Inviscid Circumbinary Discs Dynamics

• Inviscid circumbinary discs are rotating gaseous structures around binary systems characterized by negligible viscosity, inner cavities, and spiral density waves.
• Global simulations reveal that non-local gravitational torque oscillations and propagating density waves drive angular momentum transfer and instabilities.
• Hydrodynamical turbulence induced by parametric instabilities limits in situ planet formation, promoting growth at larger radii with subsequent migration.

Inviscid circumbinary discs (CBDs) are rotationally supported gaseous structures orbiting around binary systems, where molecular viscosity is negligible or explicitly set to zero in hydrodynamical models. Such discs are observed or inferred in various astrophysical contexts, including young stellar binaries, supermassive black hole pairs, cataclysmic variables, and X-ray binaries. The structural and dynamical evolution of inviscid CBDs is strongly modulated by the central binary's gravity, which sculpts an inner cavity, induces global disc eccentricity, excites spiral density waves, and drives angular momentum transport through both gravitational and hydrodynamical means. Recent developments in global simulations and theoretical models have elucidated the non-local nature of binary-excitation torque oscillations, the ubiquitous emergence of turbulence via parametric instabilities, and the implications of these behaviors for planetesimal formation and accretion processes.

1. Gravitational Torque Excitation and Radial Oscillations

The dominant torque in inviscid CBDs arises from direct gravitational coupling between the binary and the disc, quantified by the excitation torque density: $$\frac{dT_{\rm ex}}{dR}(R) = - R \int_{0}^{2\pi} \Sigma(R, \phi) [\mathbf{R} \times \nabla \Phi_{\rm b}]_z \, d\phi,$$ where $\Phi_{\rm b}$ is the binary potential and $\Sigma$ the gas surface density (Cimerman et al., 2023). Azimuthal decomposition into $m$-fold harmonics yields: $$\frac{dT_{\rm ex}}{dR}(R) = \pi G M_{\mathrm{tot}} \sum_{m=1}^{\infty} m |\delta \Sigma_m(R)| A_m(R) \sin[m\tilde{\phi}_m(R)],$$ with $M_{\mathrm{tot}}$ the binary mass, $\delta \Sigma_m$ the $m$-th density perturbation amplitude, $A_m(R)$ a Laplace-coefficient modulated function, and $\tilde{\phi}_m$ the WKB phase of the density wave (Cimerman et al., 2023).

The resulting torque density profile manifests as quasi-periodic radial oscillations. Contrary to prior expectations, these are not set by Airy-function eigenmodes around local Lindblad resonances; rather, they are determined by the freely propagating spiral density waves generated at the cavity edge and advected outward. Analytical WKB theory gives the zero-crossing periodicity: $$R_{k+1} - R_k \simeq \frac{\pi}{m} \frac{c_s(R_k)}{\Omega_{\rm b}},$$ with local sound speed $c_s$ and binary orbital frequency $\Omega_{\rm b}$. The wavelength of the oscillatory feature is

$$\lambda(R) \approx \frac{2\pi c_s(R)}{m\Omega_{\rm b}},$$

demonstrating explicit scaling with the disc's thermodynamics (via $c_s$) and the geometry of density wave excitation (via $m$).

2. Wave Excitation, Propagation, and Angular Momentum Transfer

Spiral density waves in inviscid CBDs are excited by the non-axisymmetric binary potential, particularly at the locations just outside the dominant low-order Lindblad resonances. The excited wave can be described in WKB form: $$\delta\Sigma(R,\phi) \propto \exp\left[i m (\phi - \phi_{\mathrm{b}}(t)) + i \int^R k_R(R') dR'\right],$$ where $k_R$ obeys

$$k_R^2(R) = c_s^{-2}(R)[m^2(\Omega - P)^2 - \Omega^2],$$

with $P$ the binary pattern speed (Cimerman et al., 2023). Once launched, these waves propagate through the disc, retaining a pitch angle determined by $m$ and $c_s$, and continually interact gravitationally with the binary at every radius, generating the observed global torque oscillations. The persistent torque coupling at all radii invalidates scenarios where the local resonance (e.g., at the outer $m=2$ Lindblad resonance) is the sole determinant of the torque structure.

3. Hydrodynamical Turbulence: Parametric Instabilities and Eccentric Disc Dynamics

Global 3D hydrodynamical simulations of inviscid CBDs reveal that discs naturally attain significant eccentricities ($e \sim 0.08$ at $r \sim 5 a_{\rm bin}$) due to binary-disc coupling (Pierens et al., 2020, Pierens et al., 2021). Eccentric discs are unstable to parametric instabilities, where inertial-gravity waves resonate with the disc's global $m=1$ eccentric mode. The principal resonance condition is: $\omega = \frac{\Omega}{2},$ with $\omega$ as the perturbation frequency and $\Omega$ the local angular velocity (Pierens et al., 2020, Pierens et al., 2021). The linear growth rate for the instability is

$$\sigma = \frac{3}{16}e\Omega \quad \text{(unstratified)}, \qquad \sigma = \frac{3}{4}e\Omega \quad \text{(stratified)},$$

while simulations report numerical growth rates $\sigma_{\rm num} \simeq 0.045\,\Omega$ at $R \approx 6 a_{\rm bin}$.

The nonlinear outcome is sustained hydrodynamical turbulence, with measured stress parameter $\alpha \sim 5 \times 10^{-3}$ (potentially up to $10^{-2}$ near the cavity edge) and vertical turbulent diffusion coefficient $\alpha_{\rm diff} \sim 1$--$2 \times 10^{-3}$. Adopted simulation setups include spherical $(r,\theta,\varphi)$ domains and zero explicit viscosity.

4. Vortices at the Inner Cavity Edge and Their Dynamical Effects

Strong vortensity gradients at the cavity edge trigger the Rossby-wave instability, resulting in the formation of one or more anticyclonic vortices at $R \sim 2$--$4\,b$ (Cimerman et al., 2023). These vortices rotate at their local pattern speeds ($P_{\rm vrt} = \Omega(R_{\rm vrt}) < \Omega_{\rm b}$), launching more open (higher pitch angle) spiral density waves than the binary-driven arms. Notably, they transport angular momentum radially by launching non-gravitational spiral density waves rather than contributing directly to the binary-induced gravitational torque:

• The angular momentum flux $F_J$ carried by vortex-driven spirals can exceed the cumulative binary-driven torque $T_{\rm ex}$ over the spatial range where the vortices exist.
• While the time-averaged gravitational torque from vortices vanishes in the binary frame (due to non-stationary wave patterns), their transient presence modifies disc morphology, promoting the formation of density "lumps" and an eccentric cavity, until $\alpha \gtrsim 10^{-2}$ erases vortical structures.

5. Vertical Structure, Dust Settling, and Pebble Accretion in Turbulent CBDs

Hydrodynamical turbulence impacts the vertical distribution of dust and the efficiency of pebble accretion. The dust scale height is determined by the balance between turbulent diffusion and sedimentation: $$H_d^2 = \frac{D_z/\Omega}{1+\mathrm{St}^2} = H^2 \frac{\alpha_{\rm diff}}{\alpha_{\rm diff} + \mathrm{St}},$$ where $\mathrm{St}$ is the Stokes number, $H$ the disc gas scale height, and $D_z$ the vertical diffusion coefficient (Pierens et al., 2020, Pierens et al., 2021). For small $\mathrm{St}$ ($\lesssim0.1$), dust forms a finite-thickness layer, $H_d/H \sim 0.05$--$0.3$. The pebble accretion efficiency for embedded cores is reduced by the factor $r_{\rm acc}/H_d$, leading to longer growth timescales.

Enhanced turbulence (with vertical velocity fluctuations up to $0.2\,c_s$ at the cavity edge) also increases collision velocities between dust grains, often exceeding fragmentation thresholds for silicates ($1$–$10\,$m/s), thus inhibiting grain growth and planetesimal formation via streaming instability or direct coagulation within $R \lesssim 7\,a_{\rm bin}$ (Pierens et al., 2021). Outside this region, weaker turbulence allows more efficient pebble accretion and the possibility for planetesimal seeds to reach $10 M_\oplus$ within typical disc lifetimes.

6. Implications for Planet Formation and Migration

The structure and turbulence of inviscid CBDs have profound implications for the in situ formation of circumbinary planets, especially those found near the cavity edge. In fully turbulent discs where $\alpha \sim 5 \times 10^{-3}$ and $\alpha_{\rm diff} \sim 10^{-3}$, pebble accretion rates are suppressed both by vertical dilution ($r_{\rm acc}/H_d$) and by turbulent velocity kicks for low-mass cores, yielding growth times of $6$--$20$ Myr to reach $10\,M_\oplus$, compared to disc lifetimes of a few Myr (Pierens et al., 2020, Pierens et al., 2021). In a laminar disc, growth could approach the required timescale, but turbulence is a robust feature in the inner regions.

The efficiency and viability of planet formation thus depend on the radial location: in situ planet growth close to the binary is problematic, while formation at larger radii ($R \gtrsim 8\,a_{\rm bin}$), where turbulence drops to $\alpha \sim 10^{-3}$, can yield sufficiently rapid growth. This mechanism requires subsequent migration to explain the presence of circumbinary planets like Kepler–16 b at observed locations (Pierens et al., 2021).

7. Controversies and Revisions of Classical Theories

Early models attributed the oscillatory torque structure in CBDs to Airy-function solutions localized around Lindblad resonances (Cimerman et al., 2023). However, simulation and WKB analyses demonstrate that:

• The radial periodicity and global structure of torque oscillations are set by propagating spiral density waves and their phase coupling to the binary potential, not by resonance-localized eigenfunctions.
• The oscillation wavelength set by the Airy approach ($\propto R^{-1/2}$) contradicts the near-constant or $c_s(R)$-prescribed periodicity observed over many scale-heights.
• Lindblad resonances primarily serve to launch density waves; the torque signature at large radii arises from cumulative non-local gravitational coupling between the binary and the ongoing disc wave pattern, with vortices providing a non-steady angular momentum transport channel.

A plausible implication is that broader classes of binary–disc systems (X-ray binaries, CVs) may exhibit similar non-local torque oscillations and vortex-driven structures, with consequences for variability, angular momentum evolution, and formation pathways of compact binary objects.

---

In summary, inviscid circumbinary discs are shaped by the interplay of direct gravitational torques, freely propagating spiral density waves, and turbulence arising from parametric instabilities. The global oscillatory torque structure is not a linear resonance artifact, but a manifestation of non-local gravitational coupling and dynamic wave–pattern alignment. Turbulence modulates particle dynamics, suppresses rapid accretion, and imposes strong constraints on planet formation, necessitating a reconsideration of classical in situ scenarios in favor of models incorporating outward planet formation and migration.