[7

where g is the acceleration due to gravity, and z is the vertical coordinate. The magnitude N is also called stability frequency or Brunt-Vaisala frequency (s-1), which indicates the maximum frequency (o) for internal waves that can propagate in the respective stratification. N2 indicates how much energy is required to exchange water parcels in the vertical.

As a consequence, chemical gradients can only persist for longer time periods where density gradients limit the vertical transport of dissolved substances (see Figure 10). Lake basin Niemegk of Lake Goitsche (Germany) has been neutralized by introducing buffering river water to the epilimnion. During summer 2000, the vertical transport through the temperature stratification was limited and a chemical gradient in pH could be sustained (Figure 11). However, in winter the temperature stratification vanished, vertical transport was enhanced, and consequently, chemical gradients were removed. Gradients in the pH close to the lake bed were stabilized by increased density because of higher concentration of dissolved substances.

In a stratified water column, a current shear can supply kinetic energy for producing vertical

° Rassnitzer See • Mar 1999-Feb 2000

. •..«?"

10-4 10-3 10-2 10-1 Stability frequency squared N2 [s-2]

10-4 10-3 10-2 10-1 Stability frequency squared N2 [s-2]

Figure 10 Turbulent diffusive transport of an artificial tracer (SF6) in the strongly stratified monimolimnion of Rassnitzer See, versus density gradient, N2 = -g/p dp/dz. Adapted from von Rohden and Ilmberger (2001) Aquatic Sciences 63: 417-431.

60 50

8 70

Figure 10 Turbulent diffusive transport of an artificial tracer (SF6) in the strongly stratified monimolimnion of Rassnitzer See, versus density gradient, N2 = -g/p dp/dz. Adapted from von Rohden and Ilmberger (2001) Aquatic Sciences 63: 417-431.

60 50

60 50

S 40

co co

8 70

60 50

S 40

60 50

60 50

1999

2000

2001

Figure 11 Contour plot of temperature, electrical conductance and pH value versus time and depth in mining Lake Goitsche (station XN3 in Lake basin Niemegk); period of neutralization by flooding with river water; the rising water level is represented by the increasing colored area.

1999

2000

2001

Figure 11 Contour plot of temperature, electrical conductance and pH value versus time and depth in mining Lake Goitsche (station XN3 in Lake basin Niemegk); period of neutralization by flooding with river water; the rising water level is represented by the increasing colored area.

excursions and overturns. A comparison between density gradient and current shear yields the nondi-mensional gradient Richardson number:

1 10 Gradient Richardson number[-]

Figure 12 Relation between diapycnal diffusivities and gradient Richardson number. Adapted from Imboden DM and Wüest A (1995) Physics and Chemistry of Lakes, pp. 83-138. Berlin: Springer-Verlag.

1 10 Gradient Richardson number[-]

Figure 12 Relation between diapycnal diffusivities and gradient Richardson number. Adapted from Imboden DM and Wüest A (1995) Physics and Chemistry of Lakes, pp. 83-138. Berlin: Springer-Verlag.

where u = u(z) represents the horizontal current velocity profile. The critical value of Ri = 1/4, when the shear flow supplies enough energy to sustain overturning water parcels, is found by considering the energy balance in the centre of mass frame. As a consequence, diapycnal transports rapidly increase, if Richardson numbers get close to 0.25 or even fall below this critical value (see Figure 12). Although in zones of high shear, e.g., in the bottom boundary layer, supercritical Richardson numbers can be found, they appear only sporadically in the pelagic region of lakes, at least if measured on a vertical scale of meters. The measurements in Figure 12 suggest a correlation between vertical transport coefficients and gradient Richardson number of

D = 3 x 10-9ffz'-96 + 7 x 10-6R'-13 + 1.4 x 10-7 (m2s 1 ^

if the value of molecular diffusivity of heat is included. On the basis of this, vertical diffusivities can be calculated from gradient Richardson number measurements, which are displayed in Figure 13, where gradient Richardson number was measured in pelagic waters over a depth resolution of 3-5 m.

Bulk Quantities For the bulk stability of a stratified water body, various quantities have been proposed, based on potential energy integrated from lake bottom Zb to the surface zs (e.g., Birge work). We list the definition of Schmidt stability St as the most often used reference for the work required for mixing a stratified lake:

O 60

100 10

O 60

10-6 10-5 10-Vertical turbulent diffusivity [m2/s]

Figure 13 Profiles of vertical transport coefficients in Lake Constance during the stratification period, based on gradient Richardson number measurements in the area of high shear at the Sill of Mainau (solid line), in comparison with results of the gradient flux method for the entire lake based on the evolution of temperature profiles in Uberlinger See in years 1987, 1988 or 1989. Adapted from Boehrer et al. (2000) Journal of Geophysical Research 105(C12): 28,823.

10-6 10-5 10-Vertical turbulent diffusivity [m2/s]

Figure 13 Profiles of vertical transport coefficients in Lake Constance during the stratification period, based on gradient Richardson number measurements in the area of high shear at the Sill of Mainau (solid line), in comparison with results of the gradient flux method for the entire lake based on the evolution of temperature profiles in Uberlinger See in years 1987, 1988 or 1989. Adapted from Boehrer et al. (2000) Journal of Geophysical Research 105(C12): 28,823.

where z = zV = If^ zA(z)dz is the vertical position of the centre of lake volume V, and p is the density of the hypothetically homogenized lake.

In a two-layer system, like thermally stratified lakes, it is reasonable to compare the potential energy needed for vertical excursion with the wind stress applied to the surface as done by the Wedderburn number W:

u where g = g -pp, with Ap representing the density difference between epilimnion and hypolimnion, u\ = t/p is the friction velocity resulting from the surface stress t implied by the wind, and L stands for the length of the fetch.

Although the common use of the Wedderburn number is connected to its simplicity, the more sophisticated Lake number LN = Mbc/ (zV JA tdA) compares the wind stress applied to the lake surface with the angular momentum Mbc needed for tilting the thermocline, and hence represents the integral counterpart of the Wedderburn number.

Small values of both, W and LN, indicate that wind stress can overcome restoring gravity forces because of density stratification. Under such conditions, upwell-ing of hypolimnion water is possible and intense mixing of hypolimnion water into the epilimnion can be expected. A typical consequence of ecological importance facilitated by this process is the recharging of nutrients in the epilimnion from the hypolimnion.

a = (JTj s,p thermal expansion (K 1) aref,a25 coefficient at reference temperature, at

ßn coefficient for specific density contri bution of salts g conductivity specific (potential) den sity contribution (kg m-3 mS-1 cm) G (potential) density contribution by dissolved substances (kg m-3) kref,k25 electrical conductance at reference temperature, at 25 °C (mS cm-1) p (potential) density (kg m-3)

Pin situ (potential) density (kg m-3)

pT (potential) density of pure water (kg m-3) 3

p reference density (kg m 3)

0 potential temperature (K)

t surface stress (Pa)

o wave frequency (rad/s)

Nomenclature a,ai,b,bi coefficients

A area, especially surface area of a lake

C electrical conductivity (mS cm-1)

Cn concentration of substance (g kg-1)

D vertical turbulent diffusivity (m2 s-1)

hepi thickness of epilimnion (m)

g acceleration due to gravity (m2/s)

g reduced acceleration due to gravity

L length of lake or wind fetch (m)

LN lake number

Mbc angular momentum (N m)

N stability frequency (s-1)

Ri gradient Richardson number

Ricrit = 0.25 critical gradient Richardson number [ ]ref,[ ]25 at reference temperature, mostly 25°C S salinity for fresh water or ocean con ditions (psu) St Schmidt stability (kg m)

Tmd temperature of maximum density (°C)

Tref reference temperature (°C)

u horizontal current velocity (m s-1)

u* friction velocity (m s-1)

V lake volume (m3)

W Wedderburn number z vertical coordinate (m)

zb,zs,zV vertical coordinate of lake bed, sur face, centre of volume (m)

See also: The Benthic Boundary Layer (in Rivers, Lakes, and Reservoirs); Currents in Stratified Water Bodies 1: Density-Driven Flows; Currents in Stratified Water Bodies 2: Internal Waves; Currents in Stratified Water Bodies 3: Effects of Rotation; Currents in the Upper Mixed Layer and in Unstratified Water Bodies; Effects of Climate Change on Lakes; Lakes as Ecosystems; Meromictic Lakes; Mixing Dynamics in Lakes Across Climatic Zones; Paleolimnology; Saline Inland Waters; Small-Scale Turbulence and Mixing: Energy Fluxes in Stratified Lakes.

Further Reading

Boehrer B and Schultze M (2008) Stratification of lakes. Reviews in Geophysics. 46, RG2005, doi:10.1029/2006RG000210.

Boehrer B Ilmberger J, and Miinnich K (2000) Vertical structure of currents in western Lake Constance. Journal of Geophysical Research 105(C12): 28,823-28,835.

Chen CTA and Millero FJ (1986) Precise thermodynamic properties for natural waters covering only the limnological range. Limnology and Oceanography 31: 657-662.

Gat JR (1995) Stable isotopes of fresh and saline lakes. In: Lerman A, Imboden D, and Gat J (eds.) Physics and Chemistry of Lakes, pp. 139-165. Berlin: Springer-Verlag.

Halbwachs M, Sabroux J-C, Grangeon J, Kayser G, Tochon-Danguy J-C, Felix A, Beard J-C, Vilevielle A, Vitter C, Richon P, Wuest A, and Hell J (2004) Degassing the 'Killer Lakes' Nyos and Monoun, Cameroon. EOS 85(30): 281-284.

Hongve D (2002) Endogenic Meromixis: Studies ofNordbytjernet and Other Meromictic Lakes in the Upper Romerike Area. PhD thesis, Norwegian Institute of Public Health, Oslo Norway.

Hutchinson GE (1957) A Treatise on Limnology vol. 1. New York: Wiley.

Imberger J and Patterson JC (1990) Physical limnology. Advances in Applied Mechanics 27: 303-475.

Imboden DM and Wüest A (1995) Mixing mechanisms in lakes. In: Lerman A, Imboden D, and Gat J (eds.) Physics and Chemistry Lakes, pp. 83-138. Berlin, Germany: Springer-Verlag.

ISO standard 7888 (1985) Water quality: Determination of electrical conductivity. International Organization for Standardization. www.iso.org.

Johnk KD (2000) 1D hydrodynamische Modelle in der Limnophysik - Türbülenz, Meromixis, Sauerstoff. Habilitationsschrift, Darmstadt, Germany, Technical University of Darmstadt.

Kalff J (2002) Limnology. Upper Saddle River, NJ: Prentice Hall.

Kjensmo J (1994) Internal energy, the work of the wind, and the thermal stability in Lake Tyrifjord, southeastern Norway. Hydrobiologia 286: 53-59.

Sorenson JA and Glass GE (1987) Ion and temperature dependence of electrical conductance for natural waters. Analytical Chemistry 59(13): 1594-1597.

Stevens CL and Lawrence GA (1997) Estimation of wind forced internal seiche amplitudes in lakes and reservoirs, with data from British Columbia, Canada. Aquatic Science 59: 115-134.

von Rohden C and Ilmberger J (2001) Tracer experiment with sulfur hexafluoride to quantify the vertical transport in a mero-mictic pit lake. Aquatic Sciences 63: 417-431.

Relevant Websites

http://www.ilec.or.jp/- International Lake Environment Committe;

data on various lakes on Earth. http://www.ioc.unesco.org/ - Intergovernmental Oceanographic Commission of UNESCO; provides on-line calculator for salinity following the so-called UNESCO formula. http://www.cwr.uwa.edu.au/ - Centre for Water Research (CWR) at The University of Western Australia. Research Institution focussing of physical limnology. http://www.eawag.ch/ - Swiss Federal Institute of Aquatic Science and Technology. Institution dealing with water related issues.

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