Pith. sign in

REVIEW 3 major objections 4 minor 52 references

Evidence For Turbulent Concentration In Particle-Laden Midplane Layers of Planet-Forming Disks

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows that small dust grains in settled midplane layers of planet-forming disks clump by turbulent concentration rather than by the streaming instability, with a universal effective Stokes number in the voids near 0.4–0.7.

desk verdict Genuine and direct evidence for turbulent concentration in settled midplane layers, but the St_omega 'collapse' claim is contradicted by the paper's own Table 4 and needs revision before the quantitative story holds up. read the letter →

arxiv 2508.05858 v3 pith:HFSYFEWE submitted 2025-08-07 astro-ph.EP

Orkan M. Umurhan , Debanjan Sengupta , Thomas Hartlep , Jeffrey N. Cuzzi , Paul R. Estrada This is my paper
classification astro-ph.EP
keywords turbulentconcentrationstreaminginstabilitysymmetricprotoplanetarydisksplanetesimalformationStokesnumberradialdistributionfunctionparticleclumping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that in settled, weakly turbulent midplane layers of planet-forming disks, small dust grains spontaneously clump by turbulent concentration (TC): particles are flung out of coherent vortices and pile up in high-strain filaments. Working from axisymmetric shearing-box simulations with particle Stokes numbers $\mathrm{St}_K = 0.01$–$0.04$, it finds that the effective Stokes number measured inside particle voids, $\mathrm{St}_\omega$, collapses to roughly 0.4–0.7, close to the value known to maximize TC clustering. A timescale comparison shows that streaming instability growth rates are one to two orders of magnitude slower than eddy turnover frequencies, so SI is not driving the turbulence there; the paper proposes the symmetric instability (SymI) as the driver. If true, TC is a persistent, self-generated clumping mechanism in exactly the regimes where planetesimals must form, and many small-scale Roche-density-exceeding clumps attributed to SI are actually TC operating on top of elevated filament densities.

What carries the argument

The analysis is carried by the cascade-model radial distribution function (RDF) of Hartlep et al. 2017, generalized to the non-Kolmogorov spectra of these axisymmetric runs. The scale-local Stokes number is written $\mathrm{St}_\ell = \mathrm{St}_L (\ell/L)^{(-3-n)/2}$ with $n \approx -2$ instead of the Kolmogorov $-5/3$, and the clustering onset scale is $\ell_* = \mathrm{St}_L^{2/(3+n)} L$; rescaling separations by $\ell_*$ collapses the measured RDFs onto the predicted S-curve. Complementary diagnostics are the void/filament grid classification ($X^{\{vn\}}$, $X^{\{fn\}}$) used to compute vorticity inside particle voids and the corresponding $\mathrm{St}_\omega = t_s \omega$, plus timesca

What would settle it

Run the same three parameter sets in a fully 3D shearing-box simulation and measure the peak of the $\mathrm{St}_\omega$ distribution inside particle voids, the spectral slope of the gas kinetic energy, and the linear growth rate of the fastest axisymmetric SI mode. The central claim fails if the $\mathrm{St}_\omega$ peak does not fall near 0.4–0.7, if voids no longer contain coherent vorticity bounded by high-strain filaments, or if the fastest-growing SI mode grows faster than the eddy turnover frequency at $k_{\mu\epsilon}$.

Watch

Extended reading notes

Core claim

In axisymmetric (radial–vertical plane) shearing-box simulations of a settled particle layer, the authors observe that the particle field develops filament–void complexes: particle-rich filaments bound gas-only voids filled with coherent vorticity. Particles are expelled from vortex cores and collect in regions of high gas strain rate, the hallmark of TC. Across runs with $\mathrm{St}_K = 0.01$, $0.02$, and $0.04$, the effective Stokes number $\mathrm{St}_\omega = t_s \omega$ evaluated on particle-free void grid points peaks in the range 0.4–0.7, independent of $\mathrm{St}_K$. The radial distribution functions, when separation is rescaled by $\ell_* = \mathrm{St}_L^{2/(3+n)} L$ with the mea

Load-bearing premise

The conclusions assume that the axisymmetric, two-dimensional turbulent state seen in the simulations faithfully represents the real three-dimensional midplane layer; if 3D dynamics change the strain-vorticity statistics, shorten vortex lifetimes, or let SI filaments grow, the TC attribution and the $\mathrm{St}_\omega$ collapse lose their footing.

Editorial extensions

If this is right

  • In settled midplane layers with $\mathrm{St}_K \lesssim 0.04$ and midplane dust-to-gas ratio below unity, self-generated turbulence alone can produce persistent small-scale particle clumping via TC, independent of the streaming instability.
  • The effective void Stokes number $\mathrm{St}_\omega \approx 0.4$–$0.7$ offers a scale-free diagnostic: whenever particles inside coherent vortex voids meet this condition, TC expulsion should be strongest regardless of the nominal $\mathrm{St}_K$.
  • The streaming instability cannot be assumed to be the turbulence driver or the small-scale clumping mechanism in this parameter regime, because its linear growth is overwhelmed by eddy overturn at the relevant scales.
  • Numerical simulations seeking TC-driven Roche-density exceedance must resolve the $\mathrm{St}_\ell = O(1)$ scale; simulations that do not will under-report clustering, explaining resolution-dependent clumping results in the literature.
  • Small-scale density fluctuations exceeding the Roche density inside large-scale SI filaments are reinterpreted as TC acting on top of the locally elevated particle density, rather than as direct SI nonlinear structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $\mathrm{St}_\omega$ collapse persists in fully 3D turbulence, TC could be identified in simulations by the strain-vorticity alignment of the gas field rather than by particle density alone, offering a cleaner observable diagnostic.
  • A testable extension is to run the same $\mathrm{St}_K$ and metallicity parameters in a fully 3D shearing box: if the kinetic energy spectral slope reverts to Kolmogorov $-5/3$, the RDF collapse should use the standard scaling and the predicted clustering onset would shift accordingly.
  • The SymI attribution remains suggestive; a direct measurement of SymI eigenmode growth from the simulated mean azimuthal shear profile, or identification of its energy injection band, would settle whether SymI or a SymI–SI hybrid drives the turbulence.
  • If TC alone can push local densities past the Roche threshold in higher-resolution runs of these small Stokes numbers, planetesimal formation may not require SI for the smallest particles, changing the expected sizes and locations of first planetesimals.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper analyzes three axisymmetric shearing-box simulations of settled particle layers in a globally laminar protoplanetary disk, with particle Stokes numbers St_K = 0.01, 0.02, and 0.04 and midplane metallicities Z = 0.01–0.02. It reports small-scale particle clumping attributed to turbulent concentration (TC): particle-rich filaments align with high gas strain rate and enclose gas-only voids with coherent vorticity. The authors introduce a void-based effective Stokes number St_omega = t_s omega and claim its peak collapses to ~0.4–0.7 across the St_K range. They further compare measured radial distribution functions (RDFs) with the Hartlep et al. (2017) cascade model using a generalized spectral slope n ≈ -2, compare SI linear growth rates with turbulent overturn frequencies, and suggest Symmetric Instability rather than SI drives the turbulence. They conclude that TC is a persistent, self-generated mechanism and that small-scale Roche-exceeding fluctuations in SI filament simulations may be TC operating on elevated filament densities.

Significance. If the central claim holds, the paper provides the first direct demonstration of turbulent concentration under nebula conditions and challenges the default assumption that small-scale clumping in settled, low-metallicity midplane layers is dominated by the streaming instability. The work is strengthened by direct, non-inferential diagnostics: the spatial anti-correlation between particles and vorticity and correlation with strain rate (Fig. 7), a generalized spectral rescaling of the RDF (Eq. 25b, Fig. 8), and an explicit first-principles treatment of particle–gas velocity coupling in Appendices A–B. The authors are also candid about the axisymmetric nature of the simulations. However, the quantitative centerpiece of the paper — the claimed St_omega collapse — is internally inconsistent with the paper's own Table 4, and the statistical support from single snapshots is thin. These issues must be repaired before the headline claim can be accepted.

major comments (3)
  1. [§4.2 / Table 4] The abstract states that St_omega 'collapses to values in the range ~0.4–0.7', and the Fig. 6 caption states that for z_lim = h and 0.02H the peak satisfies 0.4 < St_omega(peak) < 0.75. Table 4, however, lists St_omega(peak) = 0.35 for St_K = 0.01 at z_lim = 0.01H, h, and 0.02H. The values are 0.35, 0.44, and 0.72 for St_K = 0.01, 0.02, and 0.04 respectively at z_lim = h. The lowest-Stokes case therefore lies outside the stated range, and the sequence is monotonic with St_K rather than a collapse. Because this claimed universality is a central piece of evidence for TC and for the connection to the critical St ≈ 0.3 scale, this inconsistency must be resolved. Either the stated range must be corrected, or additional time-averaged statistics must be provided to demonstrate that the peak values are robust and not single-snapshot fluctuations.
  2. [§6] The simulations are 3D-axisymmetric: azimuthal gradients are absent, and the turbulence spectra, strain/vorticity statistics, void structure, and RDFs are all computed in the (R,z) plane. The paper itself concedes in §6 that 'Whether this behavior persists with similar clarity in fully 3D simulations remains to be seen.' This is a load-bearing limitation because the title and abstract generalize to planet-forming disks, and because the claimed universality of St_omega near 0.4–0.7 and the attribution of small-scale clumping to TC depend on the 2D turbulent state. Without a 3D test—or at minimum a careful statement that the conclusions are provisional to axisymmetric dynamics—the central claim outruns the presented evidence.
  3. [Table 2 / §4] Each simulation contributes a single snapshot: Table 2 footnote c states that the analyzed timestamp is a single time for each run. All statistics—spectra, PDFs, St_omega peaks, RDFs, and particle–gas velocity comparisons—are therefore derived from one instant per simulation. The paper repeatedly describes TC as a 'persistent' feature and stresses robustness, but no time averaging, multiple snapshots, or uncertainty estimates are given for the St_omega peaks or RDF inflection points. A single snapshot could produce a favorable configuration by chance. At minimum, the analysis should be repeated over several snapshots in the turbulent phase to support the persistence claim.
minor comments (4)
  1. [§3.6] The text says the particle density field is normalized by the mean vertical particle density profile '(equation 16)', but the relevant definition appears to be Eq. (12) or the Gaussian fit Eq. (14). The equation reference should be corrected.
  2. [§4.3] The right panel of Fig. 8 is described as collapsing the RDFs, but for St_K = 0.01 and 0.02 the simulations do not resolve scales at or below ell_* and the S-curve inflection is not captured. The qualitative agreement with the cascade model is limited to the onset location. The text should more precisely state that the collapse is only partial and that the lowest-Stokes cases provide weaker evidence.
  3. [Table 2] The St_K = 0.01 and 0.02 runs use a 0.4H box with 4096^2 resolution, while the St_K = 0.04 run uses a 0.2H box with 2048^2 resolution. Box size and resolution differ across the three simulations; this complicates direct comparisons of clustering statistics and void scales. A brief comment on how this affects the claimed universality would be helpful.
  4. [Eq. (27b)] The empirical expression for sigma_SI in the epsilon < 1 regime is stated without derivation or error estimate. Given that the SI-vs-turbulence timescale comparison is central to ruling out SI as the driver, a reference to the source of this fit and its range of validity would strengthen the argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TC evidence is measured directly; the H17/SU23 self-citations are non-load-bearing benchmarks. The internal inconsistency in the St_omega collapse range is a correctness/consistency issue, not a circularity.

full rationale

The central evidence for turbulent concentration is measured in the simulations, not imposed by the analysis: particle-strain-rate correlation and particle-vorticity anti-correlation (Fig. 7), void vorticity PDFs (Fig. 5/6), and RDFs computed from the particle density autocorrelation (Sec. 3.6, Fig. 8). The RDF comparison to the H17 cascade model is an independent benchmark: H17 was developed from separate 3D turbulence simulations (Bec et al. 2010), so citing it for the St_ell~O(0.3) intermittency peak is not load-bearing self-citation, even though some authors overlap. The generalized scaling in Eq. (25b) and the length scale L* in Eq. (26) use the measured spectral slope n and St_L, but the RDF onset is an independent observable; agreement with the model is a nontrivial test, not an input. St_omega is defined directly as t_s times local gas vorticity in void sets and is measured, not fitted; the claimed 0.4-0.7 collapse is an empirical finding. The SI growth-rate comparison relies on standard external formulas (Youdin & Goodman 2005; Squire & Hopkins 2018, 2020), and the SymI proposal is explicitly a suggestion, not the load-bearing derivation. The manuscript itself concedes the axisymmetric dimensionality and that 3D behavior remains to be seen, which is a limitation rather than circularity. One internal consistency problem is worth flagging: Table 4 lists St_omega(peak)=0.35 for St_K=0.01 at z_lim=h and 0.02H, outside the abstract's stated range of ~0.4-0.7 and contradicting the Fig. 6 caption; this weakens the universality claim as evidence but does not reduce any prediction to its inputs by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The paper leans on: cutoffs chosen after the fact (z_lim), a spectral slope read from its own runs (n), same-group benchmarks (H17, SU23), and a laminar linear-theory extrapolation to turbulent flow. These do not invalidate the qualitative TC detection, but they cap how much of the quantitative headline is independently earned.

free parameters (4)
  • z_lim (void analysis cutoff) = 0.01H, h, 0.02H, 2h; headline range uses z_lim <= 0.02H
    The quoted St_omega collapse (0.4-0.7) is obtained by restricting analysis to voids with |z| < z_lim <= 0.02H; the z_lim=2h case (peaks 0.31-0.48) is excluded as 'contamination'. The headline range is therefore a hand-chosen window over an analysis threshold.
  • spectral slope n = about -2.0 to -2.1 (read from Fig. 1); n=-2 used in the RDF rescaling and Table 5
    The generalized cascade rescaling ell* = St_L^(2/(3+n)) L (Eq. 26) and the comparison to H17 use a slope measured from the paper's own spectra; the agreement of the rescaled RDFs is not an independent test of the spectrum.
  • Gaussian fit parameters (h, rho_p0, deltaV0, Hs) = Table 3 (h/H 0.014-0.017, deltaV0/cs 0.02-0.03, etc.)
    The particle scale height h normalizes the density fluctuation field used in the RDF (Eq. 23c), and the azimuthal velocity fit enters Ri_phi (Eq. 21b) used for the SymI growth-rate estimate; these are fits to simulation output, not externally determined values.
  • St_L / R' (large-eddy Stokes number) = 0.03, 0.05, 0.155 (Table 3)
    Used to label RDF curves and to compute ell*; derived from the measured spectrum rather than predicted, so the RDF collapse is conditioned on these measured values.
assumptions (5)
  • domain assumption Axisymmetric turbulence with spectral slope n approx -2 is a valid stage for TC; H17's Kolmogorov cascade is generalized by Eq. (25b) with the measured n
    Sec. 4.3: the paper replaces Eq. (25a) with (25b) because the runs are axisymmetric and non-Kolmogorov; all RDF comparisons inherit this premise. The paper flags the 3D question as open in Sec. 6.
  • domain assumption SI growth in the turbulent, clumpy layer is bounded by laminar asymptotic linear theory (Eqs. 27a,b)
    Sec. 5.2: sigma_SI estimates assume the fastest-growing asymptotic mode of Youdin & Goodman (2005) with measured midplane epsilon values; the background turbulence and density fluctuations are not inserted into the growth-rate estimate.
  • domain assumption The H17 critical intermittency scale St_ell approx 0.3 (and its cascade RDF) is the correct external benchmark for maximal TC clumping
    Sec. 5.1 and Fig. 8: the interpretation of the St_omega peak range as 'close to' the clustering-critical value rests on H17, a same-group model (Hartlep, Cuzzi) calibrated on Bec et al. (2010) DNS.
  • domain assumption Super-particle discretization and hyperviscosity yield converged small-scale clustering statistics
    Sec. 2: 0.125-0.5 particles per grid point and sixth-order hyper-viscosity/diffusion; no resolution-convergence study is presented, and the paper states the low-St runs under-resolve the St_ell=0.3 scale.
  • standard math Standard Fourier and linear-algebra results (convolution theorem, eigendecomposition of S_ij, incomplete-beta-function integral)
    Secs. 3.5-3.6 and Appendix B: the RDF via power spectra with mask correction, strain magnitude as largest eigenvalue, and the exact integral Eq. (B3) rely on standard results.
invented entities (1)
  • St_omega (void vorticity-scaled Stokes number)
    purpose: Diagnostic quantity claimed to collapse to 0.4-0.7 inside particle voids and to indicate the operative TC regime
    It is a composite definition (ts times local gas vorticity), not a detected physical object; its salience is asserted by the clustering coincidence with H17's threshold. Whether particles actually respond to void vorticity as the timescale suggests is addressed only indirectly in Appendix A.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Evidence For Turbulent Concentration In Particle-Laden Midplane Layers of Planet-Forming Disks." pith.science (2026). https://pith.science/paper/HFSYFEWE

@misc{pith2026250805858,
  author       = {Pith},
  title        = {Pith review of: Evidence For Turbulent Concentration In Particle-Laden Midplane Layers of Planet-Forming Disks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFSYFEWE}},
  note         = {Machine review of arXiv:2508.05858}
}
abstract

We investigate the axisymmetric, weakly turbulent state of settled particle layers in a model of a globally laminar protoplanetary disk. We focus on conditions in which the large-scale axisymmetric filaments associated with the streaming instability (SI) either cannot form or have not developed yet. We observe small-scale particle clumping consistent with turbulent concentration (TC), in which particle-rich filaments align with regions of high gas strain rate and enclose gas-only voids exhibiting coherent vorticity. Across a range of particle Stokes numbers, St$_K$ ($0.01-0.04$) -- defined as stopping times relative to the Keplerian frequency -- effective Stokes number within particle voids, St$_{\omega}$, defined instead using local gas vorticity, collapses to values in the range $\sim 0.4-0.7$ for the St$_K$ considered. These values lie close to critical turbulent Stokes numbers associated with maximal clustering intermittency identified in statistical studies of TC. A timescale comparison reveals that in simulations with midplane particle-to-gas density ratios below unity and St$_K \ll 1$, SI growth rates are 1 - 2 orders of magnitude slower than turbulent overturn frequencies at the large-eddy scale, which appears to rule out SI as the primary driver of turbulence here. Instead, we suggest the Symmetric Instability (SymI) may be responsible. We show for our St$_K$ that TC is a persistent feature of our turbulent particle layers, and conclude that small scale particle density fluctuations exceeding the Roche density within large-scale axisymmetric SI filaments reported in the literature are also expressions of TC operating on top of the slightly elevated background particle densities within those large-scale structures.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

52 extracted references · 28 canonical work pages

  1. [1]

    Balachandar, S., & Eaton, J. K. 2010, Annual Review of Fluid Mechanics, 42, 111, doi: 10.1146/annurev.fluid.010908.165243

    work page arXiv 2010

  2. [2]

    A., Yang, C.-C., & Zhu, Z

    Baronett, S. A., Yang, C.-C., & Zhu, Z. 2024, MNRAS, 529, 275, doi: 10.1093/mnras/stae272

    work page doi:10.1093/mnras/stae272 2024

  3. [3]

    2010, Journal of Fluid Mechanics, 645, 497, doi: 10.1017/S0022112009992783

    Toschi, F. 2010, Journal of Fluid Mechanics, 645, 497, doi: 10.1017/S0022112009992783

    doi:10.1017/s0022112009992783 2010

  4. [4]

    2023, Annual Review of Fluid Mechanics, 56, 189, doi: 10.1146/annurev-fluid-032822-014140

    Bec, J., Gustavsson, K., & Mehlig, B. 2023, Annual Review of Fluid Mechanics, 56, 189, doi: 10.1146/annurev-fluid-032822-014140

    work page doi:10.1146/annurev-fluid-032822-014140 2023

  5. [5]

    D., & Collins, L

    Bragg, A. D., & Collins, L. R. 2014, New Journal of Physics, 16, 055013, doi: 10.1088/1367-2630/16/5/055013 Carrasco-Gonz´ alez, C., Sierra, A., Flock, M., et al. 2019, ApJ, 883, 71, doi: 10.3847/1538-4357/ab3d33

    work page doi:10.1088/1367-2630/16/5/055013 2014

  6. [6]

    2020, ApJ, 891, 132, doi: 10.3847/1538-4357/ab76ca

    Chen, K., & Lin, M.-K. 2020, ApJ, 891, 132, doi: 10.3847/1538-4357/ab76ca

    doi:10.3847/1538-4357/ab76ca 2020

  7. [7]

    2008, ApJ, 675, 1549, doi: 10.1086/527354

    Chiang, E. 2008, ApJ, 675, 1549, doi: 10.1086/527354

    work page doi:10.1086/527354 2008

  8. [8]

    N., Davis, S

    Cuzzi, J. N., Davis, S. S., & Dobrovolskis, A. R. 2003, Icarus, 166, 385, doi: 10.1016/j.icarus.2003.08.016

    doi:10.1016/j.icarus.2003.08.016 2003

Show all 52 references
  1. [9]

    N., Dobrovolskis, A

    Cuzzi, J. N., Dobrovolskis, A. R., & Champney, J. M. 1993, Icarus, 106, 102, doi: 10.1006/icar.1993.1161

    1993

  2. [10]

    N., & Hogan, R

    Cuzzi, J. N., & Hogan, R. C. 2003, Icarus, 164, 127, doi: 10.1016/S0019-1035(03)00104-0

    2003 doi

  3. [11]

    N., Hogan, R

    Cuzzi, J. N., Hogan, R. C., & Bottke, W. F. 2010, Icarus, 208, 518, doi: 10.1016/j.icarus.2010.03.005

    2010 doi

  4. [12]

    N., Hogan, R

    Cuzzi, J. N., Hogan, R. C., Paque, J. M., & Dobrovolskis, A. R. 2001, ApJ, 546, 496, doi: 10.1086/318233

    2001 doi

  5. [13]

    N., Hogan, R

    Cuzzi, J. N., Hogan, R. C., & Shariff, K. 2008, ApJ, 687, 1432, doi: 10.1086/591239

    2008 doi

  6. [14]

    R., Dacles-Mariani, J

    Dobrovolskis, A. R., Dacles-Mariani, J. S., & Cuzzi, J. N. 1999, J. Geophys. Res., 104, 30805, doi: 10.1029/1999JE001053

    1999 doi

  7. [15]

    1995, Icarus, 114, 237, doi: 10.1006/icar.1995.1058

    Dubrulle, B., Morfill, G., & Sterzik, M. 1995, Icarus, 114, 237, doi: 10.1006/icar.1995.1058

    1995

  8. [16]

    Elghobashi, S., & Truesdell, G. C. 1992, Journal of Fluid Mechanics, 242, 655, doi: 10.1017/S0022112092002532

    1992 doi

  9. [17]

    R., Cuzzi, J

    Estrada, P. R., Cuzzi, J. N., & Morgan, D. A. 2016, ApJ, 818, 200, doi: 10.3847/0004-637X/818/2/200

    2016 doi

  10. [18]

    R., Cuzzi, J

    Estrada, P. R., Cuzzi, J. N., & Umurhan, O. M. 2022, ApJ, 936, 42, doi: 10.3847/1538-4357/ac7ffd 8 For example, the leading term behavior aroundn=−2 is∝ St2 L ln StL

    2022 doi

  11. [19]

    A., M´ eheut, H., & Bec, J

    Gerosa, F. A., M´ eheut, H., & Bec, J. 2023, European Physical Journal Plus, 138, 9, doi: 10.1140/epjp/s13360-022-03585-8

    2023 doi

  12. [20]

    Hartlep, T., & Cuzzi, J. N. 2017, in LPI Contributions, Vol. 2043, Accretion: Building New Worlds Conference, ed. LPI Editorial Board, 2049

    2017

  13. [21]

    Hartlep, T., & Cuzzi, J. N. 2020, ApJ, 892, 120, doi: 10.3847/1538-4357/ab76c3

    2020 doi

  14. [22]

    N., & Weston, B

    Hartlep, T., Cuzzi, J. N., & Weston, B. 2017, PhRvE, 95, 033115, doi: 10.1103/PhysRevE.95.033115

    2017 doi

  15. [23]

    Hopkins, P. F. 2016a, MNRAS, 455, 89, doi: 10.1093/mnras/stv2226 —. 2016b, MNRAS, 456, 2383, doi: 10.1093/mnras/stv2820

    doi

  16. [24]

    F., & Lee, H

    Hopkins, P. F., & Lee, H. 2016, MNRAS, 456, 4174, doi: 10.1093/mnras/stv2745

    2016 doi

  17. [25]

    2009, arXiv e-prints, arXiv:0905.4404, doi: 10.48550/arXiv.0905.4404

    Ishitsu, N., Inutsuka, S.-i., & Sekiya, M. 2009, arXiv e-prints, arXiv:0905.4404, doi: 10.48550/arXiv.0905.4404

    2009 doi arXiv

  18. [26]

    S., Mac Low, M.-M., et al

    Johansen, A., Oishi, J. S., Mac Low, M.-M., et al. 2007, Nature, 448, 1022, doi: 10.1038/nature06086

    2007 doi

  19. [27]

    1941, Akademiia Nauk SSSR Doklady, 30, 301

    Kolmogorov, A. 1941, Akademiia Nauk SSSR Doklady, 30, 301

    1941

  20. [28]

    Li, R., & Youdin, A. N. 2021, ApJ, 919, 107, doi: 10.3847/1538-4357/ac0e9f

    2021 doi

  21. [29]

    A., Simon, J

    Lim, J., Baronett, S. A., Simon, J. B., et al. 2025a, arXiv e-prints, arXiv:2505.23902, doi: 10.48550/arXiv.2505.23902

    doi arXiv

  22. [30]

    B., Li, R., et al

    Lim, J., Simon, J. B., Li, R., et al. 2025b, ApJ, 981, 160, doi: 10.3847/1538-4357/adb311

    doi

  23. [31]

    2021, ApJ, 907, 64, doi: 10.3847/1538-4357/abcd9b

    Lin, M.-K. 2021, ApJ, 907, 64, doi: 10.3847/1538-4357/abcd9b

    2021 doi

  24. [32]

    Maxey, M. R. 1987, Journal of Fluid Mechanics, 174, 441–465, doi: 10.1017/S0022112087000193

    1987 doi

  25. [33]

    G., & Norman, M

    Pan, L., Padoan, P., Scalo, J., Kritsuk, A. G., & Norman, M. L. 2011, ApJ, 740, 6, doi: 10.1088/0004-637X/740/1/6 Pencil Code Collaboration, Brandenburg, A., Johansen, A., et al. 2021, The Journal of Open Source Software, 6, 2807, doi: 10.21105/joss.02807

    2011 doi

  26. [34]

    1998, Icarus, 133, 298, doi: 10.1006/icar.1998.5933

    Sekiya, M. 1998, Icarus, 133, 298, doi: 10.1006/icar.1998.5933

    1998

  27. [35]

    N., Umurhan, O

    Sengupta, D., Cuzzi, J. N., Umurhan, O. M., & Lyra, W. 2024, ApJ, 966, 90, doi: 10.3847/1538-4357/ad2c89

    2024 doi

  28. [36]

    Turner, N. J. 2019, ApJ, 874, 26, doi: 10.3847/1538-4357/aafc36 22Umurhan et al

    2019 doi

  29. [37]

    Sengupta, D., & Umurhan, O. M. 2023, ApJ, 942, 74, doi: 10.3847/1538-4357/ac9411

    2023 doi

  30. [38]

    I., Cuzzi, J

    Simon, J. I., Cuzzi, J. N., McCain, K. A., et al. 2018, Earth and Planetary Science Letters, 494, 69, doi: 10.1016/j.epsl.2018.04.021

    2018 doi

  31. [39]

    Squire, J., & Hopkins, P. F. 2018, MNRAS, 477, 5011, doi: 10.1093/mnras/sty854 —. 2020, MNRAS, 498, 1239, doi: 10.1093/mnras/staa2311

    2018 doi

  32. [40]

    D., & Eaton, J

    Squires, K. D., & Eaton, J. K. 1990, Physics of Fluids A, 2, 1191, doi: 10.1063/1.857620 —. 1991, Physics of Fluids A, 3, 1169, doi: 10.1063/1.858045

    1990 doi

  33. [41]

    A., & Taylor, J

    Stamper, M. A., & Taylor, J. R. 2017, Ocean Dynamics, 67, 65, doi: 10.1007/s10236-016-1011-6

    2017 doi

  34. [42]

    J., Testi, L., et al

    Tazzari, M., Clarke, C. J., Testi, L., et al. 2021, MNRAS, 506, 2804, doi: 10.1093/mnras/stab1808

    2021 doi

  35. [43]

    M., Estrada, P

    Umurhan, O. M., Estrada, P. R., & Cuzzi, J. N. 2020, ApJ, 895, 26, doi: 10.3847/1538-4357/ab899d V¨ olk, H. J., Jones, F. C., Morfill, G. E., & Roeser, S. 1980, A&A, 85, 316 V¨ olk, H. J., Morfill, G., Roeser, S., & Jones, F. C. 1978, Moon and Planets, 19, 221, doi: 10.1007/BF00896995

    2020 doi

  36. [44]

    Wang, Q., & Squires, K. D. 1996, Physics of Fluids, 8, 1207, doi: 10.1063/1.868911

    1996 doi

  37. [45]

    Weidenschilling, S. J. 1980, Icarus, 44, 172, doi: 10.1016/0019-1035(80)90064-0

    1980 doi

  38. [46]

    C., Johansen, A., & Carrera, D

    Yang, C. C., Johansen, A., & Carrera, D. 2017, A&A, 606, A80, doi: 10.1051/0004-6361/201630106

    2017 doi

  39. [47]

    Y., & Lei, U

    Yang, C. Y., & Lei, U. 1998, Journal of Fluid Mechanics, 371, 179, doi: 10.1017/S0022112098002328

    1998 doi

  40. [48]

    2007, ApJ, 662, 613, doi: 10.1086/516729

    Youdin, A., & Johansen, A. 2007, ApJ, 662, 613, doi: 10.1086/516729

    2007 doi

  41. [49]

    N., & Goodman, J

    Youdin, A. N., & Goodman, J. 2005, ApJ, 620, 459, doi: 10.1086/426895

    2005 doi

  42. [50]

    N., & Lithwick, Y

    Youdin, A. N., & Lithwick, Y. 2007, Icarus, 192, 588, doi: 10.1016/j.icarus.2007.07.012

    2007 doi

  43. [51]

    2025, arXiv e-prints, arXiv:2507.08797, doi: 10.48550/arXiv.2507.08797

    Zagaria, F., Facchini, S., Curone, P., et al. 2025, arXiv e-prints, arXiv:2507.08797, doi: 10.48550/arXiv.2507.08797

    2025 doi

  44. [52]

    I., & Alipchenkov, V

    Zaichik, L. I., & Alipchenkov, V. M. 2009, New Journal of Physics, 11, 103018, doi: 10.1088/1367-2630/11/10/103018

    2009 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.