Wave theory
Rossby and Kelvin waves
To have a quantitative understanding of the behavior of the waves during El Niņo event, we are going to consider a shallow water model, assuming that the thermocline is the interface between the two immiscible layers of fluid (warm surface waters and cold waters in the deep ocean). The equations of motion are given by :
(1.a)
(1.b)
(1.c)
where
H is the mean depth of the upper layer,
is the windstress,
is the vertical displacement of the
interface,
(
plane approximation, with
.
is the rate of rotation of the Earth and a is its radius),
reduced gravity (g gravitational acceleration,
is the density of the layer i), u
and v are the zonal and meridional velocities respectively. Note that
is the phase speed of the wave. This set of equations decribes the motion of the
interface resulting from a windstress on the upper layer.
Rossby waves
We
are considering wave-like solutions for which the restoring force is the
latitudinal gradient of Earth vorticity. A wavelike solution can be written as :
(2).
Where
k is the wave number (k>0 means an eastward phase propagation and k<0 a
westward phase propagation) and
is the frequency of the wave (defined positive ). Substitution of (2) in the
shallow water equations (1) yields in the following equation (1) :
(3)
where
.
The
solution of equation (3) is a wavelike solution in an equatorial zone of width
2Y whith exponential decay poleward of
Y. The dispersion relationship, derived from equation (3) is :
(4)
where
is the local inertial frequency. At low frequency,
, the dispersion relationship is :
(5).
As
the frequency is by definition a positive number, k must be negative. Therefore,
the Rossby waves have a westward phase propagation. The Rossby wave dispersion
diagram shows that even with westward phase speed, the Rossby waves can have
either an eastward (waves with zonal scale smaller than the radius of
deformation
) or westward group velocity (long waves with an horizontal scale greater
). Only the long wave that are nondispersive are important for the oceanic
adjustment because the short waves are not significant enough to change the wind
conditions.
Kelvin waves
Equatorial Kelvin waves have no meridional velocity fluctuations so that the shallow water equations become :
(6.a)
(6.b)
Equations
(6) admits a solution such as
, where E and F are arbitrary functions. The only physical solution is an
eastward equatorially trapped Kelvin wave :
, v=0
(7)
with
. The dispersion relationship is derived from equation (6a) :
(8).
A Kelvin wave packet is non dispersive so that the its components are always in phase with each other and can interact nonlinearly in an efficient manner. Nonlinearities modify the equation
(9)
essentially in two ways. The advection increases the phase speed from c to c+u and there is an additional change in the phase speed due to the deepening of the thermocline by the wave itself.