MA Emilio Schmidt - Revision history

Emilio.S: /* 3D Hydrodynamic Reference Test: Cylindrical coordinates */

2026-04-28T14:09:34Z

3D Hydrodynamic Reference Test: Cylindrical coordinates

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	\end{align}		\end{align}

	Thus, two candidate maxima are only counted as independent fragments if their separation is larger than $0.05\lambda_{\mathrm{frag}}(r_0)$. The motivation for relating $\Delta z_{\min}$ to $\lambda_{\mathrm{frag}}(r_0)$ is that the latter provides the natural physical length scale of the fragmentation process. Choosing the minimum separation as a fixed fraction of this scale ensures that the criterion adapts consistently when $r_0$ is varied. At the same time, the chosen fraction is small enough that genuinely distinct neighboring fragments are not merged, but large enough to suppress multiple counting within a single fragment.		Thus, two candidate maxima are only counted as independent fragments if their separation is larger than $0.05\lambda_{\mathrm{frag}}(r_0)$. The motivation for relating $\Delta z_{\min}$ to $\lambda_{\mathrm{frag}}(r_0)$ is that the latter provides the natural physical length scale of the fragmentation process. Choosing the minimum separation as a fixed fraction of this scale ensures that the criterion adapts consistently when $r_0$ is varied. At the same time, the chosen fraction is small enough that genuinely distinct neighboring fragments are not merged, but large enough to suppress multiple counting within a single fragment.

	Once the fragment positions $z_i$ have been identified in this way, the distances between neighboring fragments are computed as follows:		Once the fragment positions $z_i$ have been identified in this way, the distances between neighboring fragments are computed as follows:

Emilio.S: /* 3D Hydrodynamic Reference Test: Cylindrical coordinates */

2026-04-28T14:08:59Z

3D Hydrodynamic Reference Test: Cylindrical coordinates

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	The prefactor must satisfy a compromise. On the one hand, it must not be too small, so that maxima caused merely by noise or similar effects are excluded. On the other hand, it must not be too large, so that already developed but still weakly pronounced fragments are not excluded. For the simulations considered here, the chosen value turns out to be a good compromise.		The prefactor must satisfy a compromise. On the one hand, it must not be too small, so that maxima caused merely by noise or similar effects are excluded. On the other hand, it must not be too large, so that already developed but still weakly pronounced fragments are not excluded. For the simulations considered here, the chosen value turns out to be a good compromise.

	However, a density threshold alone is not sufficient for a robust fragment identification. In practice, a single physical fragment can produce several nearby local maxima in the axial density profile, for example due to small-scale substructure, discretization effects, or slight irregularities near the peak. Such nearby submaxima do not correspond to independent fragments. If all of ~~them~~ were counted separately, the number of identified fragments would be artificially increased ~~and the characteristic fragment spacing would be systematically underestimated~~. As a ~~consequence~~, the measured ~~separations~~ would be ~~systematically underestimated~~ and would no longer accurately represent the ~~spacing~~ between individual fragments.		However, a density threshold alone is not sufficient for a robust fragment identification. In practice, a single physical fragment can produce several nearby local maxima in the axial density profile, for example due to small-scale substructure, discretization effects, or slight irregularities near the peak. Such nearby submaxima do not correspond to independent fragments. If all of these maxima were counted separately, the number of identified fragments would be artificially increased. As a result, the measured fragment spacing would be biased toward smaller values and would no longer accurately represent the separation between individual fragments.

	To avoid this, the peak detection is supplemented by a minimum-separation criterion along the $z$-direction. More precisely, only maxima whose mutual distance exceeds a prescribed minimum value are counted as distinct fragments. In this analysis, this minimum separation is chosen as a fixed fraction of the analytically expected fragment spacing:		To avoid this, the peak detection is supplemented by a minimum-separation criterion along the $z$-direction. More precisely, only maxima whose mutual distance exceeds a prescribed minimum value are counted as distinct fragments. In this analysis, this minimum separation is chosen as a fixed fraction of the analytically expected fragment spacing:

Emilio.S: /* Structure of the numerical spectrum */

2026-04-28T14:06:40Z

Structure of the numerical spectrum

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	This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.		This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.

	This numerically observed structure of the spectrum ~~should~~ not ~~be understood as a mere numerical finding~~, but can be explained from the structure of the underlying boundary value problem. The generalized eigenvalue problem is formulated on the semi-infinite interval $x \in [0,\infty)$ and is therefore determined to a large extent by the behavior of the solutions in the two boundary regions. A physically admissible solution must satisfy both the boundary condition at $x = 0$ and the one at $x \to \infty$. It is therefore natural to examine the coupled differential equations more closely precisely in these two limiting regions. In particular, it turns out that the threshold $-\bar{k}^2$ observed in the numerical spectrum arises naturally from the behavior of the equations at large radii.		This numerically observed structure of the spectrum is not merely an isolated feature of the discretized problem, but can be explained from the structure of the underlying boundary value problem. The generalized eigenvalue problem is formulated on the semi-infinite interval $x \in [0,\infty)$ and is therefore determined to a large extent by the behavior of the solutions in the two boundary regions. A physically admissible solution must satisfy both the boundary condition at $x = 0$ and the one at $x \to \infty$. It is therefore natural to examine the coupled differential equations more closely precisely in these two limiting regions. In particular, it turns out that the threshold $-\bar{k}^2$ observed in the numerical spectrum arises naturally from the behavior of the equations at large radii.

	=== Asymptotic behavior of the solutions ===		=== Asymptotic behavior of the solutions ===

Emilio.S: /* Structure of the numerical spectrum */

2026-04-28T14:04:49Z

Structure of the numerical spectrum

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	above the threshold $-\bar{k}^2$, that is, satisfying the condition $\bar{\sigma}^2 > -\bar{k}^2$. The remaining numerically determined eigenvalues lie below this threshold. The following figure shows the corresponding numerically determined spectrum for $(m,\bar{k}) = (0,1)$. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.		above the threshold $-\bar{k}^2$, that is, satisfying the condition $\bar{\sigma}^2 > -\bar{k}^2$. The remaining numerically determined eigenvalues lie below this threshold. The following figure shows the corresponding numerically determined spectrum for $(m,\bar{k}) = (0,1)$. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.

	[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = ~~250~~$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.]]		[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = 500$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.]]

	This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.		This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.

Emilio.S: /* Separation ansatz and radial boundary value problem */

2026-04-28T14:03:43Z

Separation ansatz and radial boundary value problem

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	\end{align}		\end{align}

	Here, $\chi(r,z)$ is an arbitrary function that depends on $r$ and $z$. Since the entries of the linear operator consist of sums and products of such building blocks, and since each of them commutes with $\mathcal{T}_{\varphi_0}$, it follows that $\mathbf{L}$ also commutes with the ~~translation~~ operator as a whole.		Here, $\chi(r,z)$ is an arbitrary function that depends on $r$ and $z$. Since the entries of the linear operator consist of sums and products of such building blocks, and since each of them commutes with $\mathcal{T}_{\varphi_0}$, it follows that $\mathbf{L}$ also commutes with the rotation operator as a whole.

	It turns out that Fourier modes $\exp(i m \varphi)$ are a convenient choice for the $\varphi$-dependence. The reason is that these functions are eigenfunctions of the ~~translation~~ operator:		It turns out that Fourier modes $\exp(i m \varphi)$ are a convenient choice for the $\varphi$-dependence. The reason is that these functions are eigenfunctions of the rotation operator:

	\begin{align}		\begin{align}

Emilio.S: /* Separation ansatz and radial boundary value problem */

2026-04-28T14:01:08Z

Separation ansatz and radial boundary value problem

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	\end{align}		\end{align}

	In this representation, the Poisson equation would correspond to the ~~fourth~~ row of the system of equations. However, this assignment is not unique, but rather a matter of the chosen ordering of the variables in $\boldsymbol{w}$ or of the equations in the operator $\mathbf{L}$. The only essential point is that the assignment is kept consistent.		In this representation, the Poisson equation would correspond to the fifth row of the system of equations. However, this assignment is not unique, but rather a matter of the chosen ordering of the variables in $\boldsymbol{w}$ or of the equations in the operator $\mathbf{L}$. The only essential point is that the assignment is kept consistent.

	Since the linear operator has no explicit dependence on $\varphi$, the linearized equations are rotationally invariant in $\varphi$. This means that a rotation of a solution about the $z$-axis is again a solution of the same linear system. To show this, the rotation operator $\mathcal{T}_{\varphi_0}$ is introduced, which is defined by:		Since the linear operator has no explicit dependence on $\varphi$, the linearized equations are rotationally invariant in $\varphi$. This means that a rotation of a solution about the $z$-axis is again a solution of the same linear system. To show this, the rotation operator $\mathcal{T}_{\varphi_0}$ is introduced, which is defined by:

Emilio.S: /* Free-fall time of an an infinitely long gas cylinder */

2026-04-28T13:59:43Z

Free-fall time of an an infinitely long gas cylinder

← Older revision		Revision as of 15:59, 28 April 2026
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	\end{align}		\end{align}

	=== Free-fall time of an an infinitely long gas cylinder ===		=== Free-fall time of an infinitely long gas cylinder ===

	In the following, the free fall time $t_{\text{ff}}$ of an infinitely long gas cylinder is calculated for the previously derived density profile, with the cylinder initially at rest. The calculation is performed in the pressureless limit $P \rightarrow 0$. The dynamics of the system are then determined solely by its own gravity.		In the following, the free fall time $t_{\text{ff}}$ of an infinitely long gas cylinder is calculated for the previously derived density profile, with the cylinder initially at rest. The calculation is performed in the pressureless limit $P \rightarrow 0$. The dynamics of the system are then determined solely by its own gravity.

Emilio.S: /* Introduction */

2026-04-28T13:59:08Z

Introduction

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	This master's thesis investigates the dynamics and stability of filamentary gas structures using numerical simulations. Such filaments are of particular physical interest because they are commonly observed in star-forming regions and are closely connected to the fragmentation of molecular clouds and the formation of stars.		This master's thesis investigates the dynamics and stability of filamentary gas structures using numerical simulations. Such filaments are of particular physical interest because they are commonly observed in star-forming regions and are closely connected to the fragmentation of molecular clouds and the formation of stars.

	As a first reference case, this work considers the classical isothermal, self-gravitating gas cylinder introduced by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1056O/abstract Ostriker (1967)]. This hydrodynamic model provides an ~~analytically~~ equilibrium solution and is therefore well suited for testing the numerical treatment of self-gravity, boundary conditions and gravitational fragmentation.		As a first reference case, this work considers the classical isothermal, self-gravitating gas cylinder introduced by [https://ui.adsabs.harvard.edu/abs/1964ApJ...140.1056O/abstract Ostriker (1967)]. This hydrodynamic model provides an analytical equilibrium solution and is therefore well suited for testing the numerical treatment of self-gravity, boundary conditions and gravitational fragmentation.

	Building on this hydrodynamic reference case, the long-term goal is to extend the analysis to magnetohydrodynamic filament models. As a theoretical reference		Building on this hydrodynamic reference case, the long-term goal is to extend the analysis to magnetohydrodynamic filament models. As a theoretical reference

Emilio.S: /* Dominant mode and characteristic fragment spacing */ Nagasawa

2026-04-28T13:56:41Z

Dominant mode and characteristic fragment spacing: Nagasawa

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	[[File:MAES-MaxEigenValuesKM0Xmax25N500.png\|400px\|thumb\|center\|Maximal squared growth rate $\bar{\sigma}_{\max}^2(m=0,\bar{k})$ as a function of the dimensionless wavenumber $\bar{k}$ for the axisymmetric mode $m=0$. The red point marks the maximum of the curve, corresponding to the fastest-growing unstable mode, and the dashed vertical line indicates the associated dimensionless wavenumber.]]		[[File:MAES-MaxEigenValuesKM0Xmax25N500.png\|400px\|thumb\|center\|Maximal squared growth rate $\bar{\sigma}_{\max}^2(m=0,\bar{k})$ as a function of the dimensionless wavenumber $\bar{k}$ for the axisymmetric mode $m=0$. The red point marks the maximum of the curve, corresponding to the fastest-growing unstable mode, and the dashed vertical line indicates the associated dimensionless wavenumber.]]

	The figure shows that the growth rate of the fastest-growing unstable mode is given by:		The figure shows that the dimensionless growth rate of the fastest-growing unstable mode is given by:

	\begin{align}		\begin{align}
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	\bar{k}_{\ast} = 0.8081		\bar{k}_{\ast} = 0.8081
	\end{align}		\end{align}

			These values agree well with the unmagnetized reference results of [https://academic.oup.com/ptp/article/77/3/635/1869157 Nagasawa (1987)]. For an isothermal cylinder without magnetic field, Nagasawa finds the following dimensionless growth rate and wavenumber of the fastest-growing unstable mode:

			\begin{align}
			\bar{\sigma}_{\ast} = 0.9588 \quad \text{and} \quad \bar{k}_{\ast} = 0.8033
			\end{align}

			These reference values are in good agreement with the numerical results obtained here.

	Since the fastest-growing Fourier mode dominates the linear evolution for sufficiently large times, periodic overdensities form along the $z$-axis. The corresponding wavelength is given by:		Since the fastest-growing Fourier mode dominates the linear evolution for sufficiently large times, periodic overdensities form along the $z$-axis. The corresponding wavelength is given by:

Emilio.S: /* Linear stability analysis */ Nagasawa and some minor changes

2026-04-28T12:32:43Z

Linear stability analysis: Nagasawa and some minor changes

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	== Linear stability analysis ==		== Linear stability analysis ==

	The aim of this section is to investigate how the hydrostatic equilibrium solution of an infinitely long, isothermal, self-gravitating gas cylinder behaves under perturbations. In the following, all equilibrium variables are denoted by the index $0$:		The aim of this section is to investigate how the hydrostatic equilibrium solution of an infinitely long, isothermal, self-gravitating gas cylinder behaves under perturbations. For the hydrodynamic reference case considered here, the results can be compared directly with the analysis of [https://academic.oup.com/ptp/article/77/3/635/1869157 Nagasawa (1987)]. In the following, all equilibrium variables are denoted by the index $0$:

	\begin{align}		\begin{align}
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	Since the dimensionless Poisson equation does not contain any terms depending on $\bar{\sigma}^2$, all entries in the rows $r = N + j$ representing the Poisson equation are zero.		Since the dimensionless Poisson equation does not contain any terms depending on $\bar{\sigma}^2$, all entries in the rows $r = N + j$ representing the Poisson equation are zero.

	This generalized eigenvalue problem can now be solved numerically. In the present work, the Python library SciPy was used for this purpose, as it provides routines for solving generalized eigenvalue problems. Of particular interest are the eigenvalues, since they determine the linear temporal evolution of the perturbations.		This generalized eigenvalue problem can now be solved numerically. In the present work, the Python library [https://docs.scipy.org/doc/scipy/ SciPy] was used for this purpose, as it provides routines for solving generalized eigenvalue problems. Of particular interest are the eigenvalues, since they determine the linear temporal evolution of the perturbations.

	=== Structure of the numerical spectrum ===		=== Structure of the numerical spectrum ===
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	In the following, the spectrum of eigenvalues for the fixed pair $(m,\bar{k}) = (0,1)$ is considered as an illustrative example. The main purpose is to investigate the qualitative structure of the numerically determined spectrum. For this purpose, the numerical solution of the given equations is first carried out on a computational domain with $x_{\max} = 25$ using $N = 500$ grid points.		In the following, the spectrum of eigenvalues for the fixed pair $(m,\bar{k}) = (0,1)$ is considered as an illustrative example. The main purpose is to investigate the qualitative structure of the numerically determined spectrum. For this purpose, the numerical solution of the given equations is first carried out on a computational domain with $x_{\max} = 25$ using $N = 500$ grid points.

	For the illustrative pair $(m,\bar{k}) = (0,1)$, the numerical solution of the generalized eigenvalue problem yields a discrete spectrum of eigenvalues $\bar{\sigma}_n^2$. A characteristic structure of the spectrum becomes apparent. ~~There exists exactly~~ one eigenvalue above the threshold $-\bar{k}^2$, that is, satisfying the condition $\bar{\sigma}^2 > -\bar{k}^2$. ~~All~~ remaining numerically determined eigenvalues~~, by contrast,~~ lie below this threshold~~, that is, at $\bar{\sigma}^2 < -\bar{k}^2$~~.		For the illustrative pair $(m,\bar{k}) = (0,1)$, the numerical solution of the generalized eigenvalue problem yields a discrete spectrum of eigenvalues $\bar{\sigma}_n^2$. A characteristic structure of the spectrum becomes apparent. Within the investigated numerical setup, only one eigenvalue is found
			above the threshold $-\bar{k}^2$, that is, satisfying the condition $\bar{\sigma}^2 > -\bar{k}^2$. The remaining numerically determined eigenvalues lie below this threshold. The following figure shows the corresponding numerically determined spectrum for $(m,\bar{k}) = (0,1)$. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.
	The following figure shows the corresponding numerically determined spectrum for $(m,\bar{k}) = (0,1)$. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.

	[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = 250$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.]]		[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = 250$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.]]

	\colorbox{yellow}{Plot of the spectrum for $(m,\bar{k}) = (0,1)$, caption: Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = 250$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.}

	This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.		This observation is not restricted to the pair $(m,\bar{k}) = (0,1)$, but can also be found for other numerically investigated combinations of azimuthal mode number $m$ and dimensionless wavenumber $\bar{k}$. Numerically, it is also observed that the eigenvalues approach the threshold $-\bar{k}^2$. Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.
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	=== Asymptotic behavior of the solutions ===		=== Asymptotic behavior of the solutions ===

	In the following, the local behavior of the solutions at $x = 0$ is first examined, followed by the asymptotic behavior for $x \to \infty$. On this basis, it then becomes possible to understand why the numerically determined spectrum for a fixed pair $(m,\bar{k})$ exhibits the characteristic structure observed above.		In the following, the local behavior of the solutions at $x = 0$ is first examined, followed by the asymptotic behavior for $x \to \infty$. On this basis, it then becomes possible to understand why the numerically determined spectrum for a fixed pair $(m,\bar{k})$ exhibits the characteristic structure observed above. This analysis is used to interpret the structure of the numerically obtained spectrum. It is not intended as a mathematical proof.

	To investigate the local behavior of the solutions at $x = 0$, it is convenient to consider the continuity equation in the following form:		To investigate the local behavior of the solutions at $x = 0$, it is convenient to consider the continuity equation in the following form:

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	above the threshold \(-\bar{k}^2\), that is, satisfying the condition \(\bar{\sigma}^2 > -\bar{k}^2\). The remaining numerically determined eigenvalues lie below this threshold. The following figure shows the corresponding numerically determined spectrum for \((m,\bar{k}) = (0,1)\). The dashed horizontal line marks the threshold \(-\bar{k}^2\). The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.		above the threshold \(-\bar{k}^2\), that is, satisfying the condition \(\bar{\sigma}^2 > -\bar{k}^2\). The remaining numerically determined eigenvalues lie below this threshold. The following figure shows the corresponding numerically determined spectrum for \((m,\bar{k}) = (0,1)\). The dashed horizontal line marks the threshold \(-\bar{k}^2\). The inset shows the last ten eigenvalues of the spectrum and illustrates that only a single eigenvalue lies above this threshold, while all remaining eigenvalues stay below it.

	[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues \(\bar{\sigma}_n^2\) for the fixed pair \((m,\bar{k}) = (0,1)\), on a computational domain with \(x_{\max} = 25\) and \(N = ~~250~~\) grid points. The dashed horizontal line marks the threshold \(-\bar{k}^2\). The inset shows the last ten eigenvalues.]]		[[File:MAES-SpectrumM0K1Xmax25N500.png\|400px\|thumb\|center\|Spectrum of the numerically determined eigenvalues \(\bar{\sigma}_n^2\) for the fixed pair \((m,\bar{k}) = (0,1)\), on a computational domain with \(x_{\max} = 25\) and \(N = 500\) grid points. The dashed horizontal line marks the threshold \(-\bar{k}^2\). The inset shows the last ten eigenvalues.]]

	This observation is not restricted to the pair \((m,\bar{k}) = (0,1)\), but can also be found for other numerically investigated combinations of azimuthal mode number \(m\) and dimensionless wavenumber \(\bar{k}\). Numerically, it is also observed that the eigenvalues approach the threshold \(-\bar{k}^2\). Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.		This observation is not restricted to the pair \((m,\bar{k}) = (0,1)\), but can also be found for other numerically investigated combinations of azimuthal mode number \(m\) and dimensionless wavenumber \(\bar{k}\). Numerically, it is also observed that the eigenvalues approach the threshold \(-\bar{k}^2\). Whether this threshold actually constitutes an accumulation point of the spectrum is not proved here, however. For the arguments and observations presented in the following, this is also not necessary.