All Figures

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig1.eps} \end{figure}$ Figure 1: Instability strip for the fundamental mode in the plane $(T_{{\rm bump}},\ {\cal A})$ for $e=0.4,\ \sigma =7$ and $\nu =1$ $\times$ 10^-4. The top (at z=1) density and temperature are $\rho =2.5$ $\times$ 10 ^-3 and T=1, respectively, while gravity is g=7 everywhere. The white cross denotes a particular simulation further studied for $T_{{\rm bump}}=2.1$ and ${\cal A}=70\%$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig2.eps} \end{figure}$ Figure 2: a) Temporal power spectrum for the momentum in the $(z,\ \omega )$ -plane. b) The resulting spectrum after integrating over depth. c) Comparison between normalised momentum profiles for $n=(0,\ 2)$ modes according to the DNS power spectrum (solid black line) and to the linear-stability analysis (dotted blue line).
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{fig/9891fig3.eps} \end{figure}$ Figure 3: Real part of normalised eigenfunctions $(u,\ R,\ T')$ for the fundamental mode (solid black line) and its adjoint (dotted blue line), for the equilibrium setup denoted by a white cross in Fig. 1.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig4.eps}\hspace*{0.4cm} \includegraphics[width=8.2cm,clip]{fig/9891fig5.eps} \end{figure}$ Figure 4: Left panel: time evolution of the real part of the complex-mode amplitudes c_n(t) for $n\in \llbracket 0,\ 3\rrbracket$ modes. The theoretical growth or damping curves are overplotted in dashed blue lines. Right panel: their respective Fourier transforms. The modes eigenfrequencies are overplotted in dashed blue lines.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig6.eps}\hspace*{0.3cm} \includegraphics[width=8.2cm,clip]{fig/9891fig7.eps} \end{figure}$ Figure 5: Phase diagrams for $n\in \llbracket 0,\ 3\rrbracket$ modes. The abscissa (ordinate) corresponds to the real (imaginary) part of the complex mode amplitude c_n(t). The increasing orbits are plotted in blue, whereas the decreasing ones are in black.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{fig/9891fig8.eps} \end{figure}$ Figure 6: a) Kinetic energy ratios for $n \in \llbracket 0, \ 6\rrbracket$ in a logarithmic y-scale. b) Zoom in the n=0 (solid black line) and n=2 (dashed green line) mode energy contents.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig9.eps} \end{figure}$ Figure 7: P₂/P₀ level lines overplotted on the instability strip. Dashed white lines denote ratio P₂/P₀ <1/2, while solid white lines correspond to P₂/P₀ >1/2. The critical level P₂/P₀=1/2 is emphasised as a red line, and the dashed box delimits the region where a large number of DNS are performed to study Hertzsprung's progression (see Figs. 8, 9).
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig10.eps} \end{figure}$ Figure 8: Isocontours of $\Delta \phi$ in the $(T_{{\rm bump}},\ {\cal A})$ plane (note that we rather plot $\Delta \phi - \pi$ to highlight the critical line $\Delta \phi =\pi$ ). This figure corresponds to the dashed box in Fig. 7 and the black circles display eleven selected simulations showed in Fig. 9.
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In the text

$\begin{figure} \includegraphics[width=8.3cm,clip]{fig/9891fig11.eps} \end{figure}$ Figure 9: Rephased time evolutions of the surface radiative flux about time t=500 for the eleven selected simulations plotted as black circles in Fig. 8. The ratio of eigenperiods P₂/P₀ is indicated in each case.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig1.eps} \end{figure}$	Figure 1: Instability strip for the fundamental mode in the plane $(T_{{\rm bump}},\ {\cal A})$ for $e=0.4,\ \sigma =7$ and $\nu =1$ $\times$ 10^-4. The top (at z=1) density and temperature are $\rho =2.5$ $\times$ 10 ^-3 and T=1, respectively, while gravity is g=7 everywhere. The white cross denotes a particular simulation further studied for $T_{{\rm bump}}=2.1$ and ${\cal A}=70\%$ .
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig2.eps} \end{figure}$	Figure 2: a) Temporal power spectrum for the momentum in the $(z,\ \omega )$ -plane. b) The resulting spectrum after integrating over depth. c) Comparison between normalised momentum profiles for $n=(0,\ 2)$ modes according to the DNS power spectrum (solid black line) and to the linear-stability analysis (dotted blue line).
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{fig/9891fig3.eps} \end{figure}$	Figure 3: Real part of normalised eigenfunctions $(u,\ R,\ T')$ for the fundamental mode (solid black line) and its adjoint (dotted blue line), for the equilibrium setup denoted by a white cross in Fig. 1.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig4.eps}\hspace*{0.4cm} \includegraphics[width=8.2cm,clip]{fig/9891fig5.eps} \end{figure}$	Figure 4: Left panel: time evolution of the real part of the complex-mode amplitudes c_n(t) for $n\in \llbracket 0,\ 3\rrbracket$ modes. The theoretical growth or damping curves are overplotted in dashed blue lines. Right panel: their respective Fourier transforms. The modes eigenfrequencies are overplotted in dashed blue lines.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{fig/9891fig6.eps}\hspace*{0.3cm} \includegraphics[width=8.2cm,clip]{fig/9891fig7.eps} \end{figure}$	Figure 5: Phase diagrams for $n\in \llbracket 0,\ 3\rrbracket$ modes. The abscissa (ordinate) corresponds to the real (imaginary) part of the complex mode amplitude c_n(t). The increasing orbits are plotted in blue, whereas the decreasing ones are in black.
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{fig/9891fig8.eps} \end{figure}$	Figure 6: a) Kinetic energy ratios for $n \in \llbracket 0, \ 6\rrbracket$ in a logarithmic y-scale. b) Zoom in the n=0 (solid black line) and n=2 (dashed green line) mode energy contents.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig9.eps} \end{figure}$	Figure 7: P₂/P₀ level lines overplotted on the instability strip. Dashed white lines denote ratio P₂/P₀ <1/2, while solid white lines correspond to P₂/P₀ >1/2. The critical level P₂/P₀=1/2 is emphasised as a red line, and the dashed box delimits the region where a large number of DNS are performed to study Hertzsprung's progression (see Figs. 8, 9).
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{fig/9891fig10.eps} \end{figure}$	Figure 8: Isocontours of $\Delta \phi$ in the $(T_{{\rm bump}},\ {\cal A})$ plane (note that we rather plot $\Delta \phi - \pi$ to highlight the critical line $\Delta \phi =\pi$ ). This figure corresponds to the dashed box in Fig. 7 and the black circles display eleven selected simulations showed in Fig. 9.
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$\begin{figure} \includegraphics[width=8.3cm,clip]{fig/9891fig11.eps} \end{figure}$	Figure 9: Rephased time evolutions of the surface radiative flux about time t=500 for the eleven selected simulations plotted as black circles in Fig. 8. The ratio of eigenperiods P₂/P₀ is indicated in each case.
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