Geoscience Reference
In-Depth Information
us with an effective way of determining the mean baroclinic flow field in high gradient
regions like fronts, even in the presence of oscillatory tidal currents.
3.4
Fundamental oscillatory motions: what a water particle
does if you give it a push
...................................................................................
We next examine two simple solutions of the dynamical equations which describe
the motions that result when a particle is impulsively accelerated, i.e. when the
particle is given a short, sharp push. This applies, for instance, to the pulse of wind
stress felt by the sea surface as a storm passes. We need to consider separately the
two cases of vertical and horizontal motion, both of which involve an oscillatory
response at a fundamental frequency of the system.
3.4.1
Inertial oscillations
The ocean is again assumed to be frictionless, but this time we set the pressure
gradient terms to zero and assume that we are dealing with a slab layer in which
the motion does not vary with depth. The balance in the equations of motion is now
between the acceleration of a particle and the Coriolis force acting on it:
Du
Dt ¼
Dv
Dt ¼
fv
;
fu
ð
3
:
31
Þ
Differentiating the x equation with respect to time and substituting from the y
equation gives:
D 2 u
Dt 2 ¼
f 2 u
:
ð
3
:
32
Þ
This is an equation describing simple harmonic motion with angular frequency f.
If the particle is given an initial velocity U along the x direction at t
¼
0, the
subsequent motion is given by:
1
f
Du
Dt ¼
¼
U 0 sin ft
;
v
¼
U 0 cos ft
ð
3
:
33
Þ
u
which describes circular motion at a constant speed of U 0 and with an inertial period:
2
f ¼
2
12
sin f L
T I ¼
sin f L ¼
hours
ð
3
:
34
Þ
2
where
is the angular frequency of the Earth's rotation (see Section 3.2.1 ) . So a
particle, impulsively accelerated to a speed U, will describe circular motion with
a period set by the latitude. Such circular motions are called inertial oscillations.
The radius of the circular motion r I is the perimeter of the circle divided by 2p:
O
U 0 T I
2
U 0
f :
r I ¼
¼
ð
3
:
35
Þ
Search WWH ::




Custom Search