The Transport of Momentum The Turbulent BBL

The equation for total average shear stress t in a turbulent boundary layer is da —J—, r , t — m~r + Pu w 111

dz where m is dynamic viscosity, du/dz is the vertical velocity gradient, u' and w' are the fluctuating horizontal and vertical velocities, and the overbar denotes temporal averaging. While the first term on the right describes viscous shear, the second term is related to momentum transport by turbulent velocity fluctuations. In most aquatic systems, the Reynolds number associated with near-bottom flows is sufficiently high to sustain a turbulent boundary layer. Under such conditions, the first term on the right-hand side of eqn [1] may be negligible and turbulent shear stress is likely to dominate the shear stress computation.

On the basis of dimensional arguments, it can be assumed that the shear stress, t, on the sediment

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Figure 1 Typical near-bottom current velocities measured at various locations (depths) in a large Lake (Lake Constance) emphasizing the different periods and magnitudes of BBL forcing. (a) A schematic cross-section of the lake with the three sampling sites indicated by numbers. Near-bottom current velocities induced by surface waves at 1-m depth are shown in panel (b). Typical periods of surface waves in lakes are in the order of seconds. (c) Near-bottom current velocities measured at the depth of the seasonal thermocline (10-m depth). The observed current velocities are driven by propagating internal waves with periods between 8 and 20min. Near-bottom currents at 100-m depth (d) are mainly driven by basin-scale internal waves. The major period of about four days is associated with a Kelvin wave. Note that several other basin-scale modes of oscillation (e.g., 12 h) are superimposed on this four-day period.

Turbulent boundary layer (logarithmic layer)

Viscous sublayer

Diffusive sublayer

Sediment

Figure 2 Idealized structure of the BBL on a flat sediment surface. Note that the heights provided by the logarithmically scaled axis represent order of magnitude estimates for typical conditions found in inland water bodies.

surface is related to the current speed at a certain height above the sediment:

t = pCDUtm

where p is the density, CD an empirical constant («1.5 x 10-3), the so-called drag coefficient, and U1 m refers to the current speed measured at a height of 1 m above the sediment surface. Note that CD depends on the reference height where the current speed was measured and a standard height of 1 m is assumed from now on. The bed shear stress t is assumed to be constant throughout the boundary layer (constant stress layer) and it is used to define a turbulent velocity scale u*, the so-called friction velocity:

Dimensional analysis can then be used to deduce the velocity distribution u(z) near the sediment surface, where z is the distance from the sediment surface. For a layer far enough from the boundary so that the direct effect of molecular viscosity on the flow can be neglected (the outer layer), the analysis results in

where k « 0.41 is von Karman's constant and z0 is the roughness length, which is related to the drag coefficient in eqn [2] by

and will be discussed later.

Equation [4] is called 'law of the wall,' and by assuming a local steady-state equilibrium between production and dissipation of turbulent kinetic energy (TKE), it can be used to estimate the vertical distribution of the turbulence dissipation rate e

Figure 3 Velocity distribution above a smooth and rigid bottom (solid line). Within the viscous sublayer the velocity u increases linearly with distance from the sediment surface z. Above the viscous sublayer the velocity distribution follows the law of the wall (eqn [4]) and increases logarithmically with distance from the surface. Extrapolated continuations of the linear and logarithmic velocity distributions are shown as dashed lines.

Figure 3 Velocity distribution above a smooth and rigid bottom (solid line). Within the viscous sublayer the velocity u increases linearly with distance from the sediment surface z. Above the viscous sublayer the velocity distribution follows the law of the wall (eqn [4]) and increases logarithmically with distance from the surface. Extrapolated continuations of the linear and logarithmic velocity distributions are shown as dashed lines.

Thus, in analogy to the wind-forced surface layer, the level of turbulence increases with decreasing distance from the boundary. This increasing turbulence leads, again in analogy to the surface mixed layer, to the development of a well-mixed boundary layer of up to several meters in height.

The Viscous Sublayer

It should be noted that eqn [4] is strictly valid for turbulent flows only for which the vertical transport is governed by cascading turbulent eddies. The maximum (vertical) size of the turbulent eddies is determined by the distance from the sediment surface, and by approaching the sediment surface down to scales where overturning turbulent motions are suppressed by the effect of molecular viscosity, the momentum transport becomes governed by viscous forces (first term on the right-hand side of eqn [1]). Within this layer, which is called the viscous sublayer, current shear becomes constant and the resulting linear velocity profile can be described by

On a smooth sediment surface, the viscous sublayer extends to a height of about 10v/m», which is comparable to the Kolmogorov microscale describing the size of the smallest turbulent eddies (typically O (1 cm), cf. Figure 2).

Since viscosity is reduced to its molecular value, current shear within the viscous sublayer is greater than that in the turbulent layer above (cf. Figure 3), a fact which has major consequences for organisms living within the viscous sublayer on the sediment surface because they have to withstand these strong shearing and overturning forces. It is further interesting to note that an appreciable amount of energy entering the BBL is dissipated within this layer (about 40%).

Effects of Bottom Roughness

The roughness length z0 in eqn [3] determines the effective height above the bottom z at which the current velocity approaches zero. It is determined by the topographic structure of the sediment surface and hence by the typical height, width, and spacing of individual roughness elements on a stationary bed. When the scale of these roughness elements zS is on the order of the height of the viscous sublayer dv or less, z0 is solely determined by dv and z0 « 0.1v/u*. This flow regime is called smooth. When the size of the roughness elements exceeds dv, the flow regime is called rough and the corresponding roughness length is given by z0 « zS/30. Note that the drag coefficient CD«1.5 x 10-3 provided earlier (eqn [2]) corresponds to a roughness length z0«2.5 x 10-5m (eqn [5]) and hence is valid for smooth flows unless u* exceeds 0.4 m s-1 or U1m exceeds 10 m s-1.

In addition to the shear stress derived from viscous forces as described above (the so-called skin friction), larger-scale roughness elements can cause a form drag, which results from pressure gradients between the upstream and downstream side of particular roughness elements. Although skin friction is important for the lower part of the turbulent BBL and for the viscous sublayer, form drag resulting from, e.g., ripples, sand waves, or submerged vegetation is u u e more important for the upper part of the turbulent BBL and for the total drag on flow. When form drag is significant, the turbulent BBL may consist of more than one logarithmic layer, described by different roughness lengths z0, respectively.

Oscillatory Boundary Layers

The turbulent BBL equations derived here are based on steady-state conditions, i.e., on a local balance between production and dissipation of TKE, which is in equilibrium with the applied forcing. As described later, however, many forcing mechanisms for near-bottom flows are related to surface or internal waves and are hence associated with oscillatory flows. Well above the viscous sublayer, such oscillatory BBL show, similar to the effect of form drag, deviations of the of the velocity distribution from its steady-state logarithmic pattern. One major characteristic of oscillatory BBL is a pronounced maximum of the current speed at some decimeters or meters above the bed. The analytical solution to this problem (Rayleigh flow or Stokes' second problem) is an exponentially damped vertical oscillation of the current profile with a vertical wave number of \Jrnj2n, where oo is the forcing frequency and n the turbulent viscosity, which, however, is a function of time and distance from the sediment surface. Depending on the overall energetics of the BBL flow, characteristic current speed maxima at 2-3 m above the bed could be observed in lakes where the internal wave forcing had a period as long as 24 h. Another major consequence of oscillatory BBL is that the maximum in turbulent intensity near the bed does not coincide with the maximum of the current speed, at the top of the BBL.

Stratified BBL

Effects of Density Stratification

Density stratification affects turbulent mixing in the outer layer by causing buoyancy forces that damp or even suppress overturning turbulent eddies. The vertical distribution of velocity and turbulence described earlier for unstratified BBL may hence change significantly under stratified conditions. In addition, the vertical structure of density stratification along with the presence of sloping boundaries can introduce additional mixing phenomena in BBL of enclosed basins.

Similar to the surface mixed layer, increased production of TKE along the boundaries of a water body often leads to the generation and maintenance of a well-mixed BBL of height hmix. On a flat bottom (away from the slopes) and where a logarithmic boundary layer occurs, hmix can be estimated by applying scaling laws from the wind-mixed surface layer b ■ = 23=4

where u* is the friction velocity in the BBL (eqn [2]), N is the Brunt-Vaisala frequency, and f is the Coriolis parameter. In small- to medium-sized water bodies, where the effect of Earth's rotation is unimportant, f has to be replaced by the respective forcing frequency, e.g., the frequency of internal seiching.

In productive water bodies, in particular, the sediment can be a significant source of remineralized nutrients as a result of microbial degradation of organic matter at the sediment surface or within the sediment. The diffusion of solutes across the sediment-water interface (see Solute Transport and Sediment-Water Exchange section) has the potential to set up density stratification within the BBL, which could suppress turbulent mixing. Hence, whether or not a turbulent and mixed BBL can be developed and maintained depends not only on the amount of available TKE, which is typically extracted from basin-scale motions, but also on the buoyancy flux from the sediment that has to be overcome by turbulent mixing. Geothermal heating, in contrast, can result in unstable stratification and convective mixing in the BBL. A mean geothermal heat flux of 46 mW m~2 results in a mean vertical temperature gradient of about —8 x 10-2K m , which can be observed when the BBL is chemically stratified and turbulent mixing is suppressed.

2-Dimensional Mixing Processes in Enclosed Basins

The occurrence of mixed BBL (in terms of density) is a straight consequence of the application of a zero-flux boundary condition at the sediment surface, i.e., no exchange of heat and dissolved solids across the sediment-water interface. This boundary condition forces the isopycnals (or isotherms if density changes are mainly caused by temperature) to intersect the boundary at a right angle, leading to a mixed density or temperature profile in the vicinity of the boundary. Enhanced mixing along the boundaries is thus not a necessary requirement for the development of such mixed layers. Along the sloping boundaries of enclosed basins, these mixed BBL cause horizontal density gradients and hence drive horizontal currents - a process which is believed to have important consequences for basin-wide diapycnal transport.

Measurements, however, have revealed that hmix is not constant along the sloping boundaries, as demonstrated in Figure 4. There the upper limit of the mixed BBL can be defined by the depth of the 5 °C isotherm

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Figure 5 Isotherms illustrating the internal wave field in the hypolimnion of Toolik Lake, Alaska, prior to and after wind forcing increased to 9ms~1 (day 195.6) and the Lake number decreased to 1.5. Isotherms are at 0.1 °C intervals with uppermost isotherm 6 °C. Thermistors were 80 cm apart between 10.2 and 12.6 m depth and 2 m apart deeper in the water column. Deepest thermistor was within 50 cm of the lake bottom. Increased temperature fluctuations after LN decreases below 1.5 are indicative of turbulence either beginning or increasing in the lower water column (unpublished data, S. Maclntyre).

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Figure 5 Isotherms illustrating the internal wave field in the hypolimnion of Toolik Lake, Alaska, prior to and after wind forcing increased to 9ms~1 (day 195.6) and the Lake number decreased to 1.5. Isotherms are at 0.1 °C intervals with uppermost isotherm 6 °C. Thermistors were 80 cm apart between 10.2 and 12.6 m depth and 2 m apart deeper in the water column. Deepest thermistor was within 50 cm of the lake bottom. Increased temperature fluctuations after LN decreases below 1.5 are indicative of turbulence either beginning or increasing in the lower water column (unpublished data, S. Maclntyre).

heat and solutes is governed by molecular diffusion. Since the molecular diffusivity of solutes D is about 3 orders of magnitude smaller than the molecular viscosity n (the Schmidt number Sc, defined as Sc = n/D, is about 1000), the height of the diffusive sublayer dD is with dD = O(1 mm) considerably smaller than the height of the viscous sublayer dn (Figure 2). A typical profile of dissolved oxygen concentrations measured through the sediment-water interface is shown in Figure 6. Although turbulent transport is already suppressed within the viscous sublayer, straining of concentration gradients by viscous shear results in an efficient vertical transport of solutes and to typically well-mixed solute distributions within most of the viscous sublayer. Concentration gradients are hence compressed to the diffusive sublayer overlaying the sediment surface and the constancy of the molecular diffusivity results in a linear concentration gradient C (z) (Figure 6). At the top of the diffusive boundary layer, the concentration gradient decreases gradually to zero and the solute concentration reaches its constant bulk value CM.

Within the sediment, molecular diffusivity is reduced by the porosity (reduction of surface area) and by turtosity (increase of diffusion path length), and the concentration profile is additionally determined by chemical and microbial production and loss processes. In the case of oxygen (Figure 6),

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Figure 6 Oxygen concentration profile across the sediment-water interface measured in Lake Alpnach (Switzerland). The diffusive sublayer is characterized by the linear concentration gradient above the sediment surface where transport is governed by molecular diffusion. The effective height of the diffusive sublayer dD is defined by the intersection of the extrapolated linear concentration gradient (dashed line) with the constant oxygen concentration above the diffusive sublayer.

a reaction-diffusion model with rather simple zero-order kinetics of oxygen consumption resulting in a parabolic oxygen profile provides surprisingly good agreement with measured oxygen distributions.

The fluxes F of solutes through the diffusive boundary layer can be derived from Fick's first law:

where C0 is the solute concentration at the sediment surface. Hence, for a given concentration gradient (CM — C0) and by ignoring the temperature dependence of the molecular diffusivity, the magnitude of the flux is determined by the thickness of the diffusive sublayer dD. It has been demonstrated in numerous laboratory and field measurements that dD depends strongly on the flow regime in the turbulent BBL above. With increasing levels of turbulence (e.g., with increasing u*) the diffusive boundary layer becomes more compressed, and according to eqn [9], the fluxes increase. The thickness of the diffusive sublayer can be related empirically to u* or to the thickness if the viscous sublayer, which in turn is related to u*, as described earlier:

where the Schmidt number Sc accounts for the different kind of 'solutes' (e.g., heat or dissolved oxygen) and observed Schmidt number exponents a range between 0.33 and 0.5. As u* is not always an appropriate parameter for describing BBL turbulence (e.g., in oscillatory BBL or in the presence of form drag or density stratification), dD can also be described in terms of the Batchelor length scale, which describes the size of the smallest fluctuations of a scalar tracer in turbulent flows as a function of the turbulence dissipation rate.

It is most interesting to note that the sediment-water exchange in productive water bodies is often flux-limited by the 'bottleneck' of the diffusive sublayer and that it is actually the wind acting at the water surface that provides energy for turbulence within the BBL and hence effects the magnitude of the sediment-water fluxes by controlling the thickness of the diffusive sublayer.

Effects of Small-Scale Sediment Topography

Increased roughness of the sediment surface (e.g., due to biological activity) affects the sediment-water exchange by increasing the mass and momentum transfer as well as by increasing the surface area of the sediment-water interface. Detailed observations have demonstrated that the diffusive sublayer tends to smooth out topographic structures that are smaller than the average height of the sublayer, but smoothly follows larger roughness elements (Figure 7). The detailed structure of the oxygen distribution within the diffusive sublayer is then not only determined by diffusion (normal to the local sediment surface) but also by advection with the flow (in parallel to the local sediment surface) and it can be expected that the degree of smoothing increases with decreasing flow velocities. Detailed comparisons of measured 3-dimensional fluxes over rough topography with the fluxes calculated from the respective average diffusive sublayer heights and concentration gradients are enhanced by factors up to 49%. It must be noted, however, that fluxes estimated from concentration profiles measured at one particular location on a rough sediment surface can severely overestimate or underestimate the average flux, as demonstrated in Figure 7.

Figure 7 Horizontal transect along the direction of flow showing how the upper limit of the diffusive sublayer (solid line with data points) follows the surface topography of a microbial mat. The diffusive sublayer limit was defined by the isopleth of 90% air saturation of oxygen. Notice different vertical and horizontal scales. Flow velocity at a height of 1 cm was 4 cm s—1. Numbers indicate specific measuring positions discussed in the original publication. Reproduced from J0rgensen BB and Des Marais DJ (1990) The diffusive boundary layer of sediments: Oxygen microgradients over a microbial mat. Limnology and Oceanography 35:1343-1355, with permission from American Society of Limnology and Oceanography.

Figure 7 Horizontal transect along the direction of flow showing how the upper limit of the diffusive sublayer (solid line with data points) follows the surface topography of a microbial mat. The diffusive sublayer limit was defined by the isopleth of 90% air saturation of oxygen. Notice different vertical and horizontal scales. Flow velocity at a height of 1 cm was 4 cm s—1. Numbers indicate specific measuring positions discussed in the original publication. Reproduced from J0rgensen BB and Des Marais DJ (1990) The diffusive boundary layer of sediments: Oxygen microgradients over a microbial mat. Limnology and Oceanography 35:1343-1355, with permission from American Society of Limnology and Oceanography.

Nondiffusive Fluxes

Besides the diffusive fluxes, there exist several additional pathways for the exchange of solutes across the sediment-water interface. Convectively driven transport of pore water through the interface can occur in shallow waters where shortwave solar radiation penetrates the water column to the sediment surface and heats the sediment. Similarly, changes in temperature of the water overlaying the sediment surface, e.g., due to internal waves in stratified water bodies, have been observed to drive convective transport across the sediment-water interface. The existence of larger roughness elements, such as ripples, on permeable sediments can further result in advective pore water exchange driven by pressure differences. Higher dynamic pressure at the upstream side of such topographic structures give rise to the transport of water into the sediment, whereas the lower pressure at the downstream side sucks pore water out of the sediment.

Bioturbation and bioirrigation are processes by which benthic fauna or flora enhances the sediment-water exchange. Whereas bioturbation mainly refers to the displacement and mixing of sediment particles by, e.g., worms, bivalves, or fish, bioirrigation refers to the flushing and active ventilation of burrows with water from above the sediment surface. These processes are particularly important in oligotrophic water bodies where the sediment surface remains oxic and provides suitable conditions for a diverse benthic fauna. In more eutrophic systems the emanation of gas bubbles (mainly methane or carbon dioxide) which are formed by bio-genic production and a resulting supersaturation of pore water with these gases may have similar effects.

In Situ Flux Measurements

The sediment oxygen demand or the release of nutrients from the sediment can be of major importance for the overall productivity and for the geochemical composition of a particular water body and quantification of sediment-water fluxes is often essential for understanding biogeochemical cycles within the water column. As these fluxes depend strongly on the hydrodynamic conditions in the BBL and as these conditions have a strong spatial and temporal dynamics, in situ measurements are often desirable. The measurement of concentration profiles through the sediment-water interface capable of resolving the diffusive sublayer are one way for estimating the fluxes. From a measured profile of dissolved oxygen, as shown in Figure 6, the sediment-water flux can be readily estimated by applying eqn [9]. Such measurements are carried out using microelectrodes, which are available for a large number of solutes, mounted on a benthic lander system. However, there are two major problems associated with this method: First, although these microelectrodes have tiny tip diameters (down to 10 mm or less for oxygen sensors), they were demonstrated to disturb the concentration distribution within the diffusive boundary layer while profiling. The second and more severe problem results from the complexity of the spatial distribution of the fluxes resulting not only from the small-scale sediment topography (cf. Figure 7) but also from the strongly localized effects of advective pore water exchange and bioturbation. To overcome these problems, the flux can be measured within the turbulent BBL at some distance from the actual sediment surface. By neglecting any sources or sinks within the water column between the sampling volume and the sediment surface, this flux represents an areal average of the sediment-water flux including all nondiffusive flux contributions. The turbulent flux Fturb is determined by the cross-correlation of turbulent vertical velocity (w') and turbulent concentration (C') fluctuations:

where the overbar denotes temporal averaging.

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