TI2006 showed that
the lowest order symmetric (*k*=0) unstable modes
are caused by resonance between equatorial Kelvin modes and westward mixed
Rossby-gravity modes (table 1).
However, resonating neutral modes causing high-order symmetric
unstable modes were not examined.
In this section, we clarify the resonating neutral modes causing
high-order symmetric unstable modes by examining dispersion curves for the
value of *E* which were not discussed
in TI2006.
The results give
physical interpretations for all of zonally symmetric unstable modes
obtained by Stevens (1983)
from the viewpoint of the concept of resonance between neutral waves.
The calculation domain is
-2.00 ≤ *y* ≤ 3.00
(figure 1)
same as TI2006.

Figure 4 shows dispersion curves for log *E*=1.30, log *E*=1.70,
and log *E*=2.40.
For log *E*=1.30 (figure 4a),
there exists only one dispersion curve
which have zonally symmetric (*k*=0) unstable mode
(labeled as Kelvin + W-MRG (n=0) in the figure).
On this dispersion curve,
all modes are unstable.
These unstable modes are casued by resonance between equatorial Kelvin
modes and westward mixed Rossby-gravity modes
(TI2006).
For log *E*=1.70 (figure 4b),
there exist two dispersion curves which have zonally symmetric
unstable modes.
The new symmetric unstable mode (labeled as E-MRG + C (n=1) )
is caused by resonance between eastward
mixed Rossby-gravity modes (E-MRG) and continuous modes.
One more symmetric unstable mode (labeled as E-G + C (n=2) in figure 4c)
emerges for log *E*=2.40.
This unstable mode is caused by resonance between the lowest order
eastward inertial gravity modes (E-G) and continuous modes.
With the increas of the value of *E*,
the number of zonally symmetric unstable modes increases
(animation of figure 4).
These high-order symmetric unstable modes are caused by resonance between
high-order eastward inertial gravity modes and continuous modes.

Figure 4: Dispersion curves that have symmetric unstable modes for
(a) log *E*=1.30,
(b) log *E*=1.70, and
(c) log *E*=2.40.
Labels "Kelvin + W-MRG", "E-MRG + C", and "E-G + C" indicate
unstable modes caused by resonance of equatorial Kelvin modes and
westward mixed Rossby-gravity modes,
unstable modes caused by resonance of
eastward mixed Rossby-gravity modes and continuous modes, and
unstable modes caused by resonance of
eastward inertial gravity modes and continuous modes, respectively.
Symmetric unstable modes (*k*=0) are not shown in these figures.
Refer to the caption of table 1
for the meaning of symbols.
Click [Animation] to show the change of dispersion curves
with varying the values of *E*.

Growth rate of symmetric unstable modes obtained in this study shows good correspondence with the values obtained theoretically by Stevens (1983). With nondimensional quantities used in this study, the equation of growth rate obtained by Stevens (1983) becomes

where *n* is the order of Hermite function
which constitutes eigenfunction.
Growth rates of symmetric unstable modes obtained by equation (5) are
shown by dashed black lines in figure 5.
With the increase of mode number *n*,
critical value of *E* where each mode becomes unstable
increases.
In thie figure,
numerically obtained growth rates of symmetric unstable modes are also
shown by color circles.
Critical values and growth rates for smaller value of *E*
coincide for analytic solutions and numerical obtained values.
This result indicates that
unstable mode of *n*=1 corresponds to
numerically obtained unstable mode caused by resonance
between eastward mixed Rossby-gravity modes and continuous modes.
Unstable mode of *n*=2 corresponds to
numerically obtained unstable mode caused by resonance
between eastward inertial gravity modes and continuous modes.

Numerically obtained asymptotic values of growth rate
for *E*→∞ are smaller
than those of analytic solutions.
The difference can be considered to be resulted from
the shortage of resolution of numerical calculation.
The asymptotic values become closer to theoretical values
in the case with larger number of grid points
(figures not shown).

Figure 5: Growth rate of zonally symmetric unstable
modes as a function of *E*.
Black dashed lines indicate analytic solution of equation (5)
obtained by Stevens (1983) for
*n*=0, *n*=1, and *n*=2.
Red, green, blue circles are numerically obtained growth rates
of symmetric unstable modes.
Red circles indecate unstable modes caused by resonance
between equatorial Kelvin modes and westward mixed Rossby-gravity
modes.
Green circles indecate unstable modes caused by resonance
between eastward mixed Rossby-gravity modes (E-MRG) and continuous modes.
Blue circles indecate unstable modes caused by resonance
between eastward inertial gravity modes (E-G) and continuous modes.