CHAPTER ____ THE AIR-WATER INTERFACE: TURBULENCE AND SCALAR EXCHANGE

Sanjoy Banerjee Department of Chemical Engineering University of California, Santa Barbara Sally MacIntyre Marine Science Institute University of California, Santa Barbara Turbulence and scalar exchange at air-water interfaces have been investigated in the laboratory, in field studies, and with numerical simulations. Parameterization and phenomena are reviewed, in particular surface divergence and renewal concepts for scalar exchange predictions. Microbreaking is seen to be important over a wide range of conditions but is not well understood. The effect of heat losses on near-interface transport processes also requires further study.

1. Introduction Transport processes at gas-liquid interfaces are of importance in a number of areas, such as for equipment like absorbers, condensers and boilers, and for environmental systems. For processes at the interface between the atmosphere and terrestrial water bodies, early interest related to the need to understand gas transfer to rivers and lakes that had high biological oxygen demand due to natural causes or effluents, e.g. from paper mills. The desorption of dissolved substances, like PCBs, from inland and coastal water bodies, can also be an air quality concern and has been a subject of continuing interest. More recently, the transfer of

greenhouse gases (e.g., CO2 and methane) and moisture, at the ocean surface have become important because of their impact on global warming. It is estimated that approximately 30-40 percent of man-made CO2 is taken up by the oceans. These estimates are substantially affected by uncertainties in the prediction of gas transfer rates at the airwater interface. For example, oceanic CO2 uptake estimates vary by about a factor of three, depending on whether they are based on correlations by Liss and Merlivat (1986) or Wanninkhof and McGillis (1999)—the first yielding an uptake of 1.1 PgC /year, whereas the second gives about 3.3 PgC/year (Donelan and Wanninkhof, 2001). There is, therefore, considerable incentive to improve our state of knowledge regarding mass, heat and momentum transfer between the atmosphere and water bodies, and it is expected that the subject will continue to be an area of active research for the foreseeable future. The purpose of this review is to survey what is known about air-water transport processes and indicate areas where research would be beneficial. To set the stage for the discussion, we divide the subject into two broad categories. In the first, we will consider what is known about transport processes at an unsheared air-water interface, i.e., a situation in which the winds are light and the fluid motions and turbulence that occur near the air-water interface are generated elsewhere. For example, turbulence could be generated at the bottom boundary of a flowing stream or in the shear layer between subsurface currents flowing at different velocities, and these turbulence structures could then impinge on the air-water surface. An example is the “boils” that one sees at the surface of rivers. Turbulence can also be generated by heat losses that give rise to natural convective motions on the liquid side. In the second category, we consider situations with significant wind shear at the surface. In this case, the turbulence production occurs at the interface itself, rather than at some other, perhaps rather distant, location, giving rise to phenomena that are qualitatively different with regard to scalar exchange. Here again, there may be significant heat loss at the airwater interface. The turbulence production occurs at the interface, but since, heat loss is often largest when latent heat fluxes (evaporation) rise due to high winds, the turbulence is not only due to shear, but also due to natural convection on the liquid side.

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In all cases, the interface is deformable and can form complex patterns in response to the fluid motions. This makes problems of measurement and simulation difficult, particularly in the near-interface region that controls transport processes. For this reason, progress in our understanding of these processes has been slow compared to our understanding of such problems at solid (non-deformable) boundaries. There have been several reviews of the subject, notably of laboratory studies and simulations by Banerjee (1990), of field studies and experimental methods by MacIntyre et al. (1995), by Jaehne and Haussecker (1998) and at a recent conference—see Donelan et al. (2002). A further subdivision of transport processes at the air-water interface is possible, depending on whether the liquid side or the gas side controls the process. Typical concentration profiles for each case are shown in Figure 1. For most sparingly soluble gases like CO2, He and CH4, it is liquid side motion that controls gas transfer. This is because the transfer velocity, as discussed in Section 2, depends on the molecular diffusivity raised to some exponent between 1/2 and 2/3, turbulent motions being damped at the interface. Molecular diffusivity of most gases is about two to three orders of magnitude less in water than in air, which explains why turbulence on the liquid side controls absorption or desorption. On the other hand, processes like evaporation draw species (in this case H2O) directly from the interface and therefore are controlled by airside motions. A classification of the controlling side based on solubility and diffusivity is presented in Jaehne and Haussecker (1998). Suffice it to say here that the air-side behaves something like flow over a moving, solid (and wavy) boundary, whereas the motion on the liquid side is quite different. The exchange of sparingly soluble gases at the air-water interface will be the focus of this review, so it is the liquid side motions that will be considered in most detail here. To proceed, consider what happens as wind velocity is increased over a stretch of water. The wind imposes shear on the water surface and this, in turn, produces turbulence when the shear rate becomes high enough— which happens at relatively low wind velocities. Undulations start to form at the surface and grow into capillary and gravity waves as the wind velocity increases. When the wind velocity U10 (at 10 m above the

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surface) increases to about 3 to 5 m/s, these waves become quite steep and start to microbreak. By microbreaking we mean that plunging motions can be detected in the water below these waves, but there is no discernible air entrainment. These

Figure 1. Schematic showing concentration profiles near the air-water interface. Here α is the Henry’s Law constant that is used to calculate the equilibrium concentration of a sparingly soluble gas at the interface, when Ca is the atmospheric concentration.

microbreaking waves have amplitudes ~ O(1 cm) and length ~O(10 cm). The character of the turbulence below the waves is different from that in lighter winds. Near-surface turbulence before commencement of microbreaking appears to be generated in much the same manner as near walls, i.e., due to shear in the near-interface region, and many of its characteristics are rather similar to wall turbulence. This means we see streamwise, streaky, high-speed/low-speed regions, as well as “bursts”, which are intermittent (active) regions consisting of ejections of the fluid from the near-interface region into the bulk and sweeps in which bulk fluid is brought to the interface. These structures scale with the shear rate in a manner similar to what is observed in wall turbulence. However, on commencement of microbreaking there is a dramatic change in the near-interface turbulence structure. The surface velocity field indicates regions of strong upwelling behind the crests of microbreaking waves, with regions of convergence just in front of the

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crests, i.e. surface fluid does not appear to cross the microbreaking wave crests but to be entrained under them. Associated with this, there are intermittent bursts which plunge much deeper into the water and in this sense are “giant” bursts, compared to what one sees in the absence of microbreaking. As wind speed is further increased to about U10 ~ 12-13 m/s, breaking with air entrainment commences (wind-driven whitecapping) with water spray being entrained at the higher wind speeds. These phenomena also significantly affect gas transfer. However, while whitecapping and spray formation may be important in localized regions of high wind and near coastlines, the average wind speed over the ocean is ~7.5 m/s (Donelan and Wanninkhof (2002), which suggests that microbreaking is widespread and of the most interest for scalar exchange between the water and air. A typical early correlation from Liss and Merlivat (1986) for gas transfer rate as a function of wind velocity is shown in Figure 2, together some data from MacIntyre et al. (1995). The data are for inland water bodies, but the scatter shown is typical of data taken over oceans. Part of the scatter, of course, is due to difficulties in doing reproducible field experiments. There are a number of other factors that are important. The most important of these, as mentioned earlier, is that natural convection can arise due to the water surface being colder than the bulk liquid due to heat losses and evaporation from the water. These motions can interact with turbulence generated by shear and affect transport processes, as recently discussed by Eugster et al. (2003). Thus, wind velocity is not the only variable that affects transfer rates, but factors like cloud cover, humidity and the diurnal cycle also play a role. Another factor contributing to the scatter arises from the experiments giving gas transfer velocities averaged over some period of time, and during these times the wind may vary, causing biases in the averaged measurements, as discussed later. Other factors include surface slicks, which can damp near-interface turbulence, and the waves and wind being out of equilibrium, e.g., due to rising or falling wind conditions. In coastal regions and near shores, phenomena involving shoaling, wavebreaking and offshore flows of warm air over cold water complicate matters further.

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Figure 2. Data on gas transfer rates normalized to a Schmidt number of 600 against wind velocity at 10 m above the water surface. Liss and Merlivat’s correlation is shown as the 1.6 dotted line, and an empirical fit to the data k(cm/hr) = 0.45 U10 is shown as the solid line. (Wanninkhof et al. 1991).

In spite of these complications, and several other effects that add to the scatter, one can see from Figure 2 that there appears to be a significant increase in the gas transfer rate in the range of wind velocities at which microbreaking commences. This is captured to some extent in the Liss and Merlivat (1986) correlation by a change in its slope at a wind velocity of 3.6 m/s. There is a second change in slope at 13 m/s (not shown in the Figure) when breaking of large waves with air entrainment may commence. In other correlations, such as that of Wanninkhof and McGillis (1999), these changes in slope are captured by a cubic polynomial fit to gas transfer data in terms of the wind velocity. Because of the nonlinear nature of the gas transfer velocity vs. wind velocity curve, the averaging interval over which data is taken gives it a

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bias, since winds fluctuate. Thus a correlation based on averaging over a long period, e.g. a month, will be different from one in which averaging is done over a shorter period. This aspect must also be taken into account in interpreting data as different averaging periods for different data sets can lead to scatter. The rest of this review is organized as follows. We will first discuss modeling approaches in Section 2, followed, in Section 3, by consideration of situations in which the wind stress is light, so phenomena near the air-water interface are dominated by interactions with far-field turbulence, as well as by waves and natural convection. Laboratory studies and simulations will be considered first, followed by field experiments. In Section 4, we consider phenomena that arise when wind shear is significant, starting again with lab studies and simulations, followed by field studies. We end by summarizing the current understanding of processes at the air-water interface and the areas in which further work is desirable. 2. Scalar Exchange Models Before proceeding to discuss experimental results and simulations on scalar exchange and turbulence in the near-interface region, modeling approaches for such processes will be briefly reviewed. Early work by Lewis and Whitman (1924) proposed that transport occurred across a film of laminar fluid, of thickness δ, adjacent to the interface, with the bulk phase being well mixed and turbulent. This gave rise to mass or heat transfer coefficients, β (sometimes denoted by k in the literature, and also sometimes in this article) that were proportional to the diffusivity, D, i.e. β ∝ D. However, it became clear from subsequent experiments that at fluid-solid boundaries, β ∝ D2/3, whereas at gasliquid boundaries β ∝ D1/2, at least when Sc = ν/D was high (where ν is the kinematic viscosity). This led Higbie (1935) to speculate that turbulence brought fluid from the bulk to the interface, where unsteady absorption occurred into an essentially laminar fluid for some period, τ, after which the surface element was replenished. Lewis and Whitman’s theory gave

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β = D /δ,

(1)

whereas Higbie’s speculation gave

β = (D / (πτ ))

1/ 2

(2)

Subsequently, Danckwerts (1951) allowed for a random distribution of surface ages for the renewed elements, which was more typical of what might be expected from a turbulent fluid, giving on the liquid side,

β = (D / τ )

1/2

(3)

where τ may be thought of as the mean time between surface renewals. Note that the Higbie-Danckwerts surface renewal model predicted the dependence of the liquid-side scalar transfer rate, β, on D, as D1/2, which was also found in lab experiments. However, the quantity τ , the time between renewals, remained unspecified. A number of researchers proposed various models for τ in the 1960s, notably the “large-eddy” model of Fortescue and Pearson (1967) and the “small-eddy model” of Banerjee et al. (1968). The large-eddy model gave

τ ~ Λ /u

,

(4)

leading to a mass transfer coefficient 2 β Sc1/2 = u Re −1/ t

(4a)

where Λ (and sometimes, l ) is the turbulence integral length scale and u is the integral velocity scale, and Re t = uΛ / ν is the turbulent Reynolds numbers. On the other hand, the small-eddy (SE) model of Banerjee et al. gave

τ

~ (v / ε )

1/2

(5)

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where ε was the turbulent energy dissipation rate close to the interface and ν was the kinematic viscosity. Here ε can be measured directly or 3 calculated. For example, ε ~ u / Λ , so we have 4 β Sc 1/ 2 = u Re −1/ t

(5a)

There is some ambiguity in how to define the turbulent Reynolds numbers in (4a) and (5a). Strictly speaking, ε is the dissipation rate near the interface, and it is better to use estimates of ε directly. Be that as it may, both models agreed with limited sets of data, but gave very different results when it came to predictions of transfer rate under the same conditions. Theofanous et al. (1984) resolved this discrepancy by showing that the large- and small-eddy models gave the asymptotic behavior of the transfer coefficient at small and large turbulent Reynolds numbers ( Re t = uΛ / ν ), respectively. As Reynolds numbers in the field are variable, but often quite high, 1 10 to 105 for winds between 1 and 8 m s-1 in lakes, the small-eddy model and the surface divergence model are appropriate, and many experiments are directed towards finding the near surface energy dissipation rate, ε, as discussed later. It is also important to note that Banerjee et al’s SE model is quite general, and energy dissipation may arise from many factors, e.g. wind shear, natural convection, wave breaking, rain, etc. In fact, Banerjee et al. originally applied the model to estimating mass transfer in situations where vorticity generation by capillary waves was important. Banerjee (1990) derived a general form of the expression for the mass transfer coefficient for the case where there is no wind shear at the interface and the far-field turbulence is homogeneous and isotropic, based on the blocking theory of Hunt and Graham (1978). This theoretical result would appear to have a more restricted application than the SE model, but is worth considering. Using a result from McCready et al. (1986), Banerjee (1990) showed that for unsheared interfaces at which high Sc gas transfer occurs,

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β u

[

]

2 1/4

= Sc −1/2 Re −1/2 (∂u' /∂x + ∂v' /∂y ) t

int

(6)

where the subscript int. denotes the interface, and all quantities on the RHS of (6) have been nondimensionalized by u and Λ, the integral scales in the far field. The quantity in square brackets in (6) is the square of the surface divergence field due to the fluctuating motions. Here u’ is the fluctuating velocity in the streamwise direction x, v’ is the fluctuating velocity in the spanwise direction y—both directions being tangential to the interface. The expression arises in a straightforward way by noting that at a free surface the velocity fluctuations normal to the interface are given by

w' ~ ∂w'/∂z zint + HOT

(7)

as a result of the boundary conditions (whereas at a solid surface 2 2 2 w' ~ ∂ w' / ∂ z bound z / 2 + HOT ).

(

)

If (6) is written in dimensional terms

β ~ [Dγ ]

1/ 2

,

(6a)

where γ is the dimensional surface divergence (s-1) and D is the molecular diffusivity. The surface divergence cannot be predicted without a theory, and therefore the Hunt and Graham (1978) blocking theory was used to relate it to the far-field turbulence characteristics when they are homogeneous and isotropic. Note, however, that Banerjee’s result in (6) and (6a) is quite general and could be applied to situations where the surface divergence is governed by other phenomena, e.g. microbreaking waves which show regions of high surface divergence behind them. It could also be used for situations where surface divergence arose from natural convective motions in the liquid due to heat losses. In a sense it takes the

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place of the renewal parameter τ in Danckwerts (1951), but is more easily measured—usually by scattering particles on the liquid surface and measuring their trajectories (see Kumar et al., 1998). To proceed, for an unsheared interface with homogeneous isotropic far-field turbulence, the mass transfer coefficient was derived by Banerjee (1990) for high Sc as 2 β / u ~ Sc −1/ 2 Re −1/ {0.3[2.83 Re 3/4t − 2.14 Re 2/t 3]} t

1/4

(8)

with the proportionality constant ~ O (1). This is sometimes called the surface divergence (SD) model. The quantity within the first set of parentheses is the square of the nondimensional surface divergence. The expression was asymptotic to Ret-1/2 at small turbulent Reynolds numbers and to Ret-1/4 at large turbulent Reynolds numbers, which was in line with Fortescue and Pearson’s (1967) models and Banerjee et al.’s (1968) small-eddy model and validated Theofanous et al.’s hypothesis. Note also that the expression given above applies only to clean, unsheared interfaces with no effects due to surfactants or natural convection. If surfactants are present, the Schmidt number exponent may be somewhere between –1/2 and –2/3 (the value for a solid boundary), but the relationship between β and the surface divergence in equation 6a is preserved, with a slightly different exponent. It should also be noted that (8) applies to a rigid slip surface because Hunt and Graham’s theory can be applied to such a scenario. A free interface, however, is mobile and can deform in response to motions on the liquid side. The surface divergence with a deformable interface may be expected to be less, suggesting that the proportionality constant may be less than unity—in fact, 0.20 fits data quite well, when Ret is the turbulent Reynolds number in the bulk fluid. If wind shear is imposed, then, as mentioned earlier, the turbulence structure near the interface has characteristics somewhat similar to that of wall turbulence. The surface renewal (or surface divergence) models (3) and (6a) are still expected to apply for liquid-controlled transport processes, but the turbulence structure is now controlled by generation in the near-interface region. The appropriate scaling variables are now

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related to the wind stress imposed on the water surface and the fluid kinematic viscosity (the so-called inner variables). A complication arises at a deformable interface in comparison to a rigid boundary in that part of the energy transferred from the wind to the water goes to generate waves, and part to generate turbulence directly. Furthermore, the interfacial waves and turbulence can interact in complex ways. In any case, for situations in which waves break infrequently, the main near-interfacial turbulence structures are found to scale with the frictional drag (Df) at the interface (see De Angelis et al. (1999)). The

appropriate velocity scale is then u frict ~ (D f / ρ) , where ρ is the fluid density. Both the surface divergence, γ, and the time between renewals, τ , are expected to scale with u*frict and ν (the kinematic viscosity). *

1/ 2

Indeed, both direct numerical simulations and experiments indicate that the turbulence events leading to surface divergence and renewal are “sweeps”, i.e. events that bring bulk fluid to the interface. On the liquid side these are sometimes termed “upwellings” and on the gas side, “downdrafts”. If the time between sweeps is taken as the time between renewals, then experiments and direct numerical simulations indicate that this scales as

τ u*2 frict ~ 100 ν

(9)

leading to, with the subscript w denoting the liquid side (sometimes the subscript L is also used)

βw Sc w0.5 / u*

frict , w

~ 0.1 .

(10)

Banerjee (1990) showed that (10) agrees with the lab-scale data available at that time and De Angelis et al. (1999) found agreement with wind-wave tank data taken subsequently. We will discuss these matters further when laboratory experiments are reviewed. For a solid (as is essentially the case for the gas-side) surface, the form of the parameterization changes as turbulence is more strongly

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damped with w’ ~ z2 (where z is the distance from the surface). This leads to

βa Sc a2/3 / u*frict,a ~ 0.07

(11)

where the subscript a, or in some cases, G, denotes the gas side. Note that u*frict is based on the frictional drag component, which is the total drag only if the surface is level. With interfacial roughness due to waves, the total or effective drag includes form drag. However, the expression in (10) requires estimation of the frictional drag component from which u*frict is calculated. An expression for u* like that in Charnock will overestimate u*frict and give a value of β somewhat larger than found in experiments if (10) is used. This aspect will be discussed in more detail in Section 4. The various expressions for the transfer rate of sparingly soluble gases are summarized in Table 1, which also includes expressions from Csanady (1990) and Soloviev and Schluessel (1994). It should be mentioned that there have been several other forms of parameterization suggested, e.g. by Caussade et al. (1990) and Coantic (1986), so Table 1 does not list all the parameterizations proposed. 3. Unsheared Air-Water Interface In this section we will consider turbulence phenomena and scalar exchange when wind shear at the interface is relatively small or nonexistent. In laboratory studies, this situation is reproduced in experiments using either a stirred tank or open-channel flow. Most stirred-vessel experiments are done by using an oscillating grid to generate a relatively homogeneous isotropic turbulence field in the bulk fluid. For open-channel flows, turbulence is generated by the shear at the bottom. In some situations, waves are superimposed on open-channel flows for investigations of wave-turbulence interactions. In most of these experiments, it is difficult to keep the liquid surface free of

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surfactants, and therefore the results should be treated with caution, unless precautions have been explicitly taken to keep the surface clean. This is especially true for stirred-vessel experiments, where the liquid surface is quite stagnant. Furthermore, most laboratory experiments have not investigated the effect of heat loss (or gain) by the liquid. As we will see later, natural convection associated with heat loss is an important factor in the field and clearly an area where further laboratory research is warranted. 3.1 Laboratory Studies and Simulations Stirred tank experiments Theoretical approaches to calculating turbulence structure near a wall moving at the mean fluid velocity were proposed by Hunt and Graham (1978). In their work, a thin viscosity-dominated layer was postulated to form near the wall. In the region outside this layer, but within one integral length scale from the wall, the turbulence structure was modified compared to the far field. In particular, the scales parallel to the surface showed a slight increase, whereas the normal component was suppressed. Theoretical prediction of the damping of the normal component is possible from the theory as a function of the wave number. Brumley and Jirka (1987) conducted experiments near the free surface of grid-stirred tanks to check these results. They measured the turbulence energy spectrum as a function of distance from the wall for both the surface-normal and horizontal velocity fluctuations. They parameterized the fits using detailed calculations from the Hunt-Graham theory, and reference should be made to Brumley and Jirka’s (1987) paper or Banerjee (1990) for the form of the energy spectra that fit the detailed calculations from the Hunt-Graham theory. Hunt-Graham theory appeared to provide a reasonable means of estimating the various parameters in the region of the surface and was used subsequently by Banerjee (1990) to obtain the form of the high Schmidt number mass transfer coefficient on the liquid side shown in Eq. (8)—based on surface divergence estimates.

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Turning now to gas transfer data, Chu and Jirka (1992) measured the gas flux at the air-water interface of a grid-stirred tank. There is some concern that their data may have been affected by accumulation of surfactants at the water surface. This is discussed in more detail by McKenna et al. (1999), who also made simultaneous mass transfer and surface divergence measurements using DPIV. They found that the particles used for DPIV gave rise to surface-active effects unless they were thoroughly cleaned. In fact, the mass transfer rate data with particles that had not been washed fell below data without particles or with washed particles. The data with and without particles are compared with Chu and Jirka’s data, and the predictions of the parameterization in Eq. (8) in Figure 3. As evident from the figure, the particles reduce the gas transfer velocity, and Chu and Jirka’s data lies somewhere between the cases in McKenna et al., with and without particles. The Banerjee (1990) parameterization shown in Eq. (8) also gives a reasonable fit to the data, which is encouraging, since it was developed before the data were taken. The constant C ~ 0.20, which might be expected since Eq. (8) is based on a “rigid lid” approximation for the free surface, and in reality there will be some give which would reduce the surface divergence. Note also that the turbulent Reynolds number, Ret, in equation (8) is half the turbulent Reynolds number, ReHT, used by McKenna et al. Also, the point in McKenna’s data shown in Figure 3, where β (or k) ~ 11 cm/hr at Re HT ~ 180, is out of line with all the other data and may be an outlier. Open-channel studies We turn now to considering turbulence generated by shear at the bottom of a channel flow. The structure of the overall fluid motions and their effect on phenomena near the free surface have been the subject of a number of studies, notably those of Komori et al. (1982), Rashidi and Banerjee (1988) and Komori et al. (1989). More recently, Kumar et al. (1998) published the results of an extensive study of free-surface turbulence in channel flow using DPIV measurements of fluid velocity at the surface. They characterized the surface

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TABLE 1. Various correlations for gas transfer velocity (β or k) for liquid-side at high Schmidt number

Large eddy

kS1/c 2 = −1/2 C1 uRe t

Small eddy

C2 uRe t −1/ 4

MODEL

Two regime

Lamont & Scott, 1972 Theofanous et al., 1976

Large eddy for Re t < 500 . Small eddy applied for Re t > 500

Sc1/2 [Dγ ]

6a

C3 u[0.3(2.83 Re 3/4 t 1/4 2 − 2.14 Re 2/3 )] Re −1/ t t

8

1/ 2

Surface divergence Surface

divergence (no shear) Interfacial shear (liquid side) Interfacial shear (gas side) Eddy resolving analytical Surface processes

Eqn. no. References 4 Fortescue & Pearson, 1967 5, 5a Banerjee et al, 1968;

* frict.w

10

0.1 u

*

0.07u frict ,G Sc

−1/ 3

11

C4 p1 u*frict,w

Banerjee 1990 Banerjee 1990 Csanady, 1990

C5 p2 u*frict,w (1+ Rf / Rfcr )

1/ 4

(1 + Ke / Kecr )−1/ 2

Banerjee 1990 Banerjee, 1990

Soloviev & Schluessel, 1994

C1, C2, C3, C4, C5 are constants. p1 is the fraction of the surface undergoing intense renewal. p2 is the probability distribution of renewal events, Rf is the flux Richardson 4 number gHv / ρ C p u frict and Rf cr is the critical value ~1.5x10-4. Ke is the Keulegan 3 number ufrict gv and Kecr ~ 0.18. H is the surface heat flux obtained by summing latent, sensible and long wave radiation fluxes. (Modified from MacIntyre et al. 1995).

( )

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Figure 3. Mass transfer coefficients at the unsheared surface of a grid-stirred vessel. Particles evidently give rise to surfactant-like effects. The data with the particles absent is considered the most reliable. The line is from equation 8 with C = 0.20 (also called C3 in Table 1).

as consisting of spiral eddies, upwellings and downdrafts. They were able to show that the upwellings were caused by large active structures (bursts) that emanated from the bottom of the channel and impinged on the surface. Associated with these upwellings, and apparently formed by their action, were vortices (spiral eddies), typically attached to the free surface—something like whirlpools. These vortices persisted for long periods, as viscous dissipation was quite low in them and they were usually annihilated either by merging or by upwellings impinging on them. A typical merger of two vortices at the free surface is shown in Figure 4, and annihilation by an upwelling is shown in Figure 5, taken from Kumar et al.

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They also measured the energy spectrum of the surface parallel velocity fluctuations and showed that the spectrum split into two ranges, viz. a –5/3 power range at low wave numbers and a –3 range at high wave numbers, as shown in Figures 6 and 7 for both the streamwise and spanwise components. Also shown are some previous DNS results from Pan and Banerjee (1995) that are in agreement with the experiments. Pan and Banerjee used a pseudospectral simulation technique for openchannel flows with a no-slip wall at the bottom and a slip surface at the top. They also burned off the shear (and hence turbulence generation) at the bottom wall for some of their studies to examine the dynamics of decaying free-surface turbulence. These results have been used to develop and test subgrid scale models for turbulence near free surfaces, but we will not consider these further here. To proceed, the –5/3 power range at the low wave numbers suggests quasi two-dimensionality of turbulence in the near-surface region. There is up cascading of energy from the wave number range at which the spectrum splits to the small wave numbers. Physically, this means that the smaller structures merge to form larger structures, which is what is seen in Figure 4. These conclusions regarding quasi two dimensionality of the surface turbulence are supported by the direct numerical simulations of Pan and Banerjee. In fact, the twodimensionality can be seen from the contours of the vortical structures hanging down below the free surface in Figure 8. Several aspects need to be discussed related to these experiments. First, particles were scattered by Kumar et al on the free surface in order to conduct the DPIV. As McKenna et al. have shown, this may give rise to some surfactant effects, even though such effects would be expected to be less apparent in open-channel flow than in grid-stirred tanks. Kumar et al. were aware of this problem, however, and used rather low particle coverage of the free surface to minimize it. Details may be found in their paper. Nonetheless, these effects cannot be discounted. Second, the direct numerical simulations of Pan

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.

Figure 4. Merger of two vortices at the free surface of an open-channel flow.

Figure 5. Annihilation of a vortex at the free surface by an upwelling. The dark area in panel “d” is an upwelling where the visualizing particles have been swept outward

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Figure 6. Energy spectra for the streamwise velocity component at the free surface (xstreamwise direction, y-spanwise direction).

Figure 7. Energy spectra for the spanwise velocity component at the free surface (x – streamwise direction, y – spanwise direction).

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and Banerjee were in the “rigid lid” approximation, i.e., the free surface was replaced by a slip surface. The interface had no give in the vertical direction in the simulations, and this might not be a good assumption. Third, while the comparison in the shapes of the energy spectra between the experiments and simulations is remarkable, nonetheless the coincidence in actual values is not significant, as the integrals are normalized to the same values over the same range of wave numbers, i.e. it is only the shapes that are similar. Finally, the experiments under consideration did not have significant interfacial waves, and the Reynolds numbers were quite low, though turbulent. The low Reynolds numbers for the flow could lead to an overestimate of the importance of surface renewal motions that originate at the bottom wall. At high Reynolds numbers, the channel flow might be quite homogeneous and isotropic near the free surface and the surface-region turbulence structure might be more like that for a grid-stirred system. For details of the DPIV technique used by Kumar et al. (1998), reference should be made to the original paper. As free surface turbulence involves motions with a circle dynamic range, some special image processing considerations are involved, as discussed briefly in what follows. The methodology—based on multi-grid image processing algorithms for rigid body motion analysis, estimates the displacement vectors at discrete particle locations. The essence of this technique is to estimate large-scale motions from image intensity patterns of low spatial frequencies and small-scale motions from intensity patterns of high spatial frequencies. Cross-correlation between a pair of time-separated particle images is implemented by the hierarchical computational scheme of Burt (1981). Each image is convolved with a series of band-pass filters and subsampled to obtain a set of images progressively decreasing in resolution and size. A coarse estimate of the displacement field obtained from pairs of lower resolution images is used to obtain more accurate estimates at the next (finer) level. Processing starts at the level of lowest resolution and stops at the highest resolution level, which contains the original image pair. Due to subsampling of low-resolution images, the match template size can be kept constant for all stages of computation, thus eliminating the dependence of the largest resolvable displacement on the size of match template. In the present work, the

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search area at each level is kept constant at 3x3 pixels and the match template size at 5x5 pixels for all levels of computation. The algorithm has been implemented using simple thresholding based on the confidence level of an estimated displacement vector, as suggested by Anandan (1989). However, the confidence-level-based smoothing technique for rigid body motions (continuous velocity fields) could not be applied to displacement estimates obtained at discrete points, i.e., the particle locations. Instead, smoothing was performed over the area covered by each particle. The algorithm has been tested against direct numerical simulations of turbulent flows when the flow field is known and particle images have been generated from these with the addition of noise. Both the accuracy of motion estimation and the computation time are seen to improve as compared to conventional PIV methods.

Figure 8. Vortical structures hanging down from a free surface on which the instantaneous streamlines are shown (Pan and Banerjee, 1995). The surface is viewed from the side of a channel flow, looking up from below. The streamwise direction is 4 πh and the spanwise direction is 2 πh in length for a liquid layer thickness of 2h in these calculations.

22

Before moving on to scalar transfer results in open-channel flows, the effects of waves interacting with a turbulent stream will be considered. Early studies of mechanically-generated waves interacting with a turbulent current were those van Hoften and Karaki (1976), Iwagaki and Asaro (1980), Kemp and Simon (1982, 1983), Simon et al. (1988), and more recently, Rashidi et al. (1992), Suphratid et al. (1992), Nan et al. (1998), and Nan (2003). The most extensive work is that of Nan, in which 3D laser Doppler anemometry (LDA) measurements were made as a function of flow depth for a wide range of wave-to-current conditions, albeit at relatively low depth-based Reynolds numbers— though still resulting in fully developed turbulence. Nan’s data, over a range of Reynolds numbers and wave strength parameters, indicates turbulence enhancement in the near-surface regions, the effect increasing with the wave strength parameter ub/U which is the ratio of the waveinduced Stokes drift velocity to the mean velocity. This is shown for the streamwise, wall-normal and spanwise components in Figures 9, 10, and 11.

Figure 9. Enhancement of streamwise turbulence intensity due to waves as a function of + depth. The average position of the free surface is at y/h = 1.0. Here ub /U and ϖ are parameters that are measures of wave strength and frequency. The mean depth of the flow is h, and y is measured from the bottom of the channel. The subscript w denotes channel flow with superimposed mechanically generated waves, and the subscript c denotes channel flow without waves. u* is the friction velocity of the channel flow. Depth based Reynolds number ~5000.

23

Figure 10. Enhancement of the vertical turbulence intensity due to waves as a function of depth. Same notation as in Figure 9.

Figure 11. Enhancement of the turbulence intensity in the spanwise direction due to waves, as a function of depth. Same notation as in Figure 9.

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While there is some variation in the friction velocities measured at the bottom of the channel with and without waves, these changes are not large, so the figures do indicate the enhancement in turbulence intensities caused by the waves, even though the intensities, with and without waves, are nondimensionalized by the respective friction velocities. Note that the fluctuating velocities due to the wave motions themselves have been removed by ensemble averaging over many identical waves. Therefore, what is left are the turbulent fluctuations. Nan took data with essentially 2D waves, so that the spanwise fluctuations are not at all contaminated by the wave motion. (In fact, the spanwise wave components of motion are negligible.) Nan attributed the increase in turbulence to a wave-induced Reynolds stress, which was significant when waves interacted with the turbulent current. While this suggests that waves could have substantial effects on mass transfer, we are not aware of any experiments that have directly measured the effect of such mechanically generated waves on scalar exchange. However, there have been experiments on gas transfer in laboratory flumes and circular wind-wave facilities, as well as combinations of mechanically generated waves with wind to simulate such effects. These do indicate increases in gas transfer velocities when waves are present. We turn now to consideration of scalar transfer at the free surface of open-channel flow. A pioneering study in this regard was that of Komori et al. (1989). Komori et al. not only measured gas transfer rates (for CO2), but also made simultaneous measurements of surface renewal events which emanated as “bursts” from the bottom of the channel. The data cover a range of water velocities of 5.9 to 23.5 cm/s and depthbased Re of about 2800 to 10,000. They found a remarkable correlation between the frequency of surface renewal events (fs) and the gas transfer velocity, β. However, the measured gas transfer velocity is about a third of what would be predicted by surface renewal theory, as pointed out by Banerjee (1990), i.e. β = 0.35 (Dfs)1/2. The discrepancy may have arisen because Komori et al. did not directly measure surface divergence, but rather viewed the structures that were ejected from the bottom of the channel, not all of which may have impinged on the free surface.

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Knowlton et al. (1999) attempted to validate the surface divergence theory by calculating surface divergence directly from the velocity field measured by Kumar et al. (1998) and then using it to solve the 3D concentration field equation from which they calculated the mass transfer coefficient. The only case for which Kumar et al.’s data coincided with one of Komori’s cases was for depth-based Re = 2800. Knowlton et al. found remarkable agreement with Komori et al.’s data for this case. However, when Knowlton et al. calculated the gas transfer data based on a “rigid lid” direct numerical simulation, they found gas transfer velocities that were about 2-3 times higher. These results are also shown in Figure 12. In Table 2, we show the predictions of Eq. (8) with C = 0.20 (which was the value for C that agreed with the stirred vessel gas transfer data of McKenna et al.) It is evident that the agreement with these predictions is quite good. The velocity scale for Eq. (8) is taken to be the wall friction velocity, and the length scale was the depth. While these scales are reasonable, it is likely that the length scale is a weak function of the depth-based Reynolds number, i.e., (Λ/depth) varies as Re-1/8, which is what would be expected in the core region of pipe flow. Making such an assumption would improve the agreement between Equation (8) and the experimental data, which is rather over predicted at low Reynolds numbers and under predicted at high Reynolds numbers. Knowlton et al. concluded that a rigid lid approximation was poor for mass transfer calculations and that the “give” in the surface significantly reduced surface divergence at the air-water interface. This is also consistent with C ~ 0.20 in Eq. (8). These data have been taken in relatively shallow channels, and ejection events from the bottom boundary maintain their coherence up to the interface. For much higher Reynolds numbers and greater water depths, the bulk turbulence structure is expected to more closely approximate homogeneous, isotropic turbulence, where the surface divergence model given in Eq. (8) might again be expected to predict gas transfer velocity. In all this work the effect of heat loss from the liquid was not investigated. As demonstrated in the next section, such effects are important in the field—particularly at low wind velocities. We move on now to field experiments at relatively low wind velocities.

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Figure 12. Comparison of DNS with rigid slip surface as the interface in channel flow with experiment and calculations based on experimentally determined surface divergence field. Note that the rigid surface gives higher values for the gas transfer velocity. TABLE 2 Comparison of Komori et al.’s (1989) Experimental Data with Eq. (8), with C [ 0.2 Run

δ(cm)

Uav(cm/s) u* cm/s

experimental

β x10 5 (m / s)

Eq. 8) prediction 5 ( β x10 (m / s)

I II III IV V VI VII VIII IX

1.1 2.9 3.1 5.0 5.1 6.4 7.0 10.0 11.2

23.5 9.7 18.3 5.9 11.9 19.9 13.8 10.5 10.9

1.65 0.75 1.60 0.45 0.90 1.2 1.3 0.7 0.8

1.97 0.81 1.22 0.49 0.78 1.01 0.8 0.56 0.55

No.

1.48 0.61 1.06 0.37 0.69 1.01 0.75 0.58 0.59

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3.2

Field Studies

Field studies using tracers to determine gas transfer coefficients have shown considerable scatter at low wind speeds (see summary in MacIntyre et al. (1995); Cole and Caraco (1998); Crucius and Wanninkhof (2003). While shear will make a contribution to turbulence at low wind speeds, the contribution from heat loss and rain is significant (Shay and Gregg 1986; Anis et al. 1994; MacIntyre et al. 1995; Ho et al.1997). Eddy sizes and rates of dissipation of turbulent kinetic energy ε in the upper water column including the air-water interface can be determined with temperature-gradient microstructure profiling. Such profilers ascend through the water column in free fall mode sampling temperature, temperature gradients, conductivity, fluorescence, and photosynthetically available radiation at 100 Hz. With an ascent rate of 0.1 m s-1, mm scale resolution is obtained. While estimates of ε have been obtained in numerous oceanographic investigations, the ship’s wake confounds the values in the upper 10 m. Data obtained during deployments from small boats while the profiler is rising illustrate the turbulence at the air-water interface. Examples of such profiles in lakes include MacIntyre (1993, 1997, 1998); Robarts et al. (1998), MacIntyre et al. (1999), MacIntyre and Jellison 2001). Typically during low winds and when incoming radiation exceeds heat losses, the upper water column is stably stratified to the air-water interface or has small eddies (1 mm - 20 cm scale) at the air-water interface. When heat loss exceeds incoming radiation, eddies can be much larger, typically meter scale, and, depending on depth of the water body, can extend to the bottom or at least through the weakly stratified upper mixed layer. In those cases, temperature profiles have a characteristic shape with low temperatures near the air-water interface, slightly higher temperatures just below and then a decrease of temperatures (Anis and Moum 1994)). Rates of dissipation of turbulent kinetic energy below the air-water interface during periods of heat loss depend on buoyancy flux (Shay and Gregg 1986; Anis and Moum 1994). Profiles illustrating dissipation rates at low speeds are included in (MacIntyre 1993, 1996, 1998). During times when wind speeds were less than 3 m s-1, values ranged from detection (10-10 m2 s-3) up to 10-7 m2 s-3.

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At Mono Lake, CA, when winds were low but sometimes reached 4 m s1 , dissipation rates occasionally exceeded 10-6 m2 s-3 (MacIntyre and Jellison 2001). A recent study in the Soppensee, a small (25 ha) sheltered lake in Switzerland, employed thermistor chains, a microstructure profiler, measurements of surface meteorology, and eddy covariance techniques to determine key processes driving gas flux (Eugster et al. 2003). Wind speeds were always less than 3 m s-1. During times when heat inputs exceeded heat losses, the variance in temperature gradient in 15 minute bins, a proxy for turbulent mixing, at 4.4 m depth was negligible. In contrast, the variance was higher at times when heat losses exceeded heat gains. Similarly, bin averaged temperature gradients between 4.5 and 7 m depth increased at times of heat loss. At these times, gas flux was elevated 30% over stratified periods. Eddy covariance measurements at Toolik Lake, AK, showed gas fluxes were three times higher during periods of heat loss than during stratified periods (MacIntyre et al. 2001; Eugster et al 2003). Schladow et al. (2002) describe the formation of thermal plumes and show the dependence of the gas transfer coefficient for oxygen on heat loss. These data clearly illustrate the importance of convective motions due to heat loss for gas exchange at low wind speeds and indicate that use of a surface renewal model to calculate gas transfer coefficients would more accurately represent the turbulence at the air-water interface than a model based on wind speed alone. 4. Sheared Air-Water Interface In Section 3, turbulence structure and scalar exchange were discussed for situations in which turbulence was generated by motions in the water column in the absence of significant wind shear. Field studies indicated the importance of heat loss from the water as a mechanism for enhancing turbulent mixing and hence gas transfer at the surface. The structures in the interfacial region therefore arose from interactions with far-field turbulence and thermal plumes generated due to cooling of the surface layer. Such mixed convective flows near the air-water interface have not, however, been studied in the laboratory or in simulations. In this

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section we turn to phenomena that occur when wind shear dominates in the generation of near-surface turbulence. 4.1 Laboratory Studies and Simulations A useful starting point is to consider a simple flow situation as shown in Figure 13, which is a schematic of pressure-driven air and water streams between two horizontal plates. Flat shear interface First, let us consider the simplest situation where the air flow exerts shear on the liquid but not enough to form significant interfacial waves. Now consider what happens to the liquid near the bottom boundary, as this will serve as a reference. If the flow is visualized with lines of micro-bubble tracers which are generated by passing a pulsed current through a spanwise line, the streaky structures shown in Figure 14 are seen to form close to the bottom boundary. These structures have been known since the pioneering experiments of Kline et al. (1978), who also observed that the low-speed regions would periodically be ejected in bursts, as shown in Figure 15, which is a side view of the flow with the microbubble-generating wire placed vertically. It is known that these bursts or ejections, and the related sweeps that replace the ejected fluid, strongly influence heat and mass transfer at a wall. Bursts in boundary layers appear to be large-scale motions that contain within them several ejections away from the boundary with interspersed sweeps that bring fluid toward the boundary. A question that arises is whether such structures also occur near gasliquid interfaces on the liquid side. They can be expected on the gas side, since the liquid surface, to a flowing gas, looks much like a wall. However, to the liquid, the interface is almost a slip surface, with a mean shear being impressed due to the gas flow. The boundary conditions are very different from those at solid walls, and it is not clear what is to be expected.

30

Figure 13. Schematic of a co-current or countercurrent gas-liquid flow between two horizontal plates. This configuration defines a canonical experiment for investigations of turbulence and scalar exchange at gas-liquid interfaces. The wires are used in experiments to electrochemically generate microbubble lines that act as flow tracers—a device widely used to visualize turbulent flow structures.

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Figure 14. Visualization of turbulence structure near the bottom boundary of turbulent channel flow. The flow tracers are microbubbles electrochemically generated by pulsing current through a spanwise (horizontal) wire lying a small distance above the channel floor. The flow structure consists of high-speed regions with alternating low-speed, “streaky”, regions where the microbubbles accumulate. These regions form and reform and move around but their essential streaky character is always there.

Figure 15. Side view of a channel flow, showing development of a burst which consists of an “active” period in which the low speed region shown in Figure 14 appear to be lifted up in “ejections” followed by downdrafts of fluid called “sweeps” that bring high speed bulk fluid to the wall region. Flow is from right to left and the panels should be viewed starting on the right.

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To clarify this, Rashidi and Banerjee (1990) carried out experiments in which they visualized the flow structure in a flowing stream of liquid, with shear imposed by the air flow. Figure 16A shows structures close to the gas-liquid interface, on the liquid side, as the shear rate due to the gas motion is increased. One goes from a patchy structure at low shear to low-speed, high-speed streaky structures that are qualitatively similar to those observed near a wall. Furthermore, if the flow is viewed from the side, one sees burst-like structures emanating from the gas-liquid interface, much like those seen in wall turbulence (see Figure 16B). This led Rashidi and Banerjee (1990) to conclude that shear rate was the primary determinant of structure in such situations, and the details of the boundary condition were secondary. There are, as discussed later, important quantitative differences in the interface flow field on the liquid side from wall turbulence, though the qualitative features are similar. The criterion for transition from the patchy structures seen at low shear to the streaky structures was clarified through direct numerical simulations by Lam and Banerjee 1992). They showed that the criterion for formation of streaky regions near a boundary depends on a mean shear rate (non dimensionalized in such a way that it is effectively a turbulence production to dissipation ratio) being greater than one.

Fig. 16A. Structures at a gas-liquid interface with decreasing wind shear (a) to (d).

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Figure 16B. Side view of a channel flow with the floor at the bottom and the interface at the top of each panel. The top series of panels are for air flow countercurrent to the liquid. Qualitatively similar burst-like structures are seen at the interface, as well as at the channel floor. The bottom series of panels indicates bursts formed in cocurrent gasliquid flow. Bursts are formed at the interface due to wind shear. Flow is from right to left, and then increases in the panels from right to left.

Lombardi et al (1996) did a direct numerical simulation where the gas and liquid was coupled through continuity of velocity and stress boundary conditions at the interface. They artificially raised the surface tension to maintain a flat interface, as waves would introduce additional complexity. The interfacial plane itself shows regions of high shear stress and low shear stress, with low shear stress regions corresponding to the low-speed regions and the high shear stress to the high-speed regions. The low shear stress regions were streaky in nature with high shear stress islands. Lombardi et al. showed that the high shear stress regions occur below the sweeps on the gas side, i.e., the motions that bring high-speed fluid from the outer regions to the interfaces on the gas side. Conversely, ejections on the gas side, which take low-speed fluid away from the interface into the outer flow, strongly correlate with low shear stress regions. The liquid does not behave in this way and the ejections and sweeps do not correlate with shear stress on the interface. The difference between the gas and the liquid phases in the nearinterface region is further clarified by observing the velocity fluctuations

34

on each side of the interface as shown in Figure 17. The left panel of the figure is for the gas, whereas the right panel is for the liquid. Gas-side turbulence, as is evident, behaves much like flow over a solid wall, i.e., the fluctuations are almost identical to that at a solid boundary in all directions – streamwise, spanwise, and wall-normal. If the distance is measured from the wavy interface, then there is little effect of waves on intensity—provided that everything is nondimensionalized with the friction velocity calculated by subtracting the form drag from the total drag. On the other hand, the liquid, as evident from the bottom figure, has the largest fluctuations in the streamwise and spanwise direction right at the interface itself. It sees the interface virtually as a free slip boundary, except for the mean shear. There is a somewhat greater effect of waves, but still rather small in the nondimensionalized form shown. These calculations are done with density for the gas and the liquid typical of air and water, i.e., ~ 1:1000. The cases with waves are at conditions where there is no microbreaking.

Figure 17. Turbulence intensities on the gas side (left panel) and liquid side (right panel) as a function of nondimensional distance z+ from the interface. velocity U1-streamwise, U2-spanwise, U3-interface-normal. Case D0-flat interface; D1-wavy interface. All quantities nondimensionalized by the friction velocity, i.e., the shear velocity calculated on the basis of frictional drag(excluding form drag) at the interface.

Certain other aspects of turbulence on each side of a gas-liquid interface are worth considering. In the wall region, it is known that ejections and sweeps have a spacing in time that lies between a nondimensional time of 30-100, with 100 being the upper limit. This last

35

period corresponds to the time between assemblages of ejections and sweeps, i.e., bursts. This is shown in Figure 18 for wall turbulence (i.e., at a solid boundary) as a function of the shear Reynolds number. Rashidi and Banerjee (1990) found that on the liquid side of a sheared interface, this type of parameterization held. In Figure 18 the time between interfacial bursts is also plotted as a function of the shear Reynolds number, and the behavior at a macroscopic level is very similar to that at a solid boundary. Thus, the qualitative behavior is similar, though within the bursts themselves there are substantial differences between the gas and liquid sides.

Figure 18. Scaling of interfacial and wall bursts. The time between bursts TΒ* is scaled with the local shear stress and kinematic viscosity.

Turbulence near nonbreaking wavy surfaces The previous discussion focused on turbulence phenomena for coupled gas-liquid flows with flat interfaces—a situation close to the experiments of Rashidi and Banerjee (1990). To understand the effect of waves on the gas side, flow over a surface with two dimensional waves

36

(with spanwise crests and troughs) was investigated by direct numerical simulation (De Angelis et al. (1997)). The waves introduced a streamwise length scale, as indicated by the vortex structure shown in Figure 19.

Figure 19. Quasi-streamwise vortical structures (shaded) over a wavy boundary with flow viewed from the side. The different shadings indicate different directions of rotation.

However, when the phase average velocities are removed from the instantaneous field, then, typically, streaky regions of high speed-lowspeed flow are seen, as indicated in Figure 20. The streak spacing is ~ 100 in units nondimensionalized with the frictional drag (removing the form drag component from the total drag). This suggests that turbulence phenomena near the wavy surface may scale with frictional drag, rather than total drag. On examining the time between sweeps and bursts, a + time non-dimensionalized with the frictional drag, i.e. t = tDf / µ , is seen to be between 30 and 90, in agreement with what happens at a flat wall. Here Df is the frictional stress.

37

Figure 20. Streaky structures over a solid wavy boundary with different wave lengths. The top left-hand and right-hand panels have the mean velocity removed and clearly show the streakiness of the high and low velocity regions very near the surface. The second row shows the velocities with the mean velocity included, and the third row shows the local shear stresses.

These simulations are of interest for gas-side turbulence phenomena. However, liquid-side behavior may be expected to be substantially different, as discussed for the flat-interface case. De Angelis and Banerjee (1999) have reported some details of DNS with a nonbreaking deformable interface between turbulent air and water streams. Further details are available in De Angelis (1998). The turbulence intensities and other qualitative features, e.g. streak spacing and burst frequency, on both gas and liquid sides of the interface, were found to scale with u*frict and ν, the kinematic viscosity. Figure 21 shows an instantaneous snapshot of the interface configuration, together with the highspeed/low-speed streaky structures close to the interface on each side. It should be emphasized that these results only hold for situations where the interface does not break.

38

Figure 21 I.nstantaneous configuration of the gas-liquid interface showing spanwise wave crests. The colors indicate velocity just above the interface (top panel). Red indicates high velocity, blue indicates low velocity, with yellow and green in between. The typical streaky structures seen in experiments are evident in these simulations as well. The bottom panel is just below the interface at the same instant and shows some differences in the local velocities.

Turbulence in microbreaking When U10 exceeds 3 to 5 m/s, relatively short waves steepen and start to microbreak. This is illustrated in Figure 22, which shows a microbreaking wave viewed from the side. If microbreaking waves are viewed from the top, they are easily visualized by following the trajectories of particles scattered on the water surface. It is found that the liquid does not cross the crest of microbreaking waves, which consequently forms regions of convergence where particles gather. Therefore, an operational way of identifying microbreaking is to observe particles gathering at, and being convected with, the wave crests (see Figure 23). Non-breaking waves do not show this behavior—particles on the liquid surface pass through wave crests.

39

Figure 22. A side view of a microbreaking wave showing typical dimensions.

Figure 23. Top view of a microbreaking wave, visualized by particles scattered in the water surface. The particles are swept along by the wave crests, which are clearly discernible in the top panel. The velocity vectors calculated for particle motion (by DPIV) are shown in the bottom panel. Note the convergence zone at the crest and the divergence zone behind.

40

Regions of high surface divergence form behind microbreaking waves, as illustrated in Figure 23. This may be expected to significantly impact scalar exchange, as indicated in the “surface divergence” equation (6a). The subsurface motions associated with the convergence zones at the wave crests and the divergence zones behind remain to be clarified. At this point, we have seen plunging motions associated with microbreaking that appear much larger than those associated with bursts that develop due to wind shear in the absence of microbreaking (Leifer et al., 2003). This is illustrated in Figure 24, which shows that the plunging motions associated with microbreaking are almost an order of magnitude larger than motions associated with the “usual” bursts. The impact of microbreaking structures on scalar exchange has yet to be studied. 4.2

Laboratory Studies and Simulations — Scalar Exchange

Simulation As simulations proved useful in clarifying aspects of turbulence structure near wavy (nonbreaking) air-water interfaces, they have been extended to studies of scalar exchange (De Angelis, 1998). The calculations for Pr or Sc ~ 0(1) are straightforward once the velocity field has been calculated. However, for higher Sc, the spacing of collocation points normal to the interface must be rescaled, i.e., interface-normal mesh spacing must be reduced as a function of Sc to resolve concentration fluctuations (see De Angelis,1998 and De Angelis et al. , 2000). Contours of the instantaneous scalar fluxes at interfaces from such DNS are shown in Figures 25 and 26, and compared with the shear stress at the interface for the gas and the liquid side, respectively. It is immediately evident that the gas-side fluxes correlate well with the shear stress. This suggests that sweeps give rise to higher scalar exchange rates, as they also produce regions of high shear stress. On the other hand, the flux field on the liquid side shows a much finer structure

41

Figure 24. Bursts formed by microbreaking shown in bottom 6 panels G-L, and the usual bursts in top 6 panels A-F. Scales are the same. Microbreaking leads to “giant” bursts. The vertical dimensions in the pictures are about 8 cm, and the liquid layer is 10 cm deep (the bottom is not shown).

42

Figure 25a. The shear stress field on the gas-side of the interface. The shear is nondimensionalized by ρu*2, where u* is the average friction velocity. The streamwise dimension is 4πh and the spanwise dimension is 2πh, where the gas-layer thickness is 2h.

Figure 25b. The mass flux field on the gas side of the interface. Note close correlation of regions of high mass transfer with high shear stress.

Figure 26a. The non-dimensional shear stress field on the liquid side of the interface. The time at which the field is shown is not exactly the same as in Fig. 25a.

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Figure 26b. The mass flux field on the liquid side of the interface. Notice the lack of correlation with the shear stress field. The mass flux field is much more fine grained. The time at which the field is shown is not exactly the same as in fig. 25b.

(see Figure 26) and no such correlation exists. It is of interest, therefore, to understand what processes control the liquid-side fluxes. To clarify what happens on the liquid side, recall the earlier discussion that liquid-side bursts (sweeps and ejections) do not affect interfacial shear stress, which is mainly determined by gas-phase motions. Therefore, we must directly examine the relationship between liquid-side motions and liquid-side scalar transfer rates. The procedure is illustrated in Figure 27. Here we plot the instantaneous interface-normal velocity fluctuations, together with the instantaneous ' ' ' Reynolds stress ( u1 u3 ), and the instantaneous scalar flux ( u1 is the ' velocity fluctuation in the streamwise direction and u3 is in the interface normal direction). It is clear from the figure that the scalar flux increases ' sharply when there is a high velocity ( u3 ) towards the interface of high ' ' velocity bulk fluid (high u1 u3 ). This is a sweep. The figure illustrates that the sweep, which leads to a high surface divergence event, is strongly correlated with high liquid-side scalar flux, which decays like what might be expected of transient absorption into a batch of stagnant fluid between sweeps. This suggests a basis for the “surface renewal” model with sweeps providing the renewal events—as developed previously to parameterize the liquid-side scalar flux in equation (10).

44

' '

Figure 27. Panels showing that a sweep, with high u1 u3 gives rise to a high mass + transfer rate β L on the liquid side. Note that the surface renewal model predicts the + instantaneous and integrated value of β L quite well if the sweep is considered as a “renewing” event.

Scalar flux parameterization The previous discussion suggests a physical basis for the surface renewal theory used to parameterize scalar transfer rates on the liquid side. It is of interest to see how equation (10) compares with data and simulations. The Schmidt number dependencies in Eqs. (10) and (11) are compared with simulation results for two different Schmidt numbers in Figure 28. It is clear that the Schmidt number dependence is correctly predicted at high Schmidt numbers, but there is some deviation at low Schmidt numbers. Also, the numerical value of the RHS of the equations is roughly correct. Equation (10) is also compared with wind-wave tank data for SF6 and CO2 transfer rates from Wanninkhof and Blivens and Ocampo-Torres et al. in Figures 29 and 30, respectively. The agreement is quite good, though the prediction always lies somewhat below the data. This could be due to a small effect related to turbulence from the

45

channel bottom reaching the interface, which enhances the local turbulence. Turning now to the gas side, the gas sees the liquid much like a solid surface, as discussed earlier. So, the surface renewal theory has to be modified somewhat for such applications, as in Equation (11). This also leads to a different dependence on the Schmidt number.

Figure 28. Gas transfer velocities nondimensionalized by the frictional velocities versus Schmidt number from direct numerical simulations with a flat interface. The solid lines are from equations (11) and (10) for the gas and liquid sides, respectively. Left panel-gas side, right panel-liquid side.

Figure 29. Gas transfer velocity for SF6 desorption in the Delft wind-wave tank versus shear velocity (gas-side), from Wanninkhof and Bliven. The open triangles may have been data taken in breaking conditions. The solid lines are from Equation (10) and Equation (10) + 50%.

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Figure 30. Gas transfer velocity, non-dimensionalized by the shear velocity plotted against the shear Reynolds number (based on liquid depth and artificial shear velocity) from Ocampo-Torres, et al.) The lines are equation (10) and equation (10) + 50%.

While this is encouraging, the issue of estimating the friction velocity u remains. This is a difficult estimate to make for field data. Form drag may be expected to increase with wave amplitude and steepness, therefore estimating u* from total drag and using it in the proposed parameterization will significantly overestimate the scalar exchange rates. The effect of microbreaking has to be factored in as it is not clear whether these parameterizations will be applicable in such circumstances. Finally, as discussed in the next section, evaporation, and hence heat loss, from the water increases with wind speed, giving rise to natural convective effects that enhance wind-induced turbulence. Such effects are important in field experiments, which are now discussed. * frict

4.2 Field Studies In field settings, as wind speeds increase that increase shear, evaporation rates also increase, and at least two mechanisms for generating turbulence co-occur. Shear on the water side of the air-water interface is parameterized by the friction velocity u*w, where τ=ρwu*w2=ρaCDU2 and ρa and ρw are density of air and water respectively, CD is the drag coefficient, and U is wind speed. A similar parameter exists for heat loss, the convective velocity scale w*, and is calculated from heat gains and losses into the actively mixing surface layer (Imberger 1985). Only when heat gains exceed heat losses will the

47

surface of the water body be influenced by shear alone. More often, heat loss and shear co-occur at the air-water interface of lakes, wetlands and oceans and contribute to turbulence production. Examples of profiles showing thermal structure and dissipation rates near the air-water interface for winds up to 12 m s-1 are provided in MacIntyre (1993, 1997, 1998), Robarts et al. (1998), MacIntyre et al. (1999), MacIntyre and Jellison (2001). A full surface energy budget was not completed for those studies, but dissipation rates in Mono Lake in 1995 reached 10-5 m2 s-3 when winds were 12 m s-1 and significant white capping occurred. Surface meteorological data obtained from a meteorological station (http://ecosystems.mbl.edu/ARC/data_doc/lanwater/landwater default.htm) on Toolik Lake, Alaska, a 1.5 km2 kettle lake in the foothills of the Brooks Range, illustrate the variability in surface forcing over aquatic ecosystems (Figs. 31-33). Data are from 12 and 14 July 2000 (day of year 194 to 197). Five minute averaged wind speeds were less than 4 m s-1 at night and increased in the afternoon to highs between 6 and 8 m s-1, a range where microbreaking and shear are both important (Fig 31A). Solar insolation, which acts to stratify the lake, was barely detectible for a few hours at midnight and had daily maxima of 800 W m2 (Fig. 31B). Intermittent cloudiness caused large fluctuations in irradiance. A surface energy budget was calculated as in MacIntyre et al. (2002). Surface heat fluxes, which are the sum of heat fluxes from sensible, latent, and long wave radiation, were always negative, ranging from -100 to -400 W m-2 (Fig 31Ca). The effective heat flux (Fig. 31Cb) represents the actual heating of the upper surface layer. The surface layer is defined as the actively mixing layer and operationally as having temperatures within 0.01 oC of those measured within the upper 5 cm of the lake. When the effective heat flux is greater than zero, the uppermost part of the water column will stratify; when it is less than zero, cooling occurs and turbulence can be induced not only by shear or microbreaking waves, but also by heat loss. Heat losses occur in the surface layer for nearly 12 hours of each 24-hour period and for an even longer period on cloudy days. Wind consistently induces some shear at the air-water interface; the shear velocity u*varied from 0.002 to 0.01 m s-1 (Fig. 32A). The convective velocity scale w* is more intermittent due to solar heating.

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Values of w* are comparable to the friction velocity when winds are high and solar radiation low (e.g. day 194.8) and are zero when heat inputs exceed losses. At night, when wind speeds are low, w* is often twice as high as u*, indicating heat losses will be the major source of turbulence at the air-water interface. Surface layer depths were calculated using data from Brancker TR1050 thermistors on a taut line mooring with loggers located at 0.05, 2.55, 3.3, 3.55, 3.8, 4.05, 4.3, 4.55, 4.8, 5.05, 5.3, 5.55, 5.8, 6.55, 7.55, and 8.55 m below the air-water interface (Fig. 32B). Due to insufficient thermistors in the upper water column, the surface layer was generally assumed to be 2 m deep during times of heating. The depth of the surface layer varied due to cooling and, during periods of high wind (e.g. Day 195.7), is confounded due to tilting of the thermocline. At these times, the surface layer deepens downwind; shallows upwind. The flux of energy into the mixed layer (Fig. 32C) was calculated following Imberger (1985) as Fq = q*3/2 = (w*3 + 1.333u*3)/2, an approach that does not explicitly include the effects of microwave breaking. q represents the turbulent velocity scale due to both shear and heat loss.., The rate of dissipation of turbulent kinetic energy (ε) for the surface layer is estimated by dividing Fq by the depth of the actively mixing surface layer. Based on this approach, ε ranged from 10-10 to 10-7 m2 s-3 (Fig. 33A). Dissipation rates are underestimates during heating periods and when the thermocline is strongly tilted due to overestimates of the depth of the surface layer. Comparison of the flux of energy into the surface layer when winds dropped to 1 m s-1 (u*w = 0.002 m s-1) but the lake was either cooling (Day 196.1, w* = 0.006 m s-1) or gaining heat (Day 196.35, w* = 0) illustrates the contribution of heat loss to turbulence in the surface layer when shear is low. When cooling occurred (Day 196.35), Fq for the surface layer was 10-7 m3 s-3 and ε was 10-8 m2 s-3, whereas when the surface layer was gaining heat (Day 196.35), Fq was 10-9 m3 s3 . ε was less than 10-9 m2 s-3. Generally, winds drop at night when cooling occurs, so the heat loss leads to continued production of turbulence and often to entrainment of deeper waters into the upper mixed layer. Temperature-gradient microstructure profiles illustrate the thermal structure, rates of dissipation of turbulent kinetic energy, and eddy sizes during periods with 7 m s-1 winds and illustrate the thermal structure,

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Wind speed (m s−1)

10 8 6 4 2 0 194

A.

194.5

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Short wave In (W/m−2)

1000 800

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600 400 200 0 194

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400 b.

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0 −200 −400 194

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196.5

197

Day of Year 2000 Day of Year 2000

Figure 31. A) Five minute averaged wind speeds (m s-1), B) solar insolation (W m-2) and C) surface heat flux (a.) and effective heat flux (b.) (W m-2) into Toolik Lake from July 12-15 (day of year 194 – 197) 2000 (unpublished data, Sally MacIntyre and George Kling).

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u* and w* (m s−1)

0.015 u * w*

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0.005

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Day of Year 2000

Figure 32 A) Friction velocity in water u*and convective velocity w*, B) depth of the surface layer, and C) flux of energy into the surface layer for days of year 194-197, Toolik Lake, AK. (unpublished data, Sally MacIntyre and George Kling).

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−5

10

−6

epsilon m2 s−3

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Gas Transfer Velocity (cm hr −1)

25

20

B.

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5

0 194

Figure 33 A) Estimates of rates of dissipation of turbulent kinetic energy (ε) based on surface energy budgets and depth of the surface layer derived from thermistor chains. Arrows indicate revised estimates of ε when the depth of the surface layer was determined from high resolution temperature profiles in Figures 35, 36. B) Gas transfer coefficient calculated from surface divergence model (equation 8) (gray dots), and two empirical wind based models (Cole and Caraco 1998, gray, and Kling et al. 1992, black). Data for these models were derived from the meteorological data from Toolik Lake (unpublished data Sally MacIntyre and George Kling); calculations are described in the text. Gas transfer velocity is augmented over wind based models when heat losses are significant and reduced when surface waters gain heat (see Fig. 34A).

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Temperature (° C) 11.75 11.77 11.79 11.81 11.83 11.85 11.87 11.89 11.91 11.93 11.95 0

1

Depth (m)

2

3

4

5

6 −10

10

−9

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−8

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−7

ε (m2s−3)10

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−5

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Figure 34. Temperature profile and rates of dissipation of turbulent kinetic energy at 2014 h on 12 July 2000, Toolik Lake, AK. show the structure within the 6.2 m deep upper mixed layer at times when the mixed layer is losing heat and winds reached 7 m s-1. ε is higher near the air-water interface and typically between 10-8 m2 s-3 and 10-7 m2 s-3 in the rest of the surface layer. Dissipation rates calculated as in MacIntyre et al. (1999).

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Temperature (° C) 11.7 0

11.74 11.78 11.82 11.86

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11.94 11.98 12.02 12.06

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Figure 35. As in Figure 36 but at 2043 h on 12 July 2000. Temperature (° C) 11 0

11.14 11.28 11.42 11.56

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11.84 11.98 12.12 12.26

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Figure 36. Temperature profile and rates of dissipation of turbulent kinetic energy at 1618 h on 14 July 2000 show the structure within the 4.7 m deep upper mixed layer. ε is higher in the upper 0.8 m; shear induced eddies occur below.

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rates of dissipation of turbulent kinetic energy, and eddy sizes during periods with 7 m s-1 winds and considerable convective cooling (Fig. 34, 35) and during a period when winds were 5 m s-1 but heating was occurring (Fig. 36). The temperature profile at 2014 h 12 July 2000 showed slightly cooler temperatures near the surface, a 0.02oC increase in temperature by 4.5 m, and a gradual decrease in temperature to the base of the mixed layer at ca. 6.1 m (Fig. 34). Such a pattern is typical of conditions when convection due to heat loss occurs (Shay and Gregg 1986; Anis and Moum 1994). Temperature fluctuations of 0.01oC and smaller were ubiquitous. Below the mixed layer, temperatures decreased rapidly. Eddy sizes are computed by calculating the displacement scale, the depth to which cooler water would descend to find water of similar density. In this case, the upper 50 cm may be considered an eddy, and dissipation rates are slightly higher within it (10-7 m2 s-3) than in the water immediately below. A second eddy occurred between 0.5 and 3 m depth, and dissipation rates within it were ca. 3 x 10-8 m2 s-3. A third eddy extended from 3 to 5 m and had similar dissipation rates. Despite this structure, the near uniformity in ε and the overall mixed layer. A profile taken 15 minutes earlier had a similar temperature profile and average dissipation rate, but no accentuation of dissipation in near surface waters (not shown). While the profile at 2043 h (Fig. 35) was also obtained during a cooling period, the 6 m deep upper mixed layer does not have the classic structure of a convecting layer, but instead consists of 3 layers. In the uppermost one, temperature ranged from 11.97 to 12.01 oC and dissipation rates were 10-6 m2 s-3 at the surface to and decreased to 3 x 108 m2 s-3 at its base at 1.5 m. The higher values at the surface may have been due to the combination of shear, heat loss, and microbreaking. While the middle layer may have been an overturning eddy, the step changes associated with the top and bottom of the layers suggests lateral advection was occurring; the enhanced dissipation rates at the top and bottom of the lower layer support this interpretation. Dissipation rates estimated from the surface energy budget for the entire 6 m upper mixed layer were 10-7 m2 s-3 (Fig. 33), which is in reasonable agreement with the measured values. However, recalculating ε from the surface energy budget based on a surface layer depth of 0.80 m gives dissipation rates of

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10-6 m2 s-3 which is representative of conditions at the surface (Fig. 33A). Four microstructure casts were obtained in the period from 1978 h to 2051 h 12 July, and the measured dissipation rates in surface waters ranged from 10-8 m2 s-3 to 10-6 m2 s-3. This range reflects the dynamic nature of the processes at the air-water interface. The microstructure cast taken at 1618 h on 14 July was obtained while the 4.7 m upper mixed layer was gaining heat (Fig. 36). Surface temperatures had increased by a degree since noon; winds were 4 m s-1. While turbulence occurred throughout the 4.7 m upper mixed layer, dissipation rates were 10-6 m2 s-3 in the uppermost 0.8 m. Eddies below it ranged in size from a few centimeters to 50 cm indicating the upper mixed layer did not mix as an entity. Dissipation rates were at least an order of magnitude lower than in the surface layer. Eddies embedded in stably stratified waters indicated shear induced turbulence. Estimates of dissipation from the surface energy budget (Fig. 33A) using a surface mixing layer of 0.25 m, based on the step change in temperature at that depth, were slightly less than the value of 10-6 m2 s-3 obtained while profiling. In summary, the distribution of turbulence in the upper mixed layer differs during times with and without cooling. In nearly all cases, turbulence is highest in a shallow layer near the surface. When heating occurs and winds are moderate, the turbulence is more accentuated in surface waters and decreases considerably by the base of the mixed layer. When cooling co-occurs with wind forcing, the turbulence in the upper mixed layer tends to be more homogeneous and eddies are larger. However, even that pattern is variable, possibly due to horizontal advection of different water masses. Ho et al. (2000) have calculated the flux of kinetic energy into the mixed layer due to rainfall ranging from 13.6 to 115.2 mm h-1. The fluxes ranged from 0.09 to 1.27 W m-2. Assuming that the energy was initially introduced to a depth of 0.2 m, as supported by measurements of temperature change and SF6 distribution in experiments with rainfall of 68 and 76 mm h-1 (David Ho, personal communication), and converting to W kg-1 (equivalent to m2 s-3) by assuming a density of 1000 kg m-3, gives dissipation estimates on the order of 5x10-4 m2 s-3. These values are an order of magnitude higher than has been observed with microstructure

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profiling at low to moderate winds and indicates that intense rainstorms could dominate the surface energy budget under those conditions. Dissipation rates under breaking waves in Lake Ontario ranged from -5 10 to 10-2 m2 s-3 (Agrawal et al. 1992; Terray et al. 1996). That ε did not follow law of the wall scaling in the near surface layer was attributed to the more intense turbulence during white capping and wave breaking. 5.0 Calculation of Gas Transfer Coefficients during low and moderate wind speedsMost attempts to relate gas transfer velocities to surface meteorology have used wind as the independent parameter (Liss and Merlivat 1986; Kling et al. 1992; Cole and Caraco 1998; McGillis and Wanninkhof 1999; Crusius and Wanninkhof 2003). However, the surface renewal model takes into account the wide variety of processes occurring at the air-water interface (Crill et al. 1987; Soloviev and Schluessel 1994; MacIntyre et al. 1995; Eugster et al. 2003). As we have seen above, dissipation rates in near-surface waters calculated from surface meteorological data and depth of the surface layer obtained from high resolution profiles are in good agreement, or are less than, those obtained from temperature-gradient microstructure profiles. Estimates might be improved were the contribution of microwave breaking to energy flux into the surface layer included. Gas transfer velocities (k600) were calculated using the small eddy, SE, (Eq. 5, 5a this paper) the surface divergence, SD, (Eq. 8 this paper), and the Soloviev and Schluessel, SS, (1994) versions of the surface renewal model (see Table 1) using five minute averaged meteorological data from Toolik Lake, AK. The 600 implies the values are normalized to CO2 at 20oC. For the small eddy version of the surface renewal model, the gas transfer coefficient k = a1 D1/2 (ε/ν)1/4. D is molecular diffusivity of the gas. ε is the rate of dissipation of turbulent kinetic energy and ν is kinematic viscosity. When written in terms of the Schmidt number and turbulent Reynolds number, Ret = ul/ν, the expression becomes k Sc1/2 = c1 u Ret –1/4. ε is related to u and l through the expression ε = u3/l (Taylor, 1935). ε and l can be obtained from temperature-gradient microstructure profiles in which ε is estimated from a least squares fit of the power

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spectral densities of the temperature-gradient signal to the Batchelor spectrum and l is the overturning scale. Here ε is calculated from surface energy budgets (MacIntyre et al. 2002) in which the energy flux into the surface layer Fq = (w*3 + 1.333u*3)/2 and ε = 0.82 Fq /l and l is the depth of the surface layer (Imberger 1985; MacIntyre et al. 1995). We let c1= 0.56 as in Crill et al. (1987) but recognize that this coefficient was obtained in laboratory experiments and has not been validated in the field. To correct for the somewhat lower estimates of ε from surface energy budgets relative to microstructure profiles due to lack of sufficient self-contained temperature loggers, ε was increased five fold (e.g., Fig. 33A); mixed layer depths during low winds (≤3 m s-1) and heating (w* < 0.003 m s-1) were assumed to be 0.3 m. Following Soloviev and Schluessel (1994), k = 1.85 Ao-1 u*Sc-1/2 (1 + Rfo/Rfcr)1.4(1+Ke/Kecr)-1/2 where Ao, based on cool skin data, is 13.3, Rfo is a surface Richardson number, -αgHυ/ρcpu*4 and Rfcr is a critical Richardson number determined to be 1.5 x 10-4. H is surface heat flux obtained from summing latent and sensible heat fluxes and long wave radiation flux (e.g. Fig. 33C, curve a.). Ke is the Keulegan number, u*3/gυ whose critical value is 0.18. α is the coefficient of thermal expansion, g is gravity, υ is kinematic viscosity, and cp is specific heat. The spread in calculated gas transfer coefficients at each wind speed and for each method, as evidenced by the 9% confidence intervals (Fig. 39, MacIntyre, Shaw and Kling, to be submitted), is due in part to the enhancement of turbulence when heat loss is included and its decrease when stratification decreases heat loss (Fig. 33A). Curves for gas transfer velocities as a function of wind speed alone developed from field studies (Cole and Caraco (CC) 1998; Crusius and Wanninkhof (CW) 2003; Kling et al. (K) 1992) are plotted for comparison. CC and SE give similar estimates at all wind speeds and are low in comparison to the other approaches at moderate winds. CW was forced to zero at low winds but is similar to SS and K at moderate winds. The data on which CC and CW are based only includes winds up to 5 and 6 m s-1, respectively.

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Gas transfer coefficients obtained from experiments (Wanninkhof 1985, Clark 1995, Crusius and Wanninkhof 2003), as well as additional data of Wanninhof in MacIntyre et al. (1995), are plotted with the five curves in Fig. 38. Considerable scatter is evident in the experimental data at both low and moderate wind speeds. At wind speeds below ca. 4 m s-1, the majority of the estimates of k are less than 3 cm hr-1. In a recent laboratory study of gas flux due to heat loss, gas transfer coefficients, normalized to CO2 at 20oC using a Schmidt number of 530 for oxygen, were 0.4 and 0.9 cm h-1 for outgoing heat fluxes of 400 and 570 W m-2 (Schladow et al. 2002). These low values of k would support Crusius and Wanninkhof’s (2003) assumption of minimal flux at wind speeds less than 1 m s-1. However, even at winds between 1 and 3 m s-1, some of the data from Lake 302N and that from Sutherland Pond have values of k that exceed those predicted from CW and are within the range of estimates from SS. Field data of Cole and Caraco (not shown) indicate k ranges between 1 and 4 cm hr-1 at winds between 1 and 2 m s1 . The low values in Schladow et al. (2000) were obtained with an undisturbed surface; hence even the minor disturbances at low winds may contribute to vorticity that enhances gas flux (e.g. MacIntyre, 1984). At winds between 3 and 8 m s-1, the gas transfer coefficients from Lake 302 N, Sutherland Pond, Mono Lake, Pyramid Lake and Crowley Lake are in the range predicted from SS, SD, CW, and K. That the field data, which was not collected over time intervals defined to identify times of convection, microbreaking, or high winds, tends to follow these four models at moderate winds suggests these models better include the greater diversity of processes contributing to gas flux than do SE or CC. Clark’s (1995) results from Sutherland Pond further indicate that estimates of gas transfer coefficients based on the typical averaging schemes in tracer studies may be too low. During two of his sampling periods, Clark’s estimates of k were obtained during steady winds, 3.2 and 3.4 m s-1. These k values are better predicted by SS, SD, CW, and K than by CC or SE. Experimental results from a wide variety of lakes are well predicted by two approaches using surface renewal, SS and SD, and results with these models are similar in the mean to those obtained following Kling et al. (1992), who primarily used other data to develop their regression. To

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illustrate the importance of including a surface renewal approach for estimating gas flux, k is calculated over 3 diurnal cycles using the meteorological and time series temperature data from Toolik Lake (Fig. 33B). K and SD provide comparable estimates of k during heating periods, but K underestimates k when heat loss occurs. For this data set, CC underestimates k by ca. a factor of 2 under all conditions. These comparisons indicate the potential greater accuracy of models for predicting k that include the various mechanisms that induce turbulence at the air-water interface.

Gas Transfer Coefficient (cm hr−1)

25

20

K CW CC SD SS SE

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Figure 37. Estimates of gas transfer velocity obtained from 5 minute averaged meteorological data from Toolik Lake, Alaska, using three different surface renewal models: small eddy (SE, eqn 5a this chapter calculated as in Crill 1987), surface divergence (SD, Eq. 8 this chapter) and following Soloviev and Schluessel (SS, 1994). 95% confidence intervals are indicated. Gas transfer coefficients calculated from wind based expressions (Cole and Caraco 1998, CC, Crusius and Wanninkhof 2003, CW, Kling et al 1992, K) are shown for comparison. Curves illustrate two to three fold variations in calculations of flux are possible dependent upon model selection. The range in values for each wind speed using surface renewal is due to diurnal heating and cooling of surface waters.

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Gas Transfer Coefficient (cm hr−1)

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Mono Lake Crowley Lake Pyramid Lake Rockland Lake Sutherland Pond Lake 302N

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Wind Speed (m s−1)

Figure 38. Data from tracer experiments (see text for description) with curves of Figure 37 for comparison show that gas transfer coefficients at moderate wind speeds are well predicted by Soloviev and Schluessel (1994) which explicitly includes waves. Similarities of mean values for each wind speed of SS and SD with K and CW and field data support these surface renewal models and the need for experimental work to define gas fluxes as a function of all processes inducing turbulence at the air-water interface.

Frequently, gas flux estimates from lakes are used in regional carbon budgets are based on k values of 1 to 3 cm hr-1 (Kling et al. 1991; Cole et al. 1994; Richey et al. 2002). While many of the sites have low wind speeds, estimates are likely to be too low by a factor of at least 2. In tropical waters, k estimated by SE was two times higher than CC (MacIntyre et al. 2001). If the trends observed here apply to the tropical data, gas flux estimates from those regions may be underestimated by a factor of four. Ho et al. (2000) show gas transfer coefficients ranging from 16 to 70 cm h-1 for rainfall ranging from 13 to 115 mm h-1. Rainfall of such intensity is most likely in tropical environments and would only occur intermittently, but gas fluxes would be comparable to those anticipated when winds are high enough to induce considerable wave breaking. Clearly, consideration of the wide diversity of processes that lead to enhanced gas fluxes will have considerable impact on estimates of global carbon budgets. These approaches will also provide better

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estimates of k for biological studies estimating net community productivity (e.g. Cole et al. 2002). These results strongly indicate that further work evaluating the surface renewal model in field situations is warranted, with special attention to heat loss, rain, and microbreaking. 5. Conclusions Turbulence phenomena and scalar exchange at the air-water interface have been discussed in terms of the models used for prediction, numerical simulations, laboratory experiments and field observations. The focus of the discussion has centered on evasion of sparingly soluble gases from water bodies. These processes are controlled by liquid-side turbulence, and the main resistance to transfer lies in a very thin region at the interface. As it is difficult to make measurements and do computations of phenomena in the vicinity of a deforming, and perhaps breaking, interface, our understanding is still at a much earlier stage than for such processes at solid boundaries. Nonetheless, considerable progress has been made in understanding the mechanisms that control scalar exchange, when the interface remains continuous (does not break). In the absence of shear, and when the far-field turbulence approximates the homogeneous isotropic case, it has been found that the Hunt-Graham (1978) blocking theory applies quite well, and predictions of the near-interface damping of the normal component of turbulence, and enhancement of the tangential components, are well predicted. Banerjee (1990) has applied the Hunt-Graham theory to calculate the surface divergence and developed the so-called surface divergence (SD) model, which predicts gas transfer across unsheared interfaces rather well. It has also been shown that this model predicts at low turbulent Reynolds number the same behavior as the Fortescue and Pearson (1967) large-eddy (LE) model, and the Banerjee et al. (1968) small-eddy (SE) model at high turbulent Reynolds numbers. In situations in which the shear rate imposed by the wind is high, turbulence is generated in the vicinity of the interface, much like near solid boundaries. For situations in which the air-water interface does not break, models for gas transfer based on scaling of active periods (such as sweeps and ejections) with interfacial frictional shear, have proved to be

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successful in predicting laboratory data. However, microbreaking (breaking of waves of amplitude ~1 cm. and length ~10 cm.) starts to occur at U10 of about 3-5 meters/sec, and very little is currently understood about the effect of these phenomena on turbulence generation and scalar exchange. Laboratory experiments indicate strong regions of convergence in the surface fluid motion near wave crests and regions of divergence behind, suggesting that such waves would markedly increase the r.m.s. surface divergence and hence gas transfer. Clearly, considerable work still needs to be done to characterize such microbreakers, both in lab experiments and in the field, and to better understand their effects on turbulence and scalar exchange. Field observations also suggest that natural convective motions due to cooling of the interfacial layer play a significant role by affecting both turbulence and scalar exchange over a wide range of wind velocities. This suggests that parameterizations based on wind speed alone are inadequate for calculations of mass transfer. The field observations also suggest that surface divergence approaches are quite fruitful, as they can accommodate a variety of processes that may lead to turbulence generation. The SD model is found to predict field data rather well even at high wind seeds. This is surprising because the model was originally developed for a no-shear interface based on Hunt-Graham theory. On the other hand, the SE model significantly underpredicts gas transfer coefficients at moderate wind speeds—possibly because energy dissipation due to microbreaking was underestimated. These discrepancies indicate better estimates of energy dissipation and length scales of the turbulence from the near-surface regions are needed in the presence of microbreakers. These analyses illustrate the utility of extending the surface divergence/surface renewal approaches, which were derived on the basis of theory and laboratory experiments, to field settings where turbulence is induced by a variety of processes. Much work, however, remains to be done. In particular, the effect of natural convective motions observed in the field and microbreaking need investigation. To this end, more extensive and accurate field observations are of the highest priority, particularly in well-characterized conditions.

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