No evidence of vorticity production from initially irrotational turbulent gravitational collapse

Technical Summary: No Evidence of Vorticity Production from Initially Irrotational Turbulent Gravitational Collapse

Problem Statement
Gravitational collapse is widely believed to generate turbulence, which could potentially seed a small-scale dynamo to amplify magnetic fields below the Jeans scale. However, small-scale dynamos fundamentally require vortical (rotational) turbulence, whereas gravitational collapse predominantly produces irrotational (potential) flows. In the absence of magnetic fields or external driving forces, and when employing a barotropic equation of state (where pressure and density gradients are parallel), the baroclinic term cannot generate vorticity. Consequently, the only mechanism available to produce vorticity from an initially irrotational flow is through the interaction with viscosity. Previous studies have suggested that collapse-induced compression might drive dynamo action, but it remains unclear whether the collapse flow itself generates the necessary vorticity or if observed dynamo effects are merely the decay of pre-existing initial turbulence.

Methodology
The authors perform direct numerical simulations (DNS) of gravitational collapse using the Pencil Code, employing sixth-order centered differences and a third-order time-stepping scheme. The study utilizes a periodic cubic domain with an isentropic equation of state ($\gamma = 5/3$) and a modified isothermal Bonnor-Ebert sphere as the initial density profile.

Key methodological choices include:

• Initial Conditions: The velocity field is constructed to be strictly irrotational in Fourier space, with random phases and a specific spectral slope. No magnetic fields are included to isolate vorticity generation mechanisms.
• Viscosity: Instead of shock viscosity (which can obscure results), the authors use a constant kinematic viscosity ($\nu$).
• Resolution and Parameters: Simulations are run at resolutions ranging from $512^3$ to $2048^3$ mesh points. The authors vary the initial velocity amplitude ($u_{ini}$) and the viscosity ($\nu$) to determine numerically reliable limits.
• Analysis: The study tracks the evolution of enstrophy ($\langle \omega^2 \rangle/2$) and decomposes its time derivative into three terms: a dynamo-like term ($T_{dyn}$), a generation term driven by viscosity and density gradients ($T_{gen}$), and a dissipative term ($T_{dis}$). The authors also analyze velocity, density, and vorticity spectra and examine the scaling of vorticity with initial flow amplitude.

Key Results

1. Viscosity Dependence: The production of vorticity is found to be a secondary effect driven by the amplification of initial flow divergences via viscosity. The study finds that the ratio of vorticity generation to dissipation ($R_{gen}$) is not proportional to viscosity; rather, vorticity generation is most significant during the early phase before the main collapse, not as a consequence of the collapse flow itself.
2. Scaling with Initial Conditions: The root-mean-square (rms) vorticity ($\omega_{rms}$) scales with the square of the initial velocity amplitude ($u_{ini}^2$), while the rms velocity divergence scales linearly with $u_{ini}$. This scaling persists throughout the simulation, suggesting that the gravitational collapse does not directly drive vorticity production. Instead, the collapse amplifies initial flow divergences, which viscosity then converts into vorticity.
3. Numerical Reliability: The authors establish that results become unreliable when the mesh Reynolds number ($Re_{mesh}$) exceeds values between 0.7 and 2.0, depending on the specific run. In the reliable regime, the dynamo-like term ($T_{dyn}$) remains subdominant, contributing less than 20% to the enstrophy evolution.
4. Spectral Analysis: The vorticity spectrum exhibits a $k^4$ subinertial range, analogous to magnetic field behavior in MHD, but the authors attribute this to the Jeans instability driving random velocity fluctuations rather than a dynamo process. The velocity spectrum shows a steep inertial range without a "bottleneck" effect.
5. Critical Vorticity for Dynamo: Revisiting the critical magnetic vorticity Reynolds number ($Pr_M Re_\omega$) for dynamo onset, the authors confirm a bifurcation around a value of 300. This implies that purely irrotational flows, which cannot sustain sufficient vorticity without external driving or specific initial conditions, would not act as dynamos.

Significance and Claims
The paper concludes that gravitational collapse, under the conditions of a barotropic equation of state and initially irrotational turbulence, does not efficiently produce vorticity. The observed vorticity is a secondary effect resulting from the viscous conversion of initial flow divergences, not a direct consequence of the collapse dynamics.

The authors assert that their findings challenge previous claims of collapse-induced small-scale dynamo action. They suggest that earlier observations of magnetic field amplification in collapsing flows were likely powered by the decay of initial turbulence rather than the collapse itself. The study emphasizes the necessity of direct numerical simulations without subgrid-scale modeling to avoid artificial vorticity generation. The work implies that achieving a self-sustaining dynamo in such collapse scenarios is difficult and that the collapse flow alone is insufficient to generate the vortical turbulence required for dynamo action.