🔭 astrophysics

Centrifugal instability of compressible flows and the hydrodynamic stability of accretion disks

This paper validates a new Mach-number-dependent generalization of the Rayleigh criterion for centrifugal instability through numerical simulations of transonic rotating flows, while demonstrating that the hydrodynamic stability of accretion disks remains governed by the standard Solberg-Høiland criterion, which is independent of the Mach number.

Original authors: Serguei S. Komissarov, Konstantinos N. Gourgouliatos

Published 2026-06-11

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CC BY 4.0

Original authors: Serguei S. Komissarov, Konstantinos N. Gourgouliatos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are spinning a bucket of water. If you spin it too fast, the water wants to fly outward. In the old days of physics, scientists had a simple rule (called the Rayleigh criterion) to predict when this spinning fluid would become unstable and start churning into chaos. The rule basically said: "If the spin slows down as you move away from the center, the system is stable. If it speeds up, it's unstable."

For a long time, scientists thought this rule worked perfectly for everything, even for high-speed gas clouds in space (like those swirling around black holes) that were moving faster than the speed of sound. They assumed that because the "spin slowing down" rule wasn't broken, these cosmic whirlpools were safe and stable.

The New Discovery
This paper challenges that assumption. The authors, Komissarov and Gourgouliatos, suggest that the old rule is missing a crucial ingredient: speed. Specifically, how fast the gas is moving compared to the speed of sound (the "Mach number").

They propose a new rule (the "KG criterion") that says: Even if the spin profile looks stable, if the gas is moving fast enough (supersonically), the centrifugal force can still rip the flow apart.

Think of it like a tightrope walker.

The Old View: If the tightrope is straight and the walker is balanced, they are safe.
The New View: It doesn't matter if the rope is straight; if the walker is running at 100 miles per hour, the wind and vibration alone might knock them off, even if they are perfectly balanced.

The Experiment
To test this, the authors didn't just do math on paper; they built a "virtual wind tunnel" using powerful computers. They simulated rotating rings of gas with different speeds and densities.

They set up scenarios where the old rule said, "Everything is fine, no instability here." But according to their new rule, these same scenarios should be chaotic and unstable.

The Results
The computer simulations acted like a referee. They showed that the new rule was right.

When the gas was slow, the old rule worked, and the flow stayed smooth.
When the gas was fast (supersonic), the flow did become unstable, exactly as the new rule predicted.

Visually, the simulation showed the gas developing "fingers" or streams shooting outward from the center, which then curled up into mushroom-like shapes and eventually turned into a turbulent, mixing mess. It was as if the high speed of the gas acted like a hidden trigger, causing the stable-looking flow to explode into chaos.

What This Means for Space
The paper applies this finding to accretion disks—the giant, swirling pancakes of gas and dust that feed stars and black holes. These disks often spin at speeds far faster than sound.

For decades, astronomers have used the old rule to say, "These disks are stable because the spin slows down correctly." This paper argues that because the gas is moving so incredibly fast, the old rule is misleading. The high speed might actually be making these disks unstable, causing them to mix and churn in ways we didn't fully understand before.

The Bigger Picture
The authors also explain that their new, complex rule isn't actually a brand-new invention that breaks physics. Instead, it's a specific way of looking at an old, well-known rule called the Solberg-Høiland criterion.

Think of the Solberg-Høiland criterion as a master key that works for all situations (gravity, pressure, heat). The authors show that when you apply this master key specifically to spinning gas where pressure is the main force holding it together, it looks exactly like their new "Mach number" rule. But when you apply it to gas where gravity is the main force (like in thin disks around black holes), the "Mach number" part disappears, and it looks like the old, simple rules again.

In Summary

The Problem: We thought fast-spinning gas was stable if it followed the old "spin slowing down" rule.
The Fix: The authors found that if the gas is moving fast enough (supersonic), it becomes unstable regardless of the spin profile.
The Proof: Computer simulations confirmed that fast-moving gas breaks apart exactly as their new math predicts.
The Takeaway: High speed changes the stability of cosmic gas flows, and we need to account for that speed to understand how stars and black holes eat and spin.

Technical Summary: Centrifugal Instability of Compressible Flows and the Hydrodynamic Stability of Accretion Disks

Problem Statement
The centrifugal instability in rotating flows is traditionally governed by the Rayleigh criterion, which states that instability arises when specific angular momentum decreases with radius. This criterion was derived for inviscid, incompressible fluids. However, recent analyses of pressure-supported compressible relativistic rotation (Gourgouliatos & Komissarov 2018b) suggested a generalization of this criterion where the instability depends explicitly on the flow Mach number ( $M$ ). In the Newtonian limit, this "KG criterion" posits that a flow is unstable if $d \ln \Psi / d \ln R < M^2$ , where $\Psi = \rho (v_\phi R)^2$ . This implies that even flows stable under the classical Rayleigh criterion (where $d(v_\phi R)^2/dR > 0$ ) could become unstable if the Mach number is sufficiently high.

This theoretical prediction challenges the conventional understanding of astrophysical accretion disks. Thin Keplerian disks, which rotate at hypersonic speeds, are widely considered hydrodynamically stable because they satisfy the Rayleigh criterion. The authors question whether the high Mach numbers in these systems, combined with compressibility, might actually render them susceptible to centrifugal instability, a factor often overlooked in standard stability analyses that assume incompressibility or ignore the explicit Mach dependence.

Methodology
To validate the Newtonian version of the KG criterion, the authors performed axisymmetric numerical simulations of non-relativistic, transonic rotating flows. The simulations solved the equations of ideal compressible fluid dynamics (continuity, momentum, and energy equations) using the AMR-VAC code with an ideal gas equation of state ( $\gamma = 5/3$ ).

The simulation setup involved:

Initial Equilibrium: A configuration with uniform density ( $\rho = \rho_0$ ) and a rotation profile $v_\phi(R)$ that transitions from solid-body rotation in a central core to a power-law decay ( $v_\phi \propto R^{-n}$ ) outside the core.
Pressure Balance: The pressure distribution was derived to satisfy the radial force balance equation $dp/dR = \rho v_\phi^2 / R$ .
Perturbation: The equilibrium was perturbed with a small azimuthal velocity disturbance ( $\delta v_\phi = \sin(kz)$ ) to trigger potential instabilities.
Parameter Space: The authors varied the power-law index $n$ (ranging from 0.2 to 1.0) and the central pressure $p_0$ to create models with different Mach numbers ( $M_c$ ) at the core boundary. This allowed them to test configurations that are stable according to the original Rayleigh criterion but unstable according to the KG criterion.

Key Contributions and Results

Validation of the KG Criterion: The numerical results demonstrated perfect agreement with the theoretical KG criterion. Models predicted to be unstable (where $2(1-n) < M^2$ ) exhibited rapid growth of perturbations, while models predicted to be stable remained so. This confirms that the centrifugal instability in compressible flows is strongly dependent on the Mach number, a feature absent in the classical incompressible limit.
Nonlinear Evolution: The simulations revealed the nonlinear development of the instability. Perturbations grew fastest at the inner boundary of the unstable domain. As the instability evolved, the flow developed radial streams (fingers) moving away from the core. These streams interacted with the surrounding fluid to form mushroom-like structures and cocoons exhibiting rolls characteristic of the Kelvin-Helmholtz instability. This process led to a turbulent layer with strong mixing, which extended significantly beyond the initial linearly unstable region.
Re-evaluation of Accretion Disk Stability: The authors extended their heuristic stability analysis to include both pressure forces and the gravity of a central compact object. They derived a general instability criterion that retains the explicit dependence on the Mach number.
- In the limit of vanishing gravity, this criterion reduces to the KG criterion for pressure-supported rotation.
- In the limit of vanishing rotation, it reduces to the Schwarzschild criterion for convective instability.
- In the general case, the derived criterion is mathematically equivalent to the standard Solberg-Høiland criterion, which does not explicitly display the Mach number dependence.

Significance and Claims
The paper claims that the hydrodynamic stability of accretion disks is often explained using the original Rayleigh criterion, even when the rotation is highly supersonic. By deriving a criterion that explicitly includes the Mach number and showing its equivalence to the Solberg-Høiland criterion, the authors clarify that the stability of thin Keplerian disks is not merely a result of satisfying the incompressible Rayleigh condition, but rather a consequence of the specific balance between centrifugal and convective forces described by Solberg-Høiland.

The authors conclude that while the KG criterion provides a useful formulation for pressure-supported flows and jets (where gravity is negligible), the explicit Mach number dependence is effectively absorbed into the entropy and angular momentum gradients in the general Solberg-Høiland formulation. Consequently, for isentropic flows (where entropy is constant), the criterion reduces to the original Rayleigh criterion, consistent with Rayleigh's (1917) analysis of barotropic fluids. The study confirms that high Mach numbers can destabilize pressure-supported flows that would otherwise be stable, a finding relevant to the dynamics of astrophysical jets and the interaction of supersonic winds with surrounding plasma, though the authors note that full 3D simulations are required to fully characterize the nonlinear mixing triggered by this instability.

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