Stratification
185
8.4. Free waves
To determine the free waves of the stratified quasi-geostrophic model, we consider the unforced equation (8.29) with boundary conditions (8.33-8.34), i.e. with
r = T =0 . Furthermore, we will consider a horizontally unbounded ocean and
boundary effects due to the presence of the continents are neglected. Free waves
are again found by determining small amplitude motions with respect to the motionless state.
8.4.1. Vertical structure functions
The linearized equation (8.29) becomes
∂
∂t
(∇
2 ψ +
∂
∂z
(
1
S
∂ψ
∂z
)+β
∂ψ
∂x
)=0.
(8.35)
We search for separable solutions
ψ(x, y, z, t)=Ψ(x, y, t)Φ(z),
(8.36)
for a vertical structure function Φ(z). It appears that solutions exist if Φ satisfies
(
1
S
Φ
′ )
′ = −χΦ.
(8.37)
The boundary conditions for Φ follow from (8.33- 8.34). For a flat bottom these
become
Φ
′ (0) = Φ
′ (−1) = 0.
(8.38)
The constant χ in (8.37) is a so-called separation constant which has to be determined such that the problem above has nontrivial solutions. Because S is positive
and the problem (8.37) is self-adjoint, there exist a countable number of real eigenvalues. A special eigenvalue is χ =0with associated eigenfunction Φ=1. Since
the baroclinic vector ∇ρ ∧∇p is zero for these waves, they are called barotropic
waves. All other waves, with χ =0 , are baroclinic waves and only exist because
of the presence of the stratification.
In the special case where S is constant, the solutions of (8.37) are
Φ(z)=A 1 cos(z
χS)+A 2 sin(z
χS).
(8.39)
The boundary conditions (8.38) imply that A 2 ≡ 0 and
sin(
χS)=0⇒ χ = χ n =
n 2 π 2
S
.
(8.40)
For n =0, we find the barotropic mode and for n>0 the baroclinic modes are
given by (8.39). The vertical structure of the modes with n =0 , 1, 2 and n =3
185
8.4. Free waves
To determine the free waves of the stratified quasi-geostrophic model, we consider the unforced equation (8.29) with boundary conditions (8.33-8.34), i.e. with
r = T =0 . Furthermore, we will consider a horizontally unbounded ocean and
boundary effects due to the presence of the continents are neglected. Free waves
are again found by determining small amplitude motions with respect to the motionless state.
8.4.1. Vertical structure functions
The linearized equation (8.29) becomes
∂
∂t
(∇
2 ψ +
∂
∂z
(
1
S
∂ψ
∂z
)+β
∂ψ
∂x
)=0.
(8.35)
We search for separable solutions
ψ(x, y, z, t)=Ψ(x, y, t)Φ(z),
(8.36)
for a vertical structure function Φ(z). It appears that solutions exist if Φ satisfies
(
1
S
Φ
′ )
′ = −χΦ.
(8.37)
The boundary conditions for Φ follow from (8.33- 8.34). For a flat bottom these
become
Φ
′ (0) = Φ
′ (−1) = 0.
(8.38)
The constant χ in (8.37) is a so-called separation constant which has to be determined such that the problem above has nontrivial solutions. Because S is positive
and the problem (8.37) is self-adjoint, there exist a countable number of real eigenvalues. A special eigenvalue is χ =0with associated eigenfunction Φ=1. Since
the baroclinic vector ∇ρ ∧∇p is zero for these waves, they are called barotropic
waves. All other waves, with χ =0 , are baroclinic waves and only exist because
of the presence of the stratification.
In the special case where S is constant, the solutions of (8.37) are
Φ(z)=A 1 cos(z
χS)+A 2 sin(z
χS).
(8.39)
The boundary conditions (8.38) imply that A 2 ≡ 0 and
sin(
χS)=0⇒ χ = χ n =
n 2 π 2
S
.
(8.40)
For n =0, we find the barotropic mode and for n>0 the baroclinic modes are
given by (8.39). The vertical structure of the modes with n =0 , 1, 2 and n =3