Turbulence and Mixing in Stratified Lakes and Reservoirs

Turbulence Production in the Surface and Bottom Boundaries

There are fundamentally two mechanisms generating turbulence in the SL: (i) the action of wind causing g e

Table 1 Typical values of dissipation, stability and vertical diffusivity in stratified waters

Dissipationa e (Wkg1) Stability N2 (s 2) Diffusivity* Kz (m2s ^

Table 1 Typical values of dissipation, stability and vertical diffusivity in stratified waters

Dissipationa e (Wkg1) Stability N2 (s 2) Diffusivity* Kz (m2s ^

Ocean thermocline

io-10-io-8

~10-4

(0.3-3) x 10~5

Surface layer

10-6-10-9

0-~10-5

10-5-10-2

Lake interior only (without BBL)

10"12-10_1°

10-8-10-3

10-7-10-5

Metalimnion (basin scale)

10-10-10-8

~10-3

(0.5-50) x 10-7

Near-shore metalimnion

10-10-10-6

~10-3

(0.3-3) x 10-4

Deep hypolimnion (basin scale)

10"12-10-10

10-8-10-6

(0.03-3) x 10-4

aDuring storm events values are larger by orders of magnitudes for short.

aDuring storm events values are larger by orders of magnitudes for short.

Time after tracer release (days)

Figure 4 Vertical spreading of the tracer Uranine after injection at 25 m depth in Lake Alpnach (Switzerland). The vertical line demarcates the initial period of 7 days, during which Uranine resided in the interior of the stratified deep water. The two insets show the lake area at the surface and at the depth of the Uranine injection, as well as the horizontal distribution of the Uranine cloud (shaded in gray) after 4 and 28 days. The slow growth of the spreading in the first 7 days illustrates the quietness in the interior. The fast growth of the vertical spreading after day 7 is due to the increasing contribution of BBL mixing after the tracer has reached the sediment at 25 m depth. Reproduced from Goudsmit GH, Peeters F, Gloor M, and Wuest A (1997) Boundary versus internal diapycnal mixing in stratified natural waters. Journal of Geophysical Research 102: 27903-27914, with permission from American Geophysical Union.

Time after tracer release (days)

Figure 4 Vertical spreading of the tracer Uranine after injection at 25 m depth in Lake Alpnach (Switzerland). The vertical line demarcates the initial period of 7 days, during which Uranine resided in the interior of the stratified deep water. The two insets show the lake area at the surface and at the depth of the Uranine injection, as well as the horizontal distribution of the Uranine cloud (shaded in gray) after 4 and 28 days. The slow growth of the spreading in the first 7 days illustrates the quietness in the interior. The fast growth of the vertical spreading after day 7 is due to the increasing contribution of BBL mixing after the tracer has reached the sediment at 25 m depth. Reproduced from Goudsmit GH, Peeters F, Gloor M, and Wuest A (1997) Boundary versus internal diapycnal mixing in stratified natural waters. Journal of Geophysical Research 102: 27903-27914, with permission from American Geophysical Union.

wave breaking and shear in the top few meters of the water column and (ii) surface cooling causing the sinking of heavier water parcels. Temperature-driven mixing (case (ii)) leads to homogenization of the SL and therefore to nonstratified conditions - at least for a few hours or days before heat fluxes from/to the atmosphere restratify the SL. This process is discussed in detail elsewhere in this encyclopedia. Only in shallow ponds or basins with relatively high through-flow will turbulence have other case-specific sources.

For wind-driven mixing (case (i)), the crucial parameter governing the dynamics of turbulence in the SL is the surface shear stress t (Nm-2), the force per unit area exerted on the water by the wind. This stress is equal to the downward eddy-transport of horizontal momentum from the atmosphere. Part of t is consumed in the acceleration and maintenance of waves (tWave), whereas the remaining momentum flux tsl generates currents and turbulence in the SL. By assuming a constant stress across the air-water interface, the two momentum fluxes on the water side equal the total wind stress (t = tSL + tWave).

Immediately below the waves, the momentum flux, tsl, drives the vertical profiles of horizontal velocity u(z) in the SL. If the wind remains relatively constant for hours, quasi-steady-state conditions may develop in the SL: u(z) then depicts the Law-of-the-Wall du/dz = = (tSL/r)1/2(kz)-1, where u* = (tSL/r)1/2 is the frictional velocity and k (= 0.41) is the von Karman constant. Because the buoyancy flux in the SL (defined in eqn. [3]) is not a large contribution in eqn. [4], we can assume a balance between the production of TKE and the rate of viscous dissipation (e) of TKE. This local balance between production and dissipation of turbulence determines the turbulence intensity as a function of depth throughout the SL. Under those assumptions, the dissipation e = (tSL/p) du/dz = u*3 (kz)-1 [9]

is only a function of the wind-induced stress tsl (here expressed as u») and of depth z. Several experiments have demonstrated that dissipation is indeed inversely proportional to depth (eqn. [9]), if averaged for long enough. However, one has to be critical about the validity of eqn. [9] for two reasons: First, at the very top of the water column, breaking waves, in addition to shear stress, produce a significant part of the turbulence in the SL. This additional TKE generation at the surface can be interpreted as an injection of TKE from above. Therefore, in the uppermost layer, the turbulence exceeds the level described by eqn. [9], depending on the intensity of the wave breaking. Second,

Ice cover Vetrical diffusivity

Ice cover Vetrical diffusivity

Day of the year

Figure 5 Vertical diffusivitiesin Lake Baikal simulated with ak-epsilon model. The numbers (1-6) on the contour plot indicate the main features of the seasonal stratification and changes in diffusivity: the formation of thermal stratification with weak mixing (1) during winter under the ice and (2) during summer; (3) the formation of a convectively mixed layer in spring under the ice; the deep convective mixing in (4) June and (5) November; and (6) the formation of a mixed layer near the temperature of maximum density. Here the emphasis is on the temporal and vertical structure of the turbulent diffusivity and not on the absolute accuracy, which may be difficult to achieve with turbulence modeling better than a factor of 2-3. Reproduced from Schmid M etal. (2007) Sources and sinks of methane in Lake Baikal: A synthesis of measurements and modeling. Limnology and Oceanography 52: 1824-1837, with permission from American Society of Liminology and Oceanography.

Day of the year

Figure 5 Vertical diffusivitiesin Lake Baikal simulated with ak-epsilon model. The numbers (1-6) on the contour plot indicate the main features of the seasonal stratification and changes in diffusivity: the formation of thermal stratification with weak mixing (1) during winter under the ice and (2) during summer; (3) the formation of a convectively mixed layer in spring under the ice; the deep convective mixing in (4) June and (5) November; and (6) the formation of a mixed layer near the temperature of maximum density. Here the emphasis is on the temporal and vertical structure of the turbulent diffusivity and not on the absolute accuracy, which may be difficult to achieve with turbulence modeling better than a factor of 2-3. Reproduced from Schmid M etal. (2007) Sources and sinks of methane in Lake Baikal: A synthesis of measurements and modeling. Limnology and Oceanography 52: 1824-1837, with permission from American Society of Liminology and Oceanography.

eqn. [9] relies on quasi-steady-state conditions which may hold applicable for limited episodes only.

Despite these restrictions, eqn. [9] gives a good estimate of the diffusivity in the SL, if it is weakly stratified. Equations [8] and [9] reveal that the rate of mixing increases substantially within the SL as the surface is approached. The corresponding stability N2 decreases at the surface and maintains rapid mixing. Therefore, gradients of temperature, nutrients, and par-ticulates are usually smallest at the surface and increase with depth. During sunny days, diurnal thermoclines form with mixing reduced below them. On cloudy, windy days, the SL may mix fully and may even deepen depending upon the surface forcing. Factors that affect the depth of mixing are discussed elsewhere in this encyclopedia. It is typically a few m during the warm season and a few tens of meters during the cold season. Below, a strong density gradient (pycnocline) can develop leading to the separation between the SL and the metalimnion/hypolimnion. In the stratified interior (away from the BBL; see below), the effect of wind is shielded and the mixing regime is completely different.

As discussed in greater detail (see The Benthic Boundary Layer (in Rivers, Lakes, and Reservoirs)), turbulence generation and mixing along the bottom boundaries of water bodies can be described in analogy to the SL. Under steady-state conditions, the resulting bottom boundary layer (BBL) follows a similar vertical structure of (i) current shear (see above), (ii) rate of TKE dissipation (eqn. [9]) and (iii) rate of vertical mixing. Although the original indirect driving force for turbulence in the BBL is also the wind, it is not the direct turbulent momentum flux from the atmosphere to the water which is the cause. Rather, the mechanism is indirectly induced by wind which causes large-scale currents and basin-wide internal waves (such as seiches) which act as intermediate energy reservoirs that generate TKE by bottom friction. Along sloping boundaries in particular, the breaking of propagating internal waves and convective processes - a secondary effect of bottom friction -can produce additional TKE, leading to dissipation and mixing in excess of that predicted by eqn. [9]. As with the SL, the BBL is also usually partly (and weakly) stratified. Again, mixing (eqn. [8]) increases substantially when approaching the sediment and often a completely homogenized layer a few m thick develops at the bottom.

Internal Waves and Turbulence in the Stratified Interior

In the lake interior, away from surface and bottom boundaries (Figure 1), the water body is stratified and quiescent, and it does not feel the direct effects of the turbulence sources at the surface and above the sediment. This stratified interior consists of an upper region, the metalimnion where gradients in temperature and density are strongest, and a lower region, the hypolimnion, which is only weakly stratified and most water properties are homogeneous. Internal waves are prevalent.

The rate of mixing in the interior water body is low because (i) currents and shear are weak and the resulting turbulence production is reduced and (ii) stratification suppresses the turbulent mixing. The mechanical energy originates mainly from basin-scale internal currents and waves (see above), whereas the waves of smaller scale and higher frequencies -potentially generated at a few specific locations - are not contributing much to the energy budget of the deep-water. At the transition between small- and large-scale waves are the near-inertial currents, which can carry - especially in large lakes - a significant portion of the mechanical energy typically in the order of ~1Jm~3. Given that observed energy residence time-scales are days (small lakes) to weeks (deepest lakes), the dissipation of the internal energy is ~1Q-l2-~ 1Q-10Wkg-1 (Table 1). Considering typical values for stratification N2 (eqn. [1]) of 10 8-10 3 s~2 and gmix « 0.1 (eqn. [5]), interior diffu-sivities of 10-7-10-5 m2 s_1 can be expected (Table 1; Figure 4). The stratified interior - away from the SL and the BBL - is by far the most quite zone in lakes.

Important for the generation of small-scale mixing are local instabilities related to internal (baroclinic) motions, such as illustrated in Figure 2. Instabilities occur mostly where the usually weak background shear is enhanced by nonlinear steepening of internal waves or by superposition of the shear with small-scale propagating internal waves.

Direct observations of turbulence and mixing, using microstructure and tracer techniques, confirm that turbulence is indeed very weak in the stratified interior. Typically, only a few percent of the water column is found to be actively mixing. The occurrence of such turbulent patches is highly intermittent in space and time. During periods when the fluid is nonturbulent, we can expect laminar conditions and thus the dominance of molecular transport. The observable average diffusivity can be considered the superposition of a few turbulent events separated by molecular diffusion for most of the time. The resulting transport in the stratified interior will therefore be close to molecular. Tracer experiments and microstructure profiling conducted in small and medium-sized lakes confirm these quiet conditions in the interior and enhanced turbulence in the bottom boundary. In Figure 4 the vertical spreading of a tracer, injected into the hypolimnion, is shown for the interior (first few days) and for a basin-wide volume including the BBL (after a few days). From Figure 4 it is evident that turbulent diffusivity in the interior is at least one order of magnitude lower than in the basin-wide deep-water volume, including the bottom boundary. In addition to these spatial differences, one has to be aware of the temporal variability. During storms, turbulence can be several orders of magnitude larger for short episodes. The transition from quiescent to actively mixing occurs rapidly once winds increase above a certain threshold relative to the stratification. The internal wave field is energized and turbulence can develop. But the greatest increases occur in the benthic BBL. It is during such storms that most of the vertical flux takes place.

The turbulent patches, where vertical fluxes are generated (as exemplified in Figure 3) vary in size depending in part upon the turbulence intensity e and the stratification N2. Several length scales have been developed to characterize the sizes of turbulent eddies. One is the Ozmidov scale

and the other is the Thorpe scale, LT, which is based on direct observations of the size of unstable regions. The ratio of the two numbers varies depending upon the strength of stratification and is useful for predicting the efficiency of mixing, gmix in eqn. [5] Typical values for LO and LT range from a few centimeters to a meter but for weak stratification eddies are larger and on scales of tens of meters to 100 m as found in weakly stratified Lake Baikal.

The spatial and temporal dynamics of mixing challenges not only the experimental estimation, but also the numerical simulation of its net effect, in terms of a turbulent diffusivity Kz. Local measurements of Kz following eqn. [8] often neither resolve its spatial nor its temporal dynamics and the coarse grid sizes used in numerical simulations do not capture the small scales relevant for mixing processes in the interior.

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Responses

  • stefan newby
    When would fishh be equally stratified in a lake?
    4 months ago

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