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$\begin{figure} \par\includegraphics[width=3.5cm,clip]{3115f1a.ps}\hspace*{2mm} \includegraphics[width=3.5cm,clip]{3115f1a.ps} \end{figure}$ Figure 1: Solutions for the linear perturbation differential equation. Left panel: Antisymmetric solution for model B05. Right panel: Symmetric solution for model D20. Vertical lines stand for the perturbed wavenumbers in the numerical simulations (from left to right: k₀, k₁, k₂ and k₃). Let us note that the fundamental mode does not appear in the right panel, as its growth rates are lower than the scale of the plot.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{3115f2.ps} \end{figure}$ Figure 2: Specific modes of solutions shown in Fig. 1, symmetric solution for model D20. Dotted line: first body mode; dashed: second body mode; dash-dot: twentieth body mode; dash-triple dot: twenty-fifth body mode. Arows point to both the broad maxima and the small wavenumber peaks present in every single mode. Low wavenumber peaks of high order body modes show higher growth rates and are thus defined as (shear layer) resonances.
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In the text

$\begin{figure} \par\includegraphics[width=5cm,clip]{3115f3a.ps}\hspace*{2mm} \in... ...15f3i.ps}\hspace*{2mm} \includegraphics[width=5cm,clip]{3115f3j.ps} \end{figure}$ Figure 3: Evolution of the relative amplitudes of perturbations. Dotted line: pressure perturbation ( $(p_{\rm max}-p_0)/p_0$ ). Dashed line: longitudinal velocity perturbation in the jet reference frame ( $0.5~(v'_{z,\rm max}-v'_{z,\rm min})$ ). Dash-dotted line: perpendicular velocity perturbation in the jet reference frame ( $0.5~(v'_{x,\rm max}-v'_{x,\rm min})$ ). Note the different scales in the horizontal axis. The search for maximum ( $p_{\rm max}$ , $v'_{x,\rm max}$ , $v'_{z,\rm max}$ ) and minimum ( $v'_{x,\rm min}$ , $v'_{z,\rm min}$ ) values were restricted to those numerical zones with jet mass fraction larger than 0.5.
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In the text

$\begin{figure} \par\includegraphics[width=11.2cm,clip]{3115f4a.ps}\par\vspace*{4mm} \includegraphics[width=11.2cm,clip]{3115f4b.ps} \end{figure}$ Figure 4: Upper panels: pressure ( left) and perpendicular velocity perturbation ( right) at late stages of linear phase (model D20). Lower panels: pressure ( left) and perpendicular velocity perturbation ( right) corresponding to one resonant mode from linear analysis. The linear, grey scale is arbitrary. Amplitudes are maxima at the shear layer, hence the name of shear layers resonances given to these modes. Oblique waves in upper panels are the result of longer wavelength perturbations, not present in the bottom panels.
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In the text

$\begin{figure} \par\includegraphics[width=11.7cm,clip]{3115f5a.ps}\par\vspace*{3... ...s}\par\vspace*{3mm} \includegraphics[width=11.7cm,clip]{3115f5c.ps} \end{figure}$ Figure 5: Snapshots of logarithm of pressure ( left) and Lorentz factor ( right) for models B2.5 ( upper panels), D05 ( center panels) and D10 ( bottom panels) at $t_{\rm sat}$ , where irregular structures caused by mode competition are observed.
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{3115f6a.ps}\hspace*{3mm} \includegraphics[width=6.8cm,clip]{3115f6b.ps} \end{figure}$ Figure 6: Transversal Mach number in simulations (see text for definition). Solid line: $\gamma =2.5$ ; dotted line: $\gamma =5.0$ ; dashed line: $\gamma =10.0$ ; dash-dot line: $\gamma =20.0$ . The thinnest line is for model A and thickest for model D, with B in between, the left panel shows models A2.5, B2.5, B05, and B10, and the right panel shows models A10, B20, D2.5, D05, D10, and D20.
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{3115f7.ps} \end{figure}$ Figure 7: Evolution of the mean width of the jet/ambient mixing layer (for tracer values between 0.05 and 0.95) with time for all simulations. Lines represent the same models as in Fig. 6. A value of 5 R_j for the width of the mixing layer (horizontal dashed line) serves to classify the evolution of the different models.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115f8.ps} \par\end{figure}$ Figure 8: Evolution of the normalized total longitudinal momentum in the jet as a function of time. Lines represent the same models as in Fig. 6. The long-dashed horizontal line identifies those models transferring more than 50% of the initial jet momentum to the ambient.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115f9.ps} \par\end{figure}$ Figure 9: Evolution of the normalized total transversal momentum in the jet as a function of time for all the simulations. Lines represent the same models as in Fig. 6.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f10.ps} \end{figure}$ Figure 10: Snapshot at the last frame of the simulation of logarithmic maps of pressure, jet mass fraction and specific internal energy and non-logarithmic Lorentz factor for model A2.5.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f11.ps} \end{figure}$ Figure 11: Same as Fig. 10 for model B2.5.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f12.ps} \end{figure}$ Figure 12: Same as Fig. 10 for model D2.5.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f13.ps} \end{figure}$ Figure 13: Same as Fig. 10 for model B05.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f14.ps} \end{figure}$ Figure 14: Same as Fig. 10 for model D05.
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In the text

$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f15.ps} \end{figure}$ Figure 15: Same as Fig. 10 for model A10.
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In the text

$\begin{figure} \par\includegraphics[width=11.2cm,clip]{3115f16.ps} \end{figure}$ Figure 16: Same as Fig. 10 for model B10.
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In the text

$\begin{figure} \par\includegraphics[width=11.2cm,clip]{3115f17.ps} \end{figure}$ Figure 17: Same as Fig. 10 for model D10.
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In the text

$\begin{figure} \par\includegraphics[width=11.4cm,clip]{3115f18.ps} \end{figure}$ Figure 18: Same as Fig. 10 for model B20.
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In the text

$\begin{figure} \par\includegraphics[width=11.4cm,clip]{3115f19.ps} \end{figure}$ Figure 19: Same as Fig. 10 for model D20.
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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{3115f20a.ps}\hspace*{5mm} \i... ...f20e.ps}\hspace*{5mm} \includegraphics[width=6cm,clip]{3115f20f.ps} \end{figure}$ Figure 20: Averaged transversal structure in the final state of the jets corresponding to models A2.5 ( upper panels), D10 ( middle), and B20 ( bottom). Left panel (thermodynamical quantities): solid line, tracer; dotted line, rest mass density; dashed line, specific internal energy. Right panels (dynamical quantities): solid line, longitudinal velocity; dotted line, lorentz factor normalized to the initial value in the jet; dashed line, longitudinal momentum normalized to the initial value in the jet. Specific internal energy for model D10 was divided by 100 to fit in the scale.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115f21.ps} \end{figure}$ Figure 21: Relativistic internal jet Mach number (M_j) versus jet Lorentz factor ( $\gamma$ ) of the simulated models here and in Papers I and II. Symbols represent different non-linear behaviors: crosses stand for UST1 disrupted jets (low relativistic Mach number and low Lorentz factor); triangles for UST2 jets (moderately fast and supersonic), and squares for ST jets (highly supersonic and fast jets). Models with two different symbols are those with a different evolution in simulations presented here and those from Papers I and II (see text).
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In the text

$\begin{figure} \par\includegraphics[width=6.4cm,clip]{3115fA1a.ps}\hspace*{3mm} \includegraphics[width=6.4cm,clip]{3115fA1b.ps} \end{figure}$ Figure A.1: Left panel: evolution of the mean width of the jet/ambient mixing layer (for tracer values between 0.05 and 0.95) with time. Right panel: evolution of the normalized total longitudinal momentum in the jet as a function of time. As in Fig. 6, dotted lines stand for $\gamma =5$ and dash-dot lines stand for $\gamma =20$ , while thin lines are for B models and thick lines for D models.

In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{3115fA2.ps} \end{figure}$ Figure A.2: Snapshot at the last frame of the simulation of jet mass fraction ( left panel) and Lorentz factor ( right panel) for model B05.

In the text

$\begin{figure} \par\includegraphics[width=7.3cm,clip]{3115fA3.ps} \end{figure}$ Figure A.3: Model D05. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115fA4.ps} \end{figure}$ Figure A.4: Model B20. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=7.3cm,clip]{3115fA5.ps} \end{figure}$ Figure A.5: Model D20. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=6.5cm,clip]{3115fB1a.ps}\hspace*{4mm} \includegraphics[width=6.5cm,clip]{3115fB1b.ps} \par\end{figure}$ Figure B.1: Left panel: evolution of the mean width of the jet/ambient mixing layer (for tracer values between 0.05 and 0.95) with time. Right panel: evolution of the normalized total longitudinal momentum in the jet as a function of time. Lines represent the same models as in Fig. A.1.

In the text

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{3115fB2.ps} \end{figure}$ Figure B.2: Model B05. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{3115fB3.ps} \end{figure}$ Figure B.3: Model D05. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{3115fB4.ps} \end{figure}$ Figure B.4: Model B20. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=6.8cm,clip]{3115fB5.ps} \end{figure}$ Figure B.5: Model D20. Same as Fig. A.2.

In the text

$\begin{figure} \par\includegraphics[width=11.7cm,clip]{3115f5a.ps}\par\vspace{3... ...s}\par\vspace{3mm} \includegraphics[width=11.7cm,clip]{3115f5c.ps} \end{figure}$	Figure 5: Snapshots of logarithm of pressure ( left) and Lorentz factor ( right) for models B2.5 ( upper panels), D05 ( center panels) and D10 ( bottom panels) at $t_{\rm sat}$ , where irregular structures caused by mode competition are observed.
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$\begin{figure} \par\includegraphics[width=7.4cm,clip]{3115f7.ps} \end{figure}$	Figure 7: Evolution of the mean width of the jet/ambient mixing layer (for tracer values between 0.05 and 0.95) with time for all simulations. Lines represent the same models as in Fig. 6. A value of 5 R_j for the width of the mixing layer (horizontal dashed line) serves to classify the evolution of the different models.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115f8.ps} \par\end{figure}$	Figure 8: Evolution of the normalized total longitudinal momentum in the jet as a function of time. Lines represent the same models as in Fig. 6. The long-dashed horizontal line identifies those models transferring more than 50% of the initial jet momentum to the ambient.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3115f9.ps} \par\end{figure}$	Figure 9: Evolution of the normalized total transversal momentum in the jet as a function of time for all the simulations. Lines represent the same models as in Fig. 6.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f10.ps} \end{figure}$	Figure 10: Snapshot at the last frame of the simulation of logarithmic maps of pressure, jet mass fraction and specific internal energy and non-logarithmic Lorentz factor for model A2.5.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f11.ps} \end{figure}$	Figure 11: Same as Fig. 10 for model B2.5.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f12.ps} \end{figure}$	Figure 12: Same as Fig. 10 for model D2.5.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f13.ps} \end{figure}$	Figure 13: Same as Fig. 10 for model B05.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f14.ps} \end{figure}$	Figure 14: Same as Fig. 10 for model D05.
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$\begin{figure} \par\includegraphics[width=11.6cm,clip]{3115f15.ps} \end{figure}$	Figure 15: Same as Fig. 10 for model A10.
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$\begin{figure} \par\includegraphics[width=11.2cm,clip]{3115f16.ps} \end{figure}$	Figure 16: Same as Fig. 10 for model B10.
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$\begin{figure} \par\includegraphics[width=11.2cm,clip]{3115f17.ps} \end{figure}$	Figure 17: Same as Fig. 10 for model D10.
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$\begin{figure} \par\includegraphics[width=11.4cm,clip]{3115f18.ps} \end{figure}$	Figure 18: Same as Fig. 10 for model B20.
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$\begin{figure} \par\includegraphics[width=11.4cm,clip]{3115f19.ps} \end{figure}$	Figure 19: Same as Fig. 10 for model D20.
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