Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 127

of turbulent eddies are discussed in terms of Kolmogorov theory of energy transfer
from large eddies to progressively smaller ones through the multiple steps of a
cascade. The small scale eddy limit is reached when the energy is dissipated to heat
at the microscale (

mm). An upper limit for the scale of isotropic eddies in a
stratified water column (the Ozmidov length) is set by the competition between the
energy dissipation rate and the buoyancy forces.

∼

FURTHER READING

The Dynamics of the Upper Ocean, by Owen M. Phillips, Cambridge University Press, 1966.
[Chapters 4 and 5]
An Introduction to Ocean Turbulence, by Steve A. Thorpe, Cambridge University Press, 2007.
Marine Turbulence: Theories, Observations and Models, edited by H. Z. Baumert, et al.,
Cambridge University Press, 2005.

Problems

....................................................................................................................
4.1. Using expressions from Equation (4.9) for the particle velocities u and w of
surface waves in deep water (l

h), show that the kinetic energy of the waves

per unit area is given by

gA 0
4
where A is the wave amplitude. Find an equivalent formula for the average
potential energy per unit area V w and hence determine the total energy density of
such waves. If the wave amplitude A 0 ¼

T w ¼

1.5 metres, estimate the rate of energy
transport by the waves given that the wave period T p ¼

6s.

4.2. Consider a train of internal waves propagating horizontally in a two-layer
system like that depicted in Fig. 4.1 . Using the velocity potential functions for
the two layers (Equation 4.17) , show that maximum horizontal velocity differ-
ence between top and bottom layers at the interface is given by:

u ¼ oA 0 ð coth kh 1 þ coth kh 2 Þ:

Determine D u for waves of amplitude A 0 ¼ 5 metres, wavelength 1.2 km and

period 20 minutes if h 1 ¼ 25 metres and h 2 ¼ 65 metres.

4.3. 2D Fickian diffusion in x and y of a conservative substance with no advection
(U

0) is described by a reduced version of the advection-diffusion
Equation (4.40) , namely:

¼ V ¼

2 s 2 s

@ s

K @

x 2 þ @

t ¼ @ @ @ y 2