Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 106

it mixes water properties and how we can measure and quantify it. The Reynolds
number is also a vital part of understanding the mechanics of plankton life. In the
Reynolds number definition above, we can replace the current speed with a plankton
sinking or swimming speed, and the relevant lengthscale is the size of the plankton.
In anticipation of the discussion of the Reynolds number in Chapter 5 , let us assume
a plankton cell diameter of 10 m and a swimming speed of 0.1 mm s 1 . Setting

the kinematic viscosity of seawater at 1.1

m

10 6 m 2 s 1 yields a Reynolds number
of 10 3 . Hence life for the plankton is dominated by viscous forces; we will look at
some of the consequences of this later in Chapter 5 .
The randommotions of turbulence may be thought of as a combination of eddies of
different scales, and it is the combined effect of these eddies which is responsible for the
dispersive tendency of turbulence which brings about mixing of fluid properties. If we
label two particles in a turbulent fluid and follow their subsequent motion, we will see
that their separation, while fluctuating, tends to increase with time. The rapid disper-
sion which results from turbulence contrasts sharply with the laminar flow situation
where the transfer of fluid properties is limited to molecular diffusion, which is
generally a very slow process. Molecular diffusion is governed by Fick's law in which
the flux J of a fluid property is related to the gradient of its concentration

@ c/ @ n by:

J ¼ k m @ c =@ n ð 4 : 27 Þ

where k m is the molecular diffusivity which is generally a small quantity. For
example, the diffusion of heat in water has a molecular diffusivity k m ¼

10 7 1.4

m 2 s 1 . If there were no turbulence in a shelf sea of depth h

¼ 100 metres, and all heat

h 2 /k m ∼ exchange was by molecular diffusion, it would take a time 2260 years to

achieve equilibrium. In reality, when turbulence is involved, vertical mixing times are
usually much less than one year.
It is important to distinguish between stirring and mixing and the way that they are
linked by turbulence, a relationship often represented in the process of mixing milk
into a cup of coffee. Stirring is the action of introducing mechanical energy into the
flow (the movement of the spoon), while mixing is the irreversible inter-mingling of the
milk and the coffee fluid at the molecular level. The two are linked by the suite of
turbulent eddies which act to stretch out the interface between milk and coffee fluids
while maintaining large gradients at the interface. In this way, turbulence creates a large
interface area and high gradients which accelerates mixing by molecular diffusion.
As well as being dispersive, turbulence is also necessarily dissipative, i.e. it converts
kinetic energy of motion into heat. This happens as energy is progressively trans-
ferred from larger to smaller and smaller scales, ending up at the smallest scales
(typically millimetres) where molecular viscosity becomes important. In the smallest
eddies, there are relatively large velocity gradients, which result in correspondingly
large stresses between fluid particles given by the stress law for molecular viscosity:

∼

n @

u

t m ¼ ð 4 : 28 Þ

@ n