Neutral curve (1)

We linearize the governing equations by assuming small amplitudes for y and q. Further, we expand y and q by a series of functions satisfying the boundary conditions and truncate up to the vertical wavenumber 1 (see Appendix B). As a result, we can easily obtain the approximate expression of the neutral curve of the system.

$\begin{displaymath} R = \frac{\pi^2}{8} \left[K^4 + \frac{P^2\eta^2k^2}{(PK^2 ... ...ight)^2 \frac{k^2}{(k^2 + \frac{P\pi^2}{P+1})^2} \right]. \end{displaymath}$ (11)

The first term on the right hand side (p²/8)(k²+p²)² describes the neutral curve for the case of the non-rotating system (center graph in figure). This neutral curve has a minimum at k=0. As the rotation rate increases, the effect of the second term (right graph in figure) appears and the neutral curve goes up in the k-R plane. However, since the effect of the second term vanishes at k=0, the critical horizontal wave number is zero even with the effect of rotation. Therefore, the critical values are

$\begin{displaymath} k_c = 0, \quad \omega_c=0, \quad R_c= \frac{\pi^6}{8}\sim 120.17\DselJ{,}{.} \end{displaymath}$ (12)

Neutral curve (1)