The conjectured dispersion-relation structure for sharp stratification

Let $\delta>0$, and let $rhob_{\delta}$ and $Vb_{\delta}$ be the density and velocity profiles defined by the sharp stratification and shear-flow setup. Let $\mathcal C_{\delta}(k)$ denote the set of dispersion-relation phase velocities at frequency $k$, let $L$ be the horizontal-period parameter, and let $k_{\min,\mathrm{BL}}$ be the lower instability threshold defined in the cited bilayer result. Dispersion-relation structure conjecture. There exist $k_{\min,\delta}$ and $k_{\max,\delta}$ such that:

1. For the low frequencies,

$$\max_{c\in\mathcal C_{\delta}(k),\ |k|\leq k_{\min,\delta}}|\operatorname{Im}(c(k))|\underset{\delta\to0}{\longrightarrow}0,$$

and

$$k_{\min,\delta}\underset{\delta\to0}{\longrightarrow}k_{\min,\mathrm{BL}}.$$

2. Every frequency $k\in[k_{\min,\delta},k_{\max,\delta}]$ is unstable, and

$$\max_{c\in\mathcal C_{\delta}(k),\ |k|\in[k_{\min,\delta},k_{\max,\delta}]\cap\frac{1}{L}\mathbb N}k\operatorname{Im}(c)\approx\delta^{-1},$$

with the maximum attained at some frequency $k^*_{\delta}$ satisfying

$$k^*_{\delta}\approx\delta^{-1}.$$

3. Every frequency with $|k|>k_{\max,\delta}$ is stable, and

$$k_{\max,\delta}\approx\delta^{-1}.$$

This conjecture organizes the numerically observed dispersion relation into stable low frequencies, an intermediate Kelvin–Helmholtz-unstable band, and stable high frequencies. The claims are based on numerical computations and are not proved in the supplied source.