190
DYNAMICAL OCEANOGRAPHY
-20
-15
-10
-5
0
5
10
15
20
024681 0
f(m)
m
tan m
-(m + 1/m)
(a)
-4
-2
0
2
4
024681 0
f(µ µ
µ
µ) )
)
)
m
tanh µ µ µ
µ
1/µ µ µ
µ − −
−
− µ µ µ
µ
(b)
Figure 8.7. (a) Plot of the functions tanm and −(m +1/m). (b) Plot of the functions tanhμ
and 1/μ − μ.
Again, we look for plane wave solutions of the form
ψ(x, y, z, t)=Φ(z)e
i(kx+ly−σt) ,
(8.58)
and through substitution in (8.57), we find the following eigenvalue problem for
Φ (with eigenvalues σ)
Φ
′′ + m
2 Φ=0
(8.59a)
Φ
′ (0) = Φ
′ (−1) − S
γk
σ
Φ(−1) = 0
(8.59b)
m
2 = −S(
βk
σ
+ k
2 + l
2 ).
(8.59c)
The complication is now that for γ =0 , the eigenvalue σ is in the boundary
Ex. 8.5
condition. There are two types of solutions for β =0 , which depend on the sign
of m 2 , i.e. m 2 > 0 and m 2 < 0.W h e nm =0 , then a constant Φ does not
satisfy (8.59b) for γ =0unless k =0(but then also l =0 , due to (8.59c)) and
we find only trivial solutions. Hence, bottom topography will strongly affect the
barotropic mode.
For m 2 > 0, the solution Φ and the dispersion relation are
Φ(z)=A cos mz,
(8.60a)
m tan m =
γkS
σ
⇒ tan m = −
γ
β
(m +
S(k 2 + l 2 )
m
).
(8.60b)
For fixed S and fixed wavevector k, (8.60b) has a discrete set of roots m i
(Fig. 8.7a). Note that there is no intersection on the first branch of the function