Quantifying the vertical asymmetry in thin+thick models. Left panel: temporal evolution of A_1, z (Eq. (2)) denoting vertical asymmetry in bar region, calculated using thin-disc (blue lines), thick-disc (red lines), and thin+thick disc (black lines) for the model rthickE0.5. Right panel: temporal evolution of buckling amplitude, A_buck (Eq. (3)), calculated using thin-disc (blue lines), thick-disc (red lines), and thin+thick disc (black lines) for the model rthickE0.5. The vertical magenta dotted line denotes the onset of buckling instability (τ_buck), calculated from the peak of A_buck profile (for details, see text). Furthermore, for reference, we indicate the onset of bar formation (τ_bar, vertical maroon dotted line), calculated from the amplitude of the m = 2 Fourier moment.