254
DYNAMICAL OCEANOGRAPHY
with ˆ
u(0) being an arbitrary amplitude. The solutions which are bounded for
y →± ∞exist only when σ =+ k. Hence, the phase velocity of these waves
is positive and the waves only move eastward. These are the well-known Kelvin
waves with a dimensional wavelength and phase speed (σ/k)givenby
λ ∗ =
2πL
k
; c ∗ = c o .
(11.24)
Patterns of the thermocline field h of a Kelvin wave are plotted in Fig. 11.9 for
four stages during the propagation. The dimensionless wavenumber is chosen to
be k = π, corresponding to a wavelength of exactly twice the basin λ ∗ =2 L.
For the Kelvin wave, the dimensionless period P is 2π/σ =2and the pictures in
Fig. 11.9 are at times t =0,t =1/8,t =1/4,t =3/8, which covers a quarter of
a period. The maximum amplitude of the thermocline field for the Kelvin wave is
located just at the equator.
Ex. 11.3
Free wave solutions with ˆ
v =0also exist. In (11.19), ˆ
u and ˆ
p can be eliminated
to give a scalar equation for ˆ
v, i.e.
ˆ
v
′′ +ˆ v
ζ
2
0 (σ
2 − k
2 ) −
k
σ
− y
2
=0,
(11.25)
where the ′ indicates differentiation with respect to y. Equation (11.25) only has
bounded solutions when
ζ
2
0 (σ
2 − k
2 ) −
k
σ
=2j +1,
(11.26)
for j =0, 1, ···. These solutions of (11.26) are of the form
ˆ
v j (η)=ψ j (y)=
e
−y 2
2 H j (y)
(2 j j!π 1/2 ) 1/2 ,
(11.27)
with H j being the Hermite polynomials. The ψ j are called the Hermite functions
(see Example 11.2).
◮
Example 11.2: Hermite polynomials and Hermite functions
The Hermite polynomials H n (x) are solutions of the differential equation
y
′′ − 2xy
′ +2ny =0.
(11.28)
for a function y n (x). The first Hermite polynomials are
H 0 (x)=1;H 1 (x)=2x; H 2 (x)=4x
2 − 2; H 3 (x)=8x
3 − 12x,
(11.29)