Free waves
165
or
σ
2 −
fks
σL
C
2
0 − C
2
0 (k
2 +
n 2 π 2
L 2 +
f 2
C 2
0
)=0.
(7.36)
Equation (7.36) is a third order polynomial and hence there are three roots. For
s =0we again find the Poincar´ e waves and because now |σ| >fthese waves
are slightly modified due to the topography. With σ = O(f ), the second term
in (7.36) is small when s → 0. The only way in which this term can be of any
importance is when σ = O(s); in that case, the terms with σ 2 can be neglected.
The new class of waves, the so-called Rossby waves, therefore has a dispersion
relation given by
σ = −
skf
L
1
k 2 +
n 2 π 2
L 2 +
f 2
C 2
0
,n =1, 2, 3,...
(7.37)
and hence they are low-frequency waves (frequency much smaller than f ).
In this situation, the existence of the waves depends on the variations of the
topography. In addition, the waves occur only in a rotating fluid. In the physical
mechanism of propagation of the waves, both these elements are crucial. The
special character of the waves is that their phase speed in the x-direction (C x =
σ/k) is always negative. Because the fluid depth decreases with y, when s>0,
an observer moving with the wave always sees a deeper layer to the left.
Ex. 7.4
For a Rossby wave, it follows that | σ|≪f which confirms that the period
is much larger than 1/f . Hence, these waves are prototypes for explaining lowfrequency variability in the ocean and atmosphere. The corresponding eigenfunction ˆ
η(y) is
ˆ
η(y)=η 0 sin
nπy
L
+ O(s),
(7.38)
which follows from (7.32). The n =0case also gives a physically meaningful
wave; it follows from (7.38) that ˜
v =0and boundary conditions can be satisfied.
For Rossby waves the time derivatives are an order of magnitude smaller than
the spatial derivatives, for example ∂ 2 /∂t 2 = − σ 2 ≪ f 2 ), and the equations
(7.3a-b) reduce to
f ˜
u = −g
∂η
∂y
,
(7.39a)
f ˜
v = g
∂η
∂x
.
(7.39b)
Hence, the horizontal velocity field is in geostrophic balance, while the propagation of the wave is controlled by ageostrophic processes.
To understand the propagation mechanism, we return to the general equation
(7.9) which, for H 0 (y) and |σ|≪f reduces to
∂
∂t
(f
2 η − gH 0 ∇
2
H η − g
∂H 0
∂y
∂η
∂y
)+fg
∂H 0
∂y
∂η
∂x
=0.
(7.40)
165
or
σ
2 −
fks
σL
C
2
0 − C
2
0 (k
2 +
n 2 π 2
L 2 +
f 2
C 2
0
)=0.
(7.36)
Equation (7.36) is a third order polynomial and hence there are three roots. For
s =0we again find the Poincar´ e waves and because now |σ| >fthese waves
are slightly modified due to the topography. With σ = O(f ), the second term
in (7.36) is small when s → 0. The only way in which this term can be of any
importance is when σ = O(s); in that case, the terms with σ 2 can be neglected.
The new class of waves, the so-called Rossby waves, therefore has a dispersion
relation given by
σ = −
skf
L
1
k 2 +
n 2 π 2
L 2 +
f 2
C 2
0
,n =1, 2, 3,...
(7.37)
and hence they are low-frequency waves (frequency much smaller than f ).
In this situation, the existence of the waves depends on the variations of the
topography. In addition, the waves occur only in a rotating fluid. In the physical
mechanism of propagation of the waves, both these elements are crucial. The
special character of the waves is that their phase speed in the x-direction (C x =
σ/k) is always negative. Because the fluid depth decreases with y, when s>0,
an observer moving with the wave always sees a deeper layer to the left.
Ex. 7.4
For a Rossby wave, it follows that | σ|≪f which confirms that the period
is much larger than 1/f . Hence, these waves are prototypes for explaining lowfrequency variability in the ocean and atmosphere. The corresponding eigenfunction ˆ
η(y) is
ˆ
η(y)=η 0 sin
nπy
L
+ O(s),
(7.38)
which follows from (7.32). The n =0case also gives a physically meaningful
wave; it follows from (7.38) that ˜
v =0and boundary conditions can be satisfied.
For Rossby waves the time derivatives are an order of magnitude smaller than
the spatial derivatives, for example ∂ 2 /∂t 2 = − σ 2 ≪ f 2 ), and the equations
(7.3a-b) reduce to
f ˜
u = −g
∂η
∂y
,
(7.39a)
f ˜
v = g
∂η
∂x
.
(7.39b)
Hence, the horizontal velocity field is in geostrophic balance, while the propagation of the wave is controlled by ageostrophic processes.
To understand the propagation mechanism, we return to the general equation
(7.9) which, for H 0 (y) and |σ|≪f reduces to
∂
∂t
(f
2 η − gH 0 ∇
2
H η − g
∂H 0
∂y
∂η
∂y
)+fg
∂H 0
∂y
∂η
∂x
=0.
(7.40)