Geography Reference
In-Depth Information
This boundary condition is necessary because the approximation f
≈
βy is not
30
◦
, so that solutions must be trapped equato-
rially if they are to be good approximations to the exact solutions on the sphere.
Equation (11.38) differs from the classic equation for a harmonic oscillator in y
because the coefficient in square brackets is not a constant, but is a function of
y. For sufficiently small y this coefficient is positive and solutions oscillate in
y, whereas for large y, solutions either grow or decay in y. Only the decaying
solutions, however, can satisfy the boundary conditions.
It turns out
5
that solutions to (11.38), which satisfy the condition of decay far
from the equator, exist only when the constant part of the coefficient in square
brackets satisfies the relationship
√
gh
e
β
valid for latitudes much beyond
±
ν
2
gh
e
k
ν
β
k
2
−
−
+
=
2n
+
1
;
n
=
0,1,2, ...
(11.39)
which is a cubic dispersion equation determining the frequencies of permitted
equatorially trapped free oscillations for zonal wave number k and meridional mode
number n. These solutions can be expressed most conveniently if y is replaced by
the nondimensional meridional coordinate
β
gh
e
1/2
y
ξ
≡
Then the solution has the form
v
0
H
n
(ξ ) exp
ξ
2
/2
v (ξ )
ˆ
=
−
(11.40)
where v
o
is a constant with velocity units, and H
n
(ξ ) designates the nth
Hermite
polynomial
. The first few of these polynomials have the values
4ξ
2
H
0
=
1,
H
1
(ξ )
=
2ξ,
H
2
(ξ )
=
−
2
Thus, the index n corresponds to the number of nodes in the meridional velocity
profile in the domain
.
In general the three solutions of (11.39) can be interpreted as eastward and west-
ward moving equatorially trapped gravity waves, and westward moving equatorial
Rossby waves. The case n
|
y
|
<
∞
0 (for which the meridional velocity perturbation has
a Gaussian distribution centered at the equator) must be treated separately. In this
case the dispersion relationship (11.39) factors as
ν
√
gh
e
−
=
k
ν
k
β
ν
−
√
gh
e
+
=
0
(11.41)
5
See Matsuno (1966).
