| Planetary equatorial waves are studied with
the shallow water equations in the presence of a mean zonal thermocline
gradient. The interactions between this gradient and waves are represented by
three non-linear terms in the equations: one in the wind-forcing formulation in
the x-momentum equation, and two for the advection of mass and divergence
of the velocity field in the continuity equation. When the mean gradient is
imposed but small, these three (linearized) terms will perturb the behaviour of
the equatorial waves. This paper gives a simple analytic treatment of this
problem.
The equatorial Kelvin mode is first solved with all three
contributions, using a Wentzel-Kramers-Brillouin method. The Kelvin mode shows a
spatial or/and temporal growth when the thermocline gradient is negative which
is the usual situation in the equatorial Pacific ocean (deep thermocline in the
west and shallow in the east). The more robust and efficient contribution comes
from the advection term.
The single effect of the advection of the mean zonal
thermocline gradient is then studied for the Kelvin and planetary Rossby modes.
The Kelvin mode remains unstable (damped), while the Rossby modes appear damped
(unstable) for a negative (positive) thermocline gradient. |