Long-wave instability and growth rate of the inviscid shear flows

In this paper, we studied the long-wave instability of the shear flows. When the wavenumber of perturbation is larger than the critical value, the flow is always neutrally stable. First, we obtain a new upper bound for the neutral wavenumber $k_1\leq (p^2-1)\mu_1$, where $p>1$ and $\mu_1$ is the smallest eigenvalue of Poincar\'{e}'s problem. Second, we find a new upper bound for the imaginary part of the complex phase velocity $c_i \leq k_1 \Delta U/\sqrt{\mu_1}$, where $\Delta U$ is the variance of the velocity. The new bound is finite for all $k>0$ similar to the Howard's semicircle theorem, while the previous ones by Craik and Banerjee et al would be infinity as $k\rightarrow 0$. Third, we find a new upper bound of growth rate $\omega_i \leq (p-1) \sqrt{\mu_1} \Delta U$. All the new bounds are much more strict than the previous ones by H{\o}iland, Howard, Craik and Banerjee et al. Our results also extend the inverse energy cascade theory by Kraichnan. As shear instability is due to long-wave instability, it implies that the truncation of long-waves may change the instability of shear flows.