Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 128

where s(x, y, t) is the concentration of the substance. Show that, if we can
express s as a product of two functions s s 1 (x, t)s 2 ( y, t), the diffusion

equation can be separated into two equations, one in x only, the other in y
only, which have the same form as the 1D equation. Hence, show that the radial
spreading of a mass M s of dye dumped at x

¼

¼ y ¼ 0 is described by:

M s

4pKt e r 2

= 4Kt

s ð r ; t Þ¼

p

x 2 where r ¼ þ y 2 is distance from the origin and t is time from moment of

dye release.

4.4. A dump of 25 kg of a conservative dye is spreading, by diffusion only, at a point
in the shelf seas where the depth is 35 metres and the water column is well mixed.
Using the result of the previous problem, estimate the radius of the patch and
the maximum volume concentration 5 hours after the release if the effective
diffusivity is K 12 m 2 s 1 .

At what time will the local maximum concentration occur 2 km from the release
point, and what will be its value at this time? (Take the patch radius to be that of
a circle which contains 95% of the dye.)

¼

4.5. Two layers of uniform density in the ocean are moving relative to each other at a
velocity which increases from zero to a steady speed of 25 cm s 1 . If the interface
between the layers has a thickness of 12.5 cm and the layers differ in density by
1.5 kg m 3 , determine the relative velocity at which the initially laminar flow will
start to become unstable.
Describe the subsequent development of the motion.