Geography Reference
In-Depth Information
Then from (8.17) a disturbance consisting of these two modes can be
expressed as
ψ
m
=
A
1
exp [ik (x
−
c
1
t)]
+
A
2
exp [ik (x
−
c
2
t)]
(8.72a)
ψ
T
=
B
1
exp [ik
(
x
−
c
1
t
)
]
+
B
2
exp [ik
(
x
−
c
2
t
)
]
(8.72b)
but from (8.18)
c
1
A
1
−
U
T
B
1
=
0
;
c
2
A
2
−
U
T
B
2
=
0
So that with the aid of (8.71)
B
1
=
µA
1
;
B
2
=−
µA
2
(8.73)
For an initial disturbance confined entirely to the upper level, it is easily verified
that ψ
m
=
ψ
T
(ψ
1
=
2ψ
m
and ψ
3
=
0). Thus, initially A
1
+
A
2
=
B
1
+
B
2
, and
substituting from (8.73) gives
A
2
=−
A
1
[(1
−
µ) / (1
+
µ)]
Hence, if A
1
is real, the streamfunctions of (8.72a,b) can be expressed as
A
1
cos [k (x
µU
T
t)]
(1
−
µ)
ψ
m
(x, t)
=
−
µU
T
t)]
−
µ)
cos [k (x
+
(1
+
cos kx cos (kµU
T
t)
µ
sin kx sin (kµU
T
t)
(8.74a)
2µA
1
(1
1
=
+
+
µ)
µA
1
cos [k (x
µU
T
t)]
(1
−
µ)
ψ
T
(x, t)
=
−
µU
T
t)]
+
µ)
cos [k (x
+
(1
+
µ
cos kx cos
kµU
T
t
+
µ sin kx sin
kµU
T
t
2µA
1
1
(8.74b)
=
+
The first forms of the right side in (8.74a,b) show that for small µ the barotropic
mode initially consists of two waves of nearly equal amplitude that are 180
◦
out
of phase so that they nearly cancel, whereas the baroclinic mode initially consists
of two very weak waves that are in phase.
As time advances the two oppositely propagating barotropic modes begin to
reinforce each other, leading to a maximum amplitude trough 90
◦
to the east of the
