When a random motion is superposed upon the velocity field, a group of particles will be subject to advection-diffusion. A group of foreign particles initially arranged in a circle surrounding a singularity will be deformed into an ellipse the area of which increases or decreases with time depending on the sign of the divergence. The linear velocity field thus considered consists of the rates of stretching deformation and shearing deformation, vorticity, and divergence (or negative convergence). In the neighbourhood of the singularity, the velocity field is approximated linearly in terms of coordinates whose origin is taken at the singularity. The simplest and most important singularities in oceanography are points of convergence (or divergence), neutral points, and lines of convergence (or divergence). Except that at very short times, the distribution of advected impurities trace neither the density nor the velocity of the advecting flow, even for delta only slightly different from one.Ī singular point in a two-dimensional velocity field is a stationary point of flow. At intermediate times, the particle distribution displays complex caustic structures. For a time-periodic streamfunction, the heavy particles undergo chaotic trajectories and standard diffusion light particles may either display chaotic behavior and diffusion or regular motions ending with periodic trajectories, depending upon the values of the control parameters of the Eulerian flow.
These features bear several resemblances to the motion of fluid particles in the presence of additive noise.
For heavy impurities in weakly anisotropic velocity fields, the diffusion coefficients in the x and y directions may be different by several orders of magnitude. For isotropic velocity fields, the diffusion coefficients display a scaling dependence upon the parameter epsilon= 1-delta. Particles denser than the fluid exhibit chaotic trajectories and standard diffusion at large times. For a stationary streamfunction, impurities that are lighter than the fluid undergo regular motions and converge to the centers of the advection cells. The impurity dynamics strongly depends on the parameter delta=rho_f/rho_p. The motion of passively advected impurities with density rho_p different from the fluid density rho_f in simple models of two-dimensional velocity fields is studied. Possible origin of the so-called `vital component' (a factor that shows how much slower viable populations sink than morphologically similar senescent or dead ones) is discussed, as is the role of form resistance in evolution of high diversity of plankton morphologies. The highest value (Φ = 8.1) was established for a 20-celled Fragilaria crotonensis chain. Flagellates (Rhodomonas, Gymnodinium) had a Φ Ceratium (Φ = 1.61) proved an exception among flagellates: in most forms tested in this study (ellipsoid flagellates, Staurastrum forms with no or very short protuberances, and Cosmarium forms), Φ > 1.
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For a series of Staurastrum forms, a significant positive correlation was found between arm-length/cell-width ratio and Φ: protuberances increased form resistance. Increasing number and length of spines on Tetrastrum coenobia substantially increased Φ. However, for Pediastrum coenobia with symmetric/asymmetric fenestration, no difference was observed with respect to symmetry. In all cases, symmetric forms had considerably higher form resistance than asymmetric ones. The effect of symmetry on Φ was tested in 1–6-celled Asterionella colonies, having variable angles between the cells, and in Tetrastrum staurogeniaeforme coenobia, having different spine arrangements. For Fragilaria crotonensis chains, no such upper limit to Φ was observed in chains of up to 20 cells (longer ones were not measured). Form resistance of Asterionella colonies increased from single cells up to 6-celled colonies than remained nearly constant. Coiling decreased Φ in filamentous forms. For cylindrical forms, a positive relationship was found between Φ and length/width ratio. Form resistance factors of PVC models of phytoplankton sinking in glycerin were measured in a large aquarium (0.6 × 0.6 × 0.95 m). Form resistance (Φ) is a dimensionless number expressing how much slower or faster a particle of any form sinks in a fluid medium than the sphere of equivalent volume.
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