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The Interplay of Parametric and Magnetorotational Instabilities in Oscillatory Shear Flows

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In unstratified MHD shearing-box runs with oscillatory warp-like forcing, the parametric instability dominates vertical momentum transport above a critical forcing amplitude while the MRI suppresses it below.

desk verdict A clean and honest local study of PI/MRI competition in forced oscillatory shear flows, with a plausible threshold result that is limited by single realizations and a fit-based viscoelastic closure. read the letter →

arxiv 2506.08839 v2 pith:6WZSNF5J submitted 2025-06-10 astro-ph.SR astro-ph.HEphysics.flu-dyn

Callum W. Fairbairn , James M. Stone This is my paper
classification astro-ph.SRastro-ph.HEphysics.flu-dyn
keywords warpedaccretiondisksparametricinstabilitymagnetorotationalshearingboxsimulationselevatormodesviscoelasticstressclosuressloshingflowsMHDturbulence
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

Warped accretion disks internally generate vertically shearing, oscillatory 'sloshing' flows, and those flows are unstable to a hydrodynamic parametric instability that may compete with the MRI. This paper asks which instability controls the flow in a magnetized disk and what stress closure should replace the standard isotropic alpha viscosity. Using unstratified shearing-box simulations with a sinusoidal forcing that mimics a warp, the authors find a critical forcing amplitude psi_c ~ 0.03: above it the parametric instability dominates vertical transport through strong 'elevator' flows and tempers the sloshing, while below it the MRI suppresses the elevator modes and mediates the sloshing. In every magnetized case the vertical stresses are anisotropic and better fit by a viscoelastic relation between stress and rate of strain than by a viscous one. If correct, this means warped-disk evolution codes and analytic models need warp-amplitude-dependent, anisotropic viscoelastic closures rather than isotropic viscous ones.

What carries the argument

The machinery is the forced shearing box. The warp is replaced by a volume force f_w = Re[rho $Omega_0^{2}$ H psi_f sin(2pi z / L_z) exp(i Omega_0 t)] x-hat, which drives a sinusoidal vertical sloshing profile; a laminar model reduces the response to a forced, damped epicyclic oscillator, allowing viscosity to be extracted from measured amplitudes. Instability growth is predicted with three-mode coupling theory and Floquet analysis of the linearized modal equations, and the saturated state is diagnosed by phase-folding the sloshing amplitudes A(t), B(t) and the vertical Reynolds and Maxwell stresses. The stress closure is then generalized by promoting the viscosity to a complex number, so that the stress can lag the strain rate.

What would settle it

A stratified local shearing-box simulation with vertical gravity and the same oscillatory forcing, run with psi_f = 0.3, would settle the transferability: if elevator modes do not emerge or the measured stress phase lag becomes close to pi, then the unstratified critical amplitude and viscoelastic coefficients do not carry over to real warped disks.

Watch

Extended reading notes

Core claim

The central claim is that in a local MHD shearing box with oscillatory forcing mimicking a warp, the parametric instability and the MRI coexist with a sharp ordering set by forcing amplitude. Above psi_f ~ 0.03, the resonant sloshing grows to near-sonic amplitudes and excites vertically coherent counter-propagating 'elevator' modes, whose Reynolds stresses dominate vertical transport of horizontal momentum; the MRI partially damps these modes, so the magnetized saturated state has slightly larger sloshing amplitude than the hydrodynamic one. Below the threshold, MRI turbulence damps the elevator modes and the sloshing is mediated primarily by MRI stresses. Phase-resolved comparison of stress versus rate of strain shows the stresses lag the shear by phase differences inconsistent with a purely viscous response, and an anisotropic complex viscosity reproduces both amplitudes and phases of the sloshing, whereas isotropic alpha does not.

Load-bearing premise

The paper's conclusions depend on an unstratified, vertically periodic box with an artificial sinusoidal radial force reproducing the essential physics of internal flows in a real stratified warped disk; the authors note that the lack of stratification makes a direct quantitative link to the stratified warping regime tenuous.

Editorial extensions

If this is right

  • In strongly forced magnetized boxes, vertical transport of horizontal momentum is dominated by hydrodynamic Reynolds stresses from elevator flows, not by Maxwell stresses, so MRI-based estimates of vertical viscosity miss the main damping.
  • Because the vertical stresses lag the shear by phase differences away from pi, closure by isotropic alpha viscosity is inadequate; an anisotropic, viscoelastic relation fits both amplitude and phase of the sloshing.
  • The MRI indirectly increases sloshing amplitude by damping the elevator modes, so adding magnetic fields to a warped-disk model does not simply add damping.
  • The extracted viscoelastic coefficients grow with forcing amplitude, so a single effective viscosity cannot describe warps of different amplitudes.
  • Below psi_c ~ 0.03 the MRI quenches the parametric instability and sloshing is mediated by MRI turbulence; above it the parametric instability takes over.

Reading between the lines

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

  • Using the paper's mapping psi ~ psi_f/(8 pi^2), its fiducial psi_f = 0.3 corresponds to a small warp psi ~ 0.004; if that mapping holds, parametric-instability-dominated, viscoelastic damping would set in for warps well below H/R, before nonlinear vertical bouncing becomes important.
  • A stratified extension with vertical boundaries would likely confine elevator modes and shift the critical amplitude and the viscoelastic coefficients; if the stress phase lag persists there, viscoelasticity would appear to be a generic property of oscillatory shear turbulence.
  • The same mechanism of a turbulent cascade damping parametric channel modes may operate in other oscillatory shear contexts, such as tidally deformed stars or precessing disks, where a background turbulent state can suppress parametric growth below a threshold.
  • A testable extension would measure whether the vertical stress phase lag and the amplitude scaling |A| proportional to psi_f^0.51 survive in a stratified warped shearing box; if they do, the anisotropic viscoelastic closure could be ported directly into global warp-evolution models.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper studies the competition between the hydrodynamic parametric instability (PI) and the magnetorotational instability (MRI) in oscillatory shear flows designed to mimic the internal sloshing flows of warped disks. The authors use unstratified, vertically periodic 3D shearing-box MHD simulations with an artificial radial forcing f_w ∝ ρ Ω^2 H ψ_f sin(2π z / L_z) exp(i Ω_0 t). They first verify the linear PI against Floquet theory and three-mode coupling, then study forced hydrodynamic runs in which the PI produces vertical 'elevator' flows that temper the resonant sloshing. In MHD runs they report that above a critical forcing ψ_c ≈ 0.03 the PI dominates vertical transport, while below it the MRI suppresses the elevator modes and mediates the sloshing. By fitting the measured stresses to a laminar forced-oscillator model, they extract complex, anisotropic viscoelastic coefficients and argue that warped-disk evolution requires an anisotropic viscoelastic closure rather than an isotropic viscous α prescription.

Significance. If the central results hold, this is the first controlled local study of the steady-state interplay of PI and MRI in warped-disk-like flows, and it provides a clear numerical template for extracting effective stress closures from phase-resolved shearing-box data. The paper has real strengths: the linear benchmark in §3 is clean and the Floquet/three-mode agreement is quantitatively convincing; the hydrodynamic and MHD runs are carefully compared; and the viscous-hydrodynamic control runs in Appendix B usefully isolate the role of MRI turbulence from explicit dissipation. The authors are also transparent about the absence of stratification, a major caveat they state explicitly. However, the evidence for the paper's headline viscoelastic-closure claim is weaker than the abstract implies: the fitted coefficients reproduce the fitted quantities by construction, and the critical threshold and power-law scalings rest on single realizations with no error bars and only a heuristic damping model. The significance for real warped disks is therefore qualitative and motivational rather than established.

major comments (4)
  1. [§4.4, Eqs. (30) and (15)-(16); §5.3, Eq. (35)] The central evidence for the viscoelastic closure is constructed rather than tested. The complex coefficients ν_xz and ν_yz are solved from the measured stress amplitudes and sloshing amplitudes using Eq. (30) (and Eq. (35) for the Maxwell part), and the same coefficients are then reinserted into the laminar forced-oscillator solutions (15)-(16). Because the four real parameters are fitted to the four real outputs (|A|, |B|, and the two phases), the resulting agreement is by construction, as the authors acknowledge when they write that the agreement 'should be expected given the added flexibility'. To support the claim that the stresses are 'better described by a viscoelastic relationship', the model needs an independent test: for example, a single complex viscosity constrained simultaneously by both A and B, a one-relaxation-time viscoelastic model, or a prediction of the sloshing response at one ψ_f using coefficients extracted at another ψ_f. Without such a test, the anisotropic coefficients in Fig. 14(d) are a curve fit rather than a validated closure.
  2. [§5.4, Fig. 14 and Fig. 15] The threshold ψ_c ≈ 0.03 and the power-law scalings |A| ∝ ψ_f^0.51 and |ν_xz| ∝ ψ_f^0.57 are based on one realization per ψ_f, with no quoted error bars, no seed variation, and no assessment of time-averaging uncertainty. MRI turbulence is intermittent and box-dependent, so the 'knee' at ψ_c, its numerical value, and the fitting exponents are not statistically established. The heuristic damping model σ_d ∼ 4 k_x v_x,MRI / (2π) used to motivate the threshold contains an ad hoc factor of 4 and adopts v_x,MRI from the base state without a derivation or sensitivity study. Repeating the near-threshold runs and the fiducial run with several seeds and reporting the spread (and, if possible, the mode-identification criterion for 'elevator modes') is necessary before a critical forcing amplitude can be claimed quantitatively.
  3. [§5.1 and §5.4] The MRI base state is characterized at a single resolution (1232×200×128) and a single choice of ν = 3.2×10^-4 and η = 8.0×10^-5, with no resolution or box-size convergence test for the MRI saturation levels (α_M, α_R, and velocity dispersions). Since the central comparison is between MHD runs and hydrodynamic runs, the conclusion that the MRI suppresses the PI below ψ_c could be affected by under-resolved or under-converged MRI turbulence. At minimum, the authors should report a resolution study of the base MRI state (and ideally of the threshold behavior) or explicitly justify why the chosen resolution is sufficient for the PI–MRI competition.
  4. [§2, Eq. (5), and §6] The bridge to real warped disks is the artificial forcing f_w ∝ sin(2π z / L_z) in an unstratified, vertically periodic box, whereas a true warp drives a pressure gradient approximately linear in z in a stratified disk; the elevator modes are n = 0 channels whose structure and saturation are likely sensitive to vertical boundaries and stratification. The authors acknowledge this explicitly in §6 ('the lack of stratification in our model makes a direct quantitative link to the full stratified, warping regime tenuous' and upper boundaries 'will confine the elevator flows'), but the Abstract and Conclusion nevertheless assert the warp-amplitude-dependent viscoelastic stresses as a property of warped systems. This extrapolation is not yet supported by the present experiments; it should be framed as a qualitative motivation, or accompanied by a stratified test (e.g., the planned stratified extension, or at least different vertical boundary conditions) to show that the n = 0 elevator modes and the threshold are robust.
minor comments (3)
  1. [Header and Appendix B] The author header contains an unintended space in 'C. W. F airbairn', and Appendix B quotes 'ψ_c ∼ 0.3' where the rest of the paper uses ψ_c ≈ 0.03; this appears to be a typo.
  2. [§5.4 and Fig. 15] The relation between the sloshing amplitude S used in the Floquet/damping model of Fig. 15 and the fitted amplitude A (or the forcing ψ_f) is stated only implicitly; the comparison at ψ_f = 0.03 would be easier to follow if the conversion S ↔ A were defined explicitly.
  3. [§6] The mapping ψ ∼ ψ_f / (8π^2) in §6 is derived from Eq. (17), but the extracted viscosity in §4.3 is explicitly noted to be too large for the ν k_b^2 ≪ Ω_0 limit underlying Eq. (17); the order-of-magnitude status of this mapping should be stated more carefully.

Circularity Check

2 steps flagged · score 6.0 of 10

Viscoelastic 'agreement' is a fitted round-trip, but the PI–MRI competition is independently simulated and Floquet-verified.

  1. fitted input called prediction [Section 4.4, after Eq. (30), with Eqs. (15)-(16)]
    "Inserting our new results for νxz and νyz we find that |A|vis = 0.186 and |B|vis = 0.054 with a phase difference of arg(Bvis) − arg(Avis) = 1.03π, almost in perfect agreement with the values extracted from the simulation. This should be expected given the added flexibility of the viscoelastic model, where each coefficient νiz provides two degrees of freedom, which can then fit two amplitudes and two phases."

    The complex coefficients νxz and νyz are solved from the same measured sloshing amplitudes and Reynolds stresses via Eq. (30), and Eqs. (15)-(16) are the algebraic solution of the same laminar closure model. Substituting the fitted coefficients back into this forced-oscillator solution is the inverse of the fitting operation, so matching |A|, |B| and the phase difference is guaranteed by construction. The paper explicitly acknowledges this by noting that the four real degrees of freedom in the two complex coefficients exactly match the two amplitudes and two phases being 'reproduced.' Thus the agreement is a consistency check, not an independent prediction of the viscoelastic model.

  2. fitted input called prediction [Section 5.3, after Eqs. (34)-(35), with Eqs. (15)-(16)]
    "Combining these, we find total viscoelastic coefficients νxz = 0.0108e0.30i and νyz = 0.0336e−0.33i. Inserting the net viscoelastic coefficients into the simplified laminar model forced response, given by equations (15)-(16), gives very good agreement with the amplitudes and phases measured from the simulation."

    The net viscoelastic coefficients are obtained by inverting Eqs. (30) and (35) from the measured sloshing amplitudes A, B and the measured Reynolds and Maxwell stresses of this same run. Inserting them into the laminar forced-oscillator solutions (15)-(16) is the algebraic reversal of that inversion, so the resulting amplitudes and phases are determined by the inputs rather than independently predicted. The claimed 'very good agreement' therefore does not provide additional evidence for the viscoelastic closure; it is an identity up to the accuracy of the Fourier fits.

full rationale

The paper's principal dynamical result—that the parametric instability emerges, dominates vertical momentum transport above ψc≈0.03, and is quenched by the MRI below that threshold—is established by direct simulation and checked against an independently derived Floquet/three-mode theory in Section 3, so that part of the derivation is self-contained and not circular. The critical-forcing toy model uses v_x,MRI measured from the same base MRI run, but combines it with parameter-free Floquet growth rates; this is an interpretive model rather than a circular prediction. Where the paper does become circular is in its repeated use of fitted complex viscosities as a 'prediction': in Sections 4.4 and 5.3, νxz and νyz are solved from the measured sloshing amplitudes and stresses (Eqs. 30 and 35), then substituted into the same laminar forced-oscillator equations (15)-(16) to reproduce those amplitudes and phases. That reproduction is by construction; the authors even acknowledge in Section 4.4 that the four real degrees of freedom exactly match the two amplitudes and two phases being fitted. This does not undermine the central PI–MRI competition claim, but it means the 'viscoelastic better than viscous' validation is partly a fit comparison rather than an independent prediction. The authors' explicit caveat that 'the lack of stratification in our model makes a direct quantitative link to the full stratified, warping regime tenuous' is a limitation on external validity, not a circularity. No load-bearing self-citation or uniqueness-imported-from-authors pattern is present.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The 'elevator' flows are a pre-existing flow pattern given a name, and the viscoelastic coefficients are fitted closure parameters rather than invented entities.

free parameters (5)
  • Complex vertical viscoelastic coefficients nu_xz and nu_yz (and their Reynolds/Maxwell parts) = Hydro: nu_xz=0.015+0.001i, nu_yz=0.043-0.030i; MHD: nu_xz=0.0108e^{0.30i}, nu_yz=0.0336e^{-0.33i}
    Solved from measured stress and sloshing phases (Eqs. 30 and 35), then used in equations (15)-(16) to reproduce the same amplitudes. This is fitting to data, not an independent prediction.
  • Damping prefactor 4 in sigma_d = 4
    Chosen by hand in Section 5.4 to model momentum transport across two boundaries of an elevator channel and used to estimate the critical forcing.
  • MRI radial velocity dispersion v_x,MRI = 0.085
    Measured from the base MRI run (Fig. 10c) and used in the toy damping model for the PI-MRI threshold.
  • Power-law exponents for |A| and |nu_xz| versus psi_f = 0.51 and 0.57
    Fit to the MHD runs in Section 5.4 to characterize the sublinear response; descriptive rather than derived.
  • Explicit viscosity and resistivity = nu=3.2e-4, eta=8.0e-5
    Chosen to produce a converged zero-net-flux MRI following Fromang et al. 2007; the PI-MRI balance may depend on these values.
assumptions (5)
  • domain assumption Shearing-box approximation with periodic y and z boundaries adequately represents the local flow of a warped disk.
    Section 2 uses the standard local patch expansion and neglects disk curvature, stratification, and vertical boundaries, which limits the direct applicability to real warps.
  • ad hoc to paper The artificial forcing fw = Re[rho Omega^2 H psi_f sin(2pi z / Lz) exp(i Omega0 t)] approximates the radial pressure gradient of a warp.
    Equation (5) imposes a sinusoidal vertical structure instead of the self-consistent warped geometry; the authors state it contains the fundamental physics, but this is an assumption.
  • domain assumption Unstratified disk with no vertical gravity is sufficient to capture the PI-MRI interplay.
    Section 2 omits vertical gravity; Section 6 concedes this makes a direct quantitative link to stratified warps tenuous.
  • domain assumption A zero-net-flux vertical field MRI with beta_m = 400 is representative of magnetized warped disks.
    Section 5.1, Eq. 31: only one field geometry and one magnetization are considered; the PI-MRI threshold may differ for net-flux or stronger fields.
  • ad hoc to paper Floquet theory with periodic, constant-S sloshing applies to the transient forced simulations.
    Section 4.2 applies Floquet results to non-periodic, growing sloshing states; the authors note the theory 'maps effectively' rather than exactly.

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Pith. "Pith review of The Interplay of Parametric and Magnetorotational Instabilities in Oscillatory Shear Flows." pith.science (2026). https://pith.science/paper/6WZSNF5J

@misc{pith2026250608839,
  author       = {Pith},
  title        = {Pith review of: The Interplay of Parametric and Magnetorotational Instabilities in Oscillatory Shear Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WZSNF5J}},
  note         = {Machine review of arXiv:2506.08839}
}
read the original abstract

The evolution of warped disks is governed by internal, oscillatory shear flows driven by their distorted geometry. However, these flows are known to be vigorously unstable to a hydrodynamic parametric instability. In many warped systems, this might coexist and compete with the magnetorotational instability (MRI). The interplay of these phenomena and their combined impact on the internal flows has not been studied. To this end, we perform three-dimensional, magnetohydrodynamic unstratified shearing box simulations with an oscillatory radial forcing function to mimic the effects of a warped disk. In the hydrodynamic study, we find that the parametric instability manifests as strong, vertical `elevator' flows that resist the sloshing motion. Above a critical forcing amplitude, these also emerge in our magnetized runs and dominate the vertical stress, although they are partially weakened by the MRI, and hence the system equilibrates with larger radial sloshing flows. Below this critical forcing, the MRI effectively quenches the parametric instability. In all cases, we find that the internal stresses are anisotropic in character and better described by a viscoelastic relationship with the shearing flows. Unfortunately, these important effects are typically unresolved in global simulations of warped disks and are simplified in analytically tractable models. The incorporation of such complex, warp-amplitude-dependent, viscoelastic stresses will sensitively regulate the laminar flow response and inevitably modify the detailed spatio-temporal evolution of warped systems.

Figures

Figures reproduced from arXiv: 2506.08839 by the authors.

Figure 1
Figure 1. Global and local cartoon of a warped disk. Left: The global warped disk consists of a series of orbits with a radially varying orientation of angular momentum. Right: A zoomed-in patch of the warped disk manifests as a tilting of the midplane. This projects a component of the vertical pressure gradient in the radial direction and drives sloshing flows. this forcing are that it is linear in the vertical coordinate z … view at source ↗
Figure 2
Figure 2. Dispersion relation curves tracing the frequency vs radial wavenumber relation for locally axisymmetric iner￾tial waves supported by our unstratified shearing box. The colored lines denote different branches, labeled by the verti￾cal modal order n (0: blue, 1: orange, 2: green, 3: red, 4: purple, 5: brown). The connective dashed lines indicate the resonant couplings susceptible to parametric instability. 3.1. Summar… view at source ↗
Figure 3
Figure 3. Maximal growth rates σ, obtained via the Flo￾quet method (black line) vs three-mode coupling theory (red markers) for S = 0.01 across a range of kx. frequency as seen in the coorbital frame. When these conditions are satisfied, energy from the shear flow is efficiently pumped into the inertial waves, which grow according to σ Ω0 = S p C1C2, (18) where C1 and C2 are defined by equations (A32)-(A33) and depend on the … view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Numerical simulation verifying three-mode cou￾pling parametric instability. Upper panel: The initialized modal structure in u ′ z is shown as the heat map, overplot￾ted with white arrows denoting the meridional streamlines. Lower panel: The growth of the vertical kinet…
Figure 5
Figure 5. Figure 5: Growth of the PI for our ψf = 0.3 hydrodynamic run. Panel (a): epicylic kinetic energy measured in the sim￾ulation (black line) and the result expected for the inviscid, laminar response (red dashed line). Inset panel: the reso￾nant growth of the sloshing motion Ares d…
Figure 6
Figure 6. Figure 6: PI excited modal structure. Panel (a): merid￾ional x–z cross-section of {uz}y. Panel (b): maximal Floquet growth rates extracted as a function of the kx (at the allowed values quantized by the box) for different sloshing ampli￾tudes S = 0.5 (blue), 1.0 (black) and 1.5 …
Figure 7
Figure 7. Figure 7: Sloshing equilibrium established for our ψf = 0.3, hydrodynamic run. Panel (a): meridional cross section of {ux}y at t = 60T. Panel (b): {ux}xy profile as a function of z for different oscillation phases over the orbital period 40 < t/T < 41. Panel (c): likewise for {u…
Figure 8
Figure 8. Figure 8: Optimization of the viscosity cost function C(νxz, νyz) to fit |A| and |B|. The absolute minimum is lo￾cated at the red star where (νxz, νyz) = (0.0256, 0.0354). When constrained to an isotropic viscosity, along the dotted white line, the minimum is located at the blue…
Figure 9
Figure 9. Figure 9: The rate of strain and stress relationship for our fiducial hydrodynamic run. The upper panels show the phase-folded, vertical profiles for: (a) vx,w, (b) vy,w, (c) Rxz, and (d) Ryz. In panel (e), we extract the best-fitting, slosh￾ing amplitudes A(t) (black) and B(t) …
Figure 10
Figure 10. Figure 10: Characterization of our zero-net-flux, base MRI state. Panel (a): plots log10(βm) through the meridional slice y = 0, showing field amplification and fine turbulent structure. Panel (b): growth and saturation of the horizon￾tal Maxwell stresses αM (red), Reynolds stre…
Figure 11
Figure 11. Figure 11: The emergence of the PI atop the MRI for our magnetized ψf = 0.3 run. Panel (a): The growth of Eepi is shown for the MHD run (black solid) compared with the hydrodynamic run (blue dotted) from Section 4. Once again, the laminar, resonant prediction is shown as the red…
Figure 12
Figure 12. Figure 12: Horizontal stresses in the equilibrium sloshing state, measured between 40 < t/T < 60. αM (red line) oscillates about the averaged value of 0.0099 (dashed red). αR (blue line) oscillates about the averaged value of 0.0019 (dashed blue). ues of 0.0099 and 0.0019, respe…
Figure 13
Figure 13. Figure 13: Rate of strain and stress relationship for our fiducial MHD run (akin to [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Variation in the sloshing diagnostics with ψf. Row (a): Sloshing motion A and B. Row (b): Reynolds stresses. Row (c): Maxwell stresses. Row (d): viscoelastic constants. Column (1) visualizes the full complex information in a polar plot, whilst columns (2) and (3) plot…
Figure 15
Figure 15. Figure 15: Model for the competition between PI growth and MRI damping of elevator modes. The maximum net growth rate σnet, normalized by the sloshing amplitude S, is plotted as a function of S. The color of the markers denotes the preferred radial order nx corresponding to this…
Figure 16
Figure 16. Figure 16: The same as [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.