Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 113

converges towards the Gaussian form of Equation (4.42) . Hence tracking of particles
moving with appropriately chosen t produces a dispersion with the same

mean properties as the Gaussian patch solution of the diffusion equation. This
convergence of particle tracking and the analytical solution of the diffusion equation
is demonstrated in the supporting software at the topic website.
Notice that individual particle tracking experiments differ from each other in a
way that is more representative of the random reality of turbulent diffusion than the
diffusion equation approach. Particle tracking has the further advantage of allowing
advection by the mean flow to be readily combined with diffusion. In particular, the
random walk approach has important applications in the modelling of the move-
ments and growth of phytoplankton, as we shall see in later chapters.

D x and D

4.3.6 When is diffusion in the ocean Fickian?

The solutions of the Fickian diffusion equation and the corresponding random walk
methods discussed in the last section provide valuable reference models for the study
of mixing in the shelf seas, but their rigorous application requires that the mixing
is Fickian, i.e. it has a constant K and we know the appropriate value of K. A wide
range of studies of ocean mixing, however, indicates that the diffusivity is generally
not constant but tends to increase with the scale of the dispersing patch. In a classic
work, Okubo (Okubo, 1971 ) compiled data from a large number of horizontal dye
diffusion experiments at different scales in conditions where the diffusion was effect-
ively two-dimensional because horizontal movement was unrestricted by boundaries
and vertical diffusion was confined to the surface layer by stratification. Figure 4.12
shows Okubo's log-log plot of horizontal dispersion K H against the scale of the
diffusing patch L which indicates an increase in the diffusivity with K H /

L 1.15 .
This result is close to the scale dependence of L 4/3 proposed on empirical grounds by
Richardson (Richardson and Stommel, 1948 ) and consistent with dimensional analysis
for isotropic turbulence (see later in Section 4.4.3 ). The scale dependence of diffusion
changes the time course of dispersion considerably. Okubo found that the variance
of patch size increases as t m with m

1.
In shelf seas, there is an important exception to this picture of non-Fickian
behaviour. The principal form of horizontal dispersion in tidally dominated situ-
ations is essentially Fickian and arises from the interaction of vertical shear of the
horizontal velocity with vertical mixing. As we saw in Chapter 3 , the flow driven by
the tides moves particles around paths which are elliptical. In strong tidal flows,
particle excursions may extend tens of kilometres from their mean position. Such
movements decrease with depth because of frictional drag at the seabed, so there is
generally a strong vertical shear even when the water column is homogeneous in
density. In stratified conditions, the vertical shear may be considerably enhanced.
Vertical shear in the tidal currents interacts with vertical turbulent diffusion to bring
about horizontal dispersion. The essential mechanism was originally described for
the case of steady flow in pipes by Taylor (Taylor, 1953 ; Taylor, 1954 ) and applied to

∼ 2.3 in contrast to the Fickian case, where m ¼