Advection-Diffusion Equations and Turbulence

File last modified 2 November 1998

Horizontal motions are less inhibited by stratification. This is evident when you think of the aspect ratio of the oceans. Density contrasts which occur over a few hundred meters in depth are reflected in changes which occur over thousands of kilometers. The typical scales of motion associated with horizontal turbulence are the "eddy scale" which is of order 10-100 km. The kinds of estimates that you see for eddy diffusion range from a few m²/s to a few thousand m²/s, depending on location and scale.

The scale is important, because the kind of averaging implicit in the parameterization of the eddy diffusion, i.e. what you call u-bar and what you call u' depends on the scales over which you are diffusing your material. This can be seen in a classic paper (we recommend you read it, however old it is) by Okubo (1971: Deep-Sea Res. 18, 789-802). For a two dimensional gaussian and a constant turbulent diffusivity, you would expect the mean width to scale with the square root of time, or that the variance (the square of the width) to vary linearly with time. Comparison with observed experiments, however, is not so simple, because the distributions quickly become distorted by motions and shearing on scales comparable to the size of the patch (remember, there is not really any actual scale separation, there is always a range of motions between the u' and the u-bar ranges). The best you can do is to define some r.m.s. (root mean square) width as

i.e., the second moment of the distribution (which obviously will not be symmetric or gaussian). The argument is that in a kind of "central limit" sense, this will represent the mean spreading of the patch, and that it should behave as a gaussian in some mean sense. However, because the increasingly larger (with time) dye patch feels larger and larger motions as "turbulent", the diffusivity should increase with time.

What is actually observed is a much stronger than linear dependence of the mean square radius with time, as seen in the plot below (from Okubo's paper):

Think of the dye patch spreading in the open ocean. Clearly, when the dye patch is small, only those turbulent motions small compared to the dye patch itself can play a role in its dispersal. However, as the patch grows, the range of turbulent motion accessible to the patch for dispersal grows, so that the amount of available energy put into mixing grows with time and size. Thus there is an apparent increase in horizontal turbulent diffusivity with scale. Using the observed spreading, Okubo estimates an approximatly l^1.1 dependence of the diffusivity on length scale:

The take-home lesson is obvious. If you are looking at the dispersion of a small dye patch in the ocean, then it will exhibit small diffusivity initially (perhaps 0.1 to 1 m²/s) but as it grows in size, diffusivity grows. On ocean basin scales, you would expect diffusivities of order 100 - 1000 m²/s. Diffusivity is in the eye of the beholder.

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