Currents in lakes

Basis-Unterlagen zu den Vorlesungen: “Environmental Fluid Dynamics II“ “Transport and Mixing in Natural Waters“

Currents in lakes

Referenz: Wüest, Alfred und A. Lorke: Small-scale hydrodynamics in lakes, in: Annual review of fluid mechanics 35 (2003): p. 373-412.

Kontakt: Alfred Wüest, Applied Aquatic Ecology, EAWAG, 6047 Kastanienbaum [email protected]

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Annu. Rev. Fluid Mech. 2003. 35:373–412 doi: 10.1146/annurev.fluid.35.101101.161220 c 2003 by Annual Reviews. All rights reserved Copyright °

SMALL-SCALE HYDRODYNAMICS IN LAKES Alfred Wüest and Andreas Lorke Applied Aquatic Ecology (APEC), Limnological Research Center, EAWAG, Kastanienbaum, Switzerland; email: [email protected], [email protected]

Key Words enclosed basin, small-scale turbulence, mixing, waves, boundary layer, stratification ■ Abstract Recent small-scale turbulence observations allow the mixing regimes in lakes, reservoirs, and other enclosed basins to be categorized into the turbulent surface and bottom boundary layers as well as the comparably quiet interior. The surface layer consists of an energetic wave-affected thin zone at the very top and a law-of-thewall layer right below, where the classical logarithmic-layer characteristic applies on average. Short-term current and dissipation profiles, however, deviate strongly from any steady state. In contrast, the quasi-steady bottom boundary layer behaves almost perfectly as a logarithmic layer, although periodic seiching modifies the structure in the details. The interior stratified turbulence is extremely weak, even though much of the mechanical energy is contained in baroclinic basin-scale seiching and Kelvin waves or inertial currents (large lakes). The transformation of large-scale motions to turbulence occurs mainly in the bottom boundary and not in the interior, where the local shear remains weak and the Richardson numbers are generally large.

1. OVERVIEW More than two decades have passed since the last Annual Review of Fluid Mechanics articles by Csanady (1975) and Imberger & Hamblin (1982) on lake hydrodynamics were written. Those reviews have been a guide to many physical limnologists and engineers. Great progress has been achieved in several areas, such as turbulence observations and modeling (Section 2 and 3) as well as microsensor technology and boundary layer fluxes (Section 4), which have been reviewed in the last decade in part by Imberger & Patterson (1990), Imboden & Wüest (1995), and Imberger (1998a). The assessment of boundary versus internal mixing in enclosed water bodies is surely such a typical area of progress. Other subjects, such as internal wave dynamics (Section 3), proved to be so complex that this field of research could not significantly improve the understanding since then, although many recent papers addressed this gap. Here, laboratory studies contributed more to the advancement of the assessment of the different processes involved.

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Lakes are similar in many aspects to the ocean, and therefore the limnologic and the oceanographic literature are very much interleaving, as documented by the two key journals Limnology & Oceanography and Journal of Geophysical Research, from which we draw most information for this review. Lakes and oceans are also extremely different in many aspects such as size, forcing, and stratification, although the greatest difference is the enormous variability among the lakes. For example, the stratification N2 in lakes varies by nine orders of magnitude (Table 1) and is determined not only by temperature and salt (as in the ocean), but also by 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 concentrates on small-scale hydrodynamics. The sections are organized according to the turbulence characteristics. As shown by several publications in the past few years, the small-scale processes are distinctly different (Figure 1) between the surface boundary layer (Section 2), the interior of the density-stratified deep water (Section 3), and the bottom boundary layer (Section 4). These three sections outline the current level of observation and understanding of these three compartments of lacustrine water bodies and highlight their links to small-scale processes. Because much of the oceanographic literature is neglected here, we also refer the reader to the most recent reviews on oceanic surface boundary layers (Csanady 2001, Jones & Toba 2001), bottom boundary layers (Boudreau & Jørgensen 2001), and small-scale processes (Kantha & Clayson 2000). The repeatedly used quantities are summarized in Table 1, including typical ranges of variations, and the continuously applied abbreviations are listed in Table 2.

2. SURFACE BOUNDARY-LAYER PROCESSES 2.1. Introduction In lakes the surface boundary layer (SBL) is the most dynamic zone in many respects. Light and nutrients enable photosynthesis of phytoplankton, which provides the basis of the food web and lays the groundwork for biological interactions. Also physical and geochemical quantities undergo the strongest dynamics owing to exchange with the adjacent atmosphere and to photochemical and biological processes. Hence, for understanding the ecologically most relevant phenomena, the atmosphere/water relationship is crucial. Depending on the properties of the air, wind, water, and waves, the air/water interface creates a bottleneck for the exchange of the physical quantities such as heat, kinetic energy, momentum, and matter (gases, vapor, aerosols, etc.). These exchanges of physical and chemical properties are driven by wind- and heatflux-induced turbulence (Figure 2). We first address the effect of wind, which is substantially more complex than convection. The most crucial parameter governing the wind-driven regime is the surface shear stress τ [N m−2], the force per unit area, acting on the water surface as a

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

Property

Definition

Typical range

B

Buoyancy flux (rate of change of potential energy)

B = (g/ρ)hw0 ρ 0 i (w0 = (ε/N )1/2: vertical eddy velocity

10−11 to 10−6 W kg−1

C1

Bottom drag coefficient (for u1 at 1 m above bottom)

C1 = τ B/(ρu21 )

0.001 to 0.003

C10

Wind drag coefficient (for U10 at 10 m above bottom)

C10 = τ /(ρ aU210 )

0.001 to 0.002 (extreme: ∼0.01)

δD

Diffusive boundary layer thickness

δ D ≈ δ ν (DS/ν)1/(3 . . . 4)

0.2 to 1 mm

δν

Viscous boundary layer thickness

δ ν ≈ 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(∂u0 /∂z)2

f

Inertial frequency due to Ä (Ä = frequency of Earth rotation)

2Ä · sin(φ) (φ = latitude)

g

Gravity acceleration

γ mix

Mixing efficiency

γ mix = B/ε

0.05 to 0.25

k

Von Kármán constant

U(z) = (w∗ /k)ln(z/zo)

0.4 to 0.42

Kv

Vertical turbulent diffusivity

Kv = w0 l

10−2 to 102 cm2 s−1

LMO

Monin-Obukhov length scale

LMO = u3∗ /kB

SBL: m to several 10 m BBL: dm to 100 m

LO

Ozmidov scale, vertical (energy-containing scale of overturns in stratification N2) Thorpe scale, vertical scale of overturns (estimated from measured profiles)

LO = (ε/N3)

cm to several m

LT = hl2i1/2 l = vertical dislocation relative to equilibrium

cm to several m

LT

ν, ν a

Molecular viscosity of water, molecular viscosity of air

N2

Stability of the water column (local)

10−11 to 10−6 W kg−1 0 to 0.000145 rad s−1 9.81 m s−2

1/2

0.012 to 0.015 cm2 s−1 ∼0.15 cm2 s−1 N2 = −(g/ρ) ∂ρ/∂z

10−10 to 10−1 s−2 ∼0.02 N m−2 (extreme: 0 to 25 N m−2)

2 = ρ a C10U10

τ

Surface wind shear stress

τ

τB

Bottom shear stress

τ B = ρC1u21

∼0.001 to 0.02 N m−2

TKE

Turbulent kinetic energy

TKE = 1/2(u02+v 02+w02)

∼10−6 J kg−1

U10

Horizontal wind speed (10 m above water surface)

u∗ , w∗

Friction velocity in water, friction velocity in air

u∗ = (τ B/ρ)1/2 = C1 2 u 1/2 w∗ = (τ /ρ a)1/2 = C10 U

0.5 to 5 mm s−1 0 to 1 m s−1

ωp, λp, cp

Frequency, wavelength, and phase speed of dominant waves

cp = (gλp/2π )1/2

1 to 10 rad s−1 ∼ m several m s−1

zo

Roughness length (SBL) Roughness length (BBL)

U(z) = (w∗ /k)ln(z/zo) u(z) = (u∗ /k)ln(h/zo)

0.1 to 10 mm

Typical: 2 m s−1 (extreme: 0 to 20 m s−1) 1/

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Figure 1 Level of turbulence in a medium-sized lake, expressed by dissipation of turbulent kinetic 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 separately: the energetic surface boundary layer (Section 2), the slightly less turbulent bottom boundary (Section 4), and the strongly stratified and almost laminar interior (Section 3). Adapted from Wüest et al. (2000b).

result of the wind. The source of this stress can be interpreted as the downward eddy-transport of horizontal momentum (τ = ρ ahU0 W0 i) from the atmosphere. The concept of “constant stress” would call for the same Reynolds flux (τ SBL = ρhu0 w0 i) in the underlying water, where U,W (u,w) are the horizontal and vertical velocities of air (water) [m s−1], ρ a(ρ) is air (water) density [kg m−3], and 0 (prime) denotes fluctuations. 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 for k (TKE) and ε (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

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Figure 2 Schematic overview of the surface boundary processes, the shear stress, and the associated 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 the conservation of momentum that is relevant at the interface. Owing to the presence of waves, the momentum flux into the SBL, τ SBL, is smaller than the applied stress τ from the air (Figure 2). Part of τ is consumed by the acceleration and maintenance of waves (so-called wave stress τ Wave), whereas the remaining momentum flux τ SBL is forcing the SBL water underneath the waves (Anis & Moum 1995, Terray et al. 1996, Burchard 2001). The conservation of momentum at the interface implies that 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 momentum transfer to the water. As a consequence, the wind stress, which is usually parameterized by 2 τ = ρa C10 U10

[N m−2 ],

(2)

using the wind drag coefficient C10 [-], depends not only on the wind speed U10 (measured at standard 10-m height above surface), but also on the presence and on the state of the surface waves. In fact, the wave field is fundamental for the amount of 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 is addressed in the following two sections, before we discuss the resulting small-scale processes in the lacustrine SBL.

2.2. Surface Gravity Waves Waves produce additional roughness, thereby increasing the friction and enhancing the momentum flux from the air to the water. This effect is related to the more efficient 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 which the wind can push laterally scales with the significant wave height H1/3 [m], which is defined as the average height of the highest third of the waves (crest to trough). Consequently, also the steepness H1/3/λp [-] of the related waves influences the air/water coupling (λp [m] is the energy-containing wavelength; p indicates the peak of the wave energy spectrum). Due to the limited horizontal extent of the water bodies, the wave-induced complexity is more pronounced for lakes, reservoirs, and estuarine waters than for the open ocean, where quasi-steady conditions can build up more easily. At the upwind shore, the waves, not being in equilibrium with the wind field, are short in wavelength and small in amplitude. With increasing distance X [m] from the upwind shore, the characteristic wavelength λp, the corresponding deep-water wave phase velocity cp = (gλp/2π)1/2 [m s−1], and the significant wave height H1/3 increase, whereas the characteristic wave frequency ωp [s−1] decreases downwind (g represents gravity acceleration). Observations revealed that these developments obey, at least under constant wind stress, astonishingly well-defined empirical relations. According to Hasselmann et al. (1973) the significant wave height grows with fetch X by H1/3 ≈ 0.051(w2∗ X/g)1/2, whereas the frequency decreases as ωp ≈ 7.1(g/w∗ )(gX/w2∗ )−1/3 [w∗ = (τ /ρ a)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 of H1/3 ≈ 0.6 m only. This indicates that waves in most lakes never reach a state close to saturation, where wave growing stops. Therefore, the developed state of the open ocean is relevant only to very large lakes. In fact saturation for U10 = 2 m s−1 and 6 m s−1 is not reached before 10 km and 100 km, respectively (Wu 1994). For the countless number of not very large lakes, the wave fields will typically stay underdeveloped, and subsequently the momentum uptake will remain complex and subject to individual local effects (such as topography). Instead of the fetch, which is not a well-defined physical parameter, the nondimensional phase speed cp/U10 [-], called wave age, is often used as an adequate measure of wave development (cp/w∗ is sometimes also defined as wave age; ∼30 times larger than cp/U10). Therefore, most wave-relevant quantities are usually expressed as a function of the wave age, rather than as a function of fetch X. Applying the relations by Hasselmann et al. (1973) to the above defined key properties allows one to express them as a function of wave development: H1/3 ≈

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0.96(w2∗ /g)(cp/w∗ )3/2, H1/3/λp ≈ 0.153(cp/w∗ )−1/2, ωp ≈ (g/w∗ )(cp/w∗ )−1, and λp ≈ (2π/g)w2∗ (cp/w∗ )2 (Csanady 2001). Once the wave age cp/U10 approaches the value of ∼1.14 to 1.2 (Donelan et al. 1992), the waves travel at speeds comparable to 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 or wave age and become solely a function of wind speed, such as ωp = 0.88g/U10 and H1/3 ≈ 0.208U210 /g (Csanady 2001). All these observations of the age dependency of the wave heights, steepness, and other properties clearly demonstrate that the momentum transfer and its vertical distribution in lakes crucially depend on the state of wave development. Due to the limited extent of lakes, waves are commonly found to be young and underdeveloped (i.e., cp/U10 small) and therefore short, steep, and of high frequencies and, as a result, those waves break more frequently. In addition, growing waves also pick up momentum for their acceleration. As a combined effect, young and growing waves extract momentum, τ Wave (Equation 1), for growth and compensation of the loss by breaking. Therefore, young waves appear rougher and produce more turbulence than more mature waves. Long and saturated surface waves, in contrast, although containing much more momentum, lose their momentum and energy slowly because they are smoother and break at lower rates. Saturated waves, not growing any further, need τ Wave only to compensate for the loss by the breaking. For fully developed waves, we can expect that the SBL stress τ SBL is close to the atmospheric counterpart τ (Equation 1). Understanding the fetch-limited areas is of considerable relevance for inland and large parts of coastal waters, where air/water exchange processes have particularly strong ecological significance. Based on the discussion above, we can conclude that for a given wind speed, the observed surface stress will be higher near the upwind shore than in the open water, and it will also be larger in lakes compared to the open ocean. This may appear counterintuitive because SBL turbulence is usually lower in lakes compared to oceans. This seeming paradox is due to the fact that waves also carry momentum and energy to the shore. In small-to-mediumsized lakes these “missing terms” are significant for the momentum and turbulence balance. A second consequence of the fetch limitation is the importance of τ Wave. Except in the few very large lakes on Earth, τ Wave can reach a considerable fraction of 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 coefficient C10 [-], which quantifies the total vertical momentum flux τ (Equation 2) well above the wave-affected boundary (at standard 10 m height). A confusingly large number of measurements have been carried out in the past; these have been reviewed sporadically (Smith 1988, Donelan 1998, Csanady 2001, Jones & Toba 2001). Still today, the drag coefficient is associated with large scatter, which is partly due to the different techniques used and partly due to the difficulties quantifying

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the wave development. The measurement techniques include the profile method (fitting logarithmic vertical profiles to measured wind velocity), the direct method (measuring the stress hU0 W0 i = w2∗ ), and the dissipation method [determining dissipation ε (W kg−1) of turbulent kinetic energy (TKE) in the inertial subrange and calculating w∗ according to Equation 4 below]. Despite the large scatter, there is a relatively clear conceptual picture emerging. The drag coefficient depends to large extent 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. There are 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 as U(z) ≈ w∗ {k−1 ln(gz/w2∗ ) + K} for different wind velocities (k = 0.41 is von Kármán’s constant). The denominator w2∗ /g is the so-called waveheight scale, which represents a measure of the roughness of the surface waves. The constant 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 of C10 (Equation 2) allows determining C10 by ¡ ¢ 2 (3) + K}−2 [-] C10 ≈ {k −1 ln g10/C10 U10 (10 has the units m) for any given wind speed U10. Equation 3 is an implicit relation in C10, converging quickly after about four iterations. The results of Equation 3 (displayed in Figure 3) together with measured data from different sources, demonstrate the excellent match between Charnock’s relation and measured drag for strong winds and wave-dominated surface stress (Csanady 2001). The typical values of C10 range from 0.0011 (at U10 = 5 m s−1) to 0.0021 (at U10 = 25 m s−1). As a note of caution, we mention that in the literature the wind profiles are often parameterized by the roughness length zo, using U(z) = w∗ k−1ln(z/zo) instead of the Charnock (1955) relation [where zo 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 estimate the drag coefficient C10 from roughness length estimates determined by the profile method. However, this procedure is questionable because the conversion of zo to C10 = k2/ln2(10/zo) generates large uncertainty owing to errors in zo inherent to the profile method. Realistic C10 values correspond to a difficult-to-measure zo of ∼0.1 mm, and uncertainties of a factor of 10 in zo translate to uncertainties of a factor of 2 in C10. For weak winds ( 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 composite of interior and boundary turbulence, as calculated by the relations in Section 4.3 and by Equation 7. Adapted from Wüest et al. (2000a).

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turbulence, LMO is short and the well-mixed layer may disappear all together and the entire BBL may become stratified. Consequently the “active” turbulence criterion will be attained only in a very thin sublayer, which can become as thin as the VBL itself. In that case, BB can stratify the BBL because turbulence gets suppressed (ε < 20νN2) and the BBL becomes laminar (Wüest & Gloor 1998). It means that deep bottom currents are not energetic enough to erode the permanent biogenic stratification 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 ≈ Dc 1C/δ D = (Dc /δ D )1C

[gm−2 s−1 ],

(8)

the key parameter for the sediment/water flux of a solute C is the gradient within the DBL. Those gradients depend on the thickness δ D and the rate of consumption (or production) of solute C in the sediment, the latter affecting the concentration driving force 1C across the interface. The DBL thickness is solute-specific because δ D depends on Dc (varying by a factor of ∼2 among the different solutes) and is slightly temperature-dependent, as ν and Dc are both functions of temperature. The DBL thickness is not well defined for several reasons. Most obviously the top end of the DBL (in fact also of the VBL) is not sharp (Jørgensen & Revsbech 1 1985) because the turbulence cut-off at the Kolmogorov (1941) scale (ν 3/ε) /4 is a gradual roll-off following the turbulence spectrum ∼ (eddy size)−b with b = 3 to 4. Therefore a transition zone exists between the pure molecular and fully turbulent zones above (Figure 10), where Kt and Dc are approximately equal. To remove this ambiguity, Jørgensen & Des Marais (1990) defined the “effective” DBL by extrapolating the linear concentration gradient right above the sediment to the bulk water concentration (Figure 10). This theoretical DBL thickness provides a practical procedure to calculate the true flux through the interface based on 1C, the concentration difference between the sediment surface and the bulk water above. As a result of the turbulence in the overlying water, the vertical location of the transition zone is variable owing to sporadic intrusions of energetic eddies (Gundersen & Jørgensen 1990) and to horizontal heterogeneity over natural topographies (Jørgensen & Des Marais 1990). Solute concentrations therefore contain stochastic fluctuations, and stationary profiles can only be obtained after averaging over several minutes, defined by the DBL residence timescale δ 2D /Dc of the respective solute. Because the solute residence time in the DBL scales with −1 δ 2D ∼ δν2 ∼ u −2 ∗ ∼ τBBL , the frequency of the fluctuations (inverse of the residence timescale) is a direct indication of the bottom stress τ BBL (Gundersen & Jørgensen 1990). In lakes, where turbulence is often low and the available organic matter is usually plentiful, the microbiological activity in the sediments become flux-limited by the physical constraints of the interfacial molecular fluxes (“bottleneck” of the exchange). With the exception of ultra-oligotrophic lakes, where oxygen penetrates

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deep into the sediment (Martin et al. 1998), the sediment surfaces become an anoxic microenvironment, which explains the occurrence of microaerophilics, such as Beggiatoa or even anaerobic bacteria, although the sediments are exposed to fully oxic bulk water above. The relevance of the deep currents for the sediment/water transfer of solutes becomes more obvious by expressing the molecular flux by the mass transfer coefficient (Dc/δ D) [m s−1] as defined in Equation 8. Because the mass transfer coefficient is (Dc/δ D) ∼ δν−1 ∼ u∗ ∼ u, the interfacial flux becomes proportional to the flow acting on the BBL (Hondzo 1998, Steinberger & Hondzo 1999, Santschi et al. 1991). As a practical consequence, wind-exposed aquatic systems will show higher rates of degradation and turnover than weakly forced ones. As a global consequence, old lacustrine sediments contain more organic carbon and nutrients than relic marine sediments, where diffusion through the sediment/water interface is not regulated by the rate of oxygen uptake. Besides the molecular DBL fluxes, there are additional pathways between the sediment and water. In Lake Erie, for instance, Zebra mussels have created densely populated reefs (with several thousand mussels per m2), which have been identified for their enormous venting capacity (Ackerman et al. 2001). As another example, in shallow waters, heating of the sediments by short-wave radiation causes buoyant porewater to convect through the interface. Such nonlocal processes (Boudreau & Imboden 1987) are often more effective than molecular diffusion. In eutrophic waters, advective transport through the sediment-water interface occurs mainly by methane and carbon dioxide bubbles formed in the anoxic sediment. Whereas in oligotrophic lakes, where the sediment surfaces remain oxic, worms or other macrofauna (such as insect larva) act as conveyor belts through the interface and produce ventilation dips and mounds (Jørgensen & Revsbech 1985). The biotic invasion has a positive feedback effect: Enhanced activities of macro- and meiofauna, which move sediment around and pump oxygen-rich water into their burrows, improve the oxic conditions in the sediment and thereby improve the living conditions for more bottom biota. Bioturbation by benthic organisms, therefore, is of general importance for the distribution and flux of soluble and colloidal material in 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 and spatial heterogeneity and increase the bottom roughness, leading to an increase in the mass and momentum transfer. Jørgensen & Des Marais (1990) demonstrated 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 element in Figure 12 is an order of magnitude larger than δ D, the DBL has been found to remain along the complex surface. It is interesting to note, Jørgensen & Des Marais (1990) found that the two-dimensionally integrated flux was significantly larger than the flux as calculated from the averaged values of δ D and 1C, respectively. This flux enhancement is partly due to the increase of the interfacial surface area of an irregular topography (Ziebis et al. 1996a,b; Røy et al. 2002). For

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Figure 12 Two-dimensional diffusive boundary layer due to small-scale sediment topography. 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, owing to dead zones between large roughness elements. This indicates that a medium roughness is probably most effective for the transfer fluxes and that bottom biota can potentially generate a favorable environment themselves. Reef-building mass populations (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 development of instrumental techniques, such as the microstructure probes (Gregg 1991, Imberger & Head 1994, Prandke & Stips 1998, Luketina & Imberger 2001). Nowadays it is possible to measure not only high-resolution current and shear profiles with coherent ADCP, but also, simultaneously with the same instrument, the TKE, Reynolds’ stress, the production of turbulence, and the inertial dissipation (Lohrmann et al. 1990; Lhermitte & Lemmin 1994; Lu & Lueck 1999; Stacey et al. 1999a,b). The submersible Particle Image Velocimetry (PIV) instrument by Bertuccioli et al. (1999) performs measurements of the two-dimensional velocity vector maps down to the dissipative scales, which allows determining the spatial energy spectra and the dissipation directly from the deformation tensor (Doron et al. 2001). For measurements closer to the sediment, Unisense flow

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microsensors are now available to measure velocity profiles at a 50-µm vertical resolution through the VBL, the lowest cm of the BBL. Such instruments will also make the ultra-weak turbulence, often found in lakes, better assessable. For the quantification of DBL fluxes and early-diagenesis in the sediment (Boudreau 1997), a large number of in-situ microsensors are now available, including Fe, Mn, − + 2− uller et al. 2002). O2, CO2, NO− 3 , NO2 , N2O, NH4 , H2S, CH4, CO3 , and others (M¨ Those sensors are especially suitable in freshwaters, which contain fewer electrolytes but usually more nutrient species than oceanic water. Planar optodes have also been developed (Glud et al. 1996), resolving the details of two-dimensional oxygen distributions (Figure 12) at and below the sediment-water interface. These in-situ-measured microstructure and small-scale parameters can be compared directly with the output from turbulence models (Stips et al. 2002, Mironov et al. 2002, and many others). Different closure models are now available that can be tested and improved by observations. In the past few years especially, the two-equation k-ε model saw a strong revival: For lake applications this scheme seems very robust and favorable over the k-l model, in which turbulence tends to die out owing to weak forcing (Burchard et al. 1998, Burchard & Petersen 1999). Besides the one- to three-dimensional turbulence closure models, large eddy and direct numerical simulations are also increasingly used to study specific phenomena such as convection (Sander et al. 2000) or Kelvin-Helmholtz billows (Smyth et al. 2001) in great detail. Turbulence models allow for combining numerical investigations with field and laboratory measurements and are able to describe the complex coincidence and interaction of the different mechanisms occuring in natural systems. Another source of progress is the ongoing discovery of new processes and phenomena, such as the super-smooth SBL at low wind (Wu 1994), the wave-state dependency of the surface wind stress and the WASL (Terray et al. 1996), the “interior quietness” (Goudsmit et al. 1997), convective boundary currents induced by differential cooling (Fer et al. 2002), the straining-induced thermal BBL convection as a result of the Stokes solution (Lorke et al. 2002)—a process which is similar to the tidal straining in estuaries (Rippeth et al. 2001)—the seemingly rough BBL, and the astonishing two-dimensional structure of the VBL and DBL. The mechanisms of energy transfer through the spectrum of high-frequency internal waves resulting in the universal spectral slope should be assessed in the near future by 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 utilitarian needs 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 surely increase.

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

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Stips for critical reviews. Special thanks go to Daniel McGinnis and John Little for their advice on the proper use of the English language. A.L. was partially supported by the Swiss National Science Foundation Grant 2000-063723.00 and 2000-067091.01 and partially by EAWAG. The Annual Review of Fluid Mechanics is online at http://fluid.annualreviews.org

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