Nonlinear Saturation of the Acoustic Resonant Drag Instability

TL;DR

This study investigates the nonlinear saturation of the acoustic resonant drag instability (RDI) through high-resolution RAMSES simulations of supersonic dust streaming in a neutral gas. It reveals that saturation proceeds via a scale-by-scale balance between the linear instability growth and turbulent eddy turnover, generating a strongly anisotropic forcing range and signaling a developing isotropic inertial range at smaller scales; the resulting gas velocity spectrum in the forcing range follows a robust slope consistent with E_u(k) ∼ k^{-2} under the proposed balance. The results are reinforced by cross-code consistency with prior GIZMO-based work and show that saturation dynamics are governed by the interplay of growth and disruption across scales rather than solely by box-scale constraints. These insights provide a framework for understanding dust-driven turbulence in diverse astrophysical environments and lay groundwork for extending the analysis to magnetized RDIs and more complex dust physics.

Abstract

Resonant drag instabilities (RDIs) are a novel type of dust/fluid instability relevant to a diverse range of astrophysical environments. They are driven by a resonant interaction between streaming dust and waves in a background medium, which results in dust density fluctuations and amplification of the waves. This broad class of instabilities includes recently-proposed modes incorporating acoustic and magnetohydrodynamic waves, as well as the well-studied disk streaming instability. As the study of RDIs is at an early stage, their evolution beyond the linear regime is not well understood. In order to make inroads into the nonlinear theory of RDIs, we performed simulations of the simplest case, the acoustic RDI, in which sound waves in a gas are amplified by interaction with supersonically streaming dust. This particular instability is of interest both due its potential relevance in various poorly ionized environments, and due to its resemblance to the fast magnetosonic RDI. We find that the nonlinear growth and saturation of the instability are characterized by a balance between time scales of instability growth and turbulent eddy turnover. The simulations demonstrate a saturated state possessing an anisotropic outer forcing range in which this balance is maintained, and suggest the presence of an isotropic turbulent inertial range below this scale. By presenting a model for the nonlinear growth and saturated state of the acoustic RDI, this work provides a framework for further study of the nonlinear behavior of this and other RDIs.

Figures (18)

• Figure 1: An illustration of the proposed scaling of the eddy turnover rate $\gamma_{\rm{eddy}}$ as a function of wavenumber $k$ as the system evolves over time $t$ towards a statistically stationary state. $\gamma_{\rm{eddy}}(k)$ for successive times are shown as lines with varied color. As discussed in \ref{['sec:saturation']}, modes initially grow until eddy turnover is in balance with instability growth ($\gamma_{\rm{eddy}}\sim\gamma_{\rm{RDI}}\propto k^{1/2}$ shown as a thick black line for reference), reaching this point at smaller scales first. In addition, as proposed in \ref{['sec:inertial']}, at scales sufficiently smaller than the largest saturated scale, a turbulent direct cascade forms ($\gamma_{\rm{eddy}}\propto k^{2/3}$). The resulting inertial range grows until the RDI only remains dominant (and the turnover and growth rates balanced) in a forcing range near the outer scale. The resolutions of the simulations in this work resolve the outer range and only the beginning of the inertial range, corresponding to the leftmost portion of this illustration. The range has been extended to smaller scales in order to better demonstrate the scaling in different ranges.
• Figure 2: Growth rate of the fastest growing mode at the resonant angle $\gamma_{\rm{RDI}}$, calculated analytically for the parameters used in each simulation, as a function of wavenumber parallel to streaming $k_{||}$. The growth rate has been multiplied by a factor of $k_{||}^{-1/2}$ to make the power laws for each of the three regimes discussed in \ref{['sec:review']} clearly visible. Shaded rectangles are shown to indicate the wavelength ranges present in the simulations ($2\pi N/4L\le k_{||}\le 2\pi/L$ for resolution $N$ and box size $L$).
• Figure 3: a) Root mean squared gas velocity for hr-$w_s$2.5, showing the linear growth, nonlinear growth, and saturated phases in blue, orange, and green, respectively. b) Gas velocity energy spectra $E_\mathbf{u}(k)$ for each simulation snapshot, color-coded according to (a). Initial growth at small scales, followed by growth and saturation at larger scales, can be seen.
• Figure 4: Cross-sections in the $yz$-plane at $x/L=0.5$ of $\mathbf{u}$ for various simulations, during the linear growth (left), nonlinear growth (center), and saturated phases (right). $u_x$, $u_y$, and $u_z$ are depicted using the red, green, and blue channels respectively, normalized to the range $\pm0.025c_s$ (left) or $\pm0.5c_s$ (center, right). Ripples at the resonant angle for the instability (indicated with white lines) are visible. Times used in each cell are given in units of $L/c_s$.
• Figure 5: $\log_{10}\left<\tilde{\rho}^*\tilde{\rho}/\rho_0^2L^6\right>_{k_y}$ - Gas density fluctuations plotted in Fourier space (averaged over $k_y$) for various simulations, during the linear growth (left), nonlinear growth (center), and saturated (right) phases. Cone structures at the resonant angle (indicated with white lines) are visible. White crosses are placed at $(k_x,k_z)=(\pm 4\pi\tan({\theta_{\rm res}}),\pm 4\pi)$ to mark the outer scale as discussed in \ref{['sec:outer']}. Times used in each cell are given in units of $L/c_s$.