Turbulent extent defines the vertical and horizontal displacement of particles that fulfil the entrainment criterion (Fig. 2.16). The vertical extent of turbulent boundary layers, unconstrained by the basin morphometry or by the presence of density gradients (open turbulence), is related primarily to the kinetic energy transferred: rearranging Eq. (2.9), le « (u*)(du/dz)-1 m (2.26)
where (du/dz) is the vertical gradient of horizontal velocities (in ms-1 m-1) and le is the dimension of the largest eddies. Entries in Table 2.2 pertaining to the upper layers of the open ocean and also of a moderately large lake like the Bodensee (Germany/Switzerland) imply an increase in mixing depth of about 9 m for each increment in wind forcing of 1 ms-1. Of course, even this relationship applies only under a constant wind: an increase in wind speed necessarily invokes a restructuring, which may take many minutes to organise (see below). Similarly, the contraction of the thickness of the mixed layer following a weakening of the wind stress is gradual, pending the dissipation of inertia. The structure of turbulence under a variable, gusting wind is extremely complex!
The complexity is further magnified in small basins (Imberger and Ivey, 1991; Wuest and Lorke, 2003). Where the physical depth of the basin constrains even this degree of dissipative order, the water is fully mixed by a turbulence field which is, as already argued, made up by a finer grain of eddies. Moreover, the shallower is the water body, the lower is the wind speed representing the onset of full basin mixing likely to be. Thus, from the entries in Table 2.2, it is possible to deduce that a wind of 3.5 ms-1 might be sufficient to mix Lough Neagh, Northern Ireland (maximum depth 31 m, mean depth 8.9 m). In fact, the lake is usually well mixed by wind and is often quite turbid with particles entrained by direct shear stress on the sediment.
Density gradients, especially those due to the thermal expansion of the near-surface water subject to solar heating, also provide a significant barrier to the vertical dissipation of the kinetic energy of mixing. Although there is some outward conduction of heat from the interior of the Earth and some heat is released in the dissipation of mechanical energy, over 99% of the heat received by most water bodies comes directly from the Sun. The solar flux influences the ecology of phytoplankton in a number of ways but, in the present context, our concern is solely the direct role of surface heat exchanges upon the vertical extent of the surface boundary layer.
Starting with the case when there is no wind and solar heating brings expansion and decreasing water density (i.e. its temperature is >4 °C), a positive heat flux is attenuated beneath the water surface so that the heating is confined to a narrow near-surface layer. The heat reaching a depth, z, is expressed:
where e is the base of natural logarithms and k is an exponential coefficient of heat absorption. QT is that fraction of the net heat flux, QT which penetrates beyond the top millimeter or so. Roughly, QT averages about half the incoming short-wave radiation, QS. The effect of acquired buoyancy suppresses the downward transport of heat, save by conduction.
If the water temperature is <4 °C, or if it is > 4 °C but the heat flux is away from the still water (such as at night), the heat transfer increases the density of the surface water and causes instability. The denser water is liable to tumble through the water column, accelerating as it does so and displacing lighter water upwards, until it reaches a depth of approximate isopycny (that is, where water has the identical density). This process can continue so long as the heat imbalance between air and water persists, all the time depressing the depth of the density gradient. The energy of this penetrative convection may be expressed:
At all other times, the buoyancy acquired by the warmer water resists its downward transport through propagating eddies, whether generated by internal convection or externally, such as through the work of wind. They are instead confined to a layer of lesser thickness. Its depth, hb, tends to a point at which the kinetic energy (Jk) and buoyancy (Jb) forces are balanced. Its instantaneous value, also known as the Monin-Obukhov length, may be calculated, considering that the kinetic energy flux, in Wm-2, is given by:
while the buoyancy flux is the product of the expansion due to the net heat flux to the water (QT), also in W m-2,
where Y is the temperature-dependent coefficient of thermal expansion of water, * is its specific heat (4186 J kg-1 K-1) and g is gravitational acceleration (9.8081 ms-2).
Owing to the organisational lags and the variability in the opposing energy sources, Eq. (2.29) should be considered more illustrative than predictive. Nevertheless, simulations that recognise the complexity of the heat exchanges across the surface can give close approximations to actual events, both over the day (Imberger, 1985) and over seasons (Marti and Imboden, 1986).
The effect of wind is to distribute the heat evenly throughout its depth, hm. If the heat flux across its lower boundary is due only to conduction and negligible, the rate of temperature change of the whole mixed layer can be approximated from the net heat flux, de/dt = QT(hmPwff )-1 Ks-1 (2.32)
The extent of the turbulent mixed boundary layer may be then viewed as the outcome of a continuous 'war' between the buoyancy generating forces and the dissipative forces. The battles favour one or other of the opponents, depending mostly upon the heat income, QT, and the kinetic energy input, t , as encapsulated in the Monin-Obukhov equation (2.31). Note that if the instantaneous calculation of hb is less than the lagged, observable hm, buoyancy forces are dominant and the system will become more stable. If hb > hm, turbulence is dominant and the mixed layer should be expected to deepen. It becomes easy to appreciate how, at least in warm climates, when the water temperatures are above 20 °C for sustained periods, diurnal stratification and shrinkage of the mixed depth occurs during the morning and net cooling leads to its breakdown and extension of the mixed depth during the afternoon or evening. The example in Fig. 2.18 shows the outcome of diel fluctuations in heat- and mechanical-energy fluxes to the density structure of an Australian reservoir.
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