It has been discussed that many types of unstable modes
appear in equatorial regions of atmosphere and ocean.
Dunkerton (1981) and
Stevens (1983) showed that,
with shallow water models on an equatorial β-plane,
zonally symmetric inertially unstable modes appear
if there exists
the *inertially unstable region* defined as
a region where the
product of the Coriolis parameter and potential vorticity is negative.
Dunkerton (1983) showed the existence of
zonally nonsymmetric modes with large amplitudes in inertially unstable
regions.
Natarov and Boyd (2001) discussed that
equatorial Kelvin wave
(Matsuno, 1966)
is destabilized in equatorial shear flow.

Taniguchi and Ishiwatari (2006, hereafter TI2006) reconsidered the above mentioned unstable modes from the viewpoint of the concept of resonance between neutral waves (Hayashi and Young, 1987; Iga, 1999c). TI2006 obtained unstable modes of a linear shear flow in a shallow water on an equatorial β-plane over a wide range of nondimensional parameter , where , , , and are the meridional shear of basic zonal flow, gravitational constant, equivalent depth, and the north-south gradient of the Coriolis parameter, respectively.

TI2006 classified the resonance types
of the most unstable modes
according to the value of *E* as summarized in table 1
(see
animation figure for the cases not shown in table 1).
They showed that
the nonsymmetric unstable modes of
Dunkerton (1983)
and symmetric inertially unstable modes
(Dunkerton, 1981;
Stevens, 1983)
are the same kind of instability caused by the
resonance between equatorial Kelvin modes and westward mixed
Rossby-gravity modes
(Matsuno, 1966).
It was also shown that
the destabilized equatorial Kelvin wave obtained by
Natarov and Boyd, 2001
corresponds to unstable modes
caused by resonance between equatorial Kelvin modes
and continuous modes
(Case, 1960).

Table 1:
A summary of interpretation of most unstable modes obtained by TI2006.
Values of log *E* are
(a) -0.90,
(b) +1.30,
(c) +2.50.
Upper figure shows dispersion curves for each value of *E*.
Click figures of dispersion curves to show larger figures.
Horizontal and vertical axes are non-dimensional zonal wavenumber and
phase speed (*c*), respectively.
The labels 'Kelvin', 'E-MRG', 'W-MRG', 'Eastward Gravity', and
'Westward Gravity' indicate equatorial Kelvin modes,
eastward mixed Rossby-gravity modes,
westward mixed Rossby-gravity modes,
eastward inertial gravity modes, and
westward inertial gravity modes, respectively
(refer to Matsuno (1966) for each mode).
The label 'Kelvin + W-MRG' indicates unstable modes caused by
resonance between equatorial Kelvin modes and westward mixed
Rossby-gravity modes.
Horizontal lines in the range of 0.00 < *c* < 5.00
are dispersion curves of continuous modes.
Open red circles and double blue circles indicate unstable modes and
the most unstable modes, respectively.
Open and filled triangles indicate the position of dispersion curves
of north and south boundary Kelvin modes, respectively.
The third and the fourth rows show horizontal
structure of most unstable modes and resonance types, respectively.
The bottom row shows the corresponding previous study for each most
unstable mode.