Abstract
This chapter proposes a novel method for the modeling of turbulent dispersion in the absence of buoyancy effects. Starting from a few salient observations, properties that an effective model should possess are identified, and a model is subsequently developed to incorporate these necessary properties. The model turns out to make use of fractional calculus and leads to a non-local operator, which is challenging from a computational perspective. Applications to dispersion by turbulent jets (round and planar) and the marine Ekman layer (surface and bottom) demonstrate the usefulness of the model.
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Notes
- 1.
Because buoyancy affects the vertical dispersion of hot plumes, it is more instructive when considering mechanical dispersion alone to use the horizontal view provided by aerial or satellite imagery than ground-level visualization that inevitably looks sideways.
- 2.
Note the subtlety in notation: straight \(\texttt{k}\) for the turbulent kinetic energy, now italic k for the wavenumber magnitude, and, a bit later, bold \(\textbf{k}\) for the wavenumber vector.
- 3.
The treatment of boundaries is delicate and will be addressed later in the context of one-dimensional modeling.
- 4.
We assume here velocity distributions with zero mean. Adding a mean component \(\bar{\mathbf{{u}}}\) to \(\mathbf{{u}}\) is relatively straightforward and yields the expected advection term \(\bar{\mathbf{{u}}} \cdot \nabla c\) added to the time rate of change \(\partial c/\partial t\).
- 5.
For shear turbulence case, the value of C could perhaps be related to the ratio of the most probable velocity magnitude \(|\mathbf{{u}}|\) to \(u_*\).
- 6.
The origin of these additional terms is found in the “bounce-back” condition of the probability density function in a Boltzmann kinetics framework.
- 7.
It can be shown rather easily that there cannot be any pressure force across and down the jet as long as the pressure in the quiescent fluid away from the jet is uniform and radial acceleration is weak, thus ensuring conservation of momentum. Radial acceleration is weak because the jet is much longer than it is wide (so-called “thin jet approximation”). Note that in contrast mass is not conserved in the jet as it entrains fluid form the quiescent surroundings. The amount of tracer in the jet is conserved as long as the ambient fluid is tracer-free. The tracer concentration decreases by dilution with tracer-free fluid.
- 8.
There is evidence (Jischa and Rieke 1979) that the turbulent Prandtl (heat vs. momentum) and Schmidt (tracer vs. momentum) numbers depart from unity only when the molecular Prandtl and Schmidt numbers are much less than one, as in liquid metals, and the dependence of the departure depends on the Reynolds number of the flow, with smaller departures at high Reynolds numbers.
- 9.
Recall that a shear stress is a momentum flux.
- 10.
The word “substance” should be understood broadly here. It may be a passive scalar or an active scalar, such as temperature, or even momentum.
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Acknowledgements
The author thanks Professor Brenden Epps of Dartmouth College for his assistance with the Lévy distributions and their asymptotic behavior, and the editors of this volume for their invitation, encouragements, and support.
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Cushman-Roisin, B. (2022). Turbulent Dispersion. In: Schuttelaars, H., Heemink, A., Deleersnijder, E. (eds) The Mathematics of Marine Modelling. Mathematics of Planet Earth, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-031-09559-7_7
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