156
DYNAMICAL OCEANOGRAPHY
In the previous chapter, we only considered the steady circulation on large
spatial scales. In this and the next three chapters, we will consider timedependent problems. A prototype problem is how an ocean flow evolves
when at a certain time t 0 , the wind stress is changed. To solve this socalled adjustment problem (which is actually done in chapter 9), we focus
in this chapter on the basic ingredients of the response: the free waves.
We restrict ourselves here to the free waves that can be described by the
constant density shallow-water equations (section 7.1). In section 7.2,
gravity waves in a horizontally unbounded geometry and a zonal channel
are presented (Poincar´ e and Kelvin waves) and mechanisms of propagation are presented. The impact of bottom topography and the β-effect is
considered in the sections 7.3 and 7.4 leading to a description of Rossby
waves.
Note: All equations in this chapter are dimensional and we will therefore omit
the star subscript.
7.1. Small amplitude motions
Consider the situation of a motionless liquid layer with a constant depth
H 0 (x, y) that rotates around the vertical axis with a constant angular velocity Ω.
Suppose that the flow can be well-described by the special form of the shallowwater equations (4.30a-b) and (4.32), which we here restate for convenience
Du
dt
− fv = −g
∂h
∂x
,
(7.1a)
Dv
dt
+ fu = −g
∂h
∂y
,
(7.1b)
∂H
∂t
+
∂
∂x
(Hu)+
∂
∂y
(Hv)=0 .
(7.1c)
Here, H = h + D 0 − h b is the total depth of the layer, f =2 Ωis the Coriolis
parameter and h is the surface elevation.
In this situation small amplitude perturbations (quantities with a tilde) are assumed to be superposed on the background state ¯
H = H 0 , ¯
u =¯ v =0, i.e.,
H(x, y, t)=H 0 (x, y)+ε ˜
h(x, y, t),
(7.2a)
u(x, y, t)=ε˜ u(x, y, t); v(x, y, t)=ε˜ v(x, y, t),
(7.2b)
where ε (not the Rossby number ǫ here!) is the amplitude of the perturbations
with (ε ≪ 1).T h eO(ε) balances in (4.30a-b) and (4.32) become
∂ ˜
u
∂t
− f ˜
v = −g
∂ ˜
h
∂x
,
(7.3a)
DYNAMICAL OCEANOGRAPHY
In the previous chapter, we only considered the steady circulation on large
spatial scales. In this and the next three chapters, we will consider timedependent problems. A prototype problem is how an ocean flow evolves
when at a certain time t 0 , the wind stress is changed. To solve this socalled adjustment problem (which is actually done in chapter 9), we focus
in this chapter on the basic ingredients of the response: the free waves.
We restrict ourselves here to the free waves that can be described by the
constant density shallow-water equations (section 7.1). In section 7.2,
gravity waves in a horizontally unbounded geometry and a zonal channel
are presented (Poincar´ e and Kelvin waves) and mechanisms of propagation are presented. The impact of bottom topography and the β-effect is
considered in the sections 7.3 and 7.4 leading to a description of Rossby
waves.
Note: All equations in this chapter are dimensional and we will therefore omit
the star subscript.
7.1. Small amplitude motions
Consider the situation of a motionless liquid layer with a constant depth
H 0 (x, y) that rotates around the vertical axis with a constant angular velocity Ω.
Suppose that the flow can be well-described by the special form of the shallowwater equations (4.30a-b) and (4.32), which we here restate for convenience
Du
dt
− fv = −g
∂h
∂x
,
(7.1a)
Dv
dt
+ fu = −g
∂h
∂y
,
(7.1b)
∂H
∂t
+
∂
∂x
(Hu)+
∂
∂y
(Hv)=0 .
(7.1c)
Here, H = h + D 0 − h b is the total depth of the layer, f =2 Ωis the Coriolis
parameter and h is the surface elevation.
In this situation small amplitude perturbations (quantities with a tilde) are assumed to be superposed on the background state ¯
H = H 0 , ¯
u =¯ v =0, i.e.,
H(x, y, t)=H 0 (x, y)+ε ˜
h(x, y, t),
(7.2a)
u(x, y, t)=ε˜ u(x, y, t); v(x, y, t)=ε˜ v(x, y, t),
(7.2b)
where ε (not the Rossby number ǫ here!) is the amplitude of the perturbations
with (ε ≪ 1).T h eO(ε) balances in (4.30a-b) and (4.32) become
∂ ˜
u
∂t
− f ˜
v = −g
∂ ˜
h
∂x
,
(7.3a)