Eddy diffusion
This causes the fluid properties to homogenize on scale larger than that of eddies responsible for stirring, in a very efficient way compared to individual molecular motion.
The problem with turbulent diffusion in the atmosphere and beyond is that there is no single model drawn from fundamental physics that explains all its significant aspects.
In addition, computational approaches may be classified as continuous-motion or discontinuous-motion theories, depending on whether they assume that particles move continuously or in discrete steps.
In particular, these studies include eddy diffusion mechanisms to explain processes from aerosols deposition[9] to internal gravity waves in the upper atmosphere,[10] from deep sea eddy diffusion and buoyancy[11] to nutrient supply to the surface of the mixed layer in the Antarctic Circumpolar Current.
[12] Source:[13][14] In this section a mathematical framework based on continuity equation is developed to describe the evolution of concentration profile over time, under action of eddy diffusion.
The field measures the concentration of a passive conserved tracer species (could be a coloured dye in an experiment, salt in the sea, or water vapour in the air).
It exchanges fluid (and with it the tracer) with its surroundings via turbulent eddies, which are fluctuating currents going back and forth in a seemingly random way.
Source:[3] The subsection aims for a simple, rough and heuristic argument explaining how the mathematics of gradient diffusion arises.
This argument can be seen as a physically motivated dimensional analysis, since it uses solely the length and velocity scales of an eddy to estimate the tracer flux that it generates.
Source:[13] This subsection builds on the section on general mathematical treatment, and observes what happens when a gradient assumption is inserted.
[13] They are simple and mathematically convenient, but the underlying assumption on purely down-gradient diffusive flux is not universally valid.
The statistical theory of fluid turbulence comprises a large body of literature and its results are applied in many areas of research, from meteorology to oceanography.
[19] The statistical approach to diffusion is different from gradient based theories as, instead of studying the spacial transport at a fixed point in space, one makes use of the Lagrangian reference system and follows the particles in their motion through the fluid and tries to determine from these the statistical proprieties in order to represent diffusion.
Eddy activity that enables this mixing continuously dissipates energy, which it lost to smallest scales of motion.
In a few narrow, sporadic regions at high latitudes surface water becomes unstable enough to sink deeply and constitute the deep, southward branch of the overturning circulation [20] (see e.g. AMOC).
Eddy diffusion, mainly in the Antarctic Circumpolar Current, then enables the return upward flow of these water masses.
[22] Hence the efficiency of turbulent mixing in sub-Antarctic regions is the key element which sets the rate of the overturning circulation, and thus the transport of heat and salt across the global ocean.
Eddy diffusion also controls the upwelling of atmospheric carbon dissolved in upper ocean thousands of years prior, and thus plays an important role in Earth's climate system.
Convergent Ekman transport in subtropical gyres turns these into regions of increased floating plastic concentration (e.g. Great Pacific Garbage Patch).
[23] In addition to the large-scale (deterministic) circulations, many smaller scale processes blur the overall picture of plastic transport.
Mesoscale eddies are slowly rotating vortices with diameters of hundreds of kilometers, characterized by Rossby numbers much smaller than unity.
Anticyclonic eddies (counterclockwise in the Northern hemisphere) have an inward surface radial flow component, that causes net accumulation of floating particles in their centre.
[11] Sub-mesoscale vortices and ocean fronts are also important, but they are typically unresolved in numerical models, and contribute to the above-mentioned stochastic component of the transport.
Integration of this instantaneous-point-source solution with respect to space yields equations for instantaneous volume sources (bomb bursts, for example).
yield different solutions.As an example, K theory is widely used in atmospheric turbulent diffusion (heat conduction from the earth's surface, momentum distribution) because the fundamental differential equation involved can be considerably simplified by eliminating one or more of the space coordinates.
[27] Having said that, in planetary-boundary-layer heat conduction, the source is a sinusoidal time function and so the mathematical complexity of some of these solutions is considerable.
Calder[28] studied the applicability of the diffusion equation to the atmospheric case and concluded that the standard K theory form cannot be generally valid.
One of them is the study by Barad[26] where he a K theory of the complicated problem of diffusion of a bent-over stack plume in very stable atmospheres.
It can be seen in the animation in the introductory section that eddy-induced stirring breaks down the black area to smaller and more chaotic spatial patterns, but nowhere does any shade of grey appear.
In practice, concentration is defined using a very small but finite control volume in which particles of the relevant species are counted.
