Surface age of surface renewal in turbulent interfacial transport



[1] We use direct numerical simulation to study characteristics of interfacial transfer of gas and heat in free-surface turbulence. Using Lagrangian tracing, we are able to directly quantify surface age of surface renewal for the first time. The physical meaning of surface renewal and various representations of surface age, including the use of heat as a proxy, are discussed. Results show that the Higbie penetration theory and the Danckwerts random surface renewal model are inappropriate for gas transfer.

[2] Surface renewal is a critical process in turbulent interfacial transport that is important to many applications including sea surface temperature and atmosphere–ocean gas transfer. Scalar transport in the upper ocean is governed by the interplay of molecular diffusion at the sea surface and turbulent mixing underneath. When a surface renewal occurs, fluid is brought from the bulk towards the surface, the scalar is highly mixed, and the gradient of scalar concentration is increased at the surface to enhance interfacial diffusion.

[3] An influential model of surface renewal is the penetration theory of Higbie [1935], which stated that the surface is intermittently exposed to turbulent upwelling flows. The turbulent mixing process was considered instant, with the scalar well mixed. After this surface renewal, molecular diffusion was assumed to dominate in the scalar boundary layer till the next surface renewal is generated by the turbulence below. Take gas as an example. Its concentration c is governed by the diffusion equation: ∂c/∂t = D(∂2c/∂z2). Here D is the molecular diffusivity and z is the vertical coordinate. Subject to the boundary conditions at the surface c(z = 0, t) = c0 and in the deep region c(z = −∞, t) = cbulk, and the initial condition c(z, t = 0) = cbulk, the gas flux at the surface qg obtains as

equation image

In the above, the time elapsed since the surface renewal is called the surface age t.

[4] Danckwerts [1951] elaborated the penetration theory in his random surface renewal model by assuming that the chance of a surface element being renewed by fresh fluid from the bulk flow is independent of its surface age. Therefore, the probability density function (pdf) of the surface age governed by dp(t)/dt = −s p(t), where p is the pdf and s is the fractional rate of surface elements being renewed (assumed to be constant by Danckwerts), has an exponential solution: p(t) = s exp(−st). With this pdf, average surface age can be obtained as equation image = 1/s, and average gas flux at the surface is equation imageg = (c0cbulk)equation image.

[5] The pure diffusion solutions of Higbie [1935] and Danckwerts [1951] did not account for effects of vertical turbulent advection after t = 0. Near the surface, the advection is in the form of upwelling and can be measured by surface divergency a = ∂u/∂x + ∂v/∂y = −∂w/∂z (so that w = −az), which was considered by Ledwell [1984] and Banerjee [1990]. The relative effects of diffusion and advection can be seen in the upwelling stagnation flow model of Chan and Scriven [1970], in which the advection-diffusion equation ∂c/∂t = az(∂c/∂z) + D(∂2c/∂z2) is solved to obtain the surface gas flux based on the history of a:

equation image

For the special case of a = constant, the above solution is reduced to qg = (c0cbulk)equation imageequation image. In the limit of t → 0, it becomes the pure diffusion model (equation (1)). When t → ∞, it becomes a surface divergence model: qg = (c0cbulk)equation image.

[6] To study the surface renewal process, it is essential to obtain the surface age information. Many models have been proposed [e.g., Fortescue and Pearson, 1967; Banerjee et al., 1968; Lamont and Scott, 1970]. However, no model is universally applicable due to their simplification of the complex flow structures. To establish a physical basis for statistical and modeling studies, it would be helpful to quantify the surface age of all of the surface elements in question. In this study, we use direct numerical simulation (DNS) to investigate different approaches for surface age quantification.

[7] We perform DNS with a finite difference method of the Navier-Stokes equations for a pressure-driven turbulent open channel flow. At the channel top, there is no shear stress. The surface deformation and wind effect are neglected in this study, corresponding to many laboratory measurement conditions (cf. the analysis by Borue et al. [1995]). No-slip condition is used at the channel bottom. We start the simulation with a prescribed mean profile superposed with random fluctuations. After the turbulence has fully developed, the flow field is in equilibrium and independent of the initial condition. DNS of the advection–diffusion equation is performed for the transport of scalars with a focus on gas and heat, which are treated as passive in this study. Previous analysis [Liss and Slater, 1974; Jahne and Haussecker, 1998] showed that the solubilities and molecular diffusivities of many environmentally important gases are low in water. Therefore, the resistance of interfacial transfer is mainly at the water side and a fixed concentration condition at the surface serves as a good approximation for gas simulation. For heat, on the contrary, its interfacial flux is mainly controlled by air-side processes such as evaporation. Therefore, a prescribed flux condition is used for the study of temperature in aqueous boundary layer. These surface conditions have been shown to capture the essential characteristics of gas and temperature at the water surface in previous numerical studies [Komori et al., 1992; Handler et al., 1999; Nagaosa, 1999; Wang et al., 2005]. At the bottom, fixed concentration condition is used.

[8] In our simulation, the Reynolds number based on the channel height and the mean flow velocity at the surface is 5778. Schmidt numbers, Sc, ranging from 1 to 16 have been simulated. The simulation is performed on 1283 and 2563 grids (depending on Sc) for 280 large-eddy turnover times (defined as half channel depth divided by velocity fluctuation). While the DNS cannot directly address the high Schmidt numbers of gases in nature (because the limitation in the computing power forbids the resolution of small scalar structures down to the Batchelor scale), this is not a concern in the current study because it focuses on the surface renewal process, which is hydrodynamic and is not affected by the passive scalars. Effect of surface renewal on scalar transfer can be accounted for by Schmidt number scaling (e.g., for Sc = 1∼16, the Sc-scaled gas transfer velocity using the surface divergence model of Banerjee [1990] has less than 5% of variation; the normalized accumulative surface gas flux as a function of surface age shown in Figure 3 has less than 1% of variation.). Therefore, the Schmidt number effect is not discussed in this paper. Recently, the difference between the diffusivities of gas and heat has raised questions about the effect of penetration depth [Atmane et al., 2004], which led to renewed interest in the random eddy model of Harriott [1962]. An interesting observation from the current work is that the different free-surface boundary conditions for heat and gas also result in disparate behavior of the two, as will be illustrated next, which may add new insight into the physical problem.

[9] Figure 1 plots instantaneous surface features obtained from our DNS. In this paper, all of the quantities are normalized by the channel depth and the mean flow velocity at the surface. Figure 1a shows a strong upwelling, indicated by large surface divergence values, centered around (x = 1.7, y = 1.3). At the upwelling region, flow diverges in radial directions. Near the upwelling, we notice the existence of downwelling (with negative surface divergence). We pick a fluid particle near the upwelling center, place a small circle around it, and perform Lagrangian tracing. The particle and the circle stay at the free surface according to the kinematic boundary condition. Figure 1b shows that a half time unit later, the particle has moved away from the upwelling. The circle is deformed and much enlarged, indicating the stretch in the surface area (i.e., positive surface divergence in upwelling). Figure 1c shows that another one and half time units later, the particle and the circle enter a downwelling region. The flow converges; the circle shrinks and is greatly deformed. We note that not all fluid particles enter the nearby downwelling immediately; some may travel along the surface for a long time before arriving at a downwelling.

Figure 1.

Contours of surface divergence, a, at the free surface at different times of simulation: (a) t0, (b) t0 + 0.5, and (c) t0 + 2. Vectors represent horizontal velocity fluctuations. Fluctuations are normalized by their rms values to better illustrate the deviations. A fluid particle surrounded by a circle in an upwelling in Figure 1a is traced in time as shown in Figures 1b and 1c. Contours (at t0) of (d) surface gas flux qg, (e) surface temperature T, and (f) surface age t. The region shown is only a small portion of the free surface.

[10] Upwelling brings fluids from the bulk flow toward the surface. As a result, the surface gas flux and temperature are increased as shown in Figures 1d and 1e. Comparison between Figures 1a, 1d, and 1e shows that the gas flux responds to upwellings and downwellings more rapidly than the temperature does, because the former increases exponentially in time as governed by the strain field, while the latter takes a (longer) diffusion time [Handler et al., 1999]. Therefore, the gas flux is more intermittent than the temperature. Particularly, at the downwelling region near the upwelling, the gas flux is reduced immediately, while the change in temperature is slow.

[11] We quantify surface age t for each surface element in DNS. By backward Lagrangian tracing of an element, we obtain its surface age as the time elapsed since the last surface renewal it experienced. An issue with this approach is the definition of the surface renewal origin. Mathematically, if the upwelling is a perfect axisymmetric stagnation flow, the time a fluid particle takes to be traced back to the stagnation point would be infinite. In practice, the structure of upwelling is complex and its location and shape change over time (Figure 1). As a result, it is impractical to trace a fluid particle to the exact birth point, and at certain point of backward tracing we need to switch to an Eulerian method. Therefore, we next examine Eulerian approaches. We use the penetration theory to solve for t based on the instantaneous surface gas flux using equation (1):

equation image

of which the result of surface age pdf is plotted in Figure 2, which is large at small t. Figure 1d suggests that the advection associated with upwelling increases gas flux significantly. Therefore, at small t, the value of surface age calculated from equation (3) based on the assumption of pure diffusion is artificially reduced.

Figure 2.

Pdf of surface age obtained from surface temperature (short-dashed line), penetration theory using surface gas flux (long-dashed line), random surface renewal model (dotted line), hybrid Lagrangian tracing and temperature method (solid line), and lognormal curve fitting of hybrid Lagrangian tracing and temperature method (dash-dotted line).

[12] Another way to obtain a value for surface age is to use surface temperature based on the heat equation, which has the same form as gas, but subject to a Neumann surface boundary condition instead: D(∂T/∂z) = qh. The solution for surface temperature is T0 = Tbulk + 2qhequation image, where Tbulk is temperature in the bulk flow. Therefore, we obtain the surface age of a surface element as

equation image

The above equation has been used in experiments to connect surface age to variation of surface temperature [cf. Garbe et al., 2004]. Figure 2 plots the distribution of surface age obtained from equation (4). It shows that the pdf increases with surface age at small t, reaches a maximum value around t = 4, and then decreases.

[13] Neither of the above two methods account for the effect of vertical advection. It would be intuitive to consider the advection–diffusion solution for the scalars instead of the pure diffusion one. For gas flux, the solution is equation (2), which requires information on the history of surface divergence a(t). Figure 1 shows that qg is highly sensitive to a(t); the surface temperature, on the other hand, does not vary much with a(t). For temperature that is subject to a Neumann boundary condition, although there does not exist a general analytical solution for arbitrary a(t), we obtain a perturbation solution for small t:

equation image

with ξ = z/2equation image. Here the first term on the right hand side is of leading order and is the solution of the pure diffusion problem. The second term corresponds to the advection effect to the next order. At the free surface, T0 = T(ξ = 0) ≈ Tbulk + 2qhequation image(1 − equation image). Recall equation (4); we see that neglecting the advection effect results in a relative error of at/3 in the quantification of surface age at small t. Therefore, when surface divergence information is unavailable so that the pure diffusion solution has to be used to quantify surface age, heat is an acceptable proxy, and the error reduces as t decreases.

[14] We next consider the Lagrangian method. When we follow a surface fluid particle, the same scalar advection–diffusion equation in the vertical direction applies; meanwhile, the (relatively large) horizontal turbulence structures encountered by the fluid particle are measured by the a(t) it experiences. With the DNS data that are recorded at short time intervals (every 0.35% of large-eddy turnover time), we trace surface elements backward in time until they enter an upwelling (indicated by positive surface divergency; we recommend a threshold value of root-mean-square of surface divergency; in this neighborhood the result is not sensitive to the threshold because test over a range of one standard deviation of the surface divergence provides average surface age with variation less than 6%). The time elapsed during the Lagrangian tracing is denoted as tL. Because of the aforementioned difficulties of tracing all the way to the upwelling origin, we next switch to use the surface temperature to obtain surface age at this time, tT (the proceeding analysis shows that it is appropriate to use heat to quantify surface age at small t). Finally, the surface age is defined as t = tL + tT. Instantaneous distribution of surface age obtained from this hybrid Lagrangian tracing and temperature method is shown in Figure 1f. The pdf of surface age is shown in Figure 2.

[15] Since the hybrid Lagrangian tracing and temperature method is closely related to the original physical meaning of surface age, the result can be used as a benchmark to evaluate other methods. For example, Figure 2 shows that the gas diffusion solution based on Higbie's penetration theory is incorrect because the advection effect cannot be neglected. It is also inappropriate to use Danckwerts' exponential distribution at small t.

[16] The issue with the random surface renewal model can be understood as follows. For a surface element, rate of change of its area A is the surface divergency value, (dA/dt)/A = ∂u/∂x + ∂v/∂y = a. Statistically, the probability for the surface element to be “replaced” by fresh fluid from below is governed by (dp(t)/dt)/p(t) = −s [Danckwerts, 1951]. These two expressions actually have the same meaning. Therefore, the surface renewal rate s(t) is equal to the minus of average surface divergence, −equation image(t). Figure 3 plots variation of equation image = −s with t. For large t, equation image(t) is negative and is almost a constant; as a result, s(t), the probability of surface element diminishing, can be regarded as a positive constant, indicating that Danckwerts' assumption of equal chance for surface renewal regardless of surface age is acceptable at large t. This validation is, however, of limited use in application because gas transfer is insignificant at large t. What is more important is the stage with small t when the surface divergency is positive. This corresponds to the surface expansion during upwelling. At this small t stage, the surface element does not diminish, but grows instead. Shortly thereafter (at t ≈ 6, which is about 0.8 times of the large-eddy turnover time), the surface divergency is reduced rapidly to a negative value. Therefore, by assuming the renewal rate s to be a positive constant, the random surface renewal theory fails to capture the process at small t. In other words, (varying) vertical advection coexists with diffusion at small t. Figure 3 shows accumulative gas flux as a function of t. As expected, most of the gas transfer occurs at small t due to the enhancement of gas flux by upwelling, signifying the importance of the stage with young surface age. Figure 3 also shows that equation (1) based on Higbie's penetration theory has large error as expected.

Figure 3.

Surface age dependence of averaged surface divergence equation image (solid line) and normalized cumulative gas flux qc(t) = equation imageqg(t′) p(t′)dt′/equation imageqg(t′)p(t′)dt′ (dashed line without symbol is from Higbie's theory; dashed lines with different symbols are results of hybrid Lagrangian tracing and temperature method for different Schmidt numbers.)

[17] Our results support the use of heat in experiments to quantify surface age for the study of gas flux. Maximum likelihood analysis of the surface age obtained from the hybrid Lagrangian tracing and temperature method shows that the best fit is a lognormal distribution as plotted in Figure 2, which is indeed what has been used in some experiments [e.g., Garbe et al., 2004]. We also note that the error caused by neglecting the turbulent advection increases with time. If the surface velocity field can be measured, the hybrid Lagrangian tracing and temperature method developed in this study has the advantage of being more accurate. This new method can also be used in experiments, since recent developments in particle image velocimetry and infrared cameras have made it possible to make high resolution measurement of surface velocity and scalar [Zappa et al., 1998; Garbe et al., 2004; McKenna and McGillis, 2004]. We hope direct comparison of Lagrangian studies between experiment and simulation can be performed in the near future.