Small Scale Turbulence and Mixing Energy Fluxes in Stratified Lakes

A Wuest, Eawag, Surface Waters - Research and Management, Kastanienbaum, Switzerland A Lorke, University of Koblenz-Landau, Landau/Pfaly, Germany

© 2009 Elsevier Inc. All rights reserved.

Introduction

Density Stratification and Mixing - the Basin Scale

Nearly all lakes, reservoirs, and ponds that are deeper than a few meters, experience cycles of density stratification and destratification. Most important for this variation is the temperature-dependence of water density. During spring/summer - or the wet season in the tropics - the water is heated from above and a surface layer (SL: typically a few m thick) with warmer and hence lighter water develops on top of the cooler and heavier water below (Figure 1). In addition, although more important in saline lakes than freshwater ones, biological and hydrological processes may strengthen the density stratification by generating a vertical gradient in the concentration of dissolved substances (salinity). The resulting stratification is usually depicted by a strong density gradient (also called pycnocline), separating the SL from the deeper reaches of the water column (indicated as metalimnion and hypolimnion in Figure 1). Mixing of heavier water from greater depth with lighter water from the SL implies that water parcels of different densities are exchanged in the vertical direction (Figure 2). It is evident that mechanical energy is needed to move these water parcels against the prevailing density gradient, which forces lighter water up and heavier water down. The amount of energy needed to overcome vertical density stratifications is therefore determined by the potential energy A£pot (Figure 1) stored in the stratification. A£pot is calculated from the vertical separation of the centre of volume of the water body and its center of mass. Density stratification results in a lowering of the centre of mass by the vertical distance AhM (Figure 1) and the energy needed to overcome the stratification and to mix the entire water column is AEpot = HpgAhM (Jm~2), where H is the average depth of the water body, g is the gravitational acceleration, and p is the density. Density stratification thus imposes stability on the water column and reduces - or even suppresses - vertical mixing.

Besides convective mixing in the SL - caused by seasonal or nocturnal surface cooling - in most lakes and reservoirs, the major source of energy for vertical mixing is the wind, whereas river inflows usually play a minor role (Figure 1). As water is 800 times denser than air and as momentum is conserved across the air-water interface, SLs receive only about 3.5% of the wind energy from the atmosphere above. Surface waves transport a portion of this energy to the shore where it is dissipated; the remaining energy causes large-scale currents, with surface water flows of 1.5-3% of the wind velocity. Moreover, surface currents cause a stratified water body to pivot with warm water piling up at the downwind end (causing downwelling) and deep-water accumulating at the upwind end (causing upwelling). After the wind ceases, the water displacement relaxes and various internal waves develop - including basin-scale seiches -inducing motion even in the deepest layers.

These deep-water currents are usually one order of magnitude less energetic than those in the SL. Typical deep flows of a few centimeters per second (or — 1Jm~3) with energy dissipation of less than 1 mW m~2 are able to reduce the potential energy of the stratification by only —0.01-0.05 mWm~2. Compared to the potential energy stored in the stratification (order of 1000J m~2; Figure 1) it would take much longer than one season to entirely mix a moderately deep lake. This implies that wind energy input (Figure 1) forms the vertical hypolimnion structure at times of weak stratification (beginning of the season), whereas the wind is not able to substantially change the vertical structure once the strong stratification is established. Therefore, in most regions on Earth, only very shallow waters (less than a few meters deep (such as Lake Balaton, Hungary) are found to be entirely nonstratified, even during the summer season. The majority of lakes and reservoirs deeper than a few meters are thus only 'partially' mixed to a limited depth, which is basically defining the SL. For those lakes that show a pronounced SL, its maintenance is mostly supported by night-time cooling. In this article, we focus on the 'limited' mixing below the SL, which occurs in the metalimnion and hypolimnion (Figure 1).

Density Stratification and Mixing - the Small Scale

The same concepts of stability and mixing - described in the preceding section for the entire water body - also apply locally within the water column for small-scale vertical mixing of stratified layers. Local stability of the density stratification is quantified by the Brunt-Vaisala frequency (also buoyancy frequency) N (s_1), defined by:

Wind energy io-4 to io-2

Net heat flux -200 to +200

Wind energy io-4 to io-2

Net heat flux -200 to +200

Energy fluxes in W m-2 Energy content in J m-2

Density Stability Temperature

Figure 1 Energy fluxes (heat, wind, and river inflow; in red) into the water (Wm ) and energy content (heat, kinetic energy, potential energy; in blue) stored in the lake water body (J m~2). Note that the energy fluxes and contents related to heat are many orders of magnitude larger than those of kinetic and potential energy. The effect of mixing by the river is only local and less effective than wind. The stratified part of the lake (below surface layer) has historically been divided into the metalimnion (see large stability, right) and the deep hypolimnion (weak stratification) below. The lower water column can also be differentiated into an interior region (away from the boundaries) which is quiescent except during storms and a bottom boundary layer where turbulence is enhanced. Adopted from Imboden DM and Wuest A (1995) Mixing mechanisms in lakes. In: Lerman A, Imboden D, and Gat JR (eds.) Physics and Chemistry of Lakes, vol. 2, pp. 83-138. Berlin: Springer-Verlag.

Cold

-Temperature density —

Cold

Temperature density —

Figure 2 The effect of turbulent mixing in a stable stratification: if the vertical gradient of horizontal currents (current shear du/dz) is stronger then the stability of the water column (eqn. [1]), Kelvin Helmholtz instabilities can develop (top of middle panel) bringing warmer (lighter) and cooler (heavier) water in close proximity (bottom of middle panel). Finally, heat (or any other water constituent) is mixed by molecular diffusion across the manifold small-scale interfaces, which are generated by turbulence. The turbulent exchange of small water parcels leads to a fluctuating vertical heat flux (see example in Figure 3) which averages to a net downward heat flux. As a result, the original temperature profile (left) is modified (right): the gradient is weakened and expanded vertically with heat transported from top to bottom, and density vice versa, across the interface. Figure after the idea of Winters KB, Lombard PN, Riley JJ, and D'asaro EA(1995) Available potential-energy and mixing in density-stratified fluids. Journal of Fluid Mechanics 289: 115-128. Experiments were first performed by Thorpe SA (1973) Experiments on instability and turbulence in a stratified shear-flow. Journal of Fluid Mechanics 61: 731-751; and the phenomenon of sheared stratification in lakes was reported by Mortimer CH (1952) Water movements in lakes during summer stratification; evidence from the distribution of temperature in Windermere. Philosophical Transactions of the Royal Society of London B: Biological Sciences 236(635): 355-398; and by Thorpe SA (1977) Turbulence and Mixing in a Scottish loch. Philosophical Transactions of the Royal Society of London A: Mathematical Physics and Engineering Sciences 286(1334): 125-181.

z is the depth (positive upward). As a result of wind-forced motions, a vertical gradient of the horizontal current u (shear du/dz) is superimposed on the vertical density gradient dp/dz. Depending on the relative strength of N compared to the current shear du/dz such a stratified shear flow may eventually become unstable and develop into turbulence (Figure 2).

Although the large-scale (advective) motions are mainly horizontal, the turbulent eddies are associated with random velocity fluctuations in all three dimensions (U, V, w'). Turbulent kinetic energy (TKE) (Jkg-1) is defined as the energy per unit mass of water which is contained in these velocity fluctuations:

In stratified turbulence, the vertical velocity fluctuations w' are of particular importance as they transport water parcels and their contents in the vertical direction (Figure 3). The product of the vertical velocity fluctuations w and the associated density fluctuations p' describes an instantaneous vertical flux of density (w'p ' (kgm~2s_1)). Resulting from many irregular and uncorrelated fluctuations (Figure 3) the averaged flux w'p' leads to a net upward mass flux, which is usually expressed as a buoyancy flux Jb

Therefore, we can interpret vertical mixing as an upward flux of mass, which causes a change of the potential energy of the stratification (Figures 1 and 2), expressed as a buoyancy flux (eqn. [3]). The required energy originates from the TKE, which is itself extracted from the mean (horizontal) flow. Approximately 90% of the TKE, however, does not contribute to the buoyancy flux (and hence to vertical mixing) but is instead dissipated into heat by viscous friction, without any further effect. By defining local rates of production P (Wkg-1) and viscous dissipation e(Wkg_1) of TKE, the simplest form of TKE balance can be formulated as:

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Figure 3 Time series of O2 concentration (thin line, a) and vertical velocity w' (thin line, b; positive = upward), as measured 10 cm above the sediment in reservoir Wohlensee (Switzerland) at a frequency of 64 Hz. Red lines indicate the temporally varying averages, determined as running mean, whereas the black horizontal line marks the averages. Panel (c) shows the instantaneous eddy flux - covariance of w' and O2': The average downward O2 flux over the 30 s (~1900 data pairs) is -6.4mmol m_2day_1. Data source: Claudia Lorrai, Eawag.

15 20

Figure 3 Time series of O2 concentration (thin line, a) and vertical velocity w' (thin line, b; positive = upward), as measured 10 cm above the sediment in reservoir Wohlensee (Switzerland) at a frequency of 64 Hz. Red lines indicate the temporally varying averages, determined as running mean, whereas the black horizontal line marks the averages. Panel (c) shows the instantaneous eddy flux - covariance of w' and O2': The average downward O2 flux over the 30 s (~1900 data pairs) is -6.4mmol m_2day_1. Data source: Claudia Lorrai, Eawag.

As mentioned above, the dissipation rate is usually much larger than the buoyancy flux, and hence the mixing efficiency gmix, which is defined as the ratio

Jb e

is much smaller than 1. A number of studies in stratified lakes and reservoirs have revealed typical mixing efficiencies in the range of 10-15%.

Density Stratification and Mixing - the Turbulent Transport

The local flux of a water constituent is given by the product of the velocity times the concentration. In stratified waters, the time-averaged vertical velocity is often close to zero (negligible) and thus, the vertical fluxes stem only from the fluctuations of velocity and concentration, such as explained above for the vertical mass flux w'p' caused by the turbulence. This concept holds for any other water constituent, such as for oxygen, as exemplified in Figure 3, where the in situ measured w', O2' and the product w'O2' is shown for a 30-s-long record. Although the momentary fluxes up and down are almost of equal variations and amounts, the averaging WO2 reveals slightly larger fluxes downwards to the sediment, where the oxygen is consumed.

Until recently, direct measurements of turbulent fluxes had not been possible and therefore turbulent fluxes in stratified waters are commonly expressed using the eddy diffusivity concept. Applied to the mass flux w'p' it implies assuming that (i) a well-defined local density gradient dp/dz exists (due to the stratification) and (ii) the flux - in analogy to molecular diffusion - can be expressed by the eddy (or turbulent) diffusivity Kz (m2s-1) multiplied by this local gradient:

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