166
DYNAMICAL OCEANOGRAPHY
If ∂H 0 /∂y is small, then the third term within the time-derivative can be neglected
with respect to the other two terms. If we use (7.39) in the form
gH 0 ∇
2
H η = H 0 f (
∂˜ v
∂x
−
∂ ˜
u
∂y
)=H 0 f ˜
ζ,
(7.41)
where ˜
ζ is the z-component of the vorticity vector, then it follows
∂
∂t
˜
ζ −
fη
H 0
− ˜
v
∂H 0
∂y
f
H 0
=0.
(7.42)
This again suggests conservation of potential vorticity and this time in terms of
small amplitude motions. Indeed, if we substitute H = H 0 (y)+η, u =˜ u, v =˜ v
and ζ = ˜
ζ into the potential vorticity Π=( ζ + f )/H with DΠ/dt =0then we
find (7.42) through linearization. The general equation (7.9) can thus be derived
from a linearized form of conservation of potential vorticity. Hence, we can use
D
dt
ζ + f
H
=0,
(7.43)
to describe the physical mechanism of propagation of Rossby waves.
Three columns of water which are initially motionless at a latitude y = y 0 are
sketched in Fig. 7.6. Assume that the topography is such that the layer thickness
decreases northward and that the interface does not deform. Now at t =0, column
B is displaced northward as an initial perturbation. Because it moves to a location
where H is smaller (note that f is constant), column B must get a negative relative
vorticity because of conservation of potential vorticity. This anticyclonic movement of column B induces velocities in both columns A and C, such that A moves
northward and C southward. Column C starts to rotate cyclonic (positive vorticity) because it moves into an area where the liquid depth increases, but column
A get a negative vorticity, similar to column B. The velocities induced through
the rotation of both columns A and C on column B drive column B back to the
position y = y 0 . As it passes through this latitude, it moves southward and hence
the next half phase of the cycle can be described as above but with opposite signs
of the motions. Through the description, we see the wave moving westward.
To summarize, we have seen that in a situation of constant f , there exist
several types of gravity waves. The local acceleration is of central importance
to the propagation of ‘normal’ gravity waves and Poincar´ ew a v e s . T h e s ew a v e s
have relatively large phase speeds. In Kelvin waves the local acceleration is
not important, but these waves only exist in the presence of a boundary (and
at the equator, as we will see in chapter 11). Long Kelvin waves have a long
period, but the only waves which are possibly relevant for long time scale variability in the oceans (with frequencies much smaller than f ) are the Rossby waves.
DYNAMICAL OCEANOGRAPHY
If ∂H 0 /∂y is small, then the third term within the time-derivative can be neglected
with respect to the other two terms. If we use (7.39) in the form
gH 0 ∇
2
H η = H 0 f (
∂˜ v
∂x
−
∂ ˜
u
∂y
)=H 0 f ˜
ζ,
(7.41)
where ˜
ζ is the z-component of the vorticity vector, then it follows
∂
∂t
˜
ζ −
fη
H 0
− ˜
v
∂H 0
∂y
f
H 0
=0.
(7.42)
This again suggests conservation of potential vorticity and this time in terms of
small amplitude motions. Indeed, if we substitute H = H 0 (y)+η, u =˜ u, v =˜ v
and ζ = ˜
ζ into the potential vorticity Π=( ζ + f )/H with DΠ/dt =0then we
find (7.42) through linearization. The general equation (7.9) can thus be derived
from a linearized form of conservation of potential vorticity. Hence, we can use
D
dt
ζ + f
H
=0,
(7.43)
to describe the physical mechanism of propagation of Rossby waves.
Three columns of water which are initially motionless at a latitude y = y 0 are
sketched in Fig. 7.6. Assume that the topography is such that the layer thickness
decreases northward and that the interface does not deform. Now at t =0, column
B is displaced northward as an initial perturbation. Because it moves to a location
where H is smaller (note that f is constant), column B must get a negative relative
vorticity because of conservation of potential vorticity. This anticyclonic movement of column B induces velocities in both columns A and C, such that A moves
northward and C southward. Column C starts to rotate cyclonic (positive vorticity) because it moves into an area where the liquid depth increases, but column
A get a negative vorticity, similar to column B. The velocities induced through
the rotation of both columns A and C on column B drive column B back to the
position y = y 0 . As it passes through this latitude, it moves southward and hence
the next half phase of the cycle can be described as above but with opposite signs
of the motions. Through the description, we see the wave moving westward.
To summarize, we have seen that in a situation of constant f , there exist
several types of gravity waves. The local acceleration is of central importance
to the propagation of ‘normal’ gravity waves and Poincar´ ew a v e s . T h e s ew a v e s
have relatively large phase speeds. In Kelvin waves the local acceleration is
not important, but these waves only exist in the presence of a boundary (and
at the equator, as we will see in chapter 11). Long Kelvin waves have a long
period, but the only waves which are possibly relevant for long time scale variability in the oceans (with frequencies much smaller than f ) are the Rossby waves.