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which has the same form as the barotropic case but with the important difference that
the cross-shore length scale is now the internal Rossby radius Ro 0 = c 0 /f in which
c 0 ¼
p
g 0 h 1 h 2
is the speed of long internal waves. As c 0 =
1, Ro 0
h 1 þ
h 2 Þ
c
is much
smaller than Ro with values of order Ro 0
10 km.
With wind stress directed to the south (t w is negative, as in Fig. 3.10b ), there is a steady
offshore movement in the surface layer which results in an increase in the interface height
B at the coast and out to a distance of a few times Ro 0 . If the wind stress is maintained,
this upwelling motion will continue, eventually exposing the cold lower layer at the
surface and reducing the sea surface temperature. At the same time, nutrients, which
are usually much more abundant in the lower layer, are brought to the surface with
important implications for primary production which we will discuss in Chapter 10 .
3.6
Long waves and tidal motions
......................................................................................................................
As we noted in Section 2.5 , most tidal energy input to the shelf seas does not come
from the direct action of the tidal generating force but is delivered to the shelf from
the deep ocean in the form of shallow water waves; i.e. waves with wavelengths large
compared with the water depth. Also, we know that while the open ocean tidal range
is
1 metre, in shelf seas we see tidal ranges of up to an order of magnitude larger.
Clearly something interesting happens to the tidal waves as they progress from the
deep ocean into the shelf seas. Because of their importance in this context, we shall
develop the theory of long waves, starting from the equations of motion for the case
of waves in a non-rotating system. We shall then extend the theory to a rotating
system and show how the shelf seas respond to the incoming tidal waves from the
deep ocean. The introduction to the basic concepts of wave motions here should also
provide a useful foundation for the fuller discussion of surface and internal wave
motions which follows in Chapter 4 .
3.6.1
Long waves without rotation
We start with the relatively simple case of waves of long period T p and wavelength l
which is large compared with the depth. For such waves, the vertical acceleration
Dw/Dt is small compared with g so that the pressure is given by the hydrostatic
relation in Equation (3.15) . We shall further assume:
(i) the density of the fluid is constant
(ii) the mean water depth h is constant
(iii) the waves propagate in the x direction so that
y terms are zero
(iv) the motion is in a non-rotating frame (Coriolis terms can be neglected)
(v)
@
/
@
the motions are small, so the equations can be linearised (D/Dt
@
/
@
t)
With these assumptions, the x momentum Equation (3.13) and the continuity
Equation (3.3) reduce to:
@
u
@
g @
@
@
@
h @
u
t ¼
x ;
t ¼
x :
ð
3
:
57
Þ
@
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