Alfred Wuest and Andreas Lorke¨toffolon/didattica/dati_esercitazione/AnnRevFlume… · SMALL-SCALE...

22 Nov 2002 11:39 AR AR159-FM35-16.tex AR159-FM35-16.sgm LaTeX2e(2002/01/18)P1: IBC10.1146/annurev.fluid.35.101101.161220

Annu. Rev. Fluid Mech. 2003. 35:373–412doi: 10.1146/annurev.fluid.35.101101.161220

Copyright c© 2003 by Annual Reviews. All rights reserved

SMALL-SCALE HYDRODYNAMICS IN LAKES

Alfred Wuest and Andreas LorkeApplied Aquatic Ecology (APEC), Limnological Research Center, EAWAG,Kastanienbaum, Switzerland; email: [email protected], [email protected]

Key Words enclosed basin, small-scale turbulence, mixing, waves, boundarylayer, stratification

■ Abstract Recent small-scale turbulence observations allow the mixing regimes inlakes, reservoirs, and other enclosed basins to be categorized into the turbulent surfaceand bottom boundary layers as well as the comparably quiet interior. The surface layerconsists of an energetic wave-affected thin zone at the very top and a law-of-the-wall layer right below, where the classical logarithmic-layer characteristic applies onaverage. Short-term current and dissipation profiles, however, deviate strongly fromany steady state. In contrast, the quasi-steady bottom boundary layer behaves almostperfectly as a logarithmic layer, although periodic seiching modifies the structure in thedetails. The interior stratified turbulence is extremely weak, even though much of themechanical energy is contained in baroclinic basin-scale seiching and Kelvin waves orinertial currents (large lakes). The transformation of large-scale motions to turbulenceoccurs mainly in the bottom boundary and not in the interior, where the local shearremains weak and the Richardson numbers are generally large.

1. OVERVIEW

More than two decades have passed since the lastAnnual Review of FluidMechanicsarticles by Csanady (1975) and Imberger & Hamblin (1982) on lakehydrodynamics were written. Those reviews have been a guide to many physicallimnologists and engineers. Great progress has been achieved in several areas,such as turbulence observations and modeling (Section 2 and 3) as well as mi-crosensor technology and boundary layer fluxes (Section 4), which have beenreviewed in the last decade in part by Imberger & Patterson (1990), Imboden &Wuest (1995), and Imberger (1998a). The assessment of boundary versus internalmixing in enclosed water bodies is surely such a typical area of progress. Othersubjects, such as internal wave dynamics (Section 3), proved to be so complexthat this field of research could not significantly improve the understanding sincethen, although many recent papers addressed this gap. Here, laboratory studiescontributed more to the advancement of the assessment of the different processesinvolved.

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Lakes are similar in many aspects to the ocean, and therefore the limnologic andthe oceanographic literature are very much interleaving, as documented by the twokey journalsLimnology & OceanographyandJournal of Geophysical Research,from which we draw most information for this review. Lakes and oceans arealso extremely different in many aspects such as size, forcing, and stratification,although the greatest difference is the enormous variability among the lakes. Forexample, the stratificationN2 in lakes varies by nine orders of magnitude (Table 1)and is determined not only by temperature and salt (as in the ocean), but alsoby particles, gases (such as methane or carbon dioxide), and, in Lake Vostok,potentially even air hydrates.

Among the many physical processes occurring in lakes, this review concen-trates on small-scale hydrodynamics. The sections are organized according tothe turbulence characteristics. As shown by several publications in the past fewyears, the small-scale processes are distinctly different (Figure 1) between the sur-face boundary layer (Section 2), the interior of the density-stratified deep water(Section 3), and the bottom boundary layer (Section 4). These three sections outlinethe current level of observation and understanding of these three compartments oflacustrine water bodies and highlight their links to small-scale processes. Becausemuch of the oceanographic literature is neglected here, we also refer the readerto the most recent reviews on oceanic surface boundary layers (Csanady 2001,Jones & Toba 2001), bottom boundary layers (Boudreau & Jørgensen 2001), andsmall-scale processes (Kantha & Clayson 2000).

The repeatedly used quantities are summarized in Table 1, including typi-cal ranges of variations, and the continuously applied abbreviations are listed inTable 2.

2. SURFACE BOUNDARY-LAYER PROCESSES

2.1. Introduction

In lakes the surface boundary layer (SBL) is the most dynamic zone in manyrespects. Light and nutrients enable photosynthesis of phytoplankton, which pro-vides the basis of the food web and lays the groundwork for biological interactions.Also physical and geochemical quantities undergo the strongest dynamics owingto exchange with the adjacent atmosphere and to photochemical and biologicalprocesses. Hence, for understanding the ecologically most relevant phenomena,the atmosphere/water relationship is crucial. Depending on the properties of theair, wind, water, and waves, the air/water interface creates a bottleneck for theexchange of the physical quantities such as heat, kinetic energy, momentum, andmatter (gases, vapor, aerosols, etc.). These exchanges of physical and chemicalproperties are driven by wind- and heatflux-induced turbulence (Figure 2). We firstaddress the effect of wind, which is substantially more complex than convection.

The most crucial parameter governing the wind-driven regime is the surfaceshear stressτ [N m−2], the force per unit area, acting on the water surface as a


SMALL-SCALE HYDRODYNAMICS IN LAKES 375

TABLE 1 Key physical parameters used in text (definition and typical range)

Symbol Property Definition Typical range

B Buoyancy flux (rate of change B= (g/ρ)〈w′ρ′ 〉 10−11 to 10−6 W kg−1

of potential energy) (w′ = (ε/N )1/2: verticaleddy velocity

C1 Bottom drag coefficient (for C1= τB/(ρu21 ) 0.001 to 0.003

u1 at 1 m above bottom)

C10 Wind drag coefficient (for U10 C10= τ/(ρaU210) 0.001 to 0.002

at 10 m above bottom) (extreme:∼0.01)

δD Diffusive boundary layer thickness δD ≈ δν(DS/ν)1/(3 . . .4) 0.2 to 1 mm

δν Viscous boundary layer thickness δν ≈ 11ν/u∗ 0.3 to 2 cm

DS Molecular diffusivity of solute S 0.5 to 2× 10−5 cm2 s−1

ε Dissipation of TKE into heat ε= 7.5ν(∂u′/∂z)2 10−11 to 10−6 W kg−1

f Inertial frequency due toÄ 2Ä · sin(φ) 0 to 0.000145 rad s−1

(Ä= frequency of Earth rotation) (φ= latitude)

g Gravity acceleration 9.81 m s−2

γmix Mixing efficiency γmix=B/ε 0.05 to 0.25

k Von Karman constant U(z)= (w∗/k)ln(z/zo) 0.4 to 0.42

Kv Vertical turbulent diffusivity Kv =w′l 10−2 to 102 cm2 s−1

LMO Monin-Obukhov length scale LMO= u3∗/kB SBL: m to several 10 mBBL: dm to 100 m

LO Ozmidov scale, vertical LO= (ε/N3)1/2 cm to several m

(energy-containing scale ofoverturns in stratificationN2)

LT Thorpe scale, vertical scale of LT=〈l2〉1/2 cm to several moverturns (estimated from l= vertical dislocationmeasured profiles) relative to equilibrium

ν, νa Molecular viscosity of water, 0.012 to 0.015 cm2 s−1

molecular viscosity of air ∼0.15 cm2 s−1

N2 Stability of the water column (local) N2=−(g/ρ) ∂ρ/∂z 10−10 to 10−1 s−2

τ Surface wind shear stress τ = ρaC10U210 ∼0.02 N m−2

(extreme: 0 to 25 N m−2)

τB Bottom shear stress τB=ρC1u21 ∼0.001 to 0.02 N m−2

TKE Turbulent kinetic energy TKE= 1/2(u′2+v′2+w′2) ∼10−6 J kg−1

U10 Horizontal wind speed (10 m Typical: 2 m s−1

above water surface) (extreme: 0 to20 m s−1)

u∗, w∗ Friction velocity in water, u∗ = (τB/ρ)1/2=C1/21 u 0.5 to 5 mm s−1

friction velocity in air w∗ = (τ/ρa)1/2=C1/210 U 0 to 1 m s−1

ωp, λp, cp Frequency, wavelength, and cp= (gλp/2π )1/2 1 to 10 rad s−1∼ mphase speed of dominant waves several m s−1

zo Roughness length (SBL) U(z)= (w∗/k)ln(z/zo) 0.1 to 10 mmRoughness length (BBL) u(z)= (u∗/k)ln(h/zo)


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Figure 1 Level of turbulence in a medium-sized lake, expressed by dissipation of turbulentkinetic energy, as a function of depth (circles) and as a function of height above bottom(squares). The plot motivates characterizing the three distinctly different water bodies sep-arately: the energetic surface boundary layer (Section 2), the slightly less turbulent bottomboundary (Section 4), and the strongly stratified and almost laminar interior (Section 3).Adapted from W¨uest et al. (2000b).

result of the wind. The source of this stress can be interpreted as the downwardeddy-transport of horizontal momentum (τ = ρa〈U′W′〉) from the atmosphere. Theconcept of “constant stress” would call for the same Reynolds flux (τSBL= ρ〈u′w′〉)in the underlying water, whereU,W (u,w) are the horizontal and vertical velocitiesof air (water) [m s−1], ρa(ρ) is air (water) density [kg m−3], and′ (prime) denotesfluctuations. Whereas the “constant stress” assumption holds for quite some height

TABLE 2 Abbreviations used in text

ADCP Acoustic Doppler Current Profiler

BBL Bottom boundary layer

DBL Diffusive boundary layer

k-ε Two-equation turbulence model fork (TKE) andε (dissipation)

LOW Law-of-the-wall (synonymous to logarithmic boundary layer, also log-layer)

SBL Surface boundary layer

VBL Viscous boundary layer

WASL Wave affected surface layer



Figure 2 Schematic overview of the surface boundary processes, the shear stress, and theassociated vertical structure of the SBL. Adapted from Thorpe (1985) and Simon (1997).

in the atmosphere (Csanady 2001) and over some extent of the SBL, it is theconservation of momentum that is relevant at the interface. Owing to the presenceof waves, the momentum flux into the SBL,τSBL, is smaller than the applied stressτfrom the air (Figure 2). Part ofτ is consumed by the acceleration and maintenanceof waves (so-called wave stressτWave), whereas the remaining momentum fluxτSBL

is forcing the SBL water underneath the waves (Anis & Moum 1995, Terray et al.1996, Burchard 2001). The conservation of momentum at the interface impliesthat the two momentum fluxes on the water side add to the total wind stress

τ = τSBL+ τWave [N m−2]. (1)

This formulation indicates that waves act as a second pathway for the momen-tum transfer to the water. As a consequence, the wind stress, which is usuallyparameterized by

τ = ρaC10U210 [N m−2], (2)

using the wind drag coefficientC10 [-], depends not only on the wind speedU10

(measured at standard 10-m height above surface), but also on the presence and onthe state of the surface waves. In fact, the wave field is fundamental for the amountof momentum transferred into the water and for its vertical distribution within


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the SBL. This twofold effect of the surface waves for the momentum transfer isaddressed in the following two sections, before we discuss the resulting small-scaleprocesses in the lacustrine SBL.

2.2. Surface Gravity Waves

Waves produce additional roughness, thereby increasing the friction and enhancingthe momentum flux from the air to the water. This effect is related to the moreefficient momentum transfer at inclined faces of water (because part of the surface“feels” the wind from the side rather than from above). The thickness over whichthe wind can push laterally scales with the significant wave heightH1/3 [m], whichis defined as the average height of the highest third of the waves (crest to trough).Consequently, also the steepnessH1/3/λp [-] of the related waves influences theair/water coupling (λp [m] is the energy-containing wavelength;p indicates thepeak of the wave energy spectrum).

Due to the limited horizontal extent of the water bodies, the wave-inducedcomplexity is more pronounced for lakes, reservoirs, and estuarine waters thanfor the open ocean, where quasi-steady conditions can build up more easily. Atthe upwind shore, the waves, not being in equilibrium with the wind field, areshort in wavelength and small in amplitude. With increasing distanceX [m] fromthe upwind shore, the characteristic wavelengthλp, the corresponding deep-waterwave phase velocitycp= (gλp/2π )1/2 [m s−1], and the significant wave heightH1/3

increase, whereas the characteristic wave frequencyωp [s−1] decreases downwind(g represents gravity acceleration). Observations revealed that these developmentsobey, at least under constant wind stress, astonishingly well-defined empiricalrelations. According to Hasselmann et al. (1973) the significant wave height growswith fetchX by H1/3≈ 0.051(w2

∗X/g)1/2, whereas the frequency decreases asωp≈7.1(g/w∗)(gX/w2

∗)−1/3 [w∗ = (τ/ρa)1/2 denotes the air friction velocity (Table 1)]

(Csanady 2001). Applying the above relation to a 10 m s−1 wind at 10 km fetch(typically a medium-sized lake) yields surface waves ofH1/3 ≈ 0.6 m only. Thisindicates that waves in most lakes never reach a state close to saturation, where wavegrowing stops. Therefore, the developed state of the open ocean is relevant onlyto very large lakes. In fact saturation forU10= 2 m s−1 and 6 m s−1 is not reachedbefore 10 km and 100 km, respectively (Wu 1994). For the countless numberof not very large lakes, the wave fields will typically stay underdeveloped, andsubsequently the momentum uptake will remain complex and subject to individuallocal effects (such as topography).

Instead of the fetch, which is not a well-defined physical parameter, the nondi-mensional phase speedcp/U10 [-], called wave age, is often used as an adequatemeasure of wave development (cp/w∗ is sometimes also defined as wave age;∼30 times larger thancp/U10). Therefore, most wave-relevant quantities are usu-ally expressed as a function of the wave age, rather than as a function of fetchX. Applying the relations by Hasselmann et al. (1973) to the above defined keyproperties allows one to express them as a function of wave development:H1/3≈



0.96(w2∗/g)(cp/w∗)3/2, H1/3/λp≈ 0.153(cp/w∗)−1/2,ωp≈ (g/w∗)(cp/w∗)−1, andλp

≈ (2π/g)w2∗(cp/w∗)2 (Csanady 2001). Once the wave agecp/U10 approaches the

value of∼1.14 to 1.2 (Donelan et al. 1992), the waves travel at speeds comparableto the wind and therefore stop growing. The wave field is then fully developed and,for this level of saturation, the wave characteristics no longer depend on fetch orwave age and become solely a function of wind speed, such asωp= 0.88g/U10andH1/3≈ 0.208U2

10/g (Csanady 2001). All these observations of the age dependencyof the wave heights, steepness, and other properties clearly demonstrate that themomentum transfer and its vertical distribution in lakes crucially depend on thestate of wave development.

Due to the limited extent of lakes, waves are commonly found to be youngand underdeveloped (i.e.,cp/U10 small) and therefore short, steep, and of highfrequencies and, as a result, those waves break more frequently. In addition, grow-ing waves also pick up momentum for their acceleration. As a combined effect,young and growing waves extract momentum,τWave(Equation 1), for growth andcompensation of the loss by breaking. Therefore, young waves appear rougherand produce more turbulence than more mature waves. Long and saturated sur-face waves, in contrast, although containing much more momentum, lose theirmomentum and energy slowly because they are smoother and break at lower rates.Saturated waves, not growing any further, needτWaveonly to compensate for theloss by the breaking. For fully developed waves, we can expect that the SBL stressτSBL is close to the atmospheric counterpartτ (Equation 1).

Understanding the fetch-limited areas is of considerable relevance for inland andlarge parts of coastal waters, where air/water exchange processes have particularlystrong ecological significance. Based on the discussion above, we can concludethat for a given wind speed, the observed surface stress will be higher near theupwind shore than in the open water, and it will also be larger in lakes comparedto the open ocean. This may appear counterintuitive because SBL turbulence isusually lower in lakes compared to oceans. This seeming paradox is due to the factthat waves also carry momentum and energy to the shore. In small-to-medium-sized lakes these “missing terms” are significant for the momentum and turbulencebalance. A second consequence of the fetch limitation is the importance ofτWave.Except in the few very large lakes on Earth,τWavecan reach a considerable fractionof the total momentum transfer (Janssen 1989, Simon et al. 2002).

2.3. Wave Dependency of Wind-Induced Stress

The total surface stress is usually parameterized by the drag coefficientC10 [-],which quantifies the total vertical momentum fluxτ (Equation 2) well above thewave-affected boundary (at standard 10 m height). A confusingly large numberof measurements have been carried out in the past; these have been reviewedsporadically (Smith 1988, Donelan 1998, Csanady 2001, Jones & Toba 2001).Still today, the drag coefficient is associated with large scatter, which is partlydue to the different techniques used and partly due to the difficulties quantifying


380 WUEST ¥ LORKE

the wave development. The measurement techniques include the profile method(fitting logarithmic vertical profiles to measured wind velocity), the direct method(measuring the stress〈U′W′〉 =w2

∗), and the dissipation method [determining dis-sipationε (W kg−1) of turbulent kinetic energy (TKE) in the inertial subrange andcalculatingw∗ according to Equation 4 below]. Despite the large scatter, there is arelatively clear conceptual picture emerging. The drag coefficient depends to largeextent only on wind speed and the wave development state. From these two factors,we consider first the situation of developed waves at different wind speeds. Thereare basically two ranges to be described independently: wind larger than∼5 m s−1

and wind below∼5 m s−1.For strong winds (>5 m s−1) the surface roughness is determined by the height

of the gravity waves, and subsequently, friction is dominated by those waves.Charnock (1955) found the relation between wind speed and measurement height(z) to vary asU(z)≈w∗{k−1 ln(gz/w2

∗)+K} for different wind velocities (k= 0.41is von Karman’s constant). The denominatorw2

∗/g is the so-called waveheightscale, which represents a measure of the roughness of the surface waves. The con-stant K is relatively universal and has been determined by Smith (1988) and Yelland& Taylor (1996) to be 11.3 [surprisingly close to the original value by Charnock(1955) of 12.5]. Introducing the Charnock relation above into the definition ofC10

(Equation 2) allows determiningC10 by

C10 ≈ {k−1 ln(g10/C10U

210

)+ K}−2 [-] (3)

(10 has the units m) for any given wind speedU10. Equation 3 is an implicit relationin C10, converging quickly after about four iterations. The results of Equation3 (displayed in Figure 3) together with measured data from different sources,demonstrate the excellent match between Charnock’s relation and measured dragfor strong winds and wave-dominated surface stress (Csanady 2001). The typicalvalues ofC10 range from 0.0011 (atU10= 5 m s−1) to 0.0021 (atU10= 25 m s−1).As a note of caution, we mention that in the literature the wind profiles are oftenparameterized by the roughness lengthzo, usingU(z)=w∗k−1ln(z/zo) instead of theCharnock (1955) relation [wherezo translates to (w2

∗/g)e−kK or zo≈ 0.0097(w2∗/g)

for K= 11.3]. This parameterization has been convenient in the past to estimatethe drag coefficientC10 from roughness length estimates determined by the profilemethod. However, this procedure is questionable because the conversion ofzo toC10= k2/ln2(10/zo) generates large uncertainty owing to errors inzo inherent tothe profile method. RealisticC10 values correspond to a difficult-to-measurezo of∼0.1 mm, and uncertainties of a factor of 10 inzo translate to uncertainties of afactor of 2 inC10.

For weak winds (<5 m s−1) the influence of gravity waves on surface stresseases, and surface tension or small-scale capillary waves, generating “virtual”roughness, become increasingly important (Wu 1994). At low wind, one couldexpect the smooth lawU(z) ≈ w∗{k−1ln(w∗z/νa) + 5.7} to apply (whereνa is thekinematic viscosity of air, Table 1). Indeed, for wind velocities of∼3 to 5 m s−1

(Figure 3), experimental evidence (Jones & Toba 2001) confirms the smooth-law



Figure 3 Wind-drag coefficientsC10for developed waves as a function of wind speedU10 at standard (10 m) height above water. The figure combines data from Geernaertet al. (1988), Bradley et al. (1991), Yelland & Taylor (1996), and Simon et al. (2002).The solid line is calculated according to Equation 3 for the Charnock constant ofK= 11.3. The dashed line is a fit to the data at low wind, followingC10≈0.0044U−1.15

10 .

value forC10≈ 0.001, as calculated by the equivalent iteration procedure of Equa-tion 3 [i.e.,C10≈ {[k−1ln(10C1/2

10 U10/νa)+ 5.7}−2]. However, at even lower wind(<3 m s−1), the experimental values ofC10 (Yelland & Taylor 1996, Bradley et al.1991, Simon et al. 2002) consistently increase much faster (Figure 3) with de-creasing wind than smooth law predicts (Wu 1994). There is much speculationon the reason for this additional friction, but so far no convincing explanation hasbeen provided. From the data by Simon et al. (2002) it seems thatC10 followsbasically the inverse boundary layer Reynolds number, such as in laminar channelor pipe flows. Until the reason for this kind of “laminarization” has been resolved,we suggest the empirical parameterizationC10≈ 0.0044 (± 3) U−1.15(±9)

10 (U10 inm s−1) for wind weaker than 3 m s−1 (Figure 3). Astonishingly enough, such weakwinds have drag coefficients larger than those for 25 m s−1 winds. In lakes, thelight-wind C10 value is of great relevance because most inland waters (especiallysmall-to-medium-sized lakes) are exposed for most of the time to light windsonly. The surprising result of the comparison of the low- and high-wind regime inFigure 3 is the fact that there is a minimum drag coefficient (minimum roughnessof the surface) at approximately∼5 m s−1 (Wu 1994).


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The second important parameter, besides wind velocity, is the level of wavematuration (or steepness), quantified by the wave age (cp/w∗ or cp/U10). Becausesurface stresses depend heavily on the state of wave development, the aboveC10

approximations represent lower bounds because they are based on developed wavefields. For young, accelerating, and breaking waves, the friction, and thereby alsoC10, will be larger than in Figure 3 (Terray et al. 1996, Taylor & Yelland 2001).

2.4. Vertical Structure of Momentum and Turbulence

The total vertical fluxτ of horizontal momentum is split into two parts,τ =τSBL+ τWave (Equation 1), feeding the currents in the SBL (τSBL) and the waves(τWave) at the very top of the water column. This partitioning ofτ defines mainly thevertical structure of the SBL physics, including the small-scale turbulence, mixing,and air-water exchange. The momentum input at the surface occurs both by skinfriction directly onto the water lid as well as by wave-produced fluctuations ofpressure and interactions of wave- and drift-related currents (Anis & Moum 1995).As a result, the pick-up of momentum is concentrated to the so-called wave-affectedsurface layer (WASL), which is as thin as several times the significant wave heightH1/3 (typically less than 1 m) or approximately the inverse of the dominant wavenumber 2π/λp (Anis & Moum 1995). Within the WASL,τWave decreases withdepth (Figure 2), depending on the efficiency and the depth of wave breaking, andat the base of the WASL,τ approachesτSBL.

Below the WASL, the momentum flux,τSBL, is the Reynolds stressρ〈u′w′〉produced by the turbulent viscosity acting on the mean background shear(Figure 2). In the constant stress layer below the WASL, the steady-state verticalprofile of the horizontal velocityu(z) follows the law-of-the-wall (LOW)∂u/∂z=u∗(kz)−1= (τSBL/ρ)1/2(kz)−1, whereu∗ = (τSBL/ρ)1/2 is the water friction velocity(Table 1). Assuming a balance between TKE production due to Reynolds stressand the viscous dissipation of TKE in the log-layer leads to the LOW scaling ofturbulence:

ε = (τSBL/ρ)∂u/∂z= u3∗(kz)−1 [W kg−1]. (4)

Dillon et al. (1985) experimentally confirmed the LOW relation (Equation 4)for the first time in a reservoir through temperature microstructure measurements.Also, more detailed and extended turbulence observations by Imberger (1985) andSimon et al. (2002) in the near surface of lakes found the characteristic depth−1

dependence, which revealed LOW scaling. Although individual vertical currentprofiles ofu(z) do not follow the LOW, averaged properties, including turbulence,do (Simon et al. 2002).

Within the thin WASL, the current profiles are nonlogarithmic owing to theadditional wave-related stressτwave(Figure 2) right at the surface (Craig & Banner1994). As detailed above, part ofτwave is consumed for the increase of the wavemomentum by the growth of the waves (typically about 6%) (Csanady 2001), andsome of the wave momentum is continuously transferred to the water below via



breaking and other less important dissipative processes. As a result, the verticalshear∂u/∂z and the eddy viscosity are also enhanced in the WASL, which leadsto higher turbulence (Equation 4). The TKE generation can be interpreted as aninjection of TKE into the WASL, and subsequently dissipation exceeds LOWscaling right at the surface (Figure 4). This flux of TKE is given by the energycontent of the waves multiplied by the rate of decay. Melville (1994) showedthat up to 90% of the breaking-produced TKE is vertically uniformly dissipatedwithin a layer of one to two significant wave heights. A second contribution can beinterpreted as the interaction of the waves with the background shear. The Stokesdrift uSunder the force of the SBL stressτSBLrepresents such a source of TKE, asexpressed byτSBLuS (Skyllingstad & Denbo 1995).

This enhancement of the turbulence has been studied in several lakes, mainly inLake Ontario. The first evidence contradicting the LOW relation (Equation 4) came

Figure 4 Dissipation versus depth in the wave-affected surface layer: The SWADE(surface waves dynamics experiment) data were measured off the Atlantic coast ofMaryland, U.S.A. (Drennan et al. 1996), and the WAVES data set was obtained byTerray et al. (1996) in Lake Ontario. Dissipation and depth are scaled nondimensionallyby the significant wave heightH1/3 and the energy input perH1/3. The line representsthe relation 0.3(z/H1/3)−2. Reproduced from Drennan et al. (1996).


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with the dataset of Kitaigorodskii & Lumley (1983). They found dissipation ratesone to two orders of magnitude above those predicted by the LOW in stronglyforced wind-generated waves in Lake Ontario. Recent measurements in lakes(Terray et al. 1996, Donelan 1998) confirm that at least during conditions of strongforcing, the energy input from wave breaking is an important source term in theTKE balance in the uppermost part of the SBL. Based on the extensive observa-tion program WAVES at the shore of Lake Ontario, Agrawal et al. (1992), Terrayet al. (1996), and Drennan et al. (1996) observed intense turbulent events causedby wave breaking. Measured dissipation, exceeding LOW by one to two orders ofmagnitude near the surface, scaled well with wind and wave parameters. Terrayet al. (1996) found that under such conditions, the TKE flux into the water wasentirely via wind to waves and was therefore parameterized as an effective phasespeed multiplied by the wind stress. For the ocean, Anis & Moum (1995) found, inpart, dissipation rates up to one order of magnitude larger than those predicted byLOW. The dissipation was found to decay exponentially with depth with a shortscale on the order of (2π/λp)−1.

Measurements by Terray et al. (1996) in Lake Ontario revealed a three-layerstructure of the WASL. According to their model, approximately half of the energydissipation occurs in the very top layer, where turbulence is enhanced by directinjection of TKE owing to wave breaking. The thickness of this layer, whereε ≈constant, was estimated to be 60% of the significant wave heightH1/3, althoughthe constancy has not been proven yet. Below this depth, there is a transition layer,where the dissipation rate scales with both wave and wind parameters and whereε

decays with depth asz−2 (Figure 4). The thickness of this layer is dependent on bothH1/3 and the wave age and can be as large as 25·H1/3 for waves of intermediatedevelopment (cp/w∗ in the range 4–7) (Terray et al. 1996). Below, the transitionlayer merges with the log-layer where LOW scaling is appropriate (ε∼ z−1). Craig& Banner (1994), who used a one-dimensional, level 2.5 turbulence closure model,came up with dissipation decaying with depth to the power−3.4, which comesclose to the observations that were related to the second layer.

When the wave field reaches saturation, being achieved in large lakes only,τWavebecomes small, and the waves have only to compensate for the loss by wavebreaking, which is low for saturated waves. As a result, although the dissipationis still enhanced, the excess dissipation in the WASL is less for developed wavescompared to developing waves.

2.5. Convective Processes and Mixed Layer Build-Up

Another source of TKE in the SBL is a positive surface buoyancy fluxB0, causingconvection. Usually it is the result of surface cooling with the water surface tem-perature being greater than the temperature of maximum densityTMD, i.e., with apositive thermal expansivityα of the surface water. However, as a unique featureof lakes, a positive surface buoyancy flux can also result from surface warming forα < 0, which is restricted to salinitiesSsmaller than 24.7%. In salt lakes, evapo-ration is another source of convection.



Early measurements of convective turbulence in a freshwater reservoir(Imberger 1985) and in the ocean (Shay & Gregg 1986) showed that the dissi-pation rateε averaged over the convective mixed layer scales proportionally tothe surface buoyancy production, i.e.,〈ε〉=a ·B0 with estimates ofa≈ 0.45 and0.64, respectively. The range of observed values as well as the depth dependencyof the scaling functiona(z) was found to follow the same universal scaling as in theatmosphere (Shay & Gregg 1986, Brubaker 1987). Assuming that the buoyancyflux within the water columnB(z) decreases linearly to 0 at the base of the mixedlayer and assuming further that there is a simple balance between buoyancy pro-duction and dissipation [ε(z)=B(z)], the scaling functiona(z) would bea= 0.5for nonpenetrative and steady-state convection in an ideal mixed layer.

Recent measurements of shear-free nocturnal convection by Jonas et al. (2003)in a small lake, however, revealed a much smaller estimate ofa= 0.15. Theyattributed this discrepancy to the remnant stratification within the convectivemixed layer. Similar to the results by Imberger (1995), they found the temperaturestructure to consist of a thin super-adiabatic sublayer (of approximately 50-cmthickness) featuring increasing temperature with depth, followed by a layer ofincreasing stratification below, down to the base of the mixed layer. Jonas et al.(2003) argued that this stratification is maintained by differential entrainment ofdownward propagating convective plumes. Whereas entrainment is low in the cen-ter of the convective layer, increasing entrainment at its base allows the cold waterto “tunnel” through this layer, maintaining the slightly stable stratification. High-resolution current measurements showed that these plumes are asymmetric, withtypical velocities of 3–6 mm s−1 in the downward and 2–3 mm s−1 in the upwardplumes and that horizontal plume-diameters are on the order of 5 m (Figure 5).

The organized cellular structure of convective motions can become very obviousas patterns on the surface of lake ice. The water under the ice usually has temper-atures below the temperature of maximum density. Thus, if the ice is not coveredby snow and its transparency allows solar radiation to pass, absorption withinthe underlying water will lead to warming and hence to convection (Bengtssonet al. 1996). The distinct difference of convection due to surface cooling is thatthe buoyancy flux is not produced directly at the surface but, depending on theabsorption characteristic of the water column, within the convective layer itself.Jonas et al. (2002) showed that such an interior production of buoyancy consider-ably reduces the production of available potential energy compared to convectiondriven by surface flux, especially if the water has low turbidity. Moreover theyshowed that about 50% of the solar radiation was absorbed within a thin toplayerof a few tens of centimeters thickness, which is stably stratified and thus does notraise the production of TKE. This stable toplayer was produced by ice melting,adding lower-density fresh water on the top of the water column; thus its thicknessis defined by the background salinity and the melting rate of the ice.

Langmuir circulation is also a prominent and complex feature of the SBL oflakes (Figure 2). It is the organized near-surface circulation pattern that consists ofpairs of parallel counter-rotating vortices oriented in the wind direction (Langmuir1938). These vortices are slightly asymmetric with higher downwelling than


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Figure 5 Vertical velocity at 3.3 m depth (upper panel) and between 3.3 and 7.2 m(lower panel) in the active convective surface boundary layer of Soppensee. Adaptedfrom Jonas et al. (2003).

upwelling velocities. The downwelling velocities increase with increasing windspeed and have been observed to reach as much as 10 cm s−1 in Lake Ontario(Harris & Lott 1973). The Langmuir circulation becomes obvious on the watersurface owing to the formation of parallel streaks in the wind direction, formedby floating objects like bubbles or leaves in the convergence zone of the vortices.The cell spacingL of the streaks is assumed to expand with increasing wind speedU10 (Leibovich 1983); however, comprehensive measurements by Plueddemannet al. (1996) indicated that a broad range of scales (2–200 m) exists during peri-ods of active Langmuir circulation. It seems that the spacings evolve from smallto large scales as time progresses. In addition to their rotation, cells also propa-gate downwind with horizontal velocities comparable to the downwelling veloc-ities (Weller & Price 1988). For a detailed overview on the relevant theories ofLangmuir circulation dynamics we refer to Leibovich (1983).

Although Langmuir cells are often supposed to play an important role in sur-face layer dynamics (Li & Garrett 1997), and even a more pronounced role in the



transport of solutes, gas, or phytoplankton (Evans & Taylor 1980, Patterson 1991),their contribution to surface layer turbulence and mixing has not yet been deter-mined. Direct turbulence measurements within the cells revealed no significantlyenhanced dissipation rates in the convergence zones (Lombardo & Gregg 1989,Plueddemann et al. 1996).

Besides the classical effects of wind and convection, there are many otherprocesses that contribute—under the given circumstances—to the build-up of theSBL of lakes. Reviewing all of them is beyond the scope of this article. Mostimportant are differential heating/cooling (Horsch & Stefan 1988, Fer et al. 2002),as well as the breaking of waves and the dissipation of currents at the shore. Ifthe lake—or specifically the shores—are shallow, vegetation modifies the surfaceboundary layer locally into a classical mixing layer (Ghisalberti & Nepf 2002).

3. STRATIFIED INTERIOR PROCESSES

The interior water body in lakes is of a completely different nature than theSBL above. Interior processes are characterized by the (usually) strong strati-fication, suppressing turbulence and providing an ideal environment for inter-nal waves. With a few rare exceptions, mechanical energy stems from windforcing. Imberger (1998b) and W¨uest et al. (2000b) showed that most of themomentum and energy that passes through the SBL and enters the interior istransferred to basin-scale internal (wave) motions. This energy, typically about10% of the total wind energy input to the lake, is transformed to small-scaleturbulence and utilized for mixing by basically two paths: The major part ofthe energy is dissipated by bottom interaction (Section 4), and the minor partis dissipated in the interior by shear instabilities or breaking of internal waves.For both paths, propagating high-frequency internal waves provide an impor-tant link between large-scale motions and small-scale mixing (Imberger 1998b).In this section, we follow this energy path by first giving an overview on thebasin-scale internal motions, followed by a review of the high frequency internalwaves; and finally we conclude by describing the small-scale processes in theinterior.

3.1. Basin-Scale Waves and Shear

Although many mechanisms of forcing by wind are available [including the gener-ation of circulation (Csanady 1967), progressive internal waves, inertial currents,etc.], it is mainly the setup of the thermocline/pycnocline (Mortimer 1952) thatcouples wind momentum most efficiently into the lake interior. In general, theresponses of the pycnocline to an imposed wind stress (Equation 1, Table 1)are a complex function of geometry, stratification, and the temporal dynamicsof the wind forcing (Monismith 1986, Imberger & Patterson 1990). For sim-ply shaped and small-to-medium-sized lakes, modal-type responses are usuallyobserved (Stevens & Imberger 1996, Stevens & Lawrence 1997, MacIntyre et al.


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1999, Antenucci et al. 2000) that are predictable to some degree (and oftenparameterized by Wedderburn or Lake numbers).

Important for the small-scale hydrodynamics are the resulting currents, verticaldislocations, as well as the associated shear and instabilities related to internal(baroclinic) motions. From the fact that about 3% of the wind energy flux fromthe atmosphere ends up in the hypolimnion we can roughly estimate the mechan-ical energy content in the stratified interior. The level of excitation is typically∼1 J m−3, corresponding to several J m−2 in shallow lakes to almost 1000 J m−2

in very deep lakes (Ravens et al. 2000). Given that typical energy residencetimescales are days (shallow lakes) (Gloor et al. 2000) to tens of days (very deeplakes) (Ravens et al. 2000), the dissipation rateε of the internal energy is∼10−8

(shallow lakes) to∼10−10 W kg−1 (deep lakes). These energy transformation ratesand the modes of decay of the mechanical energy are crucial for the stratificationand mixing in the interior. Because most of the mechanical energy is contained inthe basin-scale waves, there is a great interest in their assessment and evaluationfor shear and instability (Thorpe & Jiang 1998).

For small lakes with lateral dimensions shorter than the internal Rossby radiuscI/f (cI is the phase speed of the internal wave), the effect of Earth’s rotation isnegligible and the baroclinic motions are mostly standing waves, referred to as in-ternal seiches. Given the geometry and the stratification, approximated by layers ofconstant density, the seiches can be characterized by “quantum numbers” indicat-ing the number of wave nodes in the horizontal (horizontal modes) and the vertical(vertical modes), respectively (Münnich et al. 1992). In many instances, the two-layer approximation gives exceptionally good agreement between observed andcalculated seiche periods (Mortimer 1952, Heaps & Ramsbottom 1966, Lemmin1987, Stevens & Lawrence 1997). [Several studies, however, revealed that obser-vations can be interpreted as second vertical mode oscillations if an intermediatelayer (metalimnion) is added in between (LaZerte 1980, Wiegand & Chamberlain1987, Munnich et al. 1992, Roget et al. 1997) or even four layers (third mode; asobserved in a Spanish reservoir) (E. Roget, personal communication).] Lemmin &Mortimer (1986), who considered varying bathymetry, presented an expansion ofthe rectangular two-layer model. Both effects, continuous stratification and real-istic bathymetry, can be taken into account by using two-dimensional numericalmodels as shown by Münnich (1996) and recently by Fricker & Nepf (2000).

As soon as the width of the lake is larger than its internal Rossby radiuscI/f,Earth’s rotation influences the internal wave field significantly. Most prominentare the circular inertial currents at the natural frequencyf (Table 1), which wedefine as the lower delimiter to the high-frequency waves (Section 3.2). Especiallyin large lakes, near-inertial currents are almost omnipresent. This is illustratedin Figure 6, which shows a distinct peak nearf in the frequency spectrum ofthe deep currents in Lake Baikal. But also the internal seiching, as shown byobservations and numerical modeling (Bäuerle 1994) becomes deflected by theCoriolis force, giving rise to waves with propagating phases like Kelvin or Poincaréwaves. Kelvin waves are boundary-trapped waves, which develop when seiching



Figure 6 Rotary spectra of deep currents (1386 m) in the southern basin of LakeBaikal in summer/fall 1996 (Ravens et al. 2000). For ice-free periods the internal wavespectrumS− shows a distinct and slightly “blue-shifted” peak close tof but no signal inS+, both typical for downward propagating inertial waves.S− andS+ are the clockwiseand counterclockwise rotary components, andStot is the sum of both.

is deflected by Earth’s rotation toward the right boundary (northern hemisphere).The pressure, decreasing exponentially with distance from the boundary, balancesthe Coriolis force and drives the wave along the shore of lakes or ocean basinsin a counterclockwise direction (on the northern hemisphere) with the highestamplitudes at the boundary. It is interesting to note that the current direction belowthe crest and trough of the fundamental horizontal mode (i.e., on diagonal pointsof the lake) is the same, as it is the case for linear seiching. In contrast, Poincar´ewaves are not trapped at the boundaries and can develop from crosswise seiching,


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which becomes deflected by the Coriolis force. On the northern hemisphere, theyhave clockwise rotating cellular structures with maximum velocities in the cellcenters. Analogous to nonrotating seiching, they can be characterized accordingto their modal structure. Both types of Coriolis-affected waves were observedin Lake Biwa by Saggio & Imberger (1998) and observed and modeled in LakeKinneret by Antenucci et al. (2000) and Hodges et al. (2000), respectively. Thelatter, comprehensive study in Lake Kinneret revealed the coexistence of a Kelvinwave with a period of about 24 h and 3 different modes of Poincar´e waves withperiods of about 12 h and 20 h. Lake Kinneret’s oval shape and its size, as wellas its strong diurnal forcing favors the occurrences of basin-scale Kelvin waves(Bennett 1973, Ou & Bennet 1979). A comprehensive overview of these types ofwaves is given in Mortimer (1974), Hutter (1983), and Hutter et al. (1998). A moreillustrative description can be found in Hodges et al. (2000).

Variable lake topography gives rise to another class of waves: the so-called to-pographic or rotational waves (Csanady 1976). Lateral gradients off/H, the ratioof the inertial frequency to the depthH, lead to horizontal gradients of the potentialvorticity. The conservation of potential vorticity leads to the generation of topo-graphic Rossby waves, which have strictly subinertial frequencies. Rossby waveswere observed not only in the very large Lake Michigan (Saylor et al. 1980), butalso in the medium-sized northern basin of Lake Lugano (Stocker & Hutter 1985).However, the long periods of these waves (several days to weeks for basin-widescales) and the rather small scales associated with shorter periods make excitationby direct wind forcing rather unlikely. Therefore, E. B¨auerle (personal communi-cation) proposed the excitation of small-scale topographic waves to occur on bedslopes or topographic irregularities by background seiching with appropriatelylong periods. However, observations of topographic waves in lakes are rare, andtheir potential role in the internal energy cascade is still to be shown.

3.2. High-Frequency Internal Waves

Traveling internal waves with frequenciesω betweenf (inertial) andN (buo-yancy) are relevant to small-scale processes, although their energy content is minorcompared to that of the large-scale waves. This energy reservoir is an importantintermediary between the energetic large-scale motions and the small, dissipativescales. The recent studies by Antenucci & Imberger (2001) and Saggio & Imberger(1998) showed that the energy spectra often show a well-defined peak near thebuoyancy frequencyN (Figure 7), indicating that there is a second frequencyband, besidesf (Figure 6), where energy is inserted into the wave field.

These studies also demonstrate that the source of wave generation is difficultto identify. Proposed mechanisms are manifold (Thorpe et al. 1996, Antenucci &Imberger 2001): Internal solitary waves can be generated by nonlinear steepeningof basin-scale waves, the interaction of propagating wave trains with the back-ground motion, or by the intersection of long waves with irregular bottom topo-graphy (Horn et al. 2001). As shown by Polzin et al. (1997) in the deep Brazil



Figure 7 Spectra of isotherm displacements in Lake Biwa, as estimated for 10-day periodsbefore (dotted line) and after (solid line) the passage of a typhoon, showing the universaldecay of energy∼ω−2 as well as the characteristic after-storm peak close to the buoyancyfrequencyN. Reproduced from Saggio & Imberger (1998).


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Basin, turbulent dissipation rates were elevated by one or two orders of magnitudein the interior above the rough flanks of the Mid-Atlantic Ridge, way above theusually turbulent bottom boundary layer (BBL) (Section 4). Such turbulence, faraway from the energy source at the bottom calls for high-frequency waves, whichradiate energy from the bottom into the interior. This is also confirmed by obser-vations of tidal flow over a sill and by the upstream generation of solitary waves(Farmer & Armi 1999). Whether such generation processes are important for high-frequency internal waves in lakes has to be shown because tides in inland watersare usually irrelevant (Hamblin et al. 1977). Other possible mechanisms includethe direct forcing by wind and subsequent oscillatory motions at the base of theSBL and shear instabilities of the mean SBL flow (Antenucci & Imberger 2001)as well as active thermals impinging the base of the SBL as a result of convection.

High-frequency internal waves propagate through the water column or are ad-vected by long wave motions. When a ray of internal waves encounters the slopingbottom of the lake, it is partly dissipated and partly reflected as shown in laboratoryexperiments by Ivey & Nokes (1989) and DeSilva et al. (1997) for periodical wavesand by Michallet & Ivey (1999) for solitary waves. Phillips (1977) and Imberger(1994) have shown that an increase in energy density of the ray results upon re-flection, which can become unlimited if the angle of the incident ray has a criticalvalue relative to the bottom slope. At this critical angle, the reflected wave travelsparallel to the sloping bottom, and the energy, which is trapped in the BBL, is dis-sipated close to the sediment (Section 4). The occurrence of critical frequencies,for which shoaling of internal waves takes place, depends on the bottom slope andon the buoyancy frequencyN (Eriksen 1982, Thorpe 1988). Spatial variations ofboth parameters in lakes create a rather broad frequency range of about one orderof magnitude for which an enhancement of energy can be expected. In additionto the reflection, the incident and reflected wave rays can interact with each otherand generate resonance phenomena (Thorpe 1997) or produce higher-mode wavesthat are trapped at the boundary (Javam et al. 1999).

As an effect of those interactions, the spectrum betweenω ≈ f andω ≈ Nconsists of the superposition of generated and decaying traveling wave groups.It is interesting to note that various authors (Imberger 1998b, MacIntyre et al.1999, Stevens 1999) found that the energy spectrum closely followsω−2 withinthis frequency band (Figure 7), as observed in the ocean (Garrett & Munk 1972).However, the mechanisms that create and distribute the energy throughout thisuniversal spectrum, seemingly independent of the size and location of the aquaticsystem, are not understood at all.

3.3. Internal Turbulence and Mixing

Direct observations of turbulence and mixing using microstructure and tracer tech-niques revealed that in the stratified interior of lakes, turbulence is very weak(Wuest et al. 1996, MacIntyre et al. 1999, Etemad-Shahidi & Imberger 2001).Typically only a few percent of the water column has been found to be actively



mixing. Analogous to the ocean, it was found that local diffusivities in the strati-fied interior are, on average, an order of magnitude lower than diffusivities inferredindirectly from tracer budgeting over basin-scales (W¨uest et al. 2000b). Differenttracer experiments conducted in a medium-sized lake revealed the importanceof boundary mixing for the basin-wide averaged diapycnal diffusivity (Goudsmitet al. 1997). The latter investigation showed clearly that the vertical spreading of atracer, injected into the hypolimnion, increased by approximately an order of mag-nitude after the tracer reached the lake bottom (Section 4). Ledwell & Bratkovich(1995) and Ledwell & Hickey (1995) obtained comparable results from similarstudies in enclosed ocean basins.

As discussed above, the high-frequency internal waves radiate a large part of theenergy out of the interior into the BBL, where it is finally dissipated. Again this cor-responds to a flux of energy into the BBL, thereby reducing the actual turbulence inthe interior. Because the energy spectrum of the high-frequency internal waves inthe metalimnion typically exhibits a peak near the buoyancy frequencyN(Figure 7),one should expect enhanced dissipation in regions where the metalimnion encoun-ters the sloping bottom. However, no observational evidence has been found so far.

A simplified budget for theTKE can be approximated by the four terms (thevertical TKE flux divergence is small and can be neglected):

∂(TKE)

∂t= P − ε − B [W kg−1], (5)

whereε denotes the rate of viscous dissipation andP=−〈u′w′〉∂u/∂z is the pro-duction of TKE by Reynolds’ stress. The shear∂u/∂z, originating from the high-frequency and basin-scale internal waves, links the internal waves and the stratifiedturbulence. Under steady-state conditions,∂TKE/∂t= 0, production and dissipa-tion are balanced by the buoyancy fluxB, which describes the loss ofTKE owingto mixing (increasing the potential energy of the water column). The ratio of thebuoyancy flux to the dissipation rate is defined as the mixing efficiencyγmix; i.e.,γmix=B/ε. It expresses the ratio of the mixing-induced potential energy storagein the water column (B=KvN2) per dissipated energy. This ratio is of great prac-tical importance because it is widely used to infer turbulent diffusivitiesKv frommeasured turbulent dissipation rates using the Osborn (1980) relation

Kv = γmix · ε/N2 [m2 s−1]. (6)

A number of recent studies revealed that the relationship in Equation 6 is com-plicated by the nonconstancy ofγmix (Ivey & Imberger 1991). Smyth et al. (2001)showed thatγmix changes drastically during the temporal evolution of turbulentevents. Laboratory measurements by Barry et al. (2001) showed differences ofa factor of two between directly measured and estimated (Equation 6) diffusivi-ties for weak turbulence and even more divergent values for the energetic regime.Nevertheless,γmix= 0.15 seems consistent with most observations based on largerdata sets, enabling appropriate spatial and temporal averaging (W¨uest et al. 2000b,Ravens et al. 2000).


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Figure 8 Vertical profile characteristics of a turbulent patch in Lake Kinneret, show-ing densityρ, the fluctuationsρ ′ [kg m−3] as well asw′, u′, v′ (vertical and horizontalvelocity fluctuations [m s−1], respectively). Their covariances allow determining buo-yancy flux〈ρ ′w′〉, dissipation 7.5ν(∂u′/∂z)2, andTKE, 1/2(u′2+ v′2 + w′2). Adaptedfrom Saggio & Imberger (2001).

Turbulent mixing in the interior of lakes (and in the ocean) occurs mainlyby sporadic and localized shear instability events of the Kelvin-Helmholtz type(DeSilva et al. 1996). A typical example is illustrated in Figure 8 by a sample profileof intensive microstructure measurements of Saggio & Imberger (2001) in LakeKinneret. They found that, on average, well-defined turbulent patches coveredonly 35% of the metalimnion. The turbulent diffusivity was close to moleculardiffusivity for most of the metalimnion, as observed also in other strongly stratifiedlakes. Wuest et al. (1996) found even lower percentages of active turbulence in amedium-sized lake. The intermittent occurrence of these turbulent patches in spaceand time (Baker & Gibson 1987) and the strong dependency of the turbulence levelon their state of evolution are crucial for the sampling statistics.

Direct numerical simulation by Smyth & Moum (2000) revealed the evolutionof the turbulent length scales for individual turbulent patches produced by Kelvin-Helmholtz billows. The vertical turbulent excursions of fluid particles in lakesare often characterized by the Thorpe scaleLT (Thorpe 1977), which quantifiesthe vertical displacements between the measured and the corresponding adiabaticprofile. To estimate turbulence, this scale is usually related to the Ozmidov scaleLO= (ε/N3)

1/2, which is the vertical overturning length scale at which buoyancyforces balance inertial forces. Smyth & Moum (2000) found that the ratio of theOzmidov length scale to the Thorpe scale increases monotonically from the initialphase of a turbulent event to its ultimate decay. In addition, there is an inverserelationship between this length scale ratio and the mixing efficiency (Smyth et al.2001). Although they found that the length scale ratios varied strongly with theevolution of turbulence, the ratio of the maximum displacement length scale to theThorpe scaleLT remained quite constant, implying that the governing length scaledistribution does not change much.

Strong evidence for the existence of a universal length scale distribution wasfound by Lorke & Wuest (2002) based on microstructure data collected under



Figure 9 Maximum versus root-mean-square displacement length scales within segmentsof microstructure profiles from three different lakes. The lines represent the analytical relationfor two different distributions of displacements: a boxcar-like distribution (solid line: Lmax=31/2LT) and an exponential distribution [dashed line: Lmax= (7.3/21/2)LT]. The inset showsthe same relation also for the individual estimates for data from M¨uggelsee. Adapted fromLorke & Wuest (2002).

very different stratification. As shown in Figure 9, the Thorpe scales, ranging from1 cm (in a small and strongly stratified lake) to 100 m (in the weakly stratifiedLake Baikal), were highly correlated to the maximum displacement length scale.Furthermore, they showed that the form of the probability density function of over-turning length scales is related to the inertial subrange of the turbulent displacementspectra.

There are many other processes that come into play under special circumstancesin the interior of lakes (Colomer et al. 2001). To review all of them is beyond thescope of this article. Most important are probably double diffusion and thermobaricinstability. Double diffusion is based on the large difference between the moleculardiffusivity of heat and salt. Usually both quantities have a stabilizing effect in lakes(i.e., temperature is decreasing and salinity is increasing with depth). However,geothermal heating, intrusions, or differential mixing can lead to conditions whereone of these two quantities tends to destabilize the water column and thus give riseto localized double diffusive convection. In the past decade there was hardly any


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research in lakes on this subject beyond the overview given by Imboden & W¨uest(1995). The thermobaric instability occurs at temperatures around the temperatureof maximum densityTMD and thus is restricted to deep and cold water bodies.The process is most prominent and well observable close to the depth, wherethe in-situ temperature crosses the temperature of maximum densityTMD. Theunderlying physics is the pressure-dependence of the thermal expansivity, whichchanges sign as a function of pressure atTMD and subsequently can also createlocalized convective instabilities as observed in Crater Lake (Crawford & Collier1997) and in Lake Baikal (W¨uest et al. 2000a).

4. BOTTOM BOUNDARY FLUXES AND DEEP MIXING

4.1. Introduction

The BBL, almost as dynamic as the surface layer, is the second prime site foranimals, plants, and microorganisms in lakes. The habitat for the biota is stronglyinfluenced by the bottom currents, which shape the physical and biogeochemicalenvironment. From a physical and geochemical point of view, the key processesare the exchange of momentum, heat, solutes (such as oxygen) and particles withthe sediment, the turbulent dissipation of energy from currents and waves, andthe subsequent diapycnal (vertical) mixing (Grant & Madsen 1986). In contrastto the SBL, the sediment is rigid, and consequently the current speed vanishesat the sediment/water interface. In the following, we refer to the BBL as thezone above the sediment where the currents are affected by the presence of theboundary.

Internal (including inertial) waves mainly cause the currents over the bottom,but surface waves or density currents (e.g., turbidity currents of inflowing rivers)also contribute in shallow zones. Spectral analysis of deep currents reveal thatmost of the kinetic energy in the stratified deep layers is contained in the basin-scale baroclinic motions (Section 3), such as seiching, Kelvin waves, and inertialcurrents at frequencyf (Figure 6). Although quite spectacular for some limitedepisodes, nonperiodic large-scale displacements, such as surges (Thorpe et al.1972) or other solitary-type motions (Farmer et al. 1978), are usually only relevantright after strong wind excitations. Short-scale internal waves (betweenN andf )contain only minor parts of the kinetic energy. Also barotropic motions, such assurface seiching (Malm et al. 1998) or tides (Hamblin et al. 1977), lead to weakBBL currents of only a few mm s−1.

We conclude that the currents acting on the BBL in lakes are typically large-scaleand low-period currents, allowing the BBL to reach some degree of quasi-steadystate over periods of hours or longer. Impinging short-scale internal waves interacton top of this relatively steady flow and contribute locally to shear and mixing(Section 3). The near-bottom current speeds are usually in the range of∼2 to∼10 cm s−1 at great depth (Lemmin 1987). Even the currents in the 1386-mdepth of Lake Baikal’s south basin seldom exceed 10 cm s−1 (Ravens et al. 2000).



Currents can reach more than∼20 cm s−1 during storms, especially in shallowzones. In contrast, under ice, currents can decay to as little as a few mm s−1 (Malmet al. 1998).

The transfer of solutes through the BBL influences a number of important bio-logical and geochemical processes in the upper sediments (Boudreau & Jørgensen2001), such as the dissolution of calcium carbonate, CaCO3, the oxidation of or-ganic matter and metals (iron, manganese, etc.), the removal of reactive nitrogen bydenitrification, the supply of oxygen to obligate-aerobic sediment-dwelling organ-isms, the growth of microbial mats, and the release of contaminants from pollutedsediments. The rate of oxygen transfer, for example, depends on the rate of stir-ring and the oxygen concentration in the overlaying water (Mackenthun & Stefan1998). The four key processes for sediment/water exchange, molecular diffusion,porewater convection/advection, bioturbation, and resuspension are all dependenton the horizontal currents in the BBL, and all four fluxes increase with the currentspeed (Dade 1993). Therefore we first concentrate on the BBL structure of theflow and turbulence before we address the sediment/water exchange.

4.2. Bottom Boundary-Layer Structure

The transition from the background flow, far away from the lake bottom, to theflow at the sediment/water interface is less complex than in the SBL because ofthe absence of waves, the lower variability of the forcing, and the rigid boundary.The temporal structure of the BBL is also steadier than the SBL owing to thedifferent nature of the forcing. Whereas the SBL is exposed to fluctuating windstress, the BBL is defined by friction, which damps unsteady currents and removesfluctuations. Nevertheless, there are many similarities between the BBL and theSBL. The BBL of lakes is usually actively turbulent (Figure 1), with the exceptionsof small and wind-protected lakes, where strong stratification can extend all theway down to the sediment.

In the outermost zone of the BBL, where flow veers because of the Coriolisdeflection (Saylor & Miller 1988), an Ekman layer can build up, especially inlarge lakes. Scaled byu∗/f (Table 1), the effect of Earth’s rotation in the BBLcan be several meters in vertical extent. In the lower part of the Ekman layer, theflow is predominantly affected by bottom friction. Down to a few cm above thesediment, the bottom shear stress [τBBL is relatively constant (as in the SBL)].Turbulent eddies exist with length scales proportional to the distanceh from theboundary (Schlichting 2000). These latter two assumptions (Thorpe 1988) leadagain to the logarithmic profile structure∂u/∂h= u∗(kh)−1 subsequently referredto as LOW [u∗ = (τBBL/ρ)1/2 defines the BBL friction velocity;k= 0.41 is vonKarman’s constant] (Table 1). Although new measurements in an estuary revealthat the vertical component of velocity fluctuations at energy-containing scales issignificantly damped as the bottom is approached (Doron et al. 2001), other setsof high-resolution Acoustic Doppler Current Profile (ADCP) measurements inlakes (Figure 10) seem indeed to match the LOW with stunning agreement. In the


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Figure 10 Measured seiching currents (upper left), showing a perfect BBL logarith-mic layer in a lake (Lorke et al. 2002), and deep ocean currents (lower left), exempli-fying the linear profile in the viscous sublayer (Caldwell & Chriss 1979). The oxygenprofile is on a sub-mm scale and shows the 0.5-mm-thick diffusive boundary layer ina lake (Muller et al. 2002). The schematic, reproduced from Jørgensen & Des Marais(1990), defines the outer limit (A), the true (C ), and the effective (B) diffusive boundarylayer (∼1-mm thick). The transition zone (broken line) is also perfectly identifiable inthe oxygen microgradient (lower right).

example presented in Figure 10 (Lorke et al. 2002), 8.5-min averages follow thelogarithmic profile over several meters within the BBL, although the interpretationis slightly more complex as described below.

Because the bottom flow is relatively steady, the turbulence field is usuallytreated as stationary. For well-mixed BBLs, the TKE equation reduces to thebalance between the production by Reynolds’ stress (τBBL/ρ)∂u/∂h and the dis-sipationε. This equity provides the measure for the level of turbulence by theLOW scalingε= u3

∗(kh)−1. According to Figure 1, the scaling seems an appro-priate model for the dissipation, which follows LOW within the few-meters-thickBBL (Wuest et al. 2000b). Observations at the ocean bottom also revealedh−1

dependency (Dewey & Crawford 1988, Stacey et al. 1999a), although within more



extended and more turbulent layers. Measurements by the inertial dissipationmethod near the bottom of Lake Baikal’s south basin were also consistent with theLOW (Ravens et al. 2000).

Recent measurements (Figure 10) revealed that the profile method producessignificantly higheru∗ than expected from smooth flows. In the past, there has beena general agreement that flows at lake bottoms are smooth, implying that roughnesszo in the logarithmic profileu(h)= u∗k−1ln(h/zo) is much smaller (∼0.1ν/u∗) thanthe laminar sublayer (see below). A potential source of this discrepancy couldlie in the periodicity of the forcing (of the internal seiches), which can lead tosignificant nonsteady contributions in the TKE balance even for seiching periodsas long as 24 h (Lorke et al. 2002). The perplexing similarity of the current profileu(h) in Figure 10 (including the maximum), with the theoretical Stokes’ solution ofperiodic BBL flow (Fricker & Nepf 2000), supports this argument. The sedimentmay in fact cause rough flows (Chriss & Caldwell 1982) owing to biologicalactivities at the lake bottom, such as those of fish. Spectacular, regular pillow-like structures observed in Lake Geneva, with 5- to 15-cm-deep trenches (Brandlet al. 1993) support this point of view. Measurements with high-resolution echosounding may resolve this question in the near future.

Constraining the vertical fluxes and their limitations through the bottom “bot-tleneck” is the turbulent flux (−Kv ∂C/∂z) in the BBL. For well-mixed bottomlayers the LOW scaling leads toKv = khu∗, indicating increasing diffusivity towardthe outer end of the BBL. It implies that the exchange becomes more efficient, aseddies become larger with distanceh from the sediment. In weakly forced lakesthe low turbulence level may not be able to keep the BBL well mixed. The con-tinuous release of dissolved solids from the sediment (by mineralization and othersediment processes) can stabilize the BBL (N2 > 0) and suppress mixing. In thiscase, the Osborn modelKv = γmix εN−2 is more appropriate, and the vertical (di-apycnal) diffusivity is given byKv = γmixu3

∗(N2kh)−1 as a function of the distance

from the sediment. In contrast to the well-mixed case,Kv is increasing toward thesediment until the distanceh= γ 1/2

mixu∗(Nk)−1 is reached. Below, the diffusivity isdecreasing again according to LOW and even faster close to the sediment (Shaw& Hanratty 1977). The transition from high turbulence in the BBL to weak mixingin the interior is often evidenced by the distribution of resuspended particles in theso-called nepheloid bottom layer (Sandilands & Mudroch 1983).

The diffusivityKv is the sum of the molecularDc and turbulent eddy diffusivityKt. Compiling the many parameterizations forKt reveals thatKt decreases steeplyash3 to h4 (Shaw & Hanratty 1977) close to the sediment. Subsequently, thereare two heights whereKt becomes smaller than the molecular viscosityν andmolecular diffusivityDc (Table 1), respectively. The heightδν, whereKt fallsbelow the kinematic viscosityν, defines the top of the viscous boundary layer(VBL). Within this sublayer (Figure 10), which is typically aboutδν ≈ 11ν/u∗ ≈1 cm or thinner (Caldwell & Chriss 1979, Chriss & Caldwell 1984), the viscousforces dominate the resistance to momentum transfer. Subsequently in the viscoussublayer the greatest changes in velocity occur and the horizontal flow becomes


400 WUEST ¥ LORKE

laminar (Figure 10). Analogously,Kt≈Dc defines the top of the diffusive boundarylayer (DBL) (Figure 10), which extends, according to thez-dependence ofKt,to δD ≈ δν(Dc/ν)a (a is between1/4 and 1/3; Boudreau & Jørgensen 2001) orapproximately one tenth ofδν (i.e.,δD≈1 mm) (Table 1). Within the DBL, transportby eddies becomes insignificant compared to molecular diffusion. Details of theDBL and the sediment/water exchange are discussed in Section 4.4.

The lower limit of the two sublayers (VBL and DBL), i.e., the sediment/waterinterface, is well but not perfectly defined. The uppermost sediment in lakes isquite permeable, which allows the horizontal currents to diffuse slightly into theporewater. However, this zone, the so-called Brinkman layer, has a depth scaleof the grain size (Svensson & Rahm 1991) that is usually smaller thanδD andtherefore negligible for benthic sediments. In the coastal surf zone the Brinkmanlayer may become large because there the grains are generally larger, and thewater exchange with the sediment is further forced by horizontal pressure gradi-ents in the sediment. These gradients can result from waves and from flow overroughness elements of cm to m scale (Ziebis et al. 1996b). In addition, in shallowwaters heating of the sediments can lead to porewater convection, which furtherenhances the exchange between water and the sediment. Under such conditions,the dominant sediment/water fluxes are advective, and the interface somehow losesits “molecular” meaning.

4.3. Bottom Turbulence and Basin-Scale Mixing

As schematically indicated in Figure 1 and discussed in many recent papers(Imberger 1998a, Ledwell & Bratkovich 1995, Arneborg & Liljebladh 2001)basin-scale vertical (diapycnal) mixing in stratified natural waters is the resultof two types of processes: turbulence at the boundary (area-specific process) andturbulence in the interior of the water body (volume-specific process). In small-to-medium-sized lakes, the boundary is usually the more important term throughoutthe water column (Goudsmit et al. 1997). In large basins, the interior turbulencedominates the mixing of the upper pycnocline, where the volume per sedimentsurface is large, but at greater depth the BBL processes supersede again.

How these two processes contribute to basin-scale mixing in oceans has beena subject of continuing controversial debate over the past two decades. Due to thelimited extent, the situation is less complex in lake basins. If the level of turbulencein the BBL as well as in the interior (Figure 1) is known at a certain depthz, thecomposite effect of the two processes can be compiled. However, the diffusivitieswithin the BBL and within the interior cannot simply be added (Garrett 1991). Aphysically more appropriate procedure is to integrate horizontally the buoyancyflux generated in the interior (KIN2 ≈ γmixεI; whereεI is TKE dissipation inthe interior) and the buoyancy flux generated in the BBL [(∂A/∂V )γmixρ

−1PBBL;where∂A/∂V [m−1] is the sediment surfaceA(z) per lake volumeV(z) below depthz andPBBL [W m−2] is the total rate of dissipation within the BBL]. This totalbuoyancy flux, which quantifies the rate of change of the potential energy at depth



z in the water column divided byN2 provides the basin-scale vertical (diapycnal)diffusivity (Garrett 1990)

Kv = γmix N−2[εI + (∂A/∂V)ρ−1PBBL

][m2 s−1]. (7)

Applications of Equation 7 to three different lakes (Wüest & Gloor 1998, Wüestet al. 2000a) demonstrate that the values obtained by this approach are in reasonableagreement with estimates by other methods (Figure 11).

If the BBL mixing is strong relative to the background stratification, someof the lower part of the BBL appears well mixed. The thickness is given by theMonin-Obukhov length scaleLMO= u3

∗/kBB, whereBB= (gβ/ρ) · Fsed[W kg−1]is the bottom buoyancy flux related to the dissolved sediment/water mass fluxFsed

[kg m−2 s−1] of the solutes (β is the coefficient of haline contraction, withβ ≈ 0.7to 0.8 ‰−1 depending on the type of salinity; Millero 2000). For strong bottomcurrents, the Monin-Obukhov length is long, and the BBL can potentially expand astime progresses, and the stratificationN2 within the BBL remains weak. As a result,the BBL fulfills the “active” turbulence criterionε > 20νN2 (Stillinger et al. 1983,Rohr et al. 1988), and the bottom friction can keep the BBL well mixed. For weak

Figure 11 Dissipation and diffusivity profiles in Lake Baikal, showing the compositeof interior and boundary turbulence, as calculated by the relations in Section 4.3 and byEquation 7. Adapted from Wüest et al. (2000a).


402 WUEST ¥ LORKE

turbulence,LMO is short and the well-mixed layer may disappear all together and theentire BBL may become stratified. Consequently the “active” turbulence criterionwill be attained only in a very thin sublayer, which can become as thin as the VBLitself. In that case,BB can stratify the BBL because turbulence gets suppressed (ε <

20νN2) and the BBL becomes laminar (W¨uest & Gloor 1998). It means that deepbottom currents are not energetic enough to erode the permanent biogenic strati-fication over a long period of time and long-term meromixis may build up.

4.4. The Diffusive Bottom Boundary Layer

As evidenced from the first Fickian law,

flux = Dc∂C/∂z≈ Dc1C/δD = (Dc/δD)1C [gm−2 s−1], (8)

the key parameter for the sediment/water flux of a soluteC is the gradient withinthe DBL. Those gradients depend on the thicknessδD and the rate of consumption(or production) of soluteC in the sediment, the latter affecting the concentrationdriving force1Cacross the interface. The DBL thickness is solute-specific becauseδD depends onDc (varying by a factor of∼2 among the different solutes) and isslightly temperature-dependent, asν andDc are both functions of temperature.

The DBL thickness is not well defined for several reasons. Most obviously thetop end of the DBL (in fact also of the VBL) is not sharp (Jørgensen & Revsbech1985) because the turbulence cut-off at the Kolmogorov (1941) scale (ν3/ε)

1/4 isa gradual roll-off following the turbulence spectrum∼ (eddy size)−b with b= 3to 4. Therefore a transition zone exists between the pure molecular and fully tur-bulent zones above (Figure 10), whereKt and Dc are approximately equal. Toremove this ambiguity, Jørgensen & Des Marais (1990) defined the “effective”DBL by extrapolating the linear concentration gradient right above the sedimentto the bulk water concentration (Figure 10). This theoretical DBL thickness pro-vides a practical procedure to calculate the true flux through the interface basedon1C, the concentration difference between the sediment surface and the bulkwater above. As a result of the turbulence in the overlying water, the vertical lo-cation of the transition zone is variable owing to sporadic intrusions of energeticeddies (Gundersen & Jørgensen 1990) and to horizontal heterogeneity over natu-ral topographies (Jørgensen & Des Marais 1990). Solute concentrations thereforecontain stochastic fluctuations, and stationary profiles can only be obtained afteraveraging over several minutes, defined by the DBL residence timescaleδ2

D/Dc

of the respective solute. Because the solute residence time in the DBL scales withδ2

D ∼ δ2ν ∼ u−2

∗ ∼ τ−1BBL, the frequency of the fluctuations (inverse of the residence

timescale) is a direct indication of the bottom stressτBBL (Gundersen & Jørgensen1990).

In lakes, where turbulence is often low and the available organic matter isusually plentiful, the microbiological activity in the sediments become flux-limitedby the physical constraints of the interfacial molecular fluxes (“bottleneck” of theexchange). With the exception of ultra-oligotrophic lakes, where oxygen penetrates



deep into the sediment (Martin et al. 1998), the sediment surfaces become an anoxicmicroenvironment, which explains the occurrence of microaerophilics, such asBeggiatoaor even anaerobic bacteria, although the sediments are exposed to fullyoxic bulk water above. The relevance of the deep currents for the sediment/watertransfer of solutes becomes more obvious by expressing the molecular flux by themass transfer coefficient (Dc/δD) [m s−1] as defined in Equation 8. Because themass transfer coefficient is (Dc/δD) ∼ δ−1

ν ∼ u∗ ∼ u, the interfacial flux becomesproportional to the flow acting on the BBL (Hondzo 1998, Steinberger & Hondzo1999, Santschi et al. 1991). As a practical consequence, wind-exposed aquaticsystems will show higher rates of degradation and turnover than weakly forcedones. As a global consequence, old lacustrine sediments contain more organiccarbon and nutrients than relic marine sediments, where diffusion through thesediment/water interface is not regulated by the rate of oxygen uptake.

Besides the molecular DBL fluxes, there are additional pathways between thesediment and water. In Lake Erie, for instance, Zebra mussels have created denselypopulated reefs (with several thousand mussels per m2), which have been identifiedfor their enormous venting capacity (Ackerman et al. 2001). As another example,in shallow waters, heating of the sediments by short-wave radiation causes buoyantporewater to convect through the interface. Such nonlocal processes (Boudreau &Imboden 1987) are often more effective than molecular diffusion. In eutrophicwaters, advective transport through the sediment-water interface occurs mainlyby methane and carbon dioxide bubbles formed in the anoxic sediment. Whereasin oligotrophic lakes, where the sediment surfaces remain oxic, worms or othermacrofauna (such as insect larva) act as conveyor belts through the interface andproduce ventilation dips and mounds (Jørgensen & Revsbech 1985). The bioticinvasion has a positive feedback effect: Enhanced activities of macro- and meio-fauna, which move sediment around and pump oxygen-rich water into their bur-rows, improve the oxic conditions in the sediment and thereby improve the livingconditions for more bottom biota. Bioturbation by benthic organisms, therefore, isof general importance for the distribution and flux of soluble and colloidal materialin lacustrine sediments.

Besides the direct effect on the flux of matter between sediments and water,biota have an indirect effect on the exchange by influencing the BBL roughness.Fecal pellets, tracks, trails, tubes, pits, and mounds enhance the structuring andspatial heterogeneity and increase the bottom roughness, leading to an increasein the mass and momentum transfer. Jørgensen & Des Marais (1990) demon-strated that a three-dimensional DBL is thinner at the upstream side of mounds,and thicker on the lee side and over dips (Figure 12). Although the structure ele-ment in Figure 12 is an order of magnitude larger thanδD, the DBL has beenfound to remain along the complex surface. It is interesting to note, Jørgensen &Des Marais (1990) found that the two-dimensionally integrated flux was signifi-cantly larger than the flux as calculated from the averaged values ofδD and1C,respectively. This flux enhancement is partly due to the increase of the interfacialsurface area of an irregular topography (Ziebis et al. 1996a,b; Røy et al. 2002). For


404 WUEST ¥ LORKE

Figure 12 Two-dimensional diffusive boundary layer due to small-scale sediment topog-raphy. The oxygen diffusive boundary layer limit (line) was defined by the isopleth of 90%saturation. Notice the different vertical and horizontal scales. Reproduced from Jørgensen &Des Marais (1990).

extremely rough topography, we can expect the flux again to be reduced, owingto dead zones between large roughness elements. This indicates that a mediumroughness is probably most effective for the transfer fluxes and that bottom biotacan potentially generate a favorable environment themselves. Reef-building masspopulations (such as zebra mussels) may take advantage of this effect.

5. OUTLOOK

Much of the progress on small-scale processes is linked to the enormous de-velopment of instrumental techniques, such as the microstructure probes (Gregg1991, Imberger & Head 1994, Prandke & Stips 1998, Luketina & Imberger 2001).Nowadays it is possible to measure not only high-resolution current and shearprofiles with coherent ADCP, but also, simultaneously with the same instrument,the TKE, Reynolds’ stress, the production of turbulence, and the inertial dissi-pation (Lohrmann et al. 1990; Lhermitte & Lemmin 1994; Lu & Lueck 1999;Stacey et al. 1999a,b). The submersible Particle Image Velocimetry (PIV) instru-ment by Bertuccioli et al. (1999) performs measurements of the two-dimensionalvelocity vector maps down to the dissipative scales, which allows determiningthe spatial energy spectra and the dissipation directly from the deformation tensor(Doron et al. 2001). For measurements closer to the sediment, Unisense flow



microsensors are now available to measure velocity profiles at a 50-µm verticalresolution through the VBL, the lowest cm of the BBL. Such instruments willalso make the ultra-weak turbulence, often found in lakes, better assessable. Forthe quantification of DBL fluxes and early-diagenesis in the sediment (Boudreau1997), a large number of in-situ microsensors are now available, including Fe, Mn,O2, CO2, NO−3 , NO−2 , N2O, NH+4 , H2S, CH4, CO2−

3 , and others (Müller et al. 2002).Those sensors are especially suitable in freshwaters, which contain fewer elec-trolytes but usually more nutrient species than oceanic water. Planar optodes havealso been developed (Glud et al. 1996), resolving the details of two-dimensionaloxygen distributions (Figure 12) at and below the sediment-water interface.

These in-situ-measured microstructure and small-scale parameters can be com-pared directly with the output from turbulence models (Stips et al. 2002, Mironovet al. 2002, and many others). Different closure models are now available thatcan be tested and improved by observations. In the past few years especially, thetwo-equationk-ε model saw a strong revival: For lake applications this schemeseems very robust and favorable over thek-l model, in which turbulence tends todie out owing to weak forcing (Burchard et al. 1998, Burchard & Petersen 1999).Besides the one- to three-dimensional turbulence closure models, large eddy anddirect numerical simulations are also increasingly used to study specific phenom-ena such as convection (Sander et al. 2000) or Kelvin-Helmholtz billows (Smythet al. 2001) in great detail. Turbulence models allow for combining numericalinvestigations with field and laboratory measurements and are able to describethe complex coincidence and interaction of the different mechanisms occuring innatural systems.

Another source of progress is the ongoing discovery of new processes and phe-nomena, such as the super-smooth SBL at low wind (Wu 1994), the wave-statedependency of the surface wind stress and the WASL (Terray et al. 1996), the“interior quietness” (Goudsmit et al. 1997), convective boundary currents inducedby differential cooling (Fer et al. 2002), the straining-induced thermal BBL con-vection as a result of the Stokes solution (Lorke et al. 2002)—a process which issimilar to the tidal straining in estuaries (Rippeth et al. 2001)—the seemingly roughBBL, and the astonishing two-dimensional structure of the VBL and DBL. Themechanisms of energy transfer through the spectrum of high-frequency internalwaves resulting in the universal spectral slope should be assessed in the near futureby synthesizing the vast amount of observational and analytical studies in this area.

Current and future progress in lake research will be much driven by the utilitarianneeds of society as a result of the increasing pressure on freshwater resources(especially irrigation, hydropower, and water scarcity) and the climate change.The need for limnologists and engineers to answer practical questions will surelyincrease.

ACKNOWLEDGMENTS

We thank Eliane Scharmin, Lorenz Moosmann, and Beat Müller for their helpin preparing the manuscript as well as Erich Bäuerle, Martin Schmid, and Adolf


406 WUEST ¥ LORKE

Stips for critical reviews. Special thanks go to Daniel McGinnis and John Littlefor their advice on the proper use of the English language. A.L. was partiallysupported by the Swiss National Science Foundation Grant 2000-063723.00 and2000-067091.01 and partially by EAWAG.

The Annual Review of Fluid Mechanicsis online at http://fluid.annualreviews.org

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