Hydrodynamic Plasma Cloud-Cone Model

• Hydrodynamic plasma cloud-cone model is a framework capturing the formation of conical shock structures in plasma and magnetofluid systems under supersonic dynamics.
• The model employs specialized equations (GH, MHD, and relativistic hydrodynamics) to describe conical structures in dusty plasmas, QGP, and colliding astrophysical clouds.
• Diagnostic measures such as cone opening angles and density variations connect microphysical effects like strong coupling and drag to large-scale flow phenomena.

The hydrodynamic plasma cloud-cone model encapsulates a class of phenomena and simulation frameworks wherein plasma (or, more generally, fluid) flows and structure formation exhibit pronounced conical geometries and shock-induced features, frequently arising from supersonic motion or dynamical collisions in a plasma or magnetofluid environment. Variants of this model are systematically developed across several physical contexts: strongly coupled complex (dusty) plasmas, expanding relativistic quark-gluon plasma (QGP) in heavy-ion collisions, driven laboratory dusty-plasma flows behind de Laval nozzles, and cloud-cloud collisions in astrophysical magnetized interstellar media. While each regime instantiates the “cloud-cone” concept with distinct microphysics and analytic treatments, all are united by the emergence, morphology, and evolution of shock- or compression-induced conical structures, whose parameters encode the interplay between source motion, background thermodynamics, dissipation, and collective effects such as strong coupling or self-gravity.

1. Foundational Models: Governing Equations and Regimes

Strongly Coupled Complex Plasmas

The plasma cloud-cone in a strongly coupled, unmagnetized dusty plasma is described within the generalized-hydrodynamic (GH) fluid framework. The system consists of electrons and ions (Boltzmann-distributed) and a dust component for which strong-coupling effects are encoded in a modified compressibility parameter $\mu$ and Epstein drag term $\gamma_{\rm EP}$ (Bandyopadhyay et al., 2016). The dust fluid equations, linearized about equilibrium, are: $$\begin{align*} &\text{Continuity:}\quad \frac{\partial n_d}{\partial t} + \nabla \cdot (n_d \mathbf{v}_d) = 0 \ &\text{Momentum:}\quad m_d \frac{\partial \mathbf{v}_{d1}}{\partial t} = Z_d e\, \nabla \phi_1 - \gamma_{\rm EP} m_d \mathbf{v}_{d1} - \frac{\mu T_d}{m_d n_{d0}}\nabla n_{d1} \end{align*}$$ where $$\mu=1 + \frac{E(\Gamma)}{3} + \frac{\Gamma}{9}\frac{dE}{d\Gamma}$$ incorporates the Coulomb coupling parameter $\Gamma$ and excess energy $E(\Gamma)$.

The Poisson equation includes a moving charge as a source: $$\nabla^2 \phi = \frac{e}{\varepsilon_0}\left[Z_d n_d + Z_t \delta(\mathbf{r} - \mathbf{u} t) + n_e - n_i\right]$$ Electrons and ions contribute Boltzmann responses, with self-consistent quasi-neutrality.

Expanding Relativistic Plasmas (QGP)

For QGP in heavy-ion collisions, the bulk evolution is captured by relativistic ideal hydrodynamics with localized jet energy-momentum source terms (Tachibana et al., 2015),

$$\partial_{\mu}T^{\mu\nu}(x) = J^{\nu}(x),\quad T^{\mu\nu} = (\varepsilon + p) u^\mu u^\nu - p \eta^{\mu\nu}$$

The external source $J^{\nu}$ is associated with jet energy loss, and the hydrodynamic variables are advanced in (3+1)D with an equation of state from lattice QCD.

Colliding Magnetized Clouds

In the star formation and molecular cloud context, the hydrodynamics is governed by the equations of ideal, isothermal magnetohydrodynamics (MHD) including self-gravity: $$\begin{align*} &\partial_t \rho + \nabla \cdot (\rho \mathbf{v}) = 0 \ &\partial_t(\rho\mathbf{v}) + \nabla\cdot\left[\rho \mathbf{v}\mathbf{v} + (P + \frac{B^2}{8\pi})I - \frac{\mathbf{B}\mathbf{B}}{4\pi}\right] = -\rho \nabla \Phi \ &\partial_t \mathbf{B} + \nabla \times (\mathbf{v}\times\mathbf{B}) = 0 \ &\nabla^2\Phi = 4\pi G \rho \end{align*}$$ Pressure is isothermal ($P = \rho c_s^2$), and AMR methods resolve scales down to 0.012 pc (Maity et al., 13 Aug 2024).

2. Cone Formation Mechanisms and Morphology

The emergence of the cone structure is a unifying feature across these models but is realized via distinct dynamical routes:

• Mach Cones in Complex Plasmas: A test (projectile) charge moving supersonically excites dust-acoustic waves with group velocities bounded by $c_{\rm ph}$. The classical Mach-cone condition $\sin\theta(k) = c_{\rm ph}(k)/u$ holds with $c_{\rm ph}(k)=\omega/k$ determined from the GH dielectric function (Bandyopadhyay et al., 2016). The opening angle narrows as the effective sound speed decreases due to strong coupling (i.e., more negative $\mu$), and the structure can evolve from smooth cones to “multi-cone” and then fragmented wakes as coupling increases.
• Experimental Dusty Plasmas with Nozzle Flow: In laboratory setups such as dc discharge plasma with a de Laval nozzle (Fink et al., 2013), the dusty cloud forms a conical indentation at the nozzle exit with an empirically measured half-angle $\theta\approx 65^\circ$, independent of gas flow $j$. The system establishes a single convection cell exhibiting pronounced upward and downward dust flows, with autowave acceleration localized at the cone head.
• QGP Cloud-Cones via Jet Quenching: In off-central high-energy nuclear collisions, the interplay of a jet-induced Mach cone and transverse radial expansion yields asymmetric, bent cone structures (Tachibana et al., 2015). The effective “Mach” angle is set by $\theta_M = \arccos(c_s/v_{\rm jet})$ (with $c_s\approx 1/\sqrt{3}$ for QGP), but is distorted by the radial background flow, leading to observable azimuthal dips in hadron distributions.
• Astrophysical Mass-Collecting Cones in Cloud-Cloud Collisions: Head-on collisions between a turbulent finite cloud and a denser, plane-parallel medium drive a magnetized shock. The interface adopts a time-evolving conical geometry with opening half-angle declining from $\sim 84^\circ$ to $52^\circ$ as shown at $t=0.2$–$0.7$ Myr (Maity et al., 13 Aug 2024). Subsequent shock compression and magnetic field orientation seed filament formation within the cone, which channels flow toward the vertex for mass concentration and star formation.

3. Parametric Dependencies and Quantitative Results

Parameter regimes and resulting cone characteristics are tightly constrained by system-specific quantities:

Context	Mach Angle/Opening Law	Peak Density/Amplitude	Breakdown/Instabilities
Strongly coupled plasma	$\theta(M,\mu)=\arcsin(C_d\sqrt{\mu}/u)$	Max $|{\delta n_d}|/n_{d0}$ at $\mu\approx -8$; then falls	For $\mu\lesssim -6$, cone arms break into micro-structures
Dusty plasma nozzle	$\theta\approx65^\circ$ (empirical, static nozzle)	$n_d\sim10^6$ cm$^{-3}$ nucleus	Pulsed accelerations at cloud head precede autowaves
QGP cloud-cone	$\theta_M=\arccos(c_s/v_{\rm jet})$; $\sim55^\circ$	Not applicable (focus is on spectra)	Cone bent/damped by radial flow; strong asymmetries
Colliding clouds (MHD)	$\theta$ decreases from $84^\circ$ to $52^\circ$	Mass in hub rises to $\sim 233 M_\odot$ at 0.7 Myr	Transition from MHD/turbulence to gravity-driven inflow

Strongly coupled plasma systems display non-monotonic intensity behavior: the wake amplitude rises with increasing $|\mu|$ up to a maximum, then falls off as rigidity suppresses compressional response (Bandyopadhyay et al., 2016).

In the colliding cloud context, shock compression follows $r\approx M^2$ (with $M\approx33$), producing a post-shock layer reaching $n\sim10^4-10^6$ cm$^{-3}$. The mass inflow rates through conical shells reach $3$–$7\times10^{-4}\,M_\odot$ yr$^{-1}$, with hub-formation timescales of sub-Myr (Maity et al., 13 Aug 2024).

4. Transition Phenomena and Wake Evolution

Transitions between different dynamical regimes are governed by coupling, dissipation, and large-scale flow properties:

• Dusty Plasma: As $\mu$ decreases (higher $\Gamma$), the cone angle narrows and lateral wakes pass from smooth “multi-cone” to fragmented, vortex-like structures. Drag parameter $\gamma_{\rm EP}$ enhances damping, suppresses oscillatory wakes, and reduces spatial extent (Bandyopadhyay et al., 2016).
• QGP: Radial expansion of the medium distorts the canonical Mach-cone, especially for jets off-center in the fireball. The signature evolves from symmetric double-humped azimuthal distributions to dip-like suppressions anti-correlated with the cone’s inner wing (Tachibana et al., 2015).
• Cloud-Cloud Collisions: Early phases are fundamentally MHD- and turbulence-driven, with velocity uncorrelated with gravitational acceleration. As mass accumulates at the cone’s vertex, gravity increasingly dominates, and longitudinal inflow aligns with the potential gradient, marking the onset of hierarchical collapse (Maity et al., 13 Aug 2024).

A plausible implication is that, across regimes, loss of compressibility (increased rigidity) induces breakdown of large-scale conical structures in favor of smaller scale, transient features.

5. Experimental Signatures, Observables, and Diagnostics

Distinct fingerprint observables emerge from cone cloud models:

• Laboratory Dusty Plasma: Diagnostic probes include real-space imaging of particle clouds, measurement of dust-density and velocity perturbations, and reconstruction of velocity fields, revealing autowave and convection cell topology (Fink et al., 2013).
• Heavy-Ion Collisions: The hydrodynamic plasma cloud-cone’s signature is a double dip (not a double peak) in the soft-hadron azimuthal spectrum near $\phi=\pi/2\pm\Delta$, linked to the cone orientation relative to jet path and background radial flow (Tachibana et al., 2015).
• Star-Forming Regions: In position-velocity (PV) diagrams, conical mass-collecting flows manifest as bow-tie or broad-bridge signatures, with magnetic field morphology bending toward the cone axis in polarization or Zeeman measurements. These serve as unique markers for the occurrence and geometry of cloud-cloud collisions (Maity et al., 13 Aug 2024).
• Complex Plasma Wakes: In parameter sweeps, the maximum normalized perturbed density $|{\delta n_d}|/n_{d0}$ and the Mach-cone angle $\theta$ can be directly compared with predictions, offering quantitative tests for strong-coupling effects (Bandyopadhyay et al., 2016).

6. Applications, Physical Interpretation, and Significance

The hydrodynamic plasma cloud-cone model framework provides a physically grounded, theoretically consistent means of interpreting conical responses to localized energy input or collisional compression in engineered and naturally occurring plasma and magnetofluid systems.

• In laboratory complex plasmas, it enables controlled studies of non-equilibrium phase transitions, acoustic wave dispersion, and correlational rigidity.
• For QGP, it elucidates medium response to jet quenching and informs the diagnostic exploitation of final-state hadron spectra in high-energy physics.
• In astrophysics, the model underpins current understanding of columnated accretion flows, hub-filament system emergence, and the initial stages of massive star formation.
• These frameworks also guide experimental and observational design—e.g., laboratory verification of Mach cone breakup thresholds, and high-resolution ALMA or JWST mapping of interstellar hub-filament systems.

A persistent theme is that the cone’s opening angle, structural integrity, and detectable wake features tightly encode the microscopic (e.g., compressibility, turbulence) and macroscopic (e.g., expansion, gravity) properties of the medium, enabling inversion of observed or measured cone parameters to yield insight into the governing plasma or fluid regime.

7. Limitations and Outlook

The hydrodynamic plasma cloud-cone models rely on context-specific assumptions: idealized equations of state, neglect of radiative transfer or non-ideal MHD effects, or, in the GH model, the omission of magnetization and higher-order correlations. In highly dissipative or strongly turbulent regimes, simple conical morphology may disintegrate, and diagnostics may become ambiguous. Further, in the astrophysical context, projection and resolution effects can obscure kinematic and morphological signatures, complicating unequivocal identification of cloud-cones and related mass-collecting structures.

Continued advances—in simulation (AMR MHD, kinetic methods), high-fidelity experimental diagnostics, and multiwavelength observation—are expected to sharpen the parameter-space mapping of cloud-cone phenomena and to resolve the precise interplay of shocks, magnetic tension, self-gravity, and strong coupling in setting the emergent structure of plasmas and interstellar material.