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 nearbottom flows is sufficiently high to sustain a turbulent boundary layer. Under such conditions, the first term on the righthand 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
0 10
Surface
0 10
Surface
19:30
20:00 20:30
Time (24 Aug 2005)
19:30
20:00 20:30
Time (24 Aug 2005)
16:22:10 16:22:18 16:22:26 16:22:34 (b) Time (16 Nov 2006)
0.05
0.00
ur0.05
0.05
0.00
4 days A A1 
Ll 

5 Nov Figure 1 Typical nearbottom 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 crosssection of the lake with the three sampling sites indicated by numbers. Nearbottom current velocities induced by surface waves at 1m depth are shown in panel (b). Typical periods of surface waves in lakes are in the order of seconds. (c) Nearbottom current velocities measured at the depth of the seasonal thermocline (10m depth). The observed current velocities are driven by propagating internal waves with periods between 8 and 20min. Nearbottom currents at 100m depth (d) are mainly driven by basinscale internal waves. The major period of about four days is associated with a Kelvin wave. Note that several other basinscale modes of oscillation (e.g., 12 h) are superimposed on this fourday period. Turbulent boundary layer (logarithmic layer) Viscous sublayer Diffusive sublayer SedimentFigure 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 103), the socalled 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 socalled 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 steadystate 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 windforced 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 wellmixed boundary layer of up to several meters in height. The Viscous SublayerIt 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 righthand 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 RoughnessThe 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 103 provided earlier (eqn [2]) corresponds to a roughness length z0«2.5 x 105m (eqn [5]) and hence is valid for smooth flows unless u* exceeds 0.4 m s1 or U1m exceeds 10 m s1. In addition to the shear stress derived from viscous forces as described above (the socalled skin friction), largerscale 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 LayersThe turbulent BBL equations derived here are based on steadystate 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 nearbottom 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 steadystate 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 23 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 StratificationDensity 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 wellmixed 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 windmixed surface layer b ■ = 23=4 where u* is the friction velocity in the BBL (eqn [2]), N is the BruntVaisala frequency, and f is the Coriolis parameter. In small to mediumsized 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 sedimentwater interface (see Solute Transport and SedimentWater 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 basinscale 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 102K m , which can be observed when the BBL is chemically stratified and turbulent mixing is suppressed. 2Dimensional Mixing Processes in Enclosed BasinsThe occurrence of mixed BBL (in terms of density) is a straight consequence of the application of a zeroflux boundary condition at the sediment surface, i.e., no exchange of heat and dissolved solids across the sedimentwater 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 basinwide 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 10 f a 14 20 195.3 10 f a 14 195.4 195.5 195.6 Day of year 195.7 195.8 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). 195.4 195.5 195.6 Day of year 195.7 195.8 195.9 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 sedimentwater 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 wellmixed 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),

Post a comment