Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 68

have an excess density relative to its new surroundings of D r ¼ z ( @ r/ @ z) and will

experience a buoyancy force b on a unit volume of:

g @
@

¼ ¼ : ð : Þ

b g

z z

3 36

Imagine what happens to the water particle. It experiences this downward force,
attempting to restore the particle to its stable position in the density profile.
The acceleration will result in the particle achieving its maximum downward speed
as it reaches this stable point, and so it overshoots deeper into the density profile.
Now the particle is less dense than its surroundings, and so it is forced back upward
in an attempt to reach its stable position. Again it overshoots, and the cycle repeats.
Mathematically, if this is the only force acting, the motion of the particle, which has
density r, will be described by:

D 2 z
Dt 2 ¼

Dw
Dt ¼

g @

@

N 2 z

z z

¼ : ð 3 : 37 Þ

Again, this is the equation of simple harmonic motion, this time describing how the
particle will oscillate vertically at the buoyancy frequency N given by:

s

@
@

g

N ¼ : ð 3 : 38 Þ

z

Free, unforced vertical oscillations in the water column are restricted to frequencies
lower than N and this sets an upper limit to the possible frequencies of internal
waves, which are dealt with in Section 4.2.4 . In the absence of friction the particle will
simply keep oscillating; in reality we expect the oscillation to be damped as energy is
lost to friction. The square of the buoyancy frequency N 2 is widely used as a measure
of the stability of the water column, which, as we shall see in the next chapter, plays
an important role in inhibiting turbulent motions.

3.5 Turbulent stresses and Ekman dynamics

...................................................................................
Now we introduce friction, which plays a key role in the equations of motion. We will
first approach this via a classical problem first solved by the Swedish oceanographer
VagnWalfrid Ekman in 1905 (Ekman, 1905 ). The problem arose from a question posed
by Fridtjof Nansen who had observed that icebergs in the Arctic did not move in the
same direction as the wind, but instead followed a path to the right of the wind direction.

3.5.1 Current structure and transport in the Ekman layer

Ekman asked what would be the form of the steady state current profile in an ocean
forced by a steady wind if the only forces acting are frictional stresses between layers
and the Coriolis force. In this scenario, the dynamical equations (3.13) simplify to: