Advection-Diffusion Equations and Turbulence

File last modified 2 November 1998

15.7 The Effects of Varying Turbulent Diffusivity

We rewrite the equation in the following way:

Where we have differentiated by parts the turbulent diffusion term. We have admitted the possibility that the diffusivity may vary in space because it is a property of the flow, not of the fluid.

The second term on the R.H.S. looks a lot like a velocity. In fact, it behaves exactly like one. How important is it? Well, consider first horizontal effects, since those diffusivities are largest. Values as high as 2000 m²/s have been observed in the Gulf Stream area, and seem to taper off to a few hundred m²/s in the ocean interior. Assuming this takes place over a distance of 2000 km yields apparent "velocities" of approximately (2000 m²/s)/(2x10⁶m) = .001 m/s, or about an order of magnitude or more smaller than typical velocities in the upper thermocline. So the conclusion is that this is unlikely a problem for the upper thermocline, but may be important deeper down where velocities are small.

Now how about the vertical? Well in the mixed layer, vertical diffusivities are of order 10^-3 or larger. We'll choose the lowest values for this calculation. Meanwhile, a few hundred meters down in the main thermocline, vertical diffusivities are of order 10^-5m²/s, so we calculate an effective "velocity" of order 10^-6 to 10^-5 m/s. This doesn't sound like much, but when you realize that typical ekman pumping rates (the rate at which water is pushed downward by wind stress convergence) are of order 50 m/y or about 10^-6m/s, this becomes really significant! Furthermore, abyssal upwelling velocities through the main thermocline (we'll get into that in lecture 17) are ten times lower. Thus we should be very careful when considering vertical balance models with turbulent diffusivities that change a lot with depth.

Now a footnote: the above considerations may be a little premature, since the physics associated with models that span large ranges in turbulent flows (which must drive these changes in turbulent diffusivities) is far from worked out. Our goal in pointing out these issues to you at this point is to make you aware of some of the pitfalls and foibles in such modeling.

A few papers which discuss variable diffusivity effects are:

• Armi, L. (1979) Effects of variations in eddy diffusivity on property distributions in the oceans. J. Mar. Res. 37, 515-530. (Horizontal)
• Armi, L. and D. Haidvogel (1982) Effects of variable and anisotropic diffusivities in a steady state diffusion model. J. Phys. Oceanogr. 12, 785-794. (Horizontal)
• Jenkins, W.J. (1980) Tritium and 3He in the Sargasso Sea. J. Mar. Res. 38,533-569. (Vertical).
  

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