1.2 Wave Equations in Geophysical Fluid Dynamies
15
0
fo/2
)
0
v = o.
which transfonns (1.28) to
-fo
o
0 0 c
ox
0
-fo
The characteristics for this system are the curves satisfying dx Idt =
dx ldt = O.
Wave solutions to (1.28) have the fonn
(u ' , v , , h') = ro{(
in
h) 0
±c and
(1.29)
provided that the frequency wand wave number k satisfy the dispersion relation
(1.30)
as may be demonstrated by substituting (1.29) into (1.28). Lines of constant phase ,
such as the locations of the troughs and crests, propagate at the phase speed io] k,
which from (1.30) is
W
-=±c
f o
2 ) 1/2
1 + -
k
(
c 2 k 2
A compact group ofwaves travels at the group velocity owlok, which can also be
computed from (1.30):
Bco
-=±c
ok
f o
2 )-1/2
1 + -
c 2 k 2
(
In the limit Ikl » folc, the phase speed and group velocity both approach the
slope of a characteristic along which Idxldtl = c. Nevertheless, for any finite
value of k,
and neither the lines of constant phase nor the wave groups follow trajectories
that coincide with the characteristic curves. Note that the magnitude of the group
velocity, which is the rate at which energy propagates in a wave, is bounded by c.
The maximum rate of energy propagation can therefore be detennined without
considering the zero-order coefficient matrix B in (1.28) .
The loose connection between wave propagation and the characteristics in the
preceding example can disappear altogether if the Coriolis parameter is a function of the spatial coordinate. Then a second type of wave, the Rossby wave,
may appear as an additional solution . If f increases linearly in proportion to y,
Rossby -wave solutions may exist with phase speeds in the negative-r direction
(Holton 1992; Pedlosky 1987). Neither the phase speeds nor the group velocities
of these waves have any relation to the characteristic curves . It is not surprising
15
0
fo/2
)
0
v = o.
which transfonns (1.28) to
-fo
o
0 0 c
ox
0
-fo
The characteristics for this system are the curves satisfying dx Idt =
dx ldt = O.
Wave solutions to (1.28) have the fonn
(u ' , v , , h') = ro{(
in
h) 0
±c and
(1.29)
provided that the frequency wand wave number k satisfy the dispersion relation
(1.30)
as may be demonstrated by substituting (1.29) into (1.28). Lines of constant phase ,
such as the locations of the troughs and crests, propagate at the phase speed io] k,
which from (1.30) is
W
-=±c
f o
2 ) 1/2
1 + -
k
(
c 2 k 2
A compact group ofwaves travels at the group velocity owlok, which can also be
computed from (1.30):
Bco
-=±c
ok
f o
2 )-1/2
1 + -
c 2 k 2
(
In the limit Ikl » folc, the phase speed and group velocity both approach the
slope of a characteristic along which Idxldtl = c. Nevertheless, for any finite
value of k,
and neither the lines of constant phase nor the wave groups follow trajectories
that coincide with the characteristic curves. Note that the magnitude of the group
velocity, which is the rate at which energy propagates in a wave, is bounded by c.
The maximum rate of energy propagation can therefore be detennined without
considering the zero-order coefficient matrix B in (1.28) .
The loose connection between wave propagation and the characteristics in the
preceding example can disappear altogether if the Coriolis parameter is a function of the spatial coordinate. Then a second type of wave, the Rossby wave,
may appear as an additional solution . If f increases linearly in proportion to y,
Rossby -wave solutions may exist with phase speeds in the negative-r direction
(Holton 1992; Pedlosky 1987). Neither the phase speeds nor the group velocities
of these waves have any relation to the characteristic curves . It is not surprising