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 $k_1\leq (p^2-1)μ_1$, where $p>1$ and $μ_1$ is the smallest eigenvalue of Poincaré's problem. Second, we find a new upper bound for the imaginary part of the complex phase velocity $c_i \leq k_1 ΔU/\sqrt{μ_1}$, where $ΔU$ 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 $ω_i \leq (p-1) \sqrt{μ_1} ΔU$. All the new bounds are much more strict than the previous ones by Høiland, 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.

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