7. Summary

As an extension of the discussion of TI2006, the physical interpretation of unstable modes other than the most unstable modes of a linear shear flow in shallow water on an equatorial β-plane are given. The behaviors of continuous modes and neutral modes are also examined. The results are as follows:

1. Continuous modes with 0 ≤ c ≤ 2.5 have equatorial Rossby wave like structure. This result is consistent with the speculation by TI2006 that all of equatorial Rossby modes assimilate into continuous modes. Therefore, it is considered that equatorial Rossby modes exist as assimilated modes into continuous modes in a linear shear flow on an equatorial β-plane.

2. Resonating modes causing zonally symmetric unstable modes are

   • For n=0, equatorial Kelvin modes and westward mixed Rossby-gravity modes,
   • For n=1, eastward mixed Rossby-gravity modes and continuous modes,
   • For n≥2, eastward inertial gravity modes and continuous modes,

   where n is the order of Hermite function which constitutes eigenfunction. Leading order modes (n=0) correspond to so-called inertially unstable modes, while high-order modes (n≥1) are different kind of unstable modes from inertially unstable modes.

3. Crossing modes have large amplitudes outside the inertially unstable region and emerge when the computational domain exists outside the inertially unstable region. They have eastward (or westward) inertial gravity wave like structure. Crossing modes are considered to be gravity waves in a mid-latitude β-channel rather than equatorial waves.
4. The positions and behaviors of dispersion curves of unstable modes cased by resonance between equatorial Kelvin modes and continuous modes, and dispersion curves of westward mixed Rossby-gravity modes are observed in the case with computational domains narrower than that of TI2006. It is confirmed that the phase speed of the unstable modes approaches c=2.5 with the increase of E, and that dispersion curves of the unstable modes kink before equatorial Kelvin modes and westward mixed Rossby-gravity modes resonate. This result is consistent with the discussion of TI2006.

Similar modes to crossing modes described in the above (3) are also observed in fig.14 of Iga and Matsuda (2005) who obtained unstable modes under the condition of Venusian atmosphere. In their figure, there exist dispersion curves which intersect nearby other neutral modes superimposed on dispersion curves of gravity modes. We speculate that the modes obtained by Iga and Matsuda (2005) also have the structure of gravity modes in mid-latitude channel.

The above result (2) suggests that, also on f-plane, the higher order zonally symmetric unstable modes are different kind of instability from so-called inertial instability. We speculate that, on f-plane, the lowest order zonally symmetric unstable modes are caused by resonance between eastward inertial gravity modes and westward inertial gravity modes, and that higher order zonally symmetric unstable modes are caused by resonance between eastward inertial gravity modes and continuous modes. With obtaining dispersion relations of unstable modes on f-plane, it is expected that our speculation will be able to confirmed.

7. Summary