Winters K.B., Ulloa H.N., Wüest A., Bouffard D.

Berntsen J., Thiem Ø., Avlesen H.

Terrain following ocean models are today applied in coastal areas and fjords where the topography may be very steep. Recent advances in high performance computing facilitate model studies with very high spatial resolution. In general, numerical discretization errors tend to zero with the grid size. However, in fjords and near the coast the slopes may be very steep, and the internal pressure gradient errors associated with σ-models may be significant even in high resolution studies. The internal pressure gradient errors are due to errors when estimating the density gradients in σ-models, and these errors are investigated for two idealized test cases and for the Hardanger fjord in Norway. The methods considered are the standard second order method and a recently proposed method that is balanced such that the density gradients are zero for the case ρ = ρ ( z ) where ρ is the density and z is the vertical coordinate. The results show that by using the balanced method, the errors may be reduced considerably also for slope parameters larger than the maximum suggested value of 0.2. For the Hardanger fjord case initialized with ρ = ρ ( z ) , the errors in the results produced with the balanced method are orders of magnitude smaller than the corresponding errors in the results produced with the second order method.

Kirillin G.B., Forrest A.L., Graves K.E., Fischer A., Engelhardt C., Laval B.E.

Below the temperature of maximum density (TMD) in freshwater lakes, heating at the lateral margins produces gravity currents along the bottom slope, akin to katabatic winds in the atmosphere and currents on continental shelves. We describe axisymmetric basin-scale circulation driven by heat flux at the shorelines in polar Lake Kilpisjarvi. A dense underflow originating near the shore converges toward the lake center, where it produces warm upwelling and return flow across the bulk of lake water column. The return flow, being subject to Coriolis force, creates a lake-wide anticyclonic gyre with velocities of 2–4 cm s-1. While warm underflows are common on ice-covered lakes, the key finding is the basin-scale anticyclonic gyre with warm upwelling in the core. This circulation mechanism provides a key to understanding transport processes in (semi) enclosed basins subject to negative buoyancy flux due to heating (or cooling at temperatures above TMD) at their lateral boundaries.

Mercier M.J., Ardekani A.M., Allshouse M.R., Doyle B., Peacock T.

Matthieu J. Mercier, Arezoo M. Ardekani, Michael R. Allshouse, Brian Doyle, and Thomas Peacock ENDLab, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, Indiana 47907, USA (Received 27 May 2013; published 21 May 2014)

Rizk W., Kirillin G., Leppäranta M.

A field and theoretical study of the mean circulation patterns, gravity currents, and heat fluxes in a mid-latitude, ice-covered lake is presented. Thermistor chains captured the lake-wide pressure gradients and thermal regime, acoustic instruments measured bottom and interior velocities, and a sediment temperature probe measured sediment temperatures and heat fluxes. Typical Rossby numbers showed that the mean circulation was geostrophic, in the form of either an anti-cyclonic or a cyclonic basin-scale gyre. Each gyre reached a steady state after ∼ 10–15 d, circulating at a maximum velocity of ∼ 1 mm s−1. The gyres persisted throughout most of the winter and generally decayed with time. The cyclonic gyre was generated by the lateral temperature gradients created by sediment heat release. It was not possible to determine the driver of the anti-cyclonic gyre. Because most ice-covered lakes are controlled by the Earth's rotation, anti-cyclonic and cyclonic circulation is postulated as being the most common mean interior flow pattern. The horizontal heat fluxes were of the same order as the vertical heat fluxes, which is consistent with the findings of previous research. The sediment heat flux decayed with time, stabilizing after ∼ 60 d. A scaling analysis of down-sloping advection with dynamic, seasonal-scale temperatures showed that a warmer lake and an increase in the length-to-depth geometric aspect ratio (both within a given lake) and an increase in lake size all increased advection time scales, and therefore circulation time scales, near the boundaries.

Kirillin G., Leppäranta M., Terzhevik A., Granin N., Bernhardt J., Engelhardt C., Efremova T., Golosov S., Palshin N., Sherstyankin P., Zdorovennova G., Zdorovennov R.

Recently, the attention to the ice season in lakes has been growing remarkably amongst limnological communities, in particular, due to interest in the response of mid- and high-latitude lakes to global warming. We review the present advances in understanding the governing physical processes in seasonally ice-covered lakes. Emphasis is placed on the general description of the main physical mechanisms that distinguish the ice-covered season from open water conditions. Physical properties of both ice cover and ice-covered water column are considered. For the former, growth and decay of the seasonal ice, its structure, mechanical and optical properties are discussed. The latter subject deals with circulation and mixing under ice. The relative contribution of the two major circulation drivers, namely heat release from sediment and solar heating, is used for classifying the typical circulation and mixing patterns under ice. In order to provide a physical basis for lake ice phenology, the heat transfer processes related to formation and melting of the seasonal ice cover are discussed in a separate section. Since the ice-covered period in lakes remains poorly investigated to date, this review aims at elaborating an effective strategy for future research based on modern field and modeling methods.

Dibike Y., Prowse T., Saloranta T., Ahmed R.

The formation and break-up of ice-cover are important seasonal events in mid- to high-latitude cold-region lakes. There is increasing concern regarding how climate change will affect lake-water thermal structure and lake-ice characteristics, particularly ice formation, duration, break-up, thickness, and composition. This study employs a one-dimensional process-based multi-year lake ice model, MyLake, to simulate the evolution of the Northern Hemisphere lake-ice and thermal structure patterns under a changing climate. After testing the model on Baker Lake located in Nunavut Canada, large-scale simulations were conducted over the major land masses of the Northern Hemisphere subarctic regions between 40° and 75°N using hypothetical lakes positioned at 2·5° latitude and longitude resolution. For the baseline period of 1960–1999, the lake-ice model was driven by gridded atmospheric forcings from the ERA-40 global reanalysis data set while atmospheric model forcings corresponding to future (2040–2079) climate were obtained by modifying the ERA-40 data according to the Canadian Global Climate Model projection based on the SRES A2 emissions scenario. Analysis of the modelling results indicates that lake-ice freeze-up timing will be delayed by 5–20 days and break-up will be advanced by approximately 10–30 days, thereby resulting in an overall decrease in lake-ice duration by about 15–50 days. Maximum lake-ice thickness will also be reduced by 10–50 cm. The change in maximum snow depth on the lake-ice ranges between − 15 to + 5 cm, while the change in white-ice thickness ranges between − 20 to + 10 cm depending on the geographic location and other climate parameters. The future warming will also result in an overall increase in lake-water temperature, with summer stratification starting earlier and extending later into the year. Copyright © 2011 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.

Salonen K., Shirasawa K., Solbakov V., Leppäranta M., Arkhipov B., Pulkkanen M., Huttula T.

Page M.A.

The process of diffusion isn't usually expected to be able to generate useful work. But when a neutrally buoyant wedge object is placed in a fluid with a vertical density gradient, the diffusion-driven flow of material can indeed generate a measureable horizontal propulsion.

Allshouse M.R., Barad M.F., Peacock T.

Few would expect the process of diffusion to generate useful work. But when a buoyant wedge-shaped object is placed in a fluid with a vertically stratified density gradient, the upward flow driven by diffusion can be translated into a measurable horizontal propulsion. Buoyancy-driven flow, which is flow driven by spatial variations in fluid density1, lies at the heart of a variety of physical processes, including mineral transport in rocks2, the melting of icebergs3 and the migration of tectonic plates4. Here we show that buoyancy-driven flows can also generate propulsion. Specifically, we find that when a neutrally buoyant wedge-shaped object floats in a density-stratified fluid, the diffusion-driven flow at its sloping boundaries generated by molecular diffusion produces a macroscopic sideways thrust. Computer simulations reveal that thrust results from diffusion-driven flow creating a region of low pressure at the front, relative to the rear of an object. This discovery has implications for transport processes in regions of varying fluid density, such as marine snow aggregation at ocean pycnoclines5, and wherever there is a temperature difference between immersed objects and the surrounding fluid, such as particles in volcanic clouds6.

Terzhevik A., Golosov S., Palshin N., Mitrokhov A., Zdorovennov R., Zdorovennova G., Kirillin G., Shipunova E., Zverev I.

The 5-year-long (2001–2005) studies of the winter thermal structure and the dissolved oxygen (DO) dynamics in Lake Vendyurskoe, Russia, a typical boreal shallow mesotrophic lake of glacial origin, revealed still poorly studied features of lake-wide dynamics, such as net lateral heat flux towards deeper parts of a lake and development of the anaerobic zone over the deepest points of the lake basin. We estimated magnitude of the heat transport along the bottom slope based on scaling analysis. The seasonal changes in DO concentration appear to be controlled mostly by biochemical consumption. We identify four factors controlling the extent of anoxic zones in shallow ice-covered lakes: (1) the amount of organic matter stored in the bottom layers, including the sediments surface during the autumnal bloom; (2) the length of the ice-covered period; (3) heat content of bottom sediments; and (4) the initial water temperatures at the time of the ice cover formation.

Kirillin G., Engelhardt C., Golosov S., Hintze T.

We performed high-resolution temperature measurements under ice cover in Lake Müggelsee, Germany, during the winter of 2005–2006. Intense seiche-like temperature oscillations developing after the ice-on have been encountered in a thin water layer above the sediments. The oscillations were initiated immediately after lake freezing by the release of the potential energy of the thermocline slope and existed for several weeks without appreciable external forcing. The oscillations were associated with a basin-scale internal waves existing in the lower stratified part of the water column. The weakness of the density stratification under ice ensured the long wave periods, exceeding the period of geostrophic inertial oscillations at the lake’s latitude. As a result, two frequency peaks were present in the oscillations corresponding to two rotational waves, one of Kelvin-wave type and another of Poincaré type wave. The rotational character provided long dissipation times of the waves and allowed the oscillations to persist in lake several weeks. Temperature measurements in the upper several centimeters of the sediment demonstrated that oscillations of the near-bottom temperature produced vertical density instability and pore-water convection in the upper sediments.

PAGE M.A., JOHNSON E.R.

An imposed normal temperature gradient on a sloping surface in a viscous stratified fluid can generate a slow steady flow along a thin ‘buoyancy layer’ against that surface, and in a contained fluid the associated mass flux leads to a broader-scale ‘outer flow’. Previous analysis for small values of the Wunsch–Phillips parameter R is extended to the nonlinear case in a contained fluid, when the imposed temperature gradient is comparable with the background temperature gradient. As for the linear case, a compatibility condition relates the buoyancy-layer mass flux along each sloping boundary to the outer-flow temperature gradient. This condition allows the leading-order flow to be determined throughout the container for a variety of configurations.

PAGE M.A., JOHNSON E.R.

Wunsch (1970) and Phillips (1970) (Deep-Sea Res. vol. 17, pp. 293, 435) showed that a temperature flux condition on a sloping non-slip surface in a stratified fluid can generate a slow steady upward flow along a thin ‘buoyancy layer’. Their analysis is extended here to the more-general case of steady flow in a contained fluid where buoyancy layers may expel or entrain fluid from their outer edge. A compatibility condition that relates the mass flux and temperature gradient along that edge is derived, and this allows the fluid recirculation and temperature perturbation to be determined in the broader-scale ‘outer flow’ region. The analysis applies when the Wunsch–Phillips parameter R is small, in the linear case for which the density variations are dominated by a constant vertical gradient.