Density Stratification and StabilityB Boehrer and M Schultze, UFZ – Helmholtz Centre for Environmental Research, Magdeburg, Germany

ã 2009 Elsevier Inc. All rights reserved.

Introduction

In most lakes, water properties change from the sur-face to greater depth, i.e., these lakes show a verticalstratification of their water masses at least for someextended time periods. Heat exchange with the atmos-phere and the forming of gradients of dissolved sub-stances controls internal waves and the verticalexchange of water within the lakes. This has decisiveimpact on the evolution of water quality and, as aconsequence, on the community of organisms livingin the lake. This article deals with processes contrib-uting to the stratification of lakes and the forming oflayers. The most common numerical approaches forthe quantitative evaluation of stratification-relevantphysical quantities, e.g., electrical conductivity, areincluded. The final section lists quantities for stabilityof density stratification and what conclusion can bedrawn from them.

Circulation Patterns

Surface temperatures of lakes show a pronounced tem-perature cycle over the year (Figure 1), in most lati-tudes. This is a consequence of heat exchange with theatmosphere and the seasonal variation of meteorologi-cal parameters, such as incoming solar radiation. Thetemperatures in the deep water follow the surfacetemperatures only for the time when the lake is homo-thermal, as in our example Lake Goitsche, Germany(Figure 1), during winter, from November until April.Throughout summer, temperatures vary from the

surface to the lake bed, and the lake remains strati-fied. Warmer and less dense water floats on top ofcolder, denser water (Figure 1). Thus, Lake Goitscheis called stably stratified, as overturning water parcelswould require energy. On the contrary, during winter,no density differences obstruct the vertical transport.These seasons are commonly referred to as the stag-nation period and the circulation period. Lakes thatexperience a complete overturn during the year arecalled holomictic.During the circulation period, dissolved sub-

stances, such as oxygen or nutrients, get distributedover the entire water body (Figure 2). Hence, thecirculation pattern is a decisive factor for the evolu-tion of water quality and the biocenosis of the lake. Inconclusion, the commonly used classification of lakesis according to their circulation patterns.

. Holomictic lakes overturn and homogenize at leastonce a year.

. Inmeromictic lakes, the deep recirculation does notreach the deepest point of the lake. As a conse-quence, a chemically different layer of bottomwater is formed, the monimolimnion (see below),and remains there for at least 1 year.

. Amictic lakes do not experience a deep recircula-tion. Usually permanently ice-covered lakes areincluded in this class. Lakes, however, can circulateunderneath an ice sheet by external forcing, such assolar radiation that penetrates to the lake bed andgeothermal heat flux, or salinity gradients createdwhen ice is forming on a salt lake.

. Lakes with episodic partial deep water renewal donot experience a complete overturn. The deep waterhowever is partially replaced in episodic events.

Holomictic lakes are subdivided into classes indicat-ing the frequency of complete overturn.

– Polymictic lakes are not deep enough to support acontinuous stratification period throughout sum-mer. The entire lake is mixed by sporadic strongwind events over the year or even on a daily basisin response to strong diurnal temperature variation.

– Dimictic lakes are handled as the prototype oflakes in moderate to cold climates. A closer lookat the lakes, however, reveals that in most cases anice cover or a great maximum depth is required toguarantee a stratification period during the coldseason (see Figure 2). Between ice cover and sum-mer stratification, the lake can be circulatedcompletely in the vertical, the easiest when surfacetemperatures traverse the temperature of maxi-mum density at 4 �C.

– Monomictic lakes possess one circulation period inaddition to the stratificationperiod.Many lakes in thetemperate climate zone belong into this class, if theydo not develop an ice cover during winter. Sometimessuch lakes are also referred to aswarmmonomictic todistinguish them from cold monomictic lakes, whichshow an ice cover for most of the year and circulateduring the short period without ice.

– Oligomictic lakes circulate less frequently thanonce a year, normally at irregular intervals, trig-gered by extreme weather conditions such as un-usually cold winters for the respective location.

As a consequence of the natural variability of theweather conditions between years, the circulation

583

584 Hydrodynamics and Mixing _ Density Stratification and Stability

patterns of the lakes also vary. A usually monomic-tic lake, for example, can show a dimictic circulationpattern when it freezes in an unusually cold winter. Asanother example, late during the twentieth century,Mono Lake turned meromictic for intermittent peri-ods of 5 or 7 years, respectively, because of inflowingfreshwater, but in other years showed a holomicticcirculation.

Density Differences and Formationof Layers

Temperature Stratification

Although the surface water is exposed to solar radia-tion and thermal contact with the atmosphere, the

Wind

Spring

Oxygen

Winter

Nutrients

Figure 2 Annual cycle of a dimictic lake with ice cover during winte

0

4

8

12

16

20

242005

0.5 m4.0 m7.0 m10.0 m18.0 m31.0 m

Tem

pera

ture

[°C

]

J F M A M J J A S O N D

Figure 1 Temperatures (24 h mean) on several depths in Lake

Goitsche near Bitterfeld, Germany during the year 2005.Reproduced from Boehrer B and Schultze M (2008) Stratification of

lakes. Reviews in Geophysics, 46, RG2005, doi:10.1029/

2006RG000210, with permission from American Geophysical

Union.

deeper layers are shielded from major sources ofheat. Diffusive heat transport on a molecular level isvery slow and requires a month for the transport ofheat over a vertical distance of 1m. A much moreefficient heat transport can be accomplished by tur-bulent transport. The energy for the turbulence ismainly supplied by wind stress at the lake surfaceand transferred via instabilities through friction atthe side walls and internal current shear.

Heating a lake over 4 �C at the surface results in astable stratification. As a consequence, transport ofheat to greater depths requires energy. The limitedbudget of kinetic energy available for mixing limitsthe depth to which a certain amount of heat can beforwarded over the stratification period. In suffi-ciently deep lakes, the thermal stratification holdsuntil cooler autumn and winter temperatures permita deeper circulation. The warm surface water layer iscalled epilimnion, while the colder water layerbeneath, which has not been mixed into the epilim-nion is called hypolimnion. A sharp temperature gra-dient (thermocline) separates both layers (Figure 3).

Epilimnion and atmosphere are in thermal contactand exchange volatile substances with each other. Inaddition, the epilimnion is recirculated by windevents or periods of lower temperatures during thestratification period. During those periods, dissolvedsubstances are distributed within the epilimnion. Onthe contrary, the hypolimnion is insulated fromexchange with the atmosphere during the stratificationperiod. Transport of dissolved matter across the verti-cal density gradient of the thermocline usually is small.

In general, wind determines the thickness of theepilimnion, with few exceptions, e.g., where lightpenetrates beyond the mixing depth because ofwind, or where the stratification is determined by

Wind

Wind

Epi-

LimnionHypo-

Summer

Autumn

Nutrients

Oxygen

r.

40

30

20

10

0

5 10 15 20 25

Thermocline

Epilimnion

Hypolimnion

Dep

th [m

]

Temperature [°C ]

03.11.200329.09.200311.08.2003

16.06.200312.05.200314.04.2003

Figure 3 Temperature profiles of Lake Goitsche/Germany in

station XN5 in 2003. Symbols are added for every sixteenth data

point to distinguish between acquisition dates.

102 104 106 108

Surface area [m2]

Patalas (1984)

Fee et al. (1996)Arai (1981)

Osgood (1988)

Patalas (1961)

1

10

100

Epi

limni

on th

ickn

ess

[m]

Figure 4 Graphical representation of several approximations

of epilimnion thickness zepi versus surface area of the respective

lakes. Adapted from Johnk KD (2000) 1D hydrodynamische

Modelle in der Limnophysik – Turbulenz, Meromixis, Sauerstoff.Habilitationsschrift, Technical University of Darmstadt, Germany.

40

30

20

10

0

0

0.0 0.2 0.4 0.6 0.8 1.0

0O2 [mg/l]

Dep

th [m

]

Metalimnion

Hypolimnion

Epilimnion

T [°C]

TC

O2

C [mS/cm]

4 8 12 16 20

2 4 6 8 10

Figure 5 Profiles of temperature (T), (in situ) conductivity (C),

and concentration of dissolved oxygen (O2) from 6 September

2000 in Arendsee/Germany. The boundaries between layerswere drawn along the gradients in the oxygen profiles. Oxygen

concentration numerically corrected for response time of 7.5 s of

the sensor. Adapted from Boehrer and Schultze, 2005,

Handbuch Angewandte Limnologie, Landsberg: ecomed.

Hydrodynamics and Mixing _ Density Stratification and Stability 585

inflow and water withdrawal (reservoirs). As inferredfrom Figure 3, the thickness of the epilimnion is notconstant over the stratification period. In spring, athin layer is formed, which gradually thickens overthe summer because of the cumulative input of windenergy and diurnal heating and cooling. It takes untilautumn, when colder temperatures at the lake surfacecan erode the stratification. During this later periodof thermal stratification, substances dissolved inhypolimnetic waters, such as nutrients, become avail-able in the epilimnion again. Eventually epilimnionand hypolimnion are homogenized.The epilimnion thickness is a crucial factor for

living organisms. Hence limnologists have tried tocorrelate epilimnion thickness with lake morphome-try to achieve an a priori estimate (Figure 4). Themost central regression is hepi ¼ 4:6� 10�4A 0:205,which includes the higher energy input from windsover lakes with larger surface area A.Because of its high gradients, the thermocline forms

a special habitat. Organisms controlling their densitycan position themselves in the strong density gradient.Also, inanimate particles can accumulate on their levelof neutral buoyancy and motile organisms dwell inthe thermocline to profit from both layers, epilimnionand hypolimnion. As a result, a layer of distinctiveproperties can form, called metalimnion. Especiallyin nutrient-rich lakes, the decomposition of organicmaterial can deplete oxygen resulting in a so-calledmetalimnetic oxygenminimum (Figure 5). On the con-trary, if light can penetrate to the thermocline andphotosynthesis can overcome the oxygen consumptionlocally, a metalimnetic oxygen maximum occurs.

Thermobaric Stratification

Cold water is more compressible than warmer water,in the range of temperatures encountered in lakes. Asa consequence, the temperature of maximum densityTmd decreases as pressure, i.e., depth, increases (byabout 0.2 K over 100m water depth). Hence in coldenough regions, very deep freshwater lakes can showtemperature profiles during summer stratification

100

200

300

400

500 1400

1200

1000

800

600

400

200

0

3 4 5 3 3.3 3.6 3 4 5

Tin

nsjo

200

6-06

-12

TmdTmd

0D

epth

[m]

Temperature [8C] Temperature [8C] Temperature [8C]

Tmd400

300

200

100

0

Bai

kal 1

997-

06-0

6

Shi

kots

u

2005

-04-

19

2005

-05-

21Figure 6 Temperatureprofiles of thermobarically stratified lakes:

Left panel: Tinnsj�, Norway, during (early) summer stratification;

Central panel: Lake Baikal, Siberia, Russia, during (late) winter

stratification with the vertical transition through Tmd; Right panel:Lake Shikotsu, Hokkaido, Japan, with the nearly isothermal

deep water body below the Tmd transition. Reproduced from

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

permission from American Geophysical Union.

586 Hydrodynamics and Mixing _ Density Stratification and Stability

that extend below 4 �C, though limited to the coldside by the Tmd profile (see Figure 6, left panel).During winter, surface temperature can be lowerthan Tmd, while at greater depth the temperaturesmay be above Tmd (Figure 6, central panel). Stabledensity stratification is achieved in these profiles, ifabove the Tmd intersection, colder temperatures over-lie warmer temperatures, while below this it is theopposite. At the intersection itself, the vertical tem-perature gradient must disappear. The water bodybelow the Tmd profile is not directly affected by theannual temperature cycle (Figure 6, right panel).Nevertheless, observations in such lakes (e.g., Lake

Baikal, Russia; Crater Lake, USA; Lake Shikotsu,Japan) show deep waters well supplied with oxygen.This can in part be attributed to the fact that tem-peratures are low, and productivity and depletion ofoxygen happen at a slow rate. However, it is also anindication for a considerable amount of mixingbetween mixolimnion and the deep water. As a con-sequence, chemical gradients do not appear in theselakes, and scientists have refrained from calling theselakes meromictic, although a complete overturn doesnot occur.

Salinity Stratification

A considerable portion of lakes is salty. As there is nocompelling boundary, lakes are called salt lakes, if the

salt content lies above 3 g in a kilogram of lake water;i.e., 3 g kg�1 ¼ 3%. From this concentration, humanscan clearly taste the salt, and ecological consequencesbecome obvious. The salt content in lakes can be ashigh as 300 g kg�1. However, salinities normally liebelow 0.5 g kg�1. Even smaller salinity gradients candetermine the circulation of lakes.

Many large salt lakes, e.g., Caspian Sea, Issyk-Kul,Aral Sea, Lake Van, Great Salt Lake, and the DeadSea, are located in endorheic basins, i.e., areas on theEarth without hydraulic connection to the worldocean at the surface. Salt lakes also occur outsidethese areas, as solar ponds or basins filled with seawater that lost the connection to the sea. In addition,some inland lakes are fed by saline groundwater.

Salt modifies the properties of lake water. As a quan-titative expression, the familiar magnitude of salinityhas been transferred from oceanography to limneticwater. One kilogram of ocean water contains about35g of salt. Brackish water, i.e., water mixed from seawater and freshwater, shows a similar mix of dissolvedsubstances, while the composition of salts in lakes cangreatly deviate from ocean conditions. Consequently,salinity is better replaced by total dissolved substancesTDS in the limnetic environment. For quantitativeinvestigations, usually the physical quantity of electricalconductance is suited much better (explained later).

In some lakes, dissolved substances raise the den-sity of the deep waters enough that part of the watercolumn is not recirculated at any time duringthe annual cycle. The remaining bottom layer, themonimolimnion, can show very different chemicalconditions (Figure 7). Such lakes are termed mero-mictic. Well-known examples are the meromicticlakes of Carinthia, Austria, and Lake Tanganyika,or Lake Malawi. Some small and deep maar lakesas well as natural lakes in southern Norway andFinland are permanently stratified by small concen-tration differences between mixolimnion and moni-molimnion. In addition, deep pit lakes tend to bemeromictic.

The monimolimnion is excluded from the gasexchange with the atmosphere over long time peri-ods. Diffusive and turbulent exchange across thechemocline usually is small. As a consequence, anoxiaestablishes in most cases after sufficient time. Underthese chemical conditions, nitrates and sulphatesserve as agents for the microbial oxidation of organicmaterial and substances can be produced that wouldchemically not be stable in the mixolimnion. Perma-nently exposed to the hydrostatic pressure, gases(CO2, H2S, and others) can accumulate in monimo-limnion in concentrations far beyond the concentra-tions encountered in mixolimnia (e.g., Lake Monounin Cameroun, Africa, Figure 8).

35

30

25

20

15

10

5

0

0

Temp

Temperature [°C]

Dep

th [m

]

Epilimnion

Monimolimnion

Chemocline

Hypolimnion

Salinity

Salinity [psu]

Density

Density- 1000 [kg/m³]

O2

Oxygen [mg/l]5 10 15 20 25

Figure 7 Profiles of temperature, salinity, dissolved oxygen

and density from Rassnitzer See in former mining areaMerseburg-Ost on 7th October 2003. Oxygen concentrations are

numerically corrected for a sensor response time of 7.5s.

Adapted from Boehrer and Schultze, 2005, Handbuch

Angewandte Limnologie, Landsberg: ecomed.

********

****** ***** * * * ******

*****

* **

**************

**********

* **

CH4

CO2

Sum

Hydrostatic

Hydrostatic pressureSum of the partial pressures

N2 partial pressureCH4 partial pressureCO2 partial pressure

0

0

10

20

30

40

50

60

70

80

90

1002 4 6 8

Pressure [bar]

Dep

th [m

]

N2

Figure 8 Profiles of partial pressures of dissolved gases in

deep water of Lake Monoun, Cameroun, in direct comparisonwith hydrostatic pressure (solid line). Reproduced from

Halbwachs et al. (2004) EOS 85(30): 281–288, with permission of

American Geophysical Union.

Hydrodynamics and Mixing _ Density Stratification and Stability 587

In many meromictic lakes, the recirculation of themixolimnion erodes the monimolimnion leaving asharp gradient at the end of the circulation period.The transition of all water properties happens withinfew decimeters frommixolimnetic tomonimolimnetic

values (see Figure 7). This sharp gradient is calledhalocline, chemocline, or pycnocline, depending onwhether the salinity, chemical, or density gradient isreferred to. From observations, cases of intensive col-onization with only few different species are known,where some plankton species obviously take advan-tage of such gradients (e.g., LagoCadagno in the SwissAlps, Lake Bolvod, Gek Gel, and Maral Gel).

Processes Forming Gradients of Dissolved

Substances

In general, diffusive and turbulent diffusive processesdistribute dissolved substances more equally through-out a lake over time. On the contrary, inflows ofdifferent concentration of dissolved substances orinternal processes can produce gradients within alake. Freshwaters can flow onto a salt lake or enterthe epilimnion during the stratification period andhence induce a difference between epilimnion andhypolimnion. These gradients contribute to the den-sity gradients and if strong enough they may evencontrol the circulation pattern of the lakes. Inextreme cases, saline waters have been captured indeep layers of lakes for several thousands of years(e.g., Rorhopfjord, Norway; Salsvatn, Norway;Powell Lake, Canada). Such stable stratificationshave been induced on purpose in mine lakes to con-fine heavy metals in the salty monimolimnion forfurther treatment (Island Copper Mine Lake, Van-couver Island, BC, Canada).

Also, the opposite case of lakes being exposed tohigh evaporation can encounter salinity gradients inthe water column with the saltier layer above. In thecase of Lake Svinsj�en (Norway), salty water fromdeicing roads has removed the previously presentpermanent stratification. In another case, when ice isforming on salt lakes the residual water can be highlyloaded with salts, while after ice melt, relatively freshwater floats onmore saline water (lakes in Antarctica).Also, groundwater inflows can form such gradients ofdissolved substances (e.g., Lago Cardagno in SwissAlps, Rassnitzer See; see Figure 7). Especially, lakes involcanic areas are known for the continuous rechargeof dissolved substances (e.g., Lake Nyos and LakeKivu in East Africa, Lake Monoun see Figure 8). Inparticular, the latter case is interesting, as the dis-solved gases contribute decisively to the stable densitystratification, but they also supply the buoyancy forcatastrophic limnic eruptions, in which poisonousgases escape abruptly from a lake with disastrousconsequences for living organisms in the area.

Chemical reactions and biological activity in pre-ferred layers can locally change the composition ofdissolved substances and impact on the density

588 Hydrodynamics and Mixing _ Density Stratification and Stability

structure. Although in most lakes at most depthsphysical transport mechanisms prevail, in somecases the density stratification is controlled by chemi-cal and biological transformations of dissolved mate-rial and even control the circulation pattern of lakes.For example, photosynthetically active plankton usesincoming solar radiation for the production oforganic material. In addition, allochthonous materialis carried into the lake by surface inflows and wind.A portion of the organic material settles on the lakebed. Its decomposition is facilitated by the presence ofoxygen or other oxidizing agents and the end pro-ducts CO2 and HCO�

3 dissolve in the deep layers ofthe lake where they contribute to the density. Alsoiron (and manganese) cycling, calcite precipitation,and sodium sulfate precipitation have been documen-ted to control density in lakes.In some cases, gradients are strong enough to

prevent overturns. These meromictic lakes are clas-sified according to the dominant process sustainingthe density difference between mixolimnion andmonimolimnion in ectogenically (surface inflow;e.g., Rorhopfjord and Salsvatn in Norway; PowellLake and Island Copper Mine Lake in Canada,Lower Mystic Lake in the United States), crenogeni-cally (groundwater inflow; e.g., Lago Cardagno,Rassnitzer See, Lake Nyos), or biogenically (orendogenically) meromictic lakes (decomposition oforganic material; e.g., Woerther See and Laengsseein Austria, iron cycle: Lakes in Norway, mine lakes,manganese: Lake Nordbytjernet, Norway; calcite:Lake La Cruz, Central Spain; sodium sulfate:Canadian prairie lakes). Also, basin depth andbasin shape play an important role in the erosionof a monimolimnion and the formation of deepwater renewal. Often monimolimnia are found inwell-defined depressions in a lake bed, which areonly marginally impacted by basin scale currents inthe water body above.

Episodic Partial Deep Water Renewal

A number of lakes do not experience a completeoverturn. Episodic events replace parts of the deepwater. For example, by cooling, water parcels of highdensity are formed within the mixolimnion and man-age to proceed through the surrounding waters downinto the deep water. This process is similar to the deepocean circulation. Thus, it is not surprising that someof the largest lakes undergo this process. Issyk-Kul(central Asia) recharges its deep water by surfacecooling and channelling the cold waters throughsubmerged valleys to the abyss, while in Lake Baikal,wind forcing can push water below the compensationdepth so that (temperature-dependent) compression

under high pressure raises the density enough com-pared with surrounding water that its buoyancybecomes negative; the water parcel continues to pro-ceed deeper. Issyk-Kul and Lake Baikal do not showsignificant chemical gradients and hence are nottermed meromictic. On the contrary, Lake Malawi(East Africa) shows an anoxic monimolimnion, andhence is called meromictic, although the deep waterformation is similar to that of Issyk-Kul.

In parallel to heat, concentration gradients of dis-solved substances also can force deep water renewalin perennially stratified water bodies. Salinity isincreased when water is exposed to high evaporationin a side basin (Dead Sea before 1979), or salt accu-mulates underneath a forming ice cover (Deep Lake,Antarctica). As soon as water parcels become denseenough, they can proceed along slopes into deep partsof the lake.

Quantifying Stability

Temperature

Temperatures recorded in lakes are so-called in situtemperatures. Without any further annotation, tem-perature data will be understood as such. Nearly allcalculations refer to this value, as it is the physically,chemically, and ecologically relevant magnitude.However, if detailed considerations on stability andvertical temperature gradients are envisaged, the ref-erence to potential temperature may be useful. Thislatter quantity includes the effect of energy requiredfor the expansion, when a water parcel is transferredto atmospheric pressure:

dT

dz

� �ad

¼ gaðT þ 273:15Þcp

½1�

where a is the thermal expansion coefficient for awater parcel of temperature T along the path fromdepth z to the surface. In lakes where the deep water isclose to temperatures of maximum density Tmd, thethermal expansion coefficient is very small, a � 0,and, as a consequence, the difference between in situtemperature and potential temperatures is smallY�T.In lakes with warmer deep waters, a can be consideredconstant. Figure 9 shows a monotonous potentialtemperature profile in Lake Malawi, Africa, whichindicates stable stratification by temperature only.

Salinity, Electrical Conductivity and Electrical

Conductance

Many substances in lake water are dissolved as ions.Hence electrical conductivity has been used to

22.5 23 23.5 24 24.5 25 25.5700

600

500

400

300

200

100

0

Temperature [°C]

Dep

th [m

]

T in-situPotential T

Figure 9 Profiles of (in situ) temperature T and potential

temperature near the deepest location of Lake Malawi on 13September 1997. Reproduced from Boehrer B and

Schultze M (2008) Stratification of lakes. Reviews in Geophysics,

46, RG2005, doi:10.1029/2006RG000210, with permissionfrom American Geophysical Union.

Hydrodynamics and Mixing _ Density Stratification and Stability 589

quantify dissolved substances. For compensation pur-poses, the temperature dependence of electrical con-ductivity of a water sample is recorded, whilescanning the relevant temperature interval. In mostcases, a linear regression C(T) = aT þ b is satisfactoryto define the conductance, i.e., the electrical conduc-tivity kref ¼ CðTref Þ ¼ aTref þ b at a certain referencetemperature Tref. Most commonly, 25 �C is used forthe reference:

k25 ¼ CðTÞa25ðT � 25�CÞ þ 1

where aref ¼ ðTref þ b=aÞ�1 ½2�

In most surface waters, a value close to a25 ¼ 0:02K�1

is appropriate. Electrical conductance is used for abulk measurement of concentrations of ionicallydissolved substances, quantifying transports fromchanges in the conductance profile, and to base den-sity regression curves on.Oceanography uses electrical conductivity and

temperature to calculate salinity in practical salinityunits (psu), which gives a good indication for dis-solved salt in grams per kilogram for ocean waterand brackish water (water mixed from ocean waterand fresh water). In limnetic systems, however, thecomposition of dissolved substances differs from thatof the ocean. Even within some lakes, there are pro-nounced vertical gradients. As a consequence, salinitycan only be used with reservation in limnic systems.

Density

As direct density measurements in the field are notaccurate enough, indirect methods based on easy to

measure temperature and conductivity are implemen-ted. As in most practical applications the differencebetween in situ and potential temperature is small;we use T for temperature. For lakes of low salinities(<0.6 psu), density can be approximated:

r ¼ rðS;TÞ ¼X6i¼0

aiTi þ S �

X2i¼0

biTi ½3�

using

ai ¼ ½999:8395; 6:7914 � 10�2;�9:0894 � 10�3; 1:0171 � 10�4;

�1:2846 � 10�6; 1:1592 � 10�8;�5:0125 � 10�11�

bi ¼ ½0:8181;�3:85 � 10�3; 4:96 � 10�5�In lakes of a composition of dissolved substances simi-lar to the ocean, the so-called UNESCO formula maybe applied (e.g., Rassnitzer See in Figure 6), which isapplicable for salinities above 2psu. In cases wheresalinity cannot be used, calculation of density maydirectly be based on measurements of temperatureand conductivity. For Lake Constance – Obersee, thefollowing formula was proposed, where 20 �C wasused as reference temperature for conductance k20:

r ¼ rT þ G ¼ 999:8429þ 10�3

� ð0:059385T 3 � 8:56272T 2 þ 65:4891T Þ þ G ½4�adding the conductivity contribution in separate

G ¼ gk20and g ¼ 0:67 � 10�3 kgm�3mS�1cm ½5�

Alternatively, if the dissolved substances are known,e.g., from chemical analysis, density can be calculatedby adding the separate contributions:

r ¼ rT 1þXn

bnCn

!½6�

where Cn is the concentration of the substancen (g kg�1). A short table of coefficients

bn ¼1

r@r@Cn

� �Y;p;Cm

m 6¼ n

is given in Table 1. In most limnological applications,density is used for stability considerations. Hence, den-sity refers to potential density, i.e., the density of acertain water parcel under normal atmospheric condi-tions (1013hPa). As a consequence of the (small) adia-batic compressibility of water, in situ density increaseswith pressure, i.e., water depth, by about 5� 10�10 Pa.Thismeans that at 200mdepth, (potential) density andin situ density differ by about 10�3.

Stability

Stability of a water column derives from the densityincrease in the vertical. Hence, it is a measure for the

Table 1 Contribution of dissolved or suspended substances to

the density of water

Substance bn[(kg/kg)]

Ca(HCO3)2 0.807

Mg(HCO3)2 0.861

Na(HCO3) 0.727K(HCO3) 0.669

Fe(HCO3)2 0.838

NH4(HCO3) 0.462

CO2 0.273CH4 �1.250

Air �0.090

Modified from Imboden, DM and Wuest A (1995). Physics and Chemistry

of Lakes, pp. 83–138. Berlin: Springer-Verlag.

Stability frequency squared N2 [s−2]10−4

10−8

10−7

10−6

10−3 10−2 10−1

Ver

tical

turb

ulen

t diff

usiv

ity [m

2 /s]

Mar 1999 – Feb 2000Rassnitzer See

Figure 10 Turbulent diffusive transport of an artificial tracer

(SF6) in the strongly stratified monimolimnion of Rassnitzer See,

versus density gradient, N2 ¼ �g/r dr/dz. Adapted from von

Rohden and Ilmberger (2001) Aquatic Sciences 63: 417–431.

20

15

10

T [8C]

Temperature70

60

50

40de [m

abo

ve s

ea]

590 Hydrodynamics and Mixing _ Density Stratification and Stability

potential energy required for vertical excursion andfor overturning of water parcels. Stability considera-tions can be made for an interface in the watercolumn or for the entire stratified water body as awhole. If an energy source is known, the ratio ofrequired and supplied energy yields a nondimensionalnumber.

5

0

k25 [mS/cm]

6

4

2

0

6

8

4

2

pH

pH

Electrical conductance

30

70

60

50

40

30

Alti

tuA

ltitu

de [m

abo

ve s

ea]

70

60

50

40

30Alti

tude

[m a

bove

sea

]

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 waterlevel is represented by the increasing colored area.

Differential Quantities The stability of a densitystratification is quantified by

N 2 ¼ � g

rdrdz

½7�

where g is the acceleration due to gravity, and z is thevertical coordinate. The magnitude N is also calledstability frequency or Brunt–Vaisala frequency (s�1),which indicates the maximum frequency (o) for inter-nal waves that can propagate in the respective strati-fication.N2 indicates how much energy is required toexchange water parcels in the vertical.As a consequence, chemical gradients can only

persist for longer time periods where density gradi-ents limit the vertical transport of dissolved sub-stances (see Figure 10). Lake basin Niemegk ofLake Goitsche (Germany) has been neutralized byintroducing buffering river water to the epilimnion.During summer 2000, the vertical transport throughthe temperature stratification was limited and achemical gradient in pH could be sustained (Figure11). However, in winter the temperature stratifica-tion vanished, vertical transport was enhanced, andconsequently, chemical gradients were removed.Gradients in the pH close to the lake bed were stabi-lized by increased density because of higher concen-tration of dissolved substances.In a stratified water column, a current shear

can supply kinetic energy for producing vertical

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

Ri ¼ N 2

ðdu=dzÞ2 ½8�

D~Ri−9.6

Peters et al. (1988)

Ricrit= 0.25

Münnich et al. (1992)

0.1 1 10 10010−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Ver

tical

trub

ulen

t diff

usiv

ity [m

2 /s]

Gradient Richardson number[−]

Figure 12 Relation between diapycnal diffusivities and gradient Richardson number. Adapted from Imboden DM andWuest A (1995)

Physics and Chemistry of Lakes, pp. 83–138. Berlin: Springer-Verlag.

×+ +

+

+

+

+

+

+

+

+

+

×

×

×

×

×

×

×

×

×

×

Vertical turbulent diffusivity [m2/s]

10−7100

Dep

th [m

]

80

1989 Rich.-nr.1987 grad-flux1988 grad-flux1989 grad-flux

60

40

20

0

10−6 10−5 10−4 10−3

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 atthe 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 or1989. Adapted from Boehrer et al. (2000) Journal of Geophysical

Research 105(C12): 28,823.

Hydrodynamics and Mixing _ Density Stratification and Stability 591

where u ¼ u(z) represents the horizontal currentvelocity profile. The critical value of Ri = 1/4, whenthe shear flow supplies enough energy to sustain over-turning water parcels, is found by considering theenergy balance in the centre of mass frame. As aconsequence, diapycnal transports rapidly increase,if Richardson numbers get close to 0.25 or even fallbelow this critical value (see Figure 12). Although inzones of high shear, e.g., in the bottom boundarylayer, supercritical Richardson numbers can befound, they appear only sporadically in the pelagicregion of lakes, at least if measured on a vertical scaleof meters. The measurements in Figure 12 suggest acorrelation between vertical transport coefficientsand gradient Richardson number of

D ¼ 3 � 10�9Ri�9:6 þ 7 � 10�6Ri�1:3 þ 1:4 � 10�7ðm2s�1Þ½9�

if the value of molecular diffusivity of heat isincluded. On the basis of this, vertical diffusivitiescan be calculated from gradient Richardson numbermeasurements, which are displayed in Figure 13,where gradient Richardson number was measured inpelagic waters over a depth resolution of 3–5m.

Bulk Quantities For the bulk stability of a stratifiedwater body, various quantities have been proposed,based on potential energy integrated from lake bottomzb to the surface zs (e.g., Birge work). We list the defini-tion of Schmidt stability St as the most often used refer-ence for the work required for mixing a stratified lake:

St ¼Z zs

zb

ðz� zV ÞðrðzÞ � �rÞAðzÞdz ½10�

where z ¼ zV ¼ 1V

R zszbzAðzÞdz is the vertical position

of the centre of lake volume V, and �r is the density ofthe hypothetically homogenized lake.

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

W ¼ g 0h2epiu2�L

½11�

592 Hydrodynamics and Mixing _ Density Stratification and Stability

where g 0 ¼ g Drr , withDr representing the density differ-

ence between epilimnion and hypolimnion, u2� ¼ t=ris the friction velocity resulting from the surface stresst implied by the wind, and L stands for the length ofthe fetch.Although the common use of the Wedderburn

number is connected to its simplicity, the moresophisticated Lake number LN ¼ Mbc= zV

RA tdA

� �compares the wind stress applied to the lake surfacewith the angular momentum Mbc needed for tiltingthe thermocline, and hence represents the integralcounterpart of the Wedderburn number.Small values of both, W and LN, indicate that wind

stress can overcome restoring gravity forces because ofdensity stratification. Under such conditions, upwell-ing of hypolimnion water is possible and intense mix-ing of hypolimnion water into the epilimnion can beexpected. A typical consequence of ecological impor-tance facilitated by this process is the recharging ofnutrients in the epilimnion from the hypolimnion.

Nomenclature

a,ai,b,bi

coefficients A area, especially surface area of a lake

(m2)

cp specific heat (J K�1 kg�1) 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) g0 reduced acceleration due to gravity

(m2/s)

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) T (in situ) temperature (�C) 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)

a = @r@T

� �S,p

thermal expansion (K�1)

aref,a25

coefficient at reference temperature, at25 �C (K�1)

bn

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 bydissolved substances (kg m�3)

kref,k25

electrical conductance at referencetemperature, at 25 �C (mS cm�1)

r

(potential) density (kg m�3) rin situ (potential) density (kg m�3) rT (potential) density of pure water (kg

m�3)

r¯ reference density (kg m�3) Y potential temperature (K) t surface stress (Pa) o wave frequency (rad/s)

See also: The Benthic Boundary Layer (in Rivers, Lakesand Reservoirs); Chemical Properties of Water; Currentsin 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 UnstratifiedWater Bodies; Effects of Climate Change on Lakes;Lakes as Ecosystems; Meromictic Lakes; MixingDynamics in Lakes Across Cl imat ic Zones;Paleolimnology; Physical Properties of Water; SalineInland Waters; Salinity; Small-scale Turbulence andMixing: Energy Fluxes in Stratified Lakes; The SurfaceMixed Layer in Lakes and Reservoirs.

Further Reading

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

Boehrer B Ilmberger J, and Munnich K (2000) Vertical structure ofcurrents in western Lake Constance. Journal of GeophysicalResearch 105(C12): 28,823–28,835.

Chen CTA and Millero FJ (1986) Precise thermodynamic proper-

ties 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’ Nyosand Monoun, Cameroon. EOS 85(30): 281–284.

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

Hutchinson GE (1957) ATreatise on Limnology vol. 1. New York:

Wiley.

Hydrodynamics and Mixing _ Density Stratification and Stability 593

Imberger J and Patterson JC (1990) Physical limnology. Advancesin Applied Mechanics 27: 303–475.

Imboden DM andWuest A (1995)Mixing mechanisms in lakes. In:

Lerman A, Imboden D, and Gat J (eds.) Physics and ChemistryLakes, pp. 83–138. Berlin, Germany: Springer-Verlag.

ISO standard 7888 (1985) Water quality: Determination of electri-cal conductivity. International Organization for Standardization.

www.iso.org.

Johnk KD (2000) 1D hydrodynamische Modelle in der

Limnophysik – Turbulenz, Meromixis, Sauerstoff. Habilita-tionsschrift, 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 thethermal stability in Lake Tyrifjord, southeastern Norway.

Hydrobiologia 286: 53–59.

Sorenson JA and Glass GE (1987) Ion and temperature dependenceof electrical conductance for natural waters. Analytical Chemis-try 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 OceanographicCommission 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.