All Figures

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.4cm,clip]{fig/2748fg1.eps} \end{figure}$ Figure 1: Transmission coefficient as function of frequency in logarithmic scale for isothermal layers of different temperatures. The value of $\alpha$ ranges from $\alpha=4\equiv T\approx3\times10^6~{\rm K}$ (hot) to $\alpha=15\equiv T\approx8\times10^5~{\rm K}$ (cold) in integer steps. $\beta _0=0.08$ for all the profiles (from Velli 1993).
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In the text

$\begin{figure} \par\includegraphics[height=4.2cm,width=4.2cm,clip]{fig/2748fg2.e... ...m} \includegraphics[height=4.2cm,width=4.2cm,clip]{fig/2748fg3.eps} \end{figure}$ Figure 2: Total wave action density normalized to the base value as a function of radius in $\alpha =4$ and $\alpha =10$ atmospheres, for a high ( $10^{-2}~{\rm Hz}$ , dotted line) and low ( $10^{-6}~{\rm Hz}$ , solid line) frequency wave with an initial $100{\rm ~km~s^{-1}}$ wave amplitude. For the cold atmosphere the profile for the wave having the minimum transmission coefficient (dashed line) is also plotted.
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In the text

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.6cm,clip]{fig/2748fg4.eps} \end{figure}$ Figure 3: Dissipation efficiency ( $\gamma$ ) as function of frequency for a hot ( $\alpha =4$ , continuous lines) and a cold ( $\alpha =10$ , dotted lines) corona for $z^+_{\rm c}=100{\rm ~km~s^{-1}}$ a), $z^+_{\rm c}=10{\rm ~km~s^{-1}}$ b) and $z^+_{\rm c}=1{\rm ~km~s^{-1}}$ c).
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In the text

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.8cm,clip]{fig/2748fg5.eps} \end{figure}$ Figure 4: Outgoing and ingoing wave amplitude (solid line) calculated at the base of the $\alpha =4$ atmosphere, as a function of outgoing amplitude at the critical point. The plots are obtained for $\omega=10^{-6}~{\rm Hz}$ . Dashed lines show the values obtained in the linear case (the upper and lower line refers to z⁺ and z^- respectively).
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In the text

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.6cm,clip]{fig/2748fg6.eps} \end{figure}$ Figure 5: Dissipation efficiency as function of initial outgoing wave amplitude for $\omega=10^{-6}~{\rm Hz}$ . The different plots refer to atmospheres with $\alpha =4,6,8,10$ .
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In the text

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.2cm,clip]{fig/2748fg7.e... ...m} \includegraphics[height=4.4cm,width=8.2cm,clip]{fig/2748fg8.eps} \end{figure}$ Figure 6: Flat spectrum. Left panel. Total wave action density, normalized to the base value, for the interacting frequencies in different coupling (labelled with letters, see text), as function of distance from the atmosphere's base expressed in unit of $R_\odot$ . Temperature is set to $\alpha =6$ and $z_{\rm c}^+(\omega_{\rm ref})=100~{\rm km~s^{-1}}$ . Right panel. Dissipation efficiency as function of initial outgoing wave amplitude $z_{\rm c}^+(\omega _{\rm ref})$ . The different plots refer to atmospheres with $\alpha=4,~6,~8,~10$ ; solid and dotted lines represent respectively $\gamma (\omega _0)$ and $\gamma (\omega _{\rm i})$ with $\omega_{\rm i}=10^{-2}~{\rm Hz})$ .
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In the text

$\begin{figure} \par\includegraphics[height=4.5cm,width=8cm,clip]{fig/2748fg9.eps... ...mm} \includegraphics[height=4.5cm,width=8cm,clip]{fig/2748fg10.eps} \end{figure}$ Figure 7: Same as Fig. 6 but for a power-law spectrum at the critical point.
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In the text

$\begin{figure} \par\includegraphics[height=4.2cm,width=8.8cm,clip]{fig/2748fg11.... ...} \includegraphics[height=4.2cm,width=8.8cm,clip]{fig/2748fg12.eps} \end{figure}$ Figure 8: Dissipation efficiency as function of initial outgoing wave amplitude $z_{\rm c}^+(\omega _0)$ imposed at the top of the layer in the interacting case. The left panel refers to case a ( $\omega_1=10^{-2}~{\rm Hz}$ and $\omega_2=10^{-1}~{\rm Hz}$ ), and the right panel to case b ( $\omega_1=10^{-4}~{\rm Hz}$ and $\omega_2=10^{-2}~{\rm Hz}$ ). Initial wave amplitude is scaled following a power-law spectrum (see text). The different plots refer to atmospheres with $\alpha=4,~10$ ; solid, dotted and dashed lines represent respectively $\gamma (\omega _0)$ , $\gamma (\omega _1)$ and $\gamma (\omega _2)$ .
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In the text

$\begin{figure} \par\includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg13.... ...} \includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg14.eps} \end{figure}$ Figure 9: Spectra at the top (dotted line) and the bottom (solid line) of the atmosphere for three representative initial values of $z^+_{\rm c}(\omega _0)$ ( $1{\rm ~km~s^{-1}}$ , $100{\rm ~km~s^{-1}}$ and $1000{\rm ~km~s^{-1}}$ marked with crosses, stars and diamonds respectively) and different temperature ( $\alpha=4,~10$ respectively on the left and right panel) in coupling b.
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In the text

$\begin{figure} \par\includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg15.... ...} \includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg16.eps} \end{figure}$ Figure 10: Ratio $\xi$ between the spectra at the top and the bottom of the atmosphere for $\alpha=4,~10$ (respectively left and right panel) in the case of coupling b. The plots are labeled with the initial amplitudes $z^+_{\rm c}(\omega _0)$ imposed at the top of the layer. Also shown is the ratio for the linear case (dotted line).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{fig/2748fg17.eps} \end{figure}$ Figure 11: Same as in Fig. 8 for coupling b and power-law initial spectrum except that integrations are performed using a spherically expanding turbulent length $L(r)=L_0\times r$ with L₀ fixed by the average dimension of supergranules at coronal level. Results are shown for $\alpha=4,~10$ atmospheres.
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In the text

$\begin{figure} \par\includegraphics[height=4.4cm,width=8.4cm,clip]{fig/2748fg1.eps} \end{figure}$	Figure 1: Transmission coefficient as function of frequency in logarithmic scale for isothermal layers of different temperatures. The value of $\alpha$ ranges from $\alpha=4\equiv T\approx3\times10^6~{\rm K}$ (hot) to $\alpha=15\equiv T\approx8\times10^5~{\rm K}$ (cold) in integer steps. $\beta _0=0.08$ for all the profiles (from Velli 1993).
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$\begin{figure} \par\includegraphics[height=4.4cm,width=8.6cm,clip]{fig/2748fg4.eps} \end{figure}$	Figure 3: Dissipation efficiency ( $\gamma$ ) as function of frequency for a hot ( $\alpha =4$ , continuous lines) and a cold ( $\alpha =10$ , dotted lines) corona for $z^+_{\rm c}=100{\rm ~km~s^{-1}}$ a), $z^+_{\rm c}=10{\rm ~km~s^{-1}}$ b) and $z^+_{\rm c}=1{\rm ~km~s^{-1}}$ c).
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$\begin{figure} \par\includegraphics[height=4.4cm,width=8.8cm,clip]{fig/2748fg5.eps} \end{figure}$	Figure 4: Outgoing and ingoing wave amplitude (solid line) calculated at the base of the $\alpha =4$ atmosphere, as a function of outgoing amplitude at the critical point. The plots are obtained for $\omega=10^{-6}~{\rm Hz}$ . Dashed lines show the values obtained in the linear case (the upper and lower line refers to z⁺ and z^- respectively).
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$\begin{figure} \par\includegraphics[height=4.4cm,width=8.6cm,clip]{fig/2748fg6.eps} \end{figure}$	Figure 5: Dissipation efficiency as function of initial outgoing wave amplitude for $\omega=10^{-6}~{\rm Hz}$ . The different plots refer to atmospheres with $\alpha =4,6,8,10$ .
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$\begin{figure} \par\includegraphics[height=4.5cm,width=8cm,clip]{fig/2748fg9.eps... ...mm} \includegraphics[height=4.5cm,width=8cm,clip]{fig/2748fg10.eps} \end{figure}$	Figure 7: Same as Fig. 6 but for a power-law spectrum at the critical point.
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$\begin{figure} \par\includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg13.... ...} \includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg14.eps} \end{figure}$	Figure 9: Spectra at the top (dotted line) and the bottom (solid line) of the atmosphere for three representative initial values of $z^+_{\rm c}(\omega _0)$ ( $1{\rm ~km~s^{-1}}$ , $100{\rm ~km~s^{-1}}$ and $1000{\rm ~km~s^{-1}}$ marked with crosses, stars and diamonds respectively) and different temperature ( $\alpha=4,~10$ respectively on the left and right panel) in coupling b.
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$\begin{figure} \par\includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg15.... ...} \includegraphics[height=4.3cm,width=8.7cm,clip]{fig/2748fg16.eps} \end{figure}$	Figure 10: Ratio $\xi$ between the spectra at the top and the bottom of the atmosphere for $\alpha=4,~10$ (respectively left and right panel) in the case of coupling b. The plots are labeled with the initial amplitudes $z^+_{\rm c}(\omega _0)$ imposed at the top of the layer. Also shown is the ratio for the linear case (dotted line).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{fig/2748fg17.eps} \end{figure}$	Figure 11: Same as in Fig. 8 for coupling b and power-law initial spectrum except that integrations are performed using a spherically expanding turbulent length $L(r)=L_0\times r$ with L₀ fixed by the average dimension of supergranules at coronal level. Results are shown for $\alpha=4,~10$ atmospheres.
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