are localized at extrema of the ZF velocity $U(x)$, where $x$ is the radial coordinate. By approximating $U(x)$ with a parabola, we analytically derive the TI growth rate, $\gamma_{\rm TI}$, using two different approaches: (i) by drawing an analogy between TI modes and quantum harmonic oscillators and (ii) by using the Wigner–Moyal equation (WME). Our theory shows that the TI is essentially a primary DW instability modified by the ZF “curvature” $U''$ near extrema of $U$. (The prime denotes ${\rm d}/{\rm d}x$.) In particular, the WME helps understand how the local $U''$ modifies the mode structure and reduces the TI growth rate; it also shows that the TI is *not* the Kelvin–Helmholtz (KH) instability, or KHI. Then, depending on $U''$, the TI can be suppressed, in which case ZFs are strong enough to suppress turbulence (Dimits regime), or unleashed, so ZFs are unstable and turbulence develops. This understanding is different from the traditional paradigm [@Biglari90], where turbulence is controlled by the flow shear $U'$. Finally, by letting $\gamma_{\rm TI}=0$, we obtain an analytic prediction of the Dimits shift, which agrees with our numerical simulations of the mTHE.

Admittedly, our explicit prediction of the Dimits shift is facilitated by the fact 