256
DYNAMICAL OCEANOGRAPHY
where δ kn =1when k = n and zero otherwise (the Kronecker symbol). The
Hermite functions ψ n (x), with
ψ n (x)=(n!2
n π
1/2 )
−1/2 exp(−x
2 /2)H n (x),
(11.32)
are solutions of the differential equation
y
′′ +(2n +1− x
2 )y =0,
(11.33)
From (11.31) it follows that
∞
−∞
ψ k (x)ψ n (x)dx = δ kn .
(11.34)
For the Hermite functions, the following relations apply
xψ n (x)=
n
2
ψ n−1 (x) −
n +1
2
ψ n+1 (x),
(11.35a)
ψ(x)=
n
2
ψ n−1 (x)+
n +1
2
ψ n+1 (x).
(11.35b)
◭
First, we consider the full spectrum of these waves by putting ζ o =1, which is
equivalent to using L as a meridional length scale. The dispersion relation (11.26)
can be written as
k = −
1
2σ
±
1
2
(
1
σ
− 2σ)
2 − 8j
1/2
.
(11.36)
For j>0, two real roots exist provided (1/σ − 2σ) 2 ≥ 8j in which case σ
satisfies
0 <σ<
1
√
2
((j +1)
1/2 − j
1/2 )
or
σ>
1
√
2
(j
1/2 +(j +1)
1/2 ).
The first interval of σ is in the low frequency range and the waves are called
equatorial Rossby waves. The second interval represents the high frequency socalled ‘inertia-gravity’ waves.
For the case j =0 , two roots are found from (11.36), the first one being σ =
−k which leads to a westward travelling Kelvin wave which becomes unbounded
far from the equator. The second root is
k = −
1
σ
+ σ,
(11.37)
DYNAMICAL OCEANOGRAPHY
where δ kn =1when k = n and zero otherwise (the Kronecker symbol). The
Hermite functions ψ n (x), with
ψ n (x)=(n!2
n π
1/2 )
−1/2 exp(−x
2 /2)H n (x),
(11.32)
are solutions of the differential equation
y
′′ +(2n +1− x
2 )y =0,
(11.33)
From (11.31) it follows that
∞
−∞
ψ k (x)ψ n (x)dx = δ kn .
(11.34)
For the Hermite functions, the following relations apply
xψ n (x)=
n
2
ψ n−1 (x) −
n +1
2
ψ n+1 (x),
(11.35a)
ψ(x)=
n
2
ψ n−1 (x)+
n +1
2
ψ n+1 (x).
(11.35b)
◭
First, we consider the full spectrum of these waves by putting ζ o =1, which is
equivalent to using L as a meridional length scale. The dispersion relation (11.26)
can be written as
k = −
1
2σ
±
1
2
(
1
σ
− 2σ)
2 − 8j
1/2
.
(11.36)
For j>0, two real roots exist provided (1/σ − 2σ) 2 ≥ 8j in which case σ
satisfies
0 <σ<
1
√
2
((j +1)
1/2 − j
1/2 )
or
σ>
1
√
2
(j
1/2 +(j +1)
1/2 ).
The first interval of σ is in the low frequency range and the waves are called
equatorial Rossby waves. The second interval represents the high frequency socalled ‘inertia-gravity’ waves.
For the case j =0 , two roots are found from (11.36), the first one being σ =
−k which leads to a westward travelling Kelvin wave which becomes unbounded
far from the equator. The second root is
k = −
1
σ
+ σ,
(11.37)