Advection-Diffusion Equations and Turbulence
File last modified 2 November 1998
15.5 Vertical Turbulent Diffusion
The gradient Richardson Number tells us something about the vigor of vertical turbulent diffusion in the environment. Typical estimates of vertical turbulent diffusion tend to vary inversely as the vertical density gradient. For example, consider the so called "mixed layer" of the ocean, which is generally isothermal (and hence uniform in density) and stirred effectively by wind stress. Vertical mixing rates are of order 10-3 to 10-2 m2/s. In fact, many upper ocean models tend to regard the mixed layer as so well mixed that they treat it as perfectly mixed: such models are called "bulk mixed layer models".
In the main oceanic thermocline, which is characterized by vertical
density gradients of order 10-3kg/m4 (that
is, a change in density of about 1 kg/m3 over a depth
range of approximately 1 km) turbulent diffusion is much less
vigorous, and is typically thought of to be of order 10-5m2/s.
15.5.1 The Brunt-Vaisala Frequency
Most wisdom tends to view vertical turbulent diffusivity as being an inverse function of the Brunt-Vaisala (spelling optional) frequency "N". This is a measure (in inverse time units) of the stratification or resistance to turbulence. The derivation of this concept is actually quite simple. Imagine displacing a fluid particle vertically in a stably stratified water column. If the particle were to be lifted a distance delta-Z, it would be denser than the surrounding water and experience a downward force related to the difference in density between it and the surrounding water:
which in turn would be a function of the distance of displacement and the density gradient. The equation applies for negative displacements as well, since the particle would be more buoyant than the surrounding water, and want to "cork" back up. Now this restoring force which is proportional to distance should look familiar, to those of you who have taken high school physics: it is the spring equation. If we used the fact that F=ma (a is the acceleration), then we can rewrite the equation (using the density rho in place of the mass) to have:
where the terms in front of the "z" on the right hand side of the equation can be expressed as a constant, which we'll call N2. The solution to this equation is simply
a simple harmonic oscillator with frequency N,
the Brunt-Vaisala Frequency. That is, in the absence of dissipation, if you were to "pluck" a fluid particle above its equilibrium position, it would return to its position, but overshoot, oscillating back and forth with a frequency "N".
This frequency is a measure of the water column stability, and is typically a few cycles per hour (10-3 Hz) in the main thermocline. Surveys by Sarmiento et al (1976, E.P.S.L. 32, 357-370), Gargett (1984., J. Mar. Res. 42, 359-393) and Gregg (1987, J. Geophys. Res. 92, 5249-5286) attempt to correlate apparent diffusivities with N and other parameters. The problem is extraordinarily complex, and not well resolved, but the following may be regarded as the typical ranges of diffusivities:
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