{"id":"5e60df72-58f0-4c79-be0f-19eb386dd557","arxiv_id":"2507.22672","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Linear superpositions of SARI eigenmodes in global disk models produce spiral, vertically displaced field structures and alpha values in the 0.007 to 0.08 range without nonlinear saturation.","lead":"Using a global MHD spectral code, this paper visualizes how two families of accretion disk instabilities, the axisymmetric MRI and the non-axisymmetric Super-Alfvénic Rotational Instability, deform magnetic fields and drive angular momentum transport. It finds that even purely linear SARI modes can produce turbulent-looking fields and stress levels comparable to those invoked in accretion disk models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The alpha transport claim rests on quasi-continuum SARIs being excitable by generic disk perturbations; the paper only asserts this, and Table 2's values may also reflect the chosen linear-phase amplitude threshold rather than a SARI-specific transport mechanism.","rationale":"The reader's weakest assumption correctly identifies the quasi-continuum nature of SARIs as a central risk: if these near-eigenmodes are not excited by realistic perturbations, the spiral structures and alpha estimates do not transfer to actual disks. I agree with that concern, and I add a second, independent weakness: the quantitative alpha values in Table 2 are evaluated at a linear-phase threshold where the perturbed field is comparable to the equilibrium field. Since Maxwell stresses scale quadratically with the perturbation amplitude, any growing mode pushed to that threshold will produce alpha of order B_0^2/p, which for beta = 10 is already of order 0.1. The reported values are thus consistent with a generic amplitude-normalization effect and do not by themselves demonstrate a SARI-specific transport mechanism. The visual and polarization parts of the paper are substantially better supported: the analytical Appendix A derives the polarization relations from the incompressible thin-disk assumptions, and the field-line comparisons are direct consequences of the computed eigenfunctions. The concern is therefore not with the eigenmode structure or the visualizations, but specifically with the extrapolation from a hand-selected mode ensemble to a physical accretion disk. A linearized initial-value calculation with broadband noise is the cleanest way to test whether the quasi-continuum SARI region actually dominates the response and whether the stresses are reproducible beyond the specific mode-selection and normalization choices. This concern supports the reader's CONDITIONAL verdict rather than changing it; the paper should either provide such a test, or explicitly reframe the alpha claim as an upper-bound-style consistency check rather than a transport prediction.","tokens_in":25271,"tokens_out":10339,"duration_ms":133476,"concrete_test":"Perform a linear initial-value calculation on the same equilibrium (e.g. with a pseudospectral global disk code) starting from broadband noise with total initial magnetic energy about 10^-8 times B_0^2, evolve for several rotation periods, and compare the disk-averaged Maxwell stress at the time when max|B_1| = |B_0| with Table 2. Repeat for at least 20 random noise realizations and for radial domains r in [1,2] and [1,3] with the same vertical wavenumber restriction. If the stress differs from Table 2 by more than a factor of 3, or if the response is dominated by wall-attached MRI or discrete modes rather than the selected quasi-continuum SARI region, the alpha claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in the abstract and Sec. 5.2 is that superposed, still linearly growing SARI modes can deliver alpha-values of order 0.01-0.1 needed for angular momentum transport. This claim is load-bearing on two linked premises that the paper states but does not establish. First, Sec. 1 and Sec. 3.1 define quasi-continuum SARIs as near-eigenmodes: they are not exact solutions of the ideal MHD operator and 'require a minute (e.g. below machine precision) addition of energy to the system'. The paper argues such energy is always present in numerical simulations, but a real disk is not a numerical experiment. If generic broadband noise in a global disk does not project onto these quasi-modes with the assumed amplitudes and phases, the mode ensembles constructed in Sec. 5.1 (n_omega = 111-7899 modes, randomized amplitudes 'chosen such that the total growth at the end of the linear phase is somewhat uniform') are not representative of disk dynamics, and Table 2 does not transfer. Second, even granting that the modes are excited, the stress magnitude is evaluated at the end of the linear phase, defined in Sec. 2.3 by B_pert being everywhere weaker than B_0. Because the stresses are quadratic in the perturbation, this threshold roughly caps alpha at order B_0^2/p ~ 2/beta ~ 0.2 for beta = 10. The values in Table 2 (0.006-0.08) are therefore close to a generic upper bound set by the amplitude criterion, not compelling evidence that SARI modes specifically provide the 'needed' viscosity. The claim needs either a linear initial-value demonstration that the quasi-modes dominate a generic response, or a saturation argument showing the stress is sustained.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper uses the open-source Legolas code to compute linear ideal-MHD eigenmodes of a cylindrical global accretion disk model with vertical, helical, or nearly azimuthal background fields, for both the axisymmetric MRI and non-axisymmetric SARI modes. It visualizes the complex eigenfunctions in 2D and 3D, identifies spiral morphology and phase speeds, contrasts the polarization of MRI and SARI perturbations, and supports the polarization picture with analytic expressions in Appendix A. The final part of the paper superposes many linear MRI or SARI modes, renders synthetic 'turbulent' disks, and computes Maxwell and Reynolds stresses to obtain the Shakura-Sunyaev alpha values reported in Table 2. The abstract and Section 5.2 state the central claim: even superposed, still linearly growing SARI modes can already provide the effective alpha values invoked for angular momentum transport. The paper also claims that SARI field perturbations are fundamentally different from MRI perturbations, with vertical displacements and, for opposite-m superpositions, plasmoid-like field reversals.","tokens_in":25603,"tokens_out":4860,"duration_ms":62368,"significance":"If the central claim were established, the paper would be significant: it would show that quasi-continuum, non-axisymmetric SARIs are not merely spectral curiosities but can produce transport stresses comparable to saturated MRI turbulence while naturally generating non-axisymmetric structure and plausible reconnection sites. The strength of the paper lies in its concrete, reproducible machinery: the mode lists and superposition coefficients are stored, the analysis uses the open-source Legolas code, and the visualizations are carefully tied to the underlying complex eigenfunctions. The analytic polarization appendix is a useful step beyond pure numerics and yields testable sign relations. However, the transport claim is currently not robust: the Table 2 alpha values are set by hand-chosen mode amplitudes and phases, no uncertainty or sensitivity analysis is given, and the quasi-continuum SARI modes are shown to be near-eigenmodes without a demonstration that real broadband disk perturbations excite them with the assumed amplitudes.","major_comments":[{"comment":"The alpha values in Table 2 depend on the arbitrary amplitudes and phases assigned to the superposed modes in Sec. 5.1. The text states that each mode receives a randomized complex rotation and a 'semi-randomised amplitude, chosen such that the total growth at the end of the linear phase is somewhat uniform in the entire disk', and that the total perturbation is normalized to an initial field perturbation of at most 1e-4 B0. Since the stresses in Eq. (12) are quadratic in the perturbed fields, this normalization and amplitude prescription largely determines the resulting alpha values. No sensitivity study is provided for different amplitude distributions, phase choices, or time of evaluation, so Table 2 cannot be read as evidence that SARIs specifically provide the needed transport; it shows only that perturbations at the linear-phase threshold can produce stresses of this magnitude.","section":"Sec. 5.1, Table 2, Eq. (12)"},{"comment":"The alpha values in Table 2 are close to a generic upper bound set by the linearity criterion itself. Equation (8) defines the linear phase by requiring the perturbed field to be everywhere weaker than B0, so the quadratic Maxwell stress is capped at order B0^2/p0 ~ 2/beta = 0.2 for beta = 10. The reported values, 0.006 to 0.08, are within one order of magnitude of this bound. The statement in Sec. 5.2 that the stresses 'imply that the linear dynamics may extend beyond the strictly defined linear regime' is therefore not supported: any perturbation at the amplitude threshold would produce comparable stresses. To substantiate the SARI-specific transport claim, the paper would need to show that these values are not simply a consequence of the amplitude cap, for example by comparing against random non-modal perturbations at the same amplitude.","section":"Sec. 5.2, Sec. 2.3, Eq. (8)"},{"comment":"The central transport claim relies on quasi-continuum SARIs being excitable in real disks, but the paper only asserts this. Sec. 1 and Sec. 3.1 state that these modes are near-eigenmodes that 'require a minute (e.g. below machine precision) addition of energy to the system', and that this is de facto satisfied in direct numerical simulations. However, a numerical simulation is not an astrophysical disk: no initial-value calculation or projection of realistic broadband perturbations onto the quasi-continuum is presented, so it remains unknown whether the amplitude and phase structure assumed in Sec. 5.1 is representative. The discussion in Sec. 7 acknowledges this as an open question, but the abstract and Sec. 5.2 nevertheless present the alpha result as the paper's conclusion. The claim should be explicitly conditioned on this excitability assumption, or supported by an initial-value test.","section":"Sec. 1, Sec. 3.1, Sec. 7"},{"comment":"The analytic derivation of the SARI polarization in Appendix A uses the sign relations (A12)-(A13) for the radial derivatives chi'_r and chi'_i, which are introduced as empirical associations inferred from the trailing spiral structure. These relations are then used to derive the signs in Table 1, so the 'analytical confirmation' of the SARI polarization is partly an input rather than a derivation. Since the fundamental difference between MRI and SARI polarization is one of the paper's main claims, these sign relations should either be derived from the eigenmode equations or explicitly labeled as assumptions whose validity is checked numerically. As written, the argument is more of a consistency check than a proof.","section":"Appendix A, Eqs. (A12)-(A13)"}],"minor_comments":[{"comment":"The paper quotes peak values of alpha at the radius of strongest growth rather than a radial average; a radially averaged alpha, or a statement of how the quoted peak relates to the disk-averaged transport, would make the comparison to simulation values more meaningful.","section":"Sec. 5.2"},{"comment":"The vertical-localization criterion is written as k > 70/r, which mixes a dimensionless wavenumber with a radius-dependent threshold; please clarify the units and the derivation, and state how the value 70 is obtained from L <= 2H.","section":"Sec. 2.1"},{"comment":"The quasi-continuum selection criterion 1 << |k/m| < B from Eq. (91) of GK22 is used but the quantity B is not defined in this paper; a sentence defining it would improve self-containedness.","section":"Sec. 5.1"},{"comment":"The text says the mode lists and coefficients are 'stored and reproducible' but the Data Availability statement offers them only on request; placing the data in a permanent repository would better support reproducibility.","section":"Data Availability"},{"comment":"Table 1 uses the symbols 0↑ and 0↓ without an explicit definition in the table caption; the surrounding text explains them, but a short note in the caption would prevent misreading.","section":"Sec. 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is part of a coherent series with the authors' own prior work, and the self-citations are appropriate given the subject. My main concern is that the abstract and Section 5.2 state the alpha-transport conclusion more strongly than the linear superposition analysis warrants; the excitability of quasi-continuum SARIs is not demonstrated, and the amplitude normalization largely sets the reported stress levels. The visual and spectral content is solid and publishable, but the framing should be revised to present the alpha computation as an illustration of the stress magnitudes that linear SARI superpositions can attain under an assumed amplitude model, rather than as evidence that SARIs provide the needed transport in real disks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Brughmans & Keppens. The genuinely new content is the visual morphology of SARI eigenfunctions in a global disk, the polarization table with the analytic appendix that explains it, and a first attempt at alpha estimates from superposed linear SARI modes. The polarization analysis is the strongest part: the WKB-style argument in Appendix A is a real, checkable derivation that matches the numerics, and the distinction between MRI (B_z out of phase, radial displacement) and SARI (vertical displacement, sign-dependent on m) is clearly laid out. The figures are informative, and the use of Legolas makes the mode sets reproducible in principle.\n\nThe soft spot is the alpha transport claim in Sec. 5.2. The values in Table 2 are computed at the end of the linear phase, defined by B_pert < B_0 everywhere. Since the stresses are quadratic in the perturbation, this caps alpha at order 2/beta ~ 0.2 for beta = 10. The reported values (0.006-0.08) sit close to that cap, so they are not strong evidence that SARI modes specifically supply the 'needed' viscosity; a generic linear instability at the same amplitude would produce similar stresses. The paper is appropriately hedged in places—it says 'can already provide' and notes that no saturation occurs—but the abstract and conclusion push the transport relevance a bit harder than the analysis supports.\n\nThe second soft spot is the quasi-continuum status of the SARIs. The paper is upfront that these are near-eigenmodes requiring a sub-machine-precision energy addition. Whether real disks excite them with the assumed phases and amplitudes is asserted, not demonstrated. An initial-value calculation showing that broadband noise projects onto these quasi-modes would close the gap. The radial walls at r=1,2 and the vertical localization assumption k^2 r^2 >> m^2 also shape the spectrum; the paper acknowledges this in part but does not quantify the sensitivity.\n\nOn the positive side, the paper is honest about its linear-phase limitation, and the opposite-m plasmoid-like structures are presented as 'reminiscent of' reconnection, not proof of it. Data availability is 'upon request' rather than a public release, which is a minor reproducibility ding given the large mode sets.\n\nWho is this for? Anyone working on non-axisymmetric MHD instabilities in disks. The polarization table and appendix are worth having on record. The alpha values should be cited as a linear estimate, not a saturation result.\n\nI would send this to a serious referee. The central physical picture is credible, the numerics are standard, and the analytic appendix is a genuine contribution. It needs qualification rather than rejection.","headline":"A clean visual and analytic study of SARI eigenmode structure; the linear alpha estimates in Table 2 are a useful consistency check, not a transport result.","tokens_in":26141,"tokens_out":2365,"would_cite":true,"duration_ms":25983,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Non-axisymmetric disk instabilities by themselves can provide accretion's needed effective viscosity.","keywords":["accretion discs","magnetorotational instability","super-Alfvénic rotational instability","MHD instabilities","angular momentum transport","alpha viscosity","spiral modes","dynamo"],"falsifier":"A direct numerical simulation of one of the paper's three equilibria (vertical, helical, or nearly azimuthal field with $B_0=0.01$, $\\beta=10$) in a global annulus, started from broadband noise with no wall-attached channel modes, should show localised, wall-insensitive spiral disturbances with outward-moving phase fronts and vertical field displacements, and should reach peak $\\alpha$ values near $10^{-2}$ before nonlinear saturation; if those structures or stress levels are absent, the SARI picture is falsified. A cheaper spectral test is to move the outer radial wall outward and check whether the quasi-continuum mode set and the resulting $\\alpha$ profile survive.","tokens_in":25056,"feed_emoji":"🌀","tokens_out":12579,"duration_ms":136662,"temperature":0.7,"pith_summary":"Accretion disks are thought to transport angular momentum outward through turbulence driven by the magneto-rotational instability (MRI), but this paper argues that a different, non-axisymmetric family of modes—the Super-Alfvénic Rotational Instability (SARI)—can do the same job on its own. The authors compute the full complex linear eigenmodes of a global disk annulus for vertical, helical, and nearly azimuthal magnetic fields, and show that superposing many still-growing SARI modes produces Maxwell and Reynolds stresses whose peak values of the standard alpha viscosity parameter (about $0.06$–$0.08$ for vertical and helical fields) fall in the range that accretion models invoke for angular momentum transport. They also find that MRI and SARI perturbations look and behave differently: MRI displaces magnetic field lines almost purely radially, while SARI modes displace them vertically as well, move with real phase speeds, and form trailing spiral arms. If this is right, angular momentum transport in disks does not require fully developed turbulence or even the axisymmetric MRI; purely linear, wall-insensitive modes can provide it.","feed_headline":"Spiral disk modes deliver the viscosity that disks need","feed_subtitle":"Full global eigenmode calculations show these non-axisymmetric modes alone reach the stress levels accretion models require.","key_machinery":"The carrying object is the full complex-valued linear eigenfunction set of ideal MHD in a cylindrical disk annulus, obtained from the self-adjoint spectral equation for the Lagrangian displacement $\\boldsymbol{\\xi}$, namely $\\mathcal{G}(\\boldsymbol{\\xi})-2\\rho\\tilde{\\omega}U\\boldsymbol{\\xi}+\\rho\\omega^2\\boldsymbol{\\xi}=0$. The SARI modes studied are quasi-continuum near-eigenmodes: they occupy two-dimensional regions in the complex frequency plane above the overlapping forward and backward Alfvén continua, localise around the corotation radius $r_\\ast$ where $\\omega_r=m\\Omega(r_\\ast)$, and need only a minute (below machine precision) energy addition to be excited. The analytical core is a set of approximate incompressible, thin-disk expressions for the perturbed velocity and magnetic field in terms of the radial displacement $\\chi$, from which the paper derives the polarization table: $B_r$ and $B_\\theta$ anti-phase, $B_z$ in or anti-phase with them according to the sign of $m$, $v_r$ and $v_\\theta$ in phase, and $\\mathbf{v}\\cdot\\mathbf{B}\\approx 0$ near corotation. These expressions also give the phase speeds of the spirals and support the generalisation of the anti-spiral theorem: only genuinely complex, overstable eigenmodes can display spiral structure, while purely oscillating waves cannot.","core_discovery":"The paper's central claim is that quasi-continuum SARI modes—near-eigenmodes of ideal MHD that are localised around their corotation radius and insensitive to the artificial radial boundaries—are physically consequential, not spectral curiosities. When many such still-growing modes are superposed in a global disk model, their disk-averaged Maxwell and Reynolds stresses already give peak $\\alpha$ values of order $10^{-2}$ (up to about $8\\times 10^{-2}$ for the helical-field case), comparable to the lower range of saturated MRI turbulence, and this happens while the perturbations are still linear. The field perturbations are described as fundamentally different from those of the MRI: $B_r$ and $B_\\theta$ stay in anti-phase as in the MRI cartoon, but the vertical component $B_z$ is now in phase or anti-phase with them depending on the sign of the azimuthal mode number $m$, so the eigenfunctions are truly complex, the mode has no zeroes of its envelope, and the field lines acquire vertical displacements. A linear superposition of two slightly misaligned opposite-$m$ SARI modes produces local, abruptly reversing field structures reminiscent of plasmoids and toroidal flux tubes, structures that axisymmetric MRI modes cannot create.","pith_inferences":["An extension the paper does not pursue: because shearing-box coordinates discard the resonance structure that localises SARIs, local-box simulations may systematically underestimate this class of transport; a global initial-value simulation with the same equilibria would test this directly.","The alpha values are computed in the linear regime, so a natural next calculation is whether those stresses persist or grow after nonlinear saturation; if they do, SARI transport could rival MRI turbulence.","The predicted polarization signatures (the sign of $B_z$ relative to $B_r,B_\\theta$ tied to $m$, and $\\mathbf{v}\\cdot\\mathbf{B}\\approx 0$ near corotation) could be searched for in existing global disk simulations by post-processing their Fourier modes.","If finite resistivity is added, the ideal-MHD field reversals seen in opposite-$m$ superpositions may reconnect genuinely, which would tie SARI directly to plasmoid formation and disk heating without invoking parasitic instabilities."],"forward_implications":["Superposed linear SARI modes alone produce disk-averaged alpha values in the $10^{-2}$ range, so angular momentum transport need not wait for fully developed turbulence.","SARI spiral shapes are stationary in the corotating frame and are not sheared away like transient non-axisymmetric MRI modes in shearing-box treatments.","The non-axisymmetry required for dynamo action is already present in the linear SARI stage, while MRI-dominated disks only acquire it through nonlinear evolution.","Opposite-$\\pm m$ SARI superpositions create plasmoid-like field reversals and flux-tube-like structures within ideal MHD, offering a linear seed for reconnection sites.","Each growing SARI has a damped, time-reversed twin with opposite spiral handedness, so spiral structure itself is a signature of instability rather than stable oscillation."],"supporting_citations":[{"why":"Introduces the SARI quasi-continuum, its corotation localisation, and the Alfvén-continuum resonance picture that this paper visualises and extends.","marker":"Goedbloed & Keppens 2022"},{"why":"Confirms and augments the SARI predictions with the same spectral approach, establishing the near-eigenmode status of the quasi-continuum modes.","marker":"Brughmans et al. 2024"},{"why":"Provides the standard MRI mechanism and the angular-momentum-transfer cartoon that the paper revisits in a global setting.","marker":"Balbus & Hawley 1998"},{"why":"Supplies the discrete non-axisymmetric modes attached to walls against which the wall-insensitive quasi-continuum SARIs are contrasted.","marker":"Ogilvie & Pringle 1996"},{"why":"Gives the global MRI eigenmode structure (channel versus radially varying parts) and the linear-phase criterion used for the visualisations.","marker":"Latter et al. 2015"},{"why":"States the anti-spiral theorem that the paper generalises to ideal MHD to explain SARI spiral structure.","marker":"Lynden-Bell & Ostriker 1967"},{"why":"Provides the vertical-wavelength constraint and the alpha values from MRI simulations used as baselines for the stress comparison.","marker":"Hawley et al. 1995"},{"why":"Proposes a two-dimensional continuum of unstable local modes in toroidal fields, the result Section 6 compares and contrasts with SARI quasi-continua.","marker":"Terquem & Papaloizou 1996"}],"fun_headline_variants":["Linear SARI modes already match MRI stress levels","Spiral disk modes alone supply needed viscosity","Non-axisymmetric SARI modes rival saturated MRI","SARI superpositions produce plasmoid-like fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that quasi-continuum SARI modes are genuinely present in real disks even though they are not exact eigenmodes and require a tiny, machine-precision seed of energy to be excited; if actual disks never excite these near-modes, or if the artificial radial walls at $r=1$ and $r=2$ together with the vertical-localisation assumption shape the spectrum, the spiral structures, stress values, and alpha estimates would not transfer to real accretion disks.","fun_headline_variants_meta":{"raw":{"variants":["Linear SARI modes already match MRI stress levels","Spiral disk modes alone supply needed viscosity","Non-axisymmetric SARI modes rival saturated MRI","SARI superpositions produce plasmoid-like fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000547,"raw_usage":{"total_tokens":2677,"prompt_tokens":1068,"completion_tokens":1609,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":684,"completion_tokens_details":{"reasoning_tokens":1551}},"tokens_in":684,"tokens_out":1609,"duration_ms":14025,"temperature":1.0,"reasoning_tokens":1551,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:24:03.245935+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical simulation of one of the paper's three equilibria (vertical, helical, or nearly azimuthal field with $B_0=0.01$, $\\beta=10$) in a global annulus, started from broadband noise with no wall-attached channel modes, should show localised, wall-insensitive spiral disturbances with outward-moving phase fronts and vertical field displacements, and should reach peak $\\alpha$ values near $10^{-2}$ before nonlinear saturation; if those structures or stress levels are absent, the SARI picture is falsified. A cheaper spectral test is to move the outer radial wall outward and check whether the quasi-continuum mode set and the resulting $\\alpha$ profile survive.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the SARI quasi-continuum, its corotation localisation, and the Alfvén-continuum resonance picture that this paper visualises and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Confirms and augments the SARI predictions with the same spectral approach, establishing the near-eigenmode status of the quasi-continuum modes."},{"cited_title":"N., Fromang S., Faure J., 2015, @doi [ ] 10.1093/mnras/stv1890 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.453.3257L 453, 3257","cited_arxiv_id":null,"evidence_quote":"Gives the global MRI eigenmode structure (channel versus radially varying parts) and the linear-phase criterion used for the visualisations."},{"cited_title":"C., 1996, , 279, 767","cited_arxiv_id":null,"evidence_quote":"Proposes a two-dimensional continuum of unstable local modes in toroidal fields, the result Section 6 compares and contrasts with SARI quasi-continua."}],"review_version":1}