TI2006 tries to identify dispersion curves of
resonating equatorial Kelvin modes
and mixed Rossby-gravity modes by the use of three
methods: (1) derivation of approximate dispersion relation of
equatorial Kelvin wave, (2) elimination of dispersion curves of
continuous modes with distorting basic flow (application
of the discussion by Iga, 1999c),
and (3) derivation of approximate dispersion relation of equatorial waves
by the use of the uniform Γ-plane approximation
(Boyd and Christdis, 1982).
However, positions and behaviors of dispersion curves of these
modes can not be identified clearly for larger value of *E*
because their dispersion curves are superimposed on
dispersion curves of other unstable modes.

For the calculational domain 0 < *y* < 1
used in section 5,
the range of phase speed of continuous modes
is narrower than that of TI2006.
Dispersion curves of
unstable modes caused by resonance between equatorial Kelvin modes
and continuous modes, and mixed Rossby-gravity modes can be observed
clearly.
In this section, with calculational domain of 0 < *y* < 1,
behaviors of the dispersion curves
of equatorial Kelvin modes and westward mixed Rossby-gravity modes
are examined
for the case when their dispersion curves are close.
The results for other calculational domains are shown in
Appendix B.

In the case with calculational domain of 0 < *y* < 1
for log *E* < 1.50 (figures 9a and 9b),
phase speed of
unstable modes caused by resonance between
equatorial Kelvin modes and continuous modes
approaches *c*=2.5 (the velocity of basic flow at the dynamic equator)
with the increase of the value of *E*.
This result is consistent with the discussion of
Clark and Haynes (1996)
who obtained approximate dispersion curves of
modes in equatorial shear flow with asymptotic expansion.
The asymptotical behavior of phase speed of the unstable modes
cannot be confirmed clearly with the dispersion curves
obtained by TI2006.

For larger values of *E* (figures 9c and 9d),
equatorial Kelvin modes and westward mixed
Rossby-gravity modes resonate.
In the results of TI2006
(their figure 6f), it seems that
the dispersion curve of unstable modes
caused by resonance between equatorial Kelvin modes and continuous
modes kinks
just before equatorial Kelvin modes resonate with
westward mixed Rossby-gravity modes.
In the figures obtained by TI2006,
the details of the behavior of dispersion curves
of unstable modes caused by equatorial Kelvin modes and continuous
modes cannot be observed,
since the dispersion curves of the unstable modes
are thoroughly buried among other dispersion curves.
For the case with calculational domain of 0 < *y* < 1,
kink of dispersion curves of unstable modes caused by
resonance between equatorial Kelvin modes and continuous modes
can be clearly observed
(figures 9c and 9d.
see also
an animation figure
for 1.50 ≤ log *E* ≤ 2.50).
Dispersion curves of the unstable modes begin to kink
at log *E*=1.60
for the calculational domain of 0 < *y* < 1.
In this case,
zonally symmetric (*k*=0) unstable modes emerge
at log *E*=1.65.
Therefore, as discussed by TI2006,
it is confirmed that
the dispersion curves of the unstable modes kink
when the value of *E* approaches the critical value of *E*
at which zonally symmetric (*k*=0) unstable modes emerge.

The critical value of *E* at which
zonally symmetric unstable modes emerge
is different from the value obtained by TI2006.
It seems that
a narrow calculational domain causes the difference of
the critical value.
For the calculational domain larger than -1 < *y* < 2,
zonally symmetric unstable modes (inertially unstable modes)
emerge at log *E*=1.20,
as shown by Stevens (1983) and
TI2006 (figures not shown).
However, for narrower calculational domains,
the critical value at which zonally symmetric unstable modes
emerge deviates from log *E*=1.20.

Figure 9: Dispersion curves for the calculational domain of
0.0 < *y* < 1.0.
(a) log *E*=+1.10,
(b) log *E*=+1.40,
(c) log *E*=+1.60, and
(d) log *E*=+1.63.
The label "Kelvin + C" indicates dispersion curves of unstable modes
caused by the resonance between equatorial Kelvin modes and continuous
modes.
Refer to table 1 for other symbols.
See also
animation
of dispersion curves for 1.50 ≤ log *E* ≤ 2.50.