(7.141)
390
7. Physically Insignificant Fast Waves
and that there is a mean north-south gradient in the bottom topography
equal to the mean gradient in the height of the free surface, oh/oy, so that
the mean fluid depth is a constant H . The linearized shallow-water equations for this system are
ot
+ U i-) s+
ox
18 + ßv = 0,
ot
+ Ui-)
ox
8 _ Is + ßu + g (02
h
+ 02
h) = 0,
ox 2 oy2
Bt
Bx
The terms involving ß in the preceding vorticity and divergence equations
can be approximated10 as
( - + U -
0
0 ) S+ 108+ -
ßg oh
- = 0,
(7.142)
ot
ox
loox
( - + U -
ot
0
0 ) 8 - los + g
ox
(02h - ox 2 + - 02h) = O.
oy2
(7.143)
(a) Show that waves of the form
(s, 8, h) = (so, 80, hO)e i(kxHy-wl)
are solutions to the preceding system if they satisfy the dispersion relation
where c
2
= gH .
(b) Show that if ß/ c « k
2 , the individual solutions to this dispersion relation are well-approximated by the solutions to either the inertial-gravitywave dispersion relation
or the Rossby-wave dispersion relation
w=Ukßk
k
2
+ f) + IJ/c
2
.
IOThe approximations used to obtain (7.142) and (7.143) are motivated by the desire to obtain a
clean dispersion relation rather than a straightforward scale analys is.