significant and fast non-modal growth via the Orr-mechanism at the very early stage.]{}]{} The adjoint-based method for non-modal energy growth described in @Barkley2008 and @Chai2019 is adopted for this analysis.

![\[fig:unstable\_region\_and\_nonmodal\_growth\] (a) Contours of the most unstable/least stable eigenvalue in the $\alpha$-$\beta$ wavenumber plane for the base flow $(U_x+1-y^2,0,U_z)$ at $Re=750$. The region circled by the bold black line marks the linearly unstable region. (b) The streamwise velocity of the most unstable mode $(\alpha,\beta)=(0.32,-1.96)$ is visualised in the $x$-$z$ cut plane at $y=-0.5$ (top) and in the $z$-$y$ cross-section at $x=0$ (bottom). The base flow is in the positive $x$ direction. Red color represents higher velocity and blue represents lower velocity compared to the base flow. (c-f) Contours of the non-modal energy growth in the $\alpha$-$\beta$ wavenumber plane at $t=10$, 20, 30 and 40.](Fig3a.pdf "fig:"){width="90.00000%"} ![\[fig:unstable\_region\_and\_nonmodal\_growth\] (a) Contours of the most unstable/least stable eigenvalue in the $\alpha$-$\beta$ wavenumber plane for the base flow $(U_x+1-y^2,0,U_z)$ at $Re=750$. The region circled by the bold black line marks the linearly unstable region. (b) The streamwise velocity of the most unstable mode $(\alpha,\beta)=(0.32,-1.96)$ is visualised in the $x$-$z$ cut plane at $y=-0.5$ (top) and in the $z$-$y$ cross-section at $x=0$ (bottom). The base flow 