## W p

In this formulation, Kz describes the vertical transport of density caused by turbulent velocity fluctuations w' over a typical eddy distance L' given by the level of turbulence and the strength of the stratification. Therefore, in contrast to the molecular diffusion process, eddy diffusivity is neither a function of medium (water) nor of the water constituents (particulate or dissolved), but rather a property of the turbulent flow within the stratified water itself. In particular, Kz reflects the extent of the velocity fluctuations w and the eddy sizes L': Kz can be interpreted as the statistical average w'L' of a large number of eddies, which exchange small water parcels as a result of the turbulent flow (Figures 2 and 3).

In addition to density, all other water properties -such as temperature or substances - are transported and mixed in the same way via the turbulent exchange of small eddies or parcels of water (Figure 3). The eddy diffusivity concept can be applied to any dissolved or particulate substance and the associated vertical fluxes F can be readily estimated in analogy to eqn. [6] by

where C is the appropriate concentration.

Assuming steady-state conditions, i.e., by neglecting the left-hand side of eqn. [4], and combining eqns. [1], [5], and [6] yields:

Kz gmix N2

This equation provides an expression to estimate Kz from field measurements of e and N2 and, moreover, it demonstrates the direct proportionality of Kz on the level of turbulence (e) and the inverse proportionality on the strength of stratification (N2). In the last decades, two fundamentally different approaches have been used for the estimation of Kz: (i) the microstructure method and (ii) the tracer method. Method (i), is based on eqn. [8] where the dissipation of TKE or of temperature variations are measured by usually free-falling profilers which measure either temperature or velocity over small spatial scales. For example, spectral analysis of the temperature gradient signal provides estimates of e and the local buoyancy frequency is obtained from density computed from the temperature and salinity profiles. For the application of tracer method (ii), one has to measure the three-dimensional spreading of a tracer (artificial or natural) and then infer the diffusivities (Kz) from the observations. Heat is also used as a tracer and Kz is obtained by computing a time series of the heat budget below the respective depth of a lake. Typical values in stratified natural waters are listed in Table 1 and Figures 4 and 5.

Turbulence is caused by current shear, breaking surface waves, and instabilities in the internal wave field. Currents induce shear near boundaries regardless of whether the flow is stratified. Thus, the concept of eddy diffusivity is also applied to surface mixing layers and to nonstratified systems such as rivers.

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