Advection-Diffusion Equations and Turbulence
File last modified 2 November 1998
15.3 Reynolds Decomposition
Here's where things get a little complicated. So far, what we have said is fundamentally sound and quantifiable. The problem is that we cannot know, measure or resolve all of the fluid motions that we know influence chemical properties in the ocean. Think of it this way: in order to completely describe the motion of the stuff that we have been talking about, we need to take into account movement on scales ranging from the molecular (order 10-8m) to ocean basin (order 107m), for all of these motions can contribute to changes in, say, oxygen concentrations. We are talking about 1015 orders of magnitude. This is clearly not possible. However, it is possible to describe the molecular diffusion of a dye in a glass of water (at least ideally) without knowing the individual motions of individual molecules: there are fundamental statistical mechanical (or thermodynamical) laws that can be derived to describe the behavior in a statistical sense. The hope is that we can do something along those lines.
We begin by thinking of the fluid motion as consisting of two parts: one which is the large scale, mean flow, and the other as being some small scale, randomly fluctuating component. This, we should warn you, is the fundamental, critical step in the development to which you must pay careful attention. It implies that there is some kind of scale separation that you can divide the motions into: large (mean) and small (random). It presupposes that the small scale processes are both stationary and random. This may not always be the case, so caveat emptor!. This definition can be presented mathematically as
where we have
which, naturally, requires that
(You can prove this by taking the time integral of the first of these three equations.) This conceptual separation of velocity components is referred to as "Reynolds Decomposition". Now the same thing could be argued about the concentration of stuff, i.e., there is some smoothly varying, mean concentration distribution coupled with some randomly varying component, so we do the same thing for "C" with
in a similar fashion. Now if we plug these back into our original equation we get the following
but the alert and clever student will notice that we've forgotten to put the molecular diffusive and "J" terms on the equation, but we're being a little lazy, and will put them back in a moment (it doesn't change anything here). Now, average the above equation with respect to time. We now have
Now in doing this, we've taken advantage of the fact that you can reverse the order of integration and differentiation in continuous systems, and that the mean distributions are time invariant. Note also that the integrals of the fluctuating components are by definition are zero, except the last integral. This is because there is likely to be a non-zero correlation between velocity and concentration fluctuations, since the former likely causes the latter (think about it!). If fact, it is the cross-correlation function between velocity and concentration and will in general, not be zero. We can therefore simplify the above equation to be
where the first term on the right hand side is the "macroscopic" advective flux divergence, and the second term is the divergence of the Reynolds Flux. The "overbar" refers to time averaging, as we have defined it for the velocity and concentration earlier. As a side note, if the stuff we were dealing with was momentum, then the equivalent term would be "Reynolds Stress".
Now where can be go from here? Well, as we mentioned, the cross-correlation must be causal in nature, since random displacements in the fluid result in apparent concentration anomalies. We would argue that in a randomly moving fluid, there is a characteristic space scale of displacement, which we'll call l'. This may be the mean vertical motion of a fluid parcel caused by breaking internal waves, or the horizontal movement caused by eddies sweeping by. Now this random displacement, coupled with a large scale mean gradient in concentration will result in an apparent concentration fluctuation C' governed by
Note the sign: if the slope is negative, a positive displacement results in a positive concentration anomaly, and if the slope is positive, a positive displacement results in a negative concentration anomaly. The figures below show this schematically.
We suggest you look at a paper discussing the mixing length concept by Chris Garrett (1989: J. Geophys. Res., 94, 9710-9712). Thus we have
Now this is another important step, since we are now separating the causes of random concentration fluctuations into two components: one due to the large scale distribution of C (thus a concentration dependent part) and one due to the fluid motion (and hence not related to C).
Thus we now have the following equation:
where we have now come clean and put back in the molecular diffusion and J terms. Whew! Notice that we have sneakily put the u'l' term in with the molecular diffusion term since they are functionally similar in form.
Now what is this u'l' thing? It is a property of the fluid flow (not of the fluid, and not of the stuff that we studying). It is often called the turbulent diffusivity coefficient since it appears in the equation like a diffusivity term.
We mentioned that it is a property of the fluid flow. It's related to the cross-correlation between fluid displacement and the velocity. One would expect them to be more-or-less correlated in a turbulent fluid (therefore the cross-correlation will not be zero!), but not perfectly, since that only occurs for wave motions. One might express this quantity as the product of the rms (root mean square) displacement times the rms velocity fluctuation, times the cross-correlation. Typically, the cross correlation is around ¼ or so for most turbulent fluids.
So how big is the size of this u'l' term? In the thermocline, vertical motions associated with internal waves and tides of order of meters to 10's of meters are not uncommon. This displacement occurs on time scales of hours, so associated velocities are of order 10-4 m/s. The product of these two yield of order 10-5 m2/s, which is many orders of magnitude larger than molecular diffusion. In fact, measurement of this property, by various means, yields values of order 10-5m2/s in the main thermocline, and larger in the mixed layer and the abyss.
How about horizontal motions? The disparity becomes even larger. In the open ocean, velocity fluctuations of order 1-10 cm/s are common, and the space scale of eddies is of order 10-100 km. Thus horizontal eddy diffusion coefficients are of order 100-1000 m2/s. So we may as well throw D away. Well, maybe not always, since there are cases (particularly in lakes) where vertical turbulent diffusion may be suppressed by density stratification to a level where molecular diffusion may dominate. We'll discuss this in a letter section. For the moment, however, we will ignore "D" and rewrite the equation as
where we have dropped the overbars (the averaging symbols) under the assumption that we are dealing with the large scale mean velocities and concentrations. Also we have introduced a turbulent diffusivity coefficient which is a property of the fluid flow, not of the fluid itself so that it may possibly change in space and time.
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