mixed. A flux of the tracer within the fluid can be brought about either by organized fluid motion or by molecular diffusion. These two flux processes can be written as

where F(1 and F/2 denote the flux vectors of the tracer (dimension: M/(L2T)), V the fluid velocity vector (L/T), p the density of the fluid (M/L3), q, the tracer mixing ratio (M/M), q the mass concentration of the tracer (M/L3), D the molecular diffusivity (L2/T) and V the gradient operator (L "'). The expression Vq,: denotes the vector (3qt/dx, dqt/dy, 9q,/Sz).

The continuity of tracer mass is expressed by the equation

^ = -V • F, + Q - S = -V • (Ffl + F,-2) + Q-S = -V • (Vc,) + V • (DiPpVfl) + Q-S (36)

where Q and S represent production and removal of the tracer (M/(L3T)). Here V • F, denotes the scalar quantity

If variations in fluid density and diffusivity can be neglected we have rV-

In most situations a fluid would be turbulent implying that the velocity vector, as well as the concentration c,, exhibits considerable variability on time scales smaller than those of prime interest. This situation can be described by writing these quantities as the sum of an average quantity (normally a time average) and a perturbation

From Equation (34), the transport flux F,i, then becomes f,i = (v + v')(c + c\) = vc, + vc'i + v'c, + v'c;

and its average value

Note that the averages of V' and c' are equal to zero. The continuity equation can now be written as

The first two terms on the right side of Equation (40) describe the contributions from transport by advection and by turbulent flux, respectively. The separation of the motion flux into advection and turbulent flux is somewhat arbitrary; depending upon the circumstances the averaging time can be anything from a few minutes to a year or even more.

Since in most situations the perturbation quantities (V' and c\) are not explicitly resolved, it is not possible to evaluate the turbulent flux term directly. Instead, it must be related to the distribution of averaged quantities - a process referred to as parameterization. A common assumption is to relate the turbulent flux vector to the gradient of the averaged tracer distribution, which is analogous with the molecular diffusion expression, Equation (35).

The coefficient kturb introduced in Equation (41) (dimension: L2/T) is called the turbulent, or eddy diffusivity. In the general case the eddy diffusivity is given separate values for the three spatial dimensions. It must be remembered that the eddy diffusivities are not constants in any real sense (like the molecular diffusivities) and that their numerical values are very uncertain. The assumption underlying Equation (41) is therefore open to question.

In most cases, the term expressing the divergence of the molecular flux in Equation (40) (DV2c,) can be neglected compared to the other two transport terms. Important excep tions occur, e.g. in a thin layer of the atmosphere close to the surface and in similar layers of the oceans close to the ocean floor and to the surface (viscous sublayers). Molecular diffusion is also an important transport process in the upper atmosphere, at heights above 100 km.

Order-of-magnitude values for the vertical eddy diffusivity in the atmosphere and the ocean are shown in Fig. 4-15. The values for the viscous layers represent molecular diffusivities of a typical air molecule like N2.

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V') and of the tracer concentration (c'), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of C02 above a forest canopy in Sweden. Since the mean vertical velocity (zv) has to vanish above such a flat surface, the only contribution to the vertical flux of C02 comes from the eddy-correlation term (c'w'). In order to capture the contributions from all important eddies, both the anemometer and the C02 instrument must be able to resolve fluctuations on time scales down to about 0.1 s.

A type of motion that is often very important in both the oceans and the atmosphere is convection. This is a vertical mixing process where parcels of water (or air) are rapidly transported in the vertical direction due to their buoyancy. In the oceans, this occurs when the surface water becomes denser than the underlying water - by cooling and/or increased salinity due to evaporation - aijd parcels of water sink down within days to depths of up to several km. In the atmosphere, convective motions occur when surface air is heated by conduction from the underlying surface. Air parcels having a horizontal dimension on the order of 1 km then rise and sometimes reach a height as high as 1015 km within less than an hour, especially in tropical areas. Cumulus and cumulonimbus clouds are visible manifestations of convection in the atmosphere. In some circumstances, con

Fig. 4-15 Orders of magnitude of the average vertical molecular or turbulent diffusivity (whichever is largest) through the atmosphere, oceans, and uppermost layer of ocean sediments.

vection may contribute to a very rapid vertical mixing.

4.8.2 Other Transport Processes

Under some circumstances transport processes other than fluid motion and molecular diffusion are important. One important example is sedimentation due to gravity acting on particulate matter submerged in a fluid, e.g., removal of dissolved sulfur from the atmosphere by precipitation scavenging, or transport of organic carbon from the surface waters to the deep layers and to the sediment by settling detritus. The rate of transport by sedimentation is determined essentially by the size and density of the particles and by the counteracting drag exerted by the fluid.

Geochemically significant mixing and transport can sometimes be accomplished by biological processes. An interesting example is redistribution of sediment material caused by the movements of worms and other organisms (bioturbation).

Exchange processes between the atmosphere and oceans and between the oceans and the sediments are treated below in separate sections.

4.8.3 Air-Sea Exchange Gas transfer

The magnitude and direction of the net flux density, F, of any gaseous species across an air-water interface is positive if the flux is directed from the atmosphere to the ocean. F is related to the difference in concentration (Ac), in the two phases by the relation

Here Ac = ca - KHcw with ca and cw representing the concentrations in the air and water respectively and KH the Henry's law constant. The parameter K, linking the flux and the concentration difference, has the dimension of a velocity. It is often referred to as the transfer (or piston) velocity. The reciprocal of the transfer velocity corresponds to a resistance to transfer across the surface. The total resistance (R = K"1) can be viewed as the sum of an air resistance (Ra) and a water resistance (Rw):

ka oki

The parameters ka and k\ are the transfer velocities for chemically unreactive gases through the viscous sublayers in the air and water, respectively. They relate the flux density F to the concentration gradients across the viscous sublayers through expressions similar to Equation (42):

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